Finite Group Databasebeta

← #29700497

Subgroups — #29700497

Order 15616.

1058 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
2C2×#16839866C211yesmaximal, characteristic, complemented
3C2×#16839769C211yesmaximal, characteristic
4#21448848C211yesmaximal, characteristic
5Q8×#11758198C211yesmaximal, characteristic
6#21446680C211yesmaximal, characteristic, complemented
7#21446510C211yesmaximal, characteristic, complemented
8#21446453C211yesmaximal, characteristic, complemented
9#21446251C211yesmaximal, characteristic
10#21446161C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446075C211yesmaximal, characteristic, complemented
13#21446021C211yesmaximal, characteristic
14#21445602C211yesmaximal, characteristic
15C4×#103×C61C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×Dic244C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C22×#11758189C2211yescharacteristic
20C2×#14519757C2211yescharacteristic
21C2×Q8×Dic61C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23C2×#14519735C2211yescharacteristic
24#16839919C418yescomplemented
25#16839866—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519708C2211yescharacteristic
28#16839769—24no—
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×#14519689C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37#16839668—21no—
38#16839668C2211yescharacteristic
39#16839668C2211yescharacteristic
40#16839667—21no—
41#16839667C2211yescharacteristic
42#16839661C2211yescharacteristic
43#16839654—21no—
44#16839653—21no—
45#16839647—21no—
46#16839619—21no—
47#16839619C2211yescharacteristic
48C4×C4⋊Dic61—21no—
49C4×#11758198—21no—
50C4×#11758198C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—21no—
63C22×Dic122C2311yescharacteristic
64C22×Dic122C2311yescharacteristic
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—41no—
69C2×Q8×D61—41no—
70C2×Q8×D61C2×C414yes—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×Dic244—41no—
74C2×Dic244—41no—
75C2×Dic244C2×C414yes—
76C2×#11758202—22no—
77C2×#11758202—22no—
78C2×#11758202C2×C414yes—
79C2×#11758189—22no—
80C2×#11758189—22no—
81C2×#11758189—41no—
82C22×#92418C2311yescharacteristic
83C22×#92418C2311yescharacteristic
84#14519757—22no—
85#14519757C2×C414yes—
86Q8×Dic61—22no—
87Q8×Dic61—41no—
88#14519742—22no—
89#14519742—41no—
90#14519735—22no—
91#14519735—41no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61—21no—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198D411yescharacteristic
102C2×#11758198D411yescharacteristic
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61C2311yescharacteristic
107#14519711—41no—
108#14519708—22no—
109#14519702—22no—
110#14519702—22no—
111#14519701—22no—
112#14519700—22no—
113#14519700—41no—
114#14519700C2×C414yes—
115#14519689C2×C414yes—
116#14519687—22no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125#14519684D411yescharacteristic
126#14519684D411yescharacteristic
127C22×Q8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C4×Q8×C61—22no—
131C4×Q8×C61—41no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61—21no—
138C2×C42×C61C2311yescharacteristic
139#103×C61C2×C414yes—
140#103×C61D414yes—
141C4×C8×C61—41no—
142#95×C61—21no—
143C23×D61C2×D411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C412yes—
148C2×Q8×C61C2×D412yes—
149C2×Q8×C61#4112yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164Q8×D61—28no—
165Q8×D61—44no—
166Q8×D61—44no—
167Q8×D61—44no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—24no—
173C2×C4×D61—42no—
174C2×C4×D61—42no—
175C2×C4×D61C22×C412yes—
176C2×C4×D61#4112yes—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—24no—
188C2×Dic122—42no—
189C2×Dic122—42no—
190C2×Dic122C22×C412yes—
191C2×Dic122C22×C412yes—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61C22×C412yes—
197C2×C8×C61C2×D412yes—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—21no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—42no—
203C4⋊C4×C61C22×C412yes—
204C4⋊C4×C61C2×D412yes—
205C42×C61—21no—
206C42×C61—21no—
207C42×C61—42no—
208C42×C61—42no—
209#11758202—28no—
210#11758198—24no—
211#11758198—22no—
212#11758198—41no—
213#11758198—42no—
214#11758198—42no—
215C4⋊Dic61—24no—
216C4⋊Dic61—21no—
