Finite Group Databasebeta

← #29691324

Subgroups — #29691324

Order 15616.

920 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839918C211yesmaximal, characteristic
2#21448501C211yesmaximal, characteristic
3#21446689C211yesmaximal, characteristic, complemented
4#21446679C211yesmaximal, characteristic, complemented
5#21446512C211yesmaximal, characteristic, complemented
6#21446498C211yesmaximal, characteristic, complemented
7#21446457C211yesmaximal, characteristic, complemented
8#21446450C211yesmaximal, characteristic, complemented
9#21446163C211yesmaximal, characteristic, complemented
10#21446163C211yesmaximal, characteristic, complemented
11#21446131C211yesmaximal, characteristic
12#21446018C211yesmaximal, characteristic
13#21445754C211yesmaximal, characteristic
14#21445585C211yesmaximal, characteristic
15C61×#2310C211yesmaximal, characteristic
16C22×#11758202C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20#16839918C418yescomplemented
21C2×#14519710C2211yescharacteristic
22C2×#14519708C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34C2×C4×#92418C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839657—21no—
38#16839656—21no—
39#16839650—21no—
40#16839643—21no—
41#16839622C2211yescharacteristic
42C4×C4⋊Dic61C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C23×Q8×C61C2211yescharacteristic
50C2×#117×C61C2211yescharacteristic
51C2×#117×C61C2211yescharacteristic
52C61×#391—21no—
53C61×#391—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#391C2211yescharacteristic
58C61×#381—21no—
59C61×#381—21no—
60C22×Dic122C2311yescharacteristic
61C2×#11758202—41no—
62C2×#11758202—41no—
63C2×#11758202—41no—
64C2×#11758202—41no—
65C2×#11758202C2×C414yes—
66C2×#11758202C2×C414yes—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519756—22no—
72#14519756C2×C414yes—
73C4×Dic122—41no—
74C4×Dic122—41no—
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×C22⋊Dic61—21no—
82C2×C22⋊Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85#14519710—41no—
86#14519708—41no—
87#14519701—41no—
88#14519701—41no—
89#14519701—41no—
90#14519701—41no—
91#14519701C2×C414yes—
92#14519701C2×C414yes—
93#14519700—41no—
94#14519700—41no—
95C22⋊#92418—22no—
96C22⋊#92418C2×C414yes—
97#14519688—41no—
98C4×#92418—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104C22×Q8×C61—21no—
105C22×Q8×C61—22no—
106C22×Q8×C61—41no—
107C22×Q8×C61—41no—
108C22×Q8×C61—41no—
109C22×Q8×C61—41no—
110C22×Q8×C61C2×C414yes—
111C22×Q8×C61C2311yescharacteristic
112C22×Q8×C61C2311yescharacteristic
113C23×C4×C61—21no—
114C23×C4×C61C2311yescharacteristic
115#117×C61—41no—
116#117×C61—41no—
117#117×C61—41no—
118#117×C61—41no—
119C2×C4⋊C4×C61D411yescharacteristic
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61D411yescharacteristic
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128C2×C42×C61C2311yescharacteristic
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133#95×C61—21no—
134#95×C61—21no—
135#95×C61—21no—
136#95×C61—21no—
137C24×C61C2×D411yescharacteristic
138C2×Q8×C61—24no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61—44no—
150C2×Q8×C61—44no—
151C2×Q8×C61—42no—
152C2×Q8×C61—42no—
153C2×Q8×C61—44no—
154C2×Q8×C61—44no—
155C2×Q8×C61C22×C412yes—
156C2×Q8×C61C22×C412yes—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—21no—
160C22×C4×C61—21no—
161C22×C4×C61—24no—
162C22×C4×C61—21no—
163C22×C4×C61—21no—
164C22×C4×C61—21no—
165C22×C4×C61—21no—
166C22×C4×C61—42no—
167C22×C4×C61—42no—
168C22×C4×C61C22×C412yes—
169C22×C4×C61C2×D411yescharacteristic
170C22×C4×C61C2×D411yescharacteristic
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2×D411yescharacteristic
174C22×C4×C61#4112yes—
175C22×C4×C61C2411yescharacteristic
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61—21no—
179C22×Dic61—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122C22×C412yes—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—41no—
