Finite Group Databasebeta

← #29691270

Subgroups — #29691270

Order 15616.

996 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839919C211yesmaximal, characteristic
2C4×#14519757C211yesmaximal, characteristic
3#391×D61C211yesmaximal, characteristic
4#21446688C211yesmaximal, characteristic, complemented
5#21446687C211yesmaximal, characteristic, complemented
6#21446511C211yesmaximal, characteristic, complemented
7#21446510C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446163C211yesmaximal, characteristic, complemented
10#21446159C211yesmaximal, characteristic, complemented
11#21446125C211yesmaximal, characteristic
12#21446010C211yesmaximal, characteristic
13#21445752C211yesmaximal, characteristic
14#21445580C211yesmaximal, characteristic
15C61×#1927C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×#14519757C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839919C418yescomplemented
24C2×#14519708C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519689C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35#16839669C2211yescharacteristic
36#16839668C2211yescharacteristic
37#16839667C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839666—21no—
40#16839666—21no—
41#16839661C2211yescharacteristic
42#16839647—21no—
43#16839636—21no—
44#16839619C2211yescharacteristic
45C4×C4⋊Dic61—21no—
46C4×#11758198—21no—
47#95×D61—21no—
48#95×D61—21no—
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53C2×#106×C61C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C4×C4⋊C4×C61C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#336C2211yescharacteristic
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C22×Dic122C2311yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—41no—
65C2×Q8×D61—41no—
66C2×Q8×D61—41no—
67C2×Q8×D61—41no—
68C2×Q8×D61C2×C414yes—
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61D411yescharacteristic
72C22×C4×D61D411yescharacteristic
73C22×C4×D61C2311yescharacteristic
74C2×Dic244—41no—
75C2×Dic244—41no—
76C2×Dic244C2×C414yescomplemented
77C2×#11758202—41no—
78C2×#11758202—41no—
79C2×#11758202C2×C414yescomplemented
80C22×#92418C2311yescharacteristic
81C22×#92418C2311yescharacteristic
82#14519757—22no—
83#14519757C2×C414yes—
84C4×Dic122—41no—
85C4×Dic122—41no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C4⋊C4×D61—41no—
93C4⋊C4×D61—41no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61D411yescharacteristic
98C2×C4×Dic61D411yescharacteristic
99C2×C4×Dic61C2311yescharacteristic
100C42×D61—41no—
101C42×D61—41no—
102#14519708—41no—
103#14519702—41no—
104#14519702—41no—
105#14519701—41no—
106#14519700—41no—
107#14519700—41no—
108#14519700C2×C414yescomplemented
109#14519689—22no—
110#14519689C2×C414yes—
111#14519687—41no—
112#14519687—41no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131#106×C61—41no—
132#103×C61—41no—
133#103×C61C2×C414yescomplemented
134#95×C61—21no—
135#95×C61—21no—
136#95×C61—21no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158Q8×D61—44no—
159Q8×D61—44no—
160Q8×D61—44no—
161Q8×D61—44no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—24no—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×C4×D61—41no—
171C2×C4×D61—41no—
172C2×C4×D61—41no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61C22×C412yes—
179C2×C4×D61#4112yes—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—24no—
191C2×Dic122—42no—
192C2×Dic122—42no—
193C2×Dic122C22×C412yes—
194C2×Dic122C22×C412yes—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C2×C8×C61—21no—
200C2×C8×C61C22×C412yes—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—21no—
206C4⋊C4×C61C22×C412yes—
207C42×C61—21no—
208C42×C61—21no—
209C42×C61—42no—
210#11758202—44no—
211#11758198—24no—
212#11758198—42no—
213C4⋊Dic61—24no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—42no—
219#11758196—42no—
220#11758196—42no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—41no—
224C4×Dic61—41no—
225C4×Dic61—42no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418C22×C412yes—
232Dic244—44no—
233C23×C61C22×D411yescharacteristic
