Finite Group Databasebeta

← #29700492

Subgroups — #29700492

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×#16839917C211yesmaximal, characteristic
2C2×#16839866C211yesmaximal, characteristic, complemented
3C2×#16839772C211yesmaximal, characteristic
4#21448850C211yesmaximal, characteristic
5#21448651C211yesmaximal, characteristic
6#21446677C211yesmaximal, characteristic, complemented
7#21446510C211yesmaximal, characteristic, complemented
8#21446451C211yesmaximal, characteristic, complemented
9#21446158C211yesmaximal, characteristic, complemented
10#21446157C211yesmaximal, characteristic, complemented
11#21446074C211yesmaximal, characteristic, complemented
12#21446016C211yesmaximal, characteristic
13#21445638C211yesmaximal, characteristic, complemented
14#21445600C211yesmaximal, characteristic
15C61×#1894C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C22×#11758202C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519736C2211yescharacteristic
23#16839917C418yescomplemented
24#16839866—24no—
25C2×#14519710C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27#16839772—24no—
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×C22⋊#92418C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C8×Dic61C2211yescharacteristic
37#16839669C2211yescharacteristic
38#16839667C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839666—21no—
41#16839666—21no—
42#16839661C2211yescharacteristic
43#16839655—21no—
44#16839655—21no—
45#16839648—21no—
46#16839620—21no—
47C4⋊C4×Dic61—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49C22⋊C4×Dic61—21no—
50C4×C22⋊Dic61C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×M16×C61C2211yescharacteristic
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#380C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122—22no—
63C22×Dic122—22no—
64C22×Dic122—21no—
65C22×Dic122—22no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122C2×C414yes—
69C22×Dic122C2311yescharacteristic
70C22×Dic122C2311yescharacteristic
71C23×Dic61—21no—
72C2×#11758202—22no—
73C2×#11758202—22no—
74C2×#11758202—41no—
75C2×#11758202—41no—
76C2×#11758202C2×C414yes—
77C2×#11758202C2×C414yes—
78C22×#92418C2311yescharacteristic
79C22×#92418C2311yescharacteristic
80Q8×Dic61—22no—
81Q8×Dic61—41no—
82#14519742—22no—
83#14519742—22no—
84#14519742—41no—
85#14519742—41no—
86#14519736—22no—
87#14519736—41no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61D411yescharacteristic
96C2×C22⋊Dic61D411yescharacteristic
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104#14519710—41no—
105#14519708—22no—
106#14519702—22no—
107#14519702—22no—
108#14519702—22no—
109#14519702D414yes—
110#14519700C2×C414yes—
111#14519700C2×C414yes—
112C22⋊#92418C2×C414yes—
113#14519687—22no—
114C8×Dic61—41no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684D411yescharacteristic
124#14519684D411yescharacteristic
125C22×Q8×C61C2311yescharacteristic
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61—22no—
128C2×M16×C61—22no—
129C2×M16×C61—41no—
130C22×C8×C61C2311yescharacteristic
131C22×C8×C61C2311yescharacteristic
132C22⋊Q8×C61—22no—
133C22⋊Q8×C61C2×C414yes—
134C2×C4⋊C4×C61—21no—
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C4⋊C4×C61C2311yescharacteristic
137C2×C4⋊C4×C61C2311yescharacteristic
138C2×C22⋊C4×C61—21no—
139C2×C22⋊C4×C61—21no—
140#103×C61—22no—
141#103×C61—41no—
142#95×C61—21no—
143C24×C61C2×D411yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C22×C412yes—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61C22×C412yes—
159C22×C4×C61C2×D411yescharacteristic
160C22×C4×C61#4112yes—
161C22×C4×C61C2411yescharacteristic
162C22×Dic61—21no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—42no—
168C22×Dic61—42no—
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—28no—
181C2×Dic122—24no—
182C2×Dic122—21no—
183C2×Dic122—44no—
184C2×Dic122—44no—
185C2×Dic122—44no—
186C2×Dic122—42no—
187C2×Dic122—42no—
188C2×Dic122C22×C412yes—
189C2×Dic122C22×C412yes—
190C2×Dic122C2×D412yes—
191C2×Dic122#4112yes—
192M16×C61—44no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61C2×D412yes—
199C4⋊C4×C61—24no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61C22×C412yes—
206C22⋊C4×C61—24no—
207C22⋊C4×C61—42no—
208C22⋊Dic61—22no—