217C4⋊Dic61—21no—
218C4⋊Dic61—21no—
219C4⋊Dic61—21no—
220C4⋊Dic61—42no—
221C4⋊Dic61—42no—
222#11758196—42no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418C22×C412yes—
234Dic244—44no—
235#11758189—44no—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61C4×D412yes—
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61—22no—
247Q8×C61C4×D414yes—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61#13712yes—
266C2×C4×C61C23×C411yescharacteristic
267C2×C4×C61C22×D411yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61H(5,𝔽2)11yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—21no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61C4×D412yes—
293C4×D61—22no—
294C4×D61—22no—
295C4×D61—22no—
296C4×D61—22no—
297C4×D61—48no—
298Dic122—22no—
299Dic122—22no—
300Dic122—22no—
301Dic122—22no—
302Dic122—48no—
303Dic122—41no—
304Dic122—41no—
305Dic122—41no—
306Dic122—41no—
307C8×C61—22no—
308C8×C61—22no—
309#92418—22no—
310#92418—41no—
311#22869—611nomaximal, Sylow
312C22×C61C2×C4×D411yescharacteristic
313C22×C61D4211yescharacteristic
314C22×C61#54611yescharacteristic
315C22×C61C2×#13711yescharacteristic
316C22×C61#57511yescharacteristic
317C22×C61#57811yescharacteristic
318C22×C61#57811yescharacteristic
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—24no—
327C4×C61—22no—
328C4×C61—42no—
329C4×C61—42no—
330C4×C61—41no—
331C4×C61—81no—
332C4×C61C2×C4×D411yescharacteristic
333C4×C61#51811yescharacteristic
334C4×C61#54611yescharacteristic
335C4×C61#55111yescharacteristic
336Dic61—22no—
337Dic61—42no—
338Dic61—42no—
339Dic61—41no—
340Dic61—41no—
341Dic61—81no—
342Dic61—81no—
343C2×#451—611no—
344C2×#441—611no—
345C2×#419—611no—
346#2362—611no—
347Q8×C22⋊C4—611no—
348#1969—611no—
349#1935—611no—
350#1927—611no—
351#1894—611no—
352#1878—611no—
353#1878—611no—
354#1863—611no—
355#1854—611no—
356C4×#103D6111yescharacteristic, complemented
357#1753—611no—
358C2×C61#235811yescharacteristic
359C2×C61C4×#13711yescharacteristic
360C2×C61#295011yescharacteristic
361C2×C61#327611yescharacteristic
362C2×C61#327711yescharacteristic
363C2×C61#331211yescharacteristic
364C2×C61#331311yescharacteristic
365D61—42no—
366C23×Q8—611no—
367C22×Q16—611no—
368C22×Q16—611no—
369C22×M16—611no—
370C2×C22⋊Q8—611no—
371C2×C4×Q8—611no—
372C2×C4×Q8—611no—
373C2×C4×Q8C2×D6111yescharacteristic
374C2×#117—611no—
375#451—618no—
376#441—1224no—
377C2×#107—611no—
378C2×#106—611no—
379#419—1224no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103C2×D6111yescharacteristic
387C2×#103C2×D6111yescharacteristic
388C2×C22⋊C8—611no—
389C2×#97—611no—
390C2×C4×C8C2×D6111yescharacteristic
391#391—611no—
392#391—611no—
393#391—611no—
394#391—1221no—
395#391—1221no—
396#389—611no—
397#388—1221no—
398#388—1221no—
399#387—1221no—
400#380—611no—
401#380—1221no—
402C4×C4⋊C4—1221no—
403C4×C4⋊C4C2×D6111yescharacteristic
404C4×C22⋊C4—611no—
405C4×C22⋊C4—1221no—
406#340—611no—
407#340—611no—
408#340C2×D6111yescharacteristic
409#340C2×D6111yescharacteristic
410#336—611no—
411#336—611no—
412C61#2286911yescharacteristic, complemented, Sylow
413C22×Q8—614no—
414C22×Q8—611no—
415C22×Q8—611no—
416C22×Q8—1221no—
417C22×Q8—1222no—
418C22×Q8—1222no—
419C22×Q8—1222no—
420C22×Q8—2441no—
421C22×Q8—2441no—
422C22×Q8C22×D6111yescharacteristic
423C23×C4—611no—
424C23×C4—1221no—
425C2×Q16—614no—
426C2×Q16—614no—
427C2×Q16—1222no—
428C2×Q16—1222no—
429C2×Q16—2441no—
430C2×Q16—2441no—
431C2×M16—1222no—
432C2×M16—1222no—
433C2×M16—2441no—
434C22×C8—611no—
435C22×C8—611no—
436C22×C8C22×D6111yescharacteristic
437C22×C8C22×D6111yescharacteristic
438C22⋊Q8—614no—
439C22⋊Q8—1222no—
440C4×Q8—22no—
441C4×Q8—41no—
442C4×Q8—1222no—
443C4×Q8—1222no—
444C4×Q8—2441no—
445C4×Q8—2441no—
446#117—1222no—
447#117—2441no—
448C2×C4⋊C4—21no—
449C2×C4⋊C4—611no—
450C2×C4⋊C4—611no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C22⋊C4—611no—
459C2×C22⋊C4—611no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—21no—
464C2×C42—21no—
465C2×C42—611no—
466C2×C42—1221no—
467C2×C42—1221no—
468C2×C42—1221no—
469C2×C42C22×D6111yescharacteristic
470#107—2441no—
471#106—1222no—
472#103—614no—
473#103—1222no—
474#103—1222no—
475#103—1222no—
476#103—1222no—