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61C4×D412yes—
239C22×D61#13712yes—
240C22×D61#13712yes—
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61#13712yes—
263C2×C4×C61C23×C411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×C4×C61H(5,𝔽2)11yescharacteristic
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61C4×D412yes—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—22no—
293C4×D61—22no—
294C4×D61—44no—
295C4×D61—44no—
296C4×D61—48no—
297C4×D61—84no—
298Dic122—22no—
299Dic122—22no—
300Dic122—22no—
301Dic122—22no—
302Dic122—48no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306Dic122—41no—
307C8×C61—22no—
308C8×C61—41no—
309#92418—22no—
310#92418—41no—
311#22050—611nomaximal, Sylow
312C22×C61C2×C4×D411yescharacteristic
313C22×C61#53411yescharacteristic
314C22×C61#53511yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61C2×#13711yescharacteristic
317C22×C61C2×#13711yescharacteristic
318C22×C61#57511yescharacteristic
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61—22no—
324C2×D61—22no—
325C2×D61#44114yes—
326C4×C61—22no—
327C4×C61—42no—
328C4×C61—41no—
329C4×C61—42no—
330C4×C61—81no—
331C4×C61—81no—
332C4×C61#51811yescharacteristic
333C4×C61#51811yescharacteristic
334C4×C61#53811yescharacteristic
335C4×C61#53811yescharacteristic
336Dic61—22no—
337Dic61—42no—
338Dic61—42no—
339Dic61—41no—
340Dic61—41no—
341Dic61—81no—
342Dic61—81no—
343C2×#451—611no—
344C4×C22⋊Q8—611no—
345C2×#391—611no—
346#1971—611no—
347#1971—611no—
348#1935—611no—
349#1935—611no—
350#1927—611no—
351#1927D6111yescharacteristic, complemented
352#1878—611no—
353#1878—611no—
354#1872—611no—
355#1853—611no—
356#1787—611no—
357#1749—611no—
358C2×C61#232111yescharacteristic
359C2×C61C2×#44111yescharacteristic
360C2×C61C4×#13711yescharacteristic
361C2×C61#319511yescharacteristic
362C2×C61#319611yescharacteristic
363C2×C61#321511yescharacteristic
364C2×C61#321811yescharacteristic
365D61—24no—
366C23×Q8—611no—
367C22×Q16—611no—
368C22×Q16—611no—
369C2×C22⋊Q8—611no—
370C2×C4×Q8—611no—
371C22×C4⋊C4—611no—
372C22×C42—611no—
373#451—618no—
374C2×#106—611no—
375C2×#106C2×D6111yescharacteristic
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103C2×D6111yescharacteristic
383C2×#103C2×D6111yescharacteristic
384C2×C22⋊C8—611no—
385C2×C22⋊C8—611no—
386C2×#97—611no—
387C2×#97—611no—
388#391—611no—
389#391—611no—
390#391—611no—
391#391—611no—
392#391—1221no—
393#391—1221no—
394#391C2×D6111yescharacteristic
395#389—611no—
396#387—1221no—
397#385—1221no—
398#380—611no—
399C4×C4⋊C4—1221no—
400C4×C4⋊C4C2×D6111yescharacteristic
401C4×C22⋊C4—1221no—
402C2×#95—1221no—
403C2×#95—1221no—
404#340—611no—
405#340—611no—
406#340—611no—
407#340C2×D6111yescharacteristic
408#336—611no—
409#336C2×D6111yescharacteristic
410C61#2205011yescharacteristic, complemented, Sylow
411C22×Q8—614no—
412C22×Q8—611no—
413C22×Q8—611no—
414C22×Q8—1221no—
415C22×Q8—1222no—
416C22×Q8—2441no—
417C22×Q8—2441no—
418C22×Q8—2441no—
419C22×Q8—2441no—
420C22×Q8C22×D6111yescharacteristic
421C23×C4—611no—
422C23×C4—611no—
423C23×C4—611no—
424C23×C4—1221no—
425C23×C4—1221no—
426C2×Q16—614no—
427C2×Q16—614no—
428C2×Q16—2441no—
429C2×Q16—2441no—
430C2×Q16—2441no—
431C2×Q16—2441no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8C22×D6111yescharacteristic
435C22×C8C22×D6111yescharacteristic
436C22⋊Q8—614no—
437C22⋊Q8—1222no—
438C4×Q8—2441no—
439C4×Q8—2441no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—2441no—
447C2×C4⋊C4—2441no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42—21no—
454C2×C42—611no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—1221no—
458C2×C42—2441no—
459C2×C42—2441no—
460C2×C42C22×D6111yescharacteristic
461#106—41no—
462#106—2441no—
463#103—41no—
464#103—614no—
465#103—2441no—
466#103—2441no—
467#103—2441no—
468#103—2441no—
469#103—2441no—
470#103C4×D6114yescomplemented
471C22⋊C8—614no—
472C22⋊C8—1222no—
473#97—2441no—
474#97—2441no—
475#95—21no—
476#95—21no—
477#95—21no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488C24—611no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—612no—