209C22⋊Dic61—41no—
210C22⋊Dic61—42no—
211#11758202—28no—
212#11758202—44no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—42no—
217C4⋊Dic61—42no—
218C4⋊Dic61C2×D412yes—
219#11758196—42no—
220#11758196—42no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—42no—
226C4×Dic61—42no—
227C4×Dic61—42no—
228C4×Dic61—42no—
229C4×Dic61—42no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418C22×C412yes—
235C2×#92418C22×C412yes—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61—21no—
240C23×C61—21no—
241C23×C61—21no—
242C23×C61C4×D412yes—
243C23×C61C22×D411yescharacteristic
244Q8×C61—22no—
245Q8×C61—22no—
246Q8×C61—41no—
247Q8×C61—41no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—22no—
254C2×C4×C61—22no—
255C2×C4×C61—22no—
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—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61C4×D412yes—
267C2×C4×C61C23×C411yescharacteristic
268C2×C4×C61C22×D411yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×C4×C61H(5,𝔽2)11yescharacteristic
272C2×C4×C61H(5,𝔽2)11yescharacteristic
273C2×Dic61—22no—
274C2×Dic61—22no—
275C2×Dic61—21no—
276C2×Dic61—22no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—21no—
280C2×Dic61—21no—
281C2×Dic61—22no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—21no—
285C2×Dic61—21no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—21no—
289C2×Dic61—48no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61#13712yes—
298Dic122—22no—
299Dic122—22no—
300Dic122—22no—
301Dic122—22no—
302Dic122—22no—
303Dic122—48no—
304Dic122—41no—
305Dic122—41no—
306Dic122C4×D414yes—
307C8×C61—22no—
308C8×C61—41no—
309#92418—22no—
310#92418—22no—
311#22869—611nomaximal, Sylow
312C22×C61—22no—
313C22×C61—22no—
314C22×C61—22no—
315C22×C61—41no—
316C22×C61—41no—
317C22×C61—41no—
318C22×C61—41no—
319C22×C61C2×C4×D411yescharacteristic
320C22×C61D4211yescharacteristic
321C22×C61#54611yescharacteristic
322C22×C61C2×#13711yescharacteristic
323C22×C61#57511yescharacteristic
324C22×C61#57811yescharacteristic
325C22×C61#57811yescharacteristic
326C4×C61—22no—
327C4×C61—42no—
328C4×C61—42no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61C2×C4×D411yescharacteristic
332C4×C61#51811yescharacteristic
333C4×C61#54611yescharacteristic
334C4×C61#55111yescharacteristic
335Dic61—24no—
336Dic61—22no—
337Dic61—42no—
338Dic61—41no—
339Dic61—42no—
340Dic61—41no—
341Dic61—41no—
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#1894D6111yescharacteristic
352#1878—611no—
353#1878—611no—
354#1863—611no—
355#1854—611no—
356C4×#103—611no—
357#1753—611no—
358C2×C61—42no—
359C2×C61#235811yescharacteristic
360C2×C61C4×#13711yescharacteristic
361C2×C61#295011yescharacteristic
362C2×C61#327611yescharacteristic
363C2×C61#327711yescharacteristic
364C2×C61#331211yescharacteristic
365C2×C61#331311yescharacteristic
366C23×Q8—611no—
367C22×Q16—611no—
368C22×Q16—611no—
369C22×M16C2×D6111yescharacteristic
370C2×C22⋊Q8C2×D6111yescharacteristic
371C2×C4×Q8—611no—
372C2×C4×Q8—611no—
373C2×C4×Q8—611no—
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×C8—611no—
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—1221no—
401#380C2×D6111yescharacteristic
402C4×C4⋊C4—611no—
403C4×C4⋊C4—1221no—
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—1222no—
417C22×Q8—1222no—
418C22×Q8—1221no—
419C22×Q8—1222no—
420C22×Q8—2441no—
421C22×Q8—2441no—
422C22×Q8C22×D6111yescharacteristic
423C23×C4—1221no—
424C23×C4C22×D6111yescharacteristic
425C2×Q16—614no—
426C2×Q16—614no—
427C2×Q16—1222no—
428C2×Q16—1222no—
429C2×Q16—2441no—
430C2×Q16—2441no—
431C2×M16—22no—
432C2×M16—22no—
433C2×M16—41no—
434C22×C8—611no—
435C22×C8—611no—
436C22×C8C22×D6111yescharacteristic
437C22×C8C22×D6111yescharacteristic
438C22⋊Q8—22no—
439C22⋊Q8C4×D6114yes—
440C4×Q8—1222no—
441C4×Q8—1222no—
442C4×Q8—1222no—
443C4×Q8—2441no—
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—21no—
459C2×C22⋊C4—21no—
460C2×C22⋊C4—611no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—1221no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—1221no—
466C2×C42—1221no—
467C2×C42—1221no—
468C2×C42—1221no—
469C2×C42—1221no—
470#107—2441no—
471#106—1222no—
472#103—22no—
473#103—41no—
474#103—614no—
475#103—614no—
476#103—614no—