477#103—2441no—
478#103C4×D6114yes—
479#103D24414yes—
480C22⋊C8—614no—
481#97—1222no—
482C4×C8—41no—
483#95—21no—
484#95—611no—
485#95—611no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491#95—1221no—
492#95—1221no—
493#95—1221no—
494C24—611no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—612no—
499C2×Q8—612no—
500C2×Q8—1224no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—1228no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8—1221no—
509C2×Q8—1221no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—2442no—
513C2×Q8—2442no—
514C2×Q8—2444no—
515C2×Q8—2444no—
516C2×Q8—2444no—
517C2×Q8C2×C4×D6112yes—
518C2×Q8C2×D24412yes—
519C2×Q8#1175822212yes—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—611no—
525C22×C4—612no—
526C22×C4—612no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—1224no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1221no—
536C22×C4—1221no—
537C22×C4—1221no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543Q16—1228no—
544Q16—2444no—
545M16—2444no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—612no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8C2×C4×D6112yes—
557C2×C8C2×D24412yes—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—42no—
563C4⋊C4—1224no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4C2×C4×D6112yes—
572C4⋊C4C2×D24412yes—
573C22⋊C4—1224no—
574C22⋊C4—1222no—
575C22⋊C4—2441no—
576C22⋊C4—2442no—
577C22⋊C4—2442no—
578C42—21no—
579C42—21no—
580C42—42no—
581C42—42no—
582C42—1221no—
583C42—1221no—
584C42—2442no—
585C42—2442no—
586C42—2442no—
587C23—612no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23C2×D4×D6111yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—22no—
598Q8—1222no—
599Q8—1222no—
600Q8—1222no—
601Q8—1222no—
602Q8—2448no—
603Q8—2441no—
604Q8—2441no—
605Q8—2441no—
606Q8—2441no—
607Q8C4×D24414yes—
608C2×C4—22no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—21no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—21no—
615C2×C4—21no—
616C2×C4—22no—
617C2×C4—21no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—612no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1221no—
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×C4C22×D24411yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4C2×#1175822211yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982311yescharacteristic
658C8—22no—
659C8—22no—
660C8—1222no—
661C8—2441no—
662C22—1222no—
663C22—1222no—
664C22—1222no—
665C22—2441no—
666C22—2441no—
667C22—2441no—
668C22—2441no—
669C22C4×D4×D6111yescharacteristic
670C22D4×D24411yescharacteristic
671C22#1684040211yescharacteristic
672C22C2×#1451980611yescharacteristic
673C22#1684055911yescharacteristic
674C22#1684057511yescharacteristic
675C22#1684057611yescharacteristic
676C4—24no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—41no—
681C4—81no—
682C4—1222no—
683C4—4881no—
684C4—4881no—
685C4—2441no—
686C4—2441no—
687C4—2442no—
688C4—2442no—
689C4C2×C4×D24411yescharacteristic
690C4#1684023411yescharacteristic
691C4#1684040711yescharacteristic
692C4#1684043611yescharacteristic
693C2—2442no—
694C2#2144880111yescharacteristic
695C2C4×#1451980611yescharacteristic
696C2#2145468211yescharacteristic
697C2#2145746611yescharacteristic
698C2#2145747511yescharacteristic
699C2#2145786611yescharacteristic
700C2#2145788111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC4×#103×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenterC4×#103D61
hypocenterC61#22869
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C4×#103

Complements

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

NH ≅ G/Ntype
2. C2×#16839866C2semidirect
6. #21446680C2semidirect
7. #21446510C2semidirect
8. #21446453C2semidirect
10. #21446161C2semidirect
11. #21446160C2semidirect
12. #21446075C2semidirect
15. C4×#103×C61C2semidirect
24. #16839919C4semidirect
356. C4×#103D61semidirect
412. C61#22869semidirect

Maximal subgroups

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

subgroupindex
C2×#168399192
C2×#168398662
C2×#168397692
#214488482
Q8×#117581982
#214466802
#214465102
#214464532
#214462512
#214461612
#214461602
#214460752
#214460212
#214456022
C4×#103×C612
#2286961