495C2×Q8—612no—
496C2×Q8—1224no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—2442no—
508C2×Q8—2442no—
509C2×Q8—2444no—
510C2×Q8—2444no—
511C2×Q8—2444no—
512C2×Q8—2444no—
513C2×Q8C2×C4×D6112yes—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—611no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—1222no—
524C22×C4—1222no—
525C22×C4—1224no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1222no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—2441no—
536C22×C4—2441no—
537C22×C4—2441no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4—2442no—
541C22×C4—2442no—
542C22×C4—2442no—
543C22×C4D4×D6111yescharacteristic
544C22×C4D4×D6111yescharacteristic
545C22×C4C23×D6111yescharacteristic
546Q16—2444no—
547Q16—2444no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—21no—
552C2×C8—21no—
553C2×C8—612no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—1221no—
557C2×C8—1221no—
558C2×C8—1221no—
559C2×C8C2×C4×D6112yes—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—21no—
565C4⋊C4—1224no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—1221no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C4⋊C4—2442no—
573C4⋊C4C2×C4×D6112yes—
574C22⋊C4—1224no—
575C22⋊C4—2442no—
576C42—21no—
577C42—21no—
578C42—42no—
579C42—1221no—
580C42—1221no—
581C42—2441no—
582C42—2441no—
583C42—2442no—
584C23—612no—
585C23—612no—
586C23—612no—
587C23—1221no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23C2×D4×D6111yescharacteristic
592Q8—22no—
593Q8—22no—
594Q8—41no—
595Q8—41no—
596Q8—1222no—
597Q8—1222no—
598Q8—1222no—
599Q8—1222no—
600Q8—2448no—
601Q8—2441no—
602Q8—2441no—
603Q8—2441no—
604Q8—2441no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—612no—
623C2×C4—1221no—
624C2×C4—1221no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—4884no—
638C2×C4—2444no—
639C2×C4—2444no—
640C2×C4—2448no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4#1451980612yes—
652C2×C4C22×C4×D6111yescharacteristic
653C2×C4#41×D6111yescharacteristic
654C2×C4#1451982211yescharacteristic
655C2×C4#1451982311yescharacteristic
656C2×C4#1451982311yescharacteristic
657C2×C4#1451982311yescharacteristic
658C8—22no—
659C8—41no—
660C8—1222no—
661C8—2441no—
662C22—614no—
663C22—1222no—
664C22—1222no—
665C22—1222no—
666C22—1222no—
667C22—1222no—
668C22—1222no—
669C22C4×D4×D6111yescharacteristic
670C22#1684032311yescharacteristic
671C22#1684032811yescharacteristic
672C22#137×D6111yescharacteristic
673C22#137×D6111yescharacteristic
674C22C2×#1451980611yescharacteristic
675C22#1684055911yescharacteristic
676C4—22no—
677C4—41no—
678C4—42no—
679C4—42no—
680C4—81no—
681C4—81no—
682C4—1222no—
683C4—4881no—
684C4—4881no—
685C4—2441no—
686C4—2441no—
687C4—2442no—
688C4—2442no—
689C4#1684023411yescharacteristic
690C4#1684023611yescharacteristic
691C4#1684034611yescharacteristic
692C4#1684035111yescharacteristic
693C2—1224no—
694C2#2144848111yescharacteristic
695C2#441×D6111yescharacteristic
696C2C4×#1451980611yescharacteristic
697C2#2145664211yescharacteristic
698C2#2145665311yescharacteristic
699C2#2145682011yescharacteristic
700C2#2145684511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1927C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1927D61
hypocenterC61#22050
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1927

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #21446688C2semidirect
5. #21446687C2semidirect
6. #21446511C2semidirect
7. #21446510C2semidirect
8. #21446453C2semidirect
9. #21446163C2semidirect
10. #21446159C2semidirect
15. C61×#1927C2semidirect
23. #16839919C4semidirect
76. C2×Dic244C2×C4semidirect
79. C2×#11758202C2×C4semidirect
108. #14519700C2×C4semidirect
133. #103×C61C2×C4semidirect
351. #1927D61semidirect
410. C61#22050semidirect
470. #103C4×D61semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168399192
C4×#145197572
#391×D612
#214466882
#214466872
#214465112
#214465102
#214464532
#214461632
#214461592
#214461252
#214460102
#214457522
#214455802
C61×#19272
#2205061