477#103—1222no—
478#103—1222no—
479#103—1222no—
480C22⋊C8—614no—
481#97—1222no—
482C4×C8—2441no—
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—
494C24D4×D6111yescharacteristic
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—21no—
498C2×Q8—21no—
499C2×Q8—21no—
500C2×Q8—612no—
501C2×Q8—612no—
502C2×Q8—612no—
503C2×Q8—612no—
504C2×Q8—1228no—
505C2×Q8—1221no—
506C2×Q8—1221no—
507C2×Q8—1221no—
508C2×Q8—1221no—
509C2×Q8—1224no—
510C2×Q8—1221no—
511C2×Q8—1221no—
512C2×Q8—1221no—
513C2×Q8—1221no—
514C2×Q8—2442no—
515C2×Q8—2442no—
516C2×Q8—2444no—
517C2×Q8—2444no—
518C2×Q8—2444no—
519C2×Q8C2×C4×D6112yes—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—21no—
523C22×C4—21no—
524C22×C4—21no—
525C22×C4—21no—
526C22×C4—21no—
527C22×C4—21no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—611no—
531C22×C4—611no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4—1224no—
536C22×C4—1221no—
537C22×C4—2442no—
538C22×C4—2442no—
539C22×C4C2×C4×D6112yes—
540C22×C4D4×D6111yescharacteristic
541C22×C4#1175822412yes—
542C22×C4C23×D6111yescharacteristic
543Q16—1228no—
544Q16—2444no—
545M16—44no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—21no—
551C2×C8—612no—
552C2×C8—612no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8—1221no—
557C2×C8D4×D6112yes—
558C4⋊C4—21no—
559C4⋊C4—24no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—21no—
563C4⋊C4—21no—
564C4⋊C4—612no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—2442no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C4⋊C4C2×C4×D6112yes—
573C22⋊C4—24no—
574C22⋊C4—42no—
575C22⋊C4—1222no—
576C22⋊C4—2441no—
577C22⋊C4—2442no—
578C42—1221no—
579C42—1221no—
580C42—1221no—
581C42—1221no—
582C42—2442no—
583C42—2442no—
584C42—2442no—
585C42—2442no—
586C42—2442no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—21no—
591C23—21no—
592C23—21no—
593C23#1451973712yes—
594C23C2×D4×D6111yescharacteristic
595Q8—22no—
596Q8—22no—
597Q8—41no—
598Q8—41no—
599Q8—614no—
600Q8—1222no—
601Q8—1222no—
602Q8—1222no—
603Q8—1222no—
604Q8—1222no—
605Q8—2448no—
606Q8—2441no—
607Q8—2441no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—612no—
627C2×C4—1222no—
628C2×C4—1221no—
629C2×C4—1222no—
630C2×C4—1221no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—1221no—
638C2×C4—1221no—
639C2×C4—1221no—
640C2×C4—1221no—
641C2×C4—1222no—
642C2×C4—1221no—
643C2×C4—2448no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4#1451973712yes—
652C2×C4C22×C4×D6111yescharacteristic
653C2×C4C2×D4×D6111yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4C2×#1175822411yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982311yescharacteristic
658C8—22no—
659C8—41no—
660C8—1222no—
661C8—1222no—
662C22—22no—
663C22—22no—
664C22—22no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22C2×#1451973711yescharacteristic
670C22#1684039611yescharacteristic
671C22#1684040111yescharacteristic
672C22#137×D6111yescharacteristic
673C22#1684055811yescharacteristic
674C22#1684057811yescharacteristic
675C22#1684057811yescharacteristic
676C4—22no—
677C4—42no—
678C4—42no—
679C4—81no—
680C4—81no—
681C4—1224no—
682C4—1222no—
683C4—4881no—
684C4—2441no—
685C4—2442no—
686C4—2441no—
687C4—2442no—
688C4—2441no—
689C4C4×D4×D6111yescharacteristic
690C4#1684023411yescharacteristic
691C4#1684039911yescharacteristic
692C4#1684043511yescharacteristic
693C2—42no—
694C2#2144879911yescharacteristic
695C2#2145464311yescharacteristic
696C2#2145468311yescharacteristic
697C2#2145746211yescharacteristic
698C2#2145747311yescharacteristic
699C2#2145786511yescharacteristic
700C2#2145788011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1894C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1894D61
hypocenterC61#22869
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
2. C2×#16839866C2semidirect
6. #21446677C2semidirect
7. #21446510C2semidirect
8. #21446451C2semidirect
9. #21446158C2semidirect
10. #21446157C2semidirect
11. #21446074C2semidirect
13. #21445638C2semidirect
23. #16839917C4semidirect
412. C61#22869semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168398662
C2×#168397722
#214488502
#214486512
#214466772
#214465102
#214464512
#214461582
#214461572
#214460742
#214460162
#214456382
#214456002
C61×#18942
#2286961