191C4⋊C4×C61—41no—
192C22⋊C4×C61—41no—
193C22⋊C4×C61—41no—
194C22⋊C4×C61—42no—
195C42×C61—21no—
196C42×C61—21no—
197C42×C61—21no—
198C42×C61—21no—
199C42×C61—42no—
200C22⋊Dic61—24no—
201C22⋊Dic61—81no—
202#11758202—44no—
203#11758202—44no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61C22×C412yes—
210#11758196—24no—
211#11758196—42no—
212#11758196—81no—
213C4×Dic61—42no—
214C4×Dic61—81no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C22×C412yes—
226C2×#92418C22×C412yes—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61C4×D412yes—
234C23×C61C22×D411yescharacteristic
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—48no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61—41no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—21no—
263C2×C4×C61—21no—
264C2×C4×C61—48no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61—41no—
275C2×C4×C61C4×D412yes—
276C2×C4×C61C23×C411yescharacteristic
277C2×C4×C61C2×#4111yescharacteristic
278C2×C4×C61H(5,𝔽2)11yescharacteristic
279C2×C4×C61H(5,𝔽2)11yescharacteristic
280C2×C4×C61H(5,𝔽2)11yescharacteristic
281C2×C4×C61H(5,𝔽2)11yescharacteristic
282C2×Dic61—22no—
283C2×Dic61—22no—
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×Dic61—41no—
293C2×Dic61—41no—
294Dic122—22no—
295Dic122—22no—
296Dic122—41no—
297Dic122—41no—
298#92418—22no—
299#92418—22no—
300#92418—41no—
301#92418—41no—
302#22054—611nomaximal, Sylow
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61C2×C4×D411yescharacteristic
311C22×C61#53411yescharacteristic
312C22×C61#53511yescharacteristic
313C22×C61#57711yescharacteristic
314C22×C61#57811yescharacteristic
315C22×C61#57811yescharacteristic
316C22×C61#57811yescharacteristic
317C4×C61—22no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61#51811yescharacteristic
327C4×C61#51811yescharacteristic
328C4×C61#53811yescharacteristic
329C4×C61#53811yescharacteristic
330Dic61—42no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334C2×#451—611no—
335#2323—611no—
336#2310D6111yescharacteristic
337#1971—611no—
338#1969—611no—
339#1935—611no—
340#1933—611no—
341#1928—611no—
342#1927—611no—
343#1878—611no—
344#1878—611no—
345#1873—611no—
346#1854—611no—
347#1787—611no—
348#1750—611no—
349C2×C61—42no—
350C2×C61#232111yescharacteristic
351C2×C61#294711yescharacteristic
352C2×C61#295011yescharacteristic
353C2×C61#320111yescharacteristic
354C2×C61#320211yescharacteristic
355C2×C61#322311yescharacteristic
356C2×C61#322511yescharacteristic
357C23×Q8C2×D6111yescharacteristic
358C22×Q16—611no—
359C22×Q16—611no—
360C2×C22⋊Q8—611no—
361C2×C4×Q8—611no—
362C2×#117C2×D6111yescharacteristic
363C2×#117C2×D6111yescharacteristic
364#451—618no—
365C2×#107—611no—
366C2×#106—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×C22⋊C8—611no—
376C2×C22⋊C8—611no—
377C2×#97—611no—
378C2×C4×C8—611no—
379#391—21no—
380#391—21no—
381#391—611no—
382#391C2×D6111yescharacteristic
383#391C2×D6111yescharacteristic
384#391C2×D6111yescharacteristic
385#391C2×D6111yescharacteristic
386#389—611no—
387#389—1221no—
388#388—1221no—
389#388—1221no—
390#387—1221no—
391#381—21no—
392#381—21no—
393#380—611no—
394C4×C4⋊C4—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#336—611no—
400#336—611no—
401C61#2205411yescharacteristic, complemented, Sylow
402C22×Q8—22no—
403C22×Q8—21no—
404C22×Q8—41no—
405C22×Q8—41no—
406C22×Q8—41no—
407C22×Q8—41no—
408C22×Q8—611no—
409C22×Q8C4×D6114yes—
410C22×Q8C22×D6111yescharacteristic
411C22×Q8C22×D6111yescharacteristic
412C23×C4—21no—
413C23×C4C22×D6111yescharacteristic
414C2×Q16—614no—
415C2×Q16—614no—
416C2×Q16—2441no—
417C2×Q16—2441no—
418C2×Q16—2441no—
419C2×Q16—2441no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8—611no—
423C22×C8—611no—
424C22⋊Q8—614no—
425C22⋊Q8—1222no—
426C4×Q8—2441no—
427C4×Q8—2441no—
428#117—41no—
429#117—41no—
430#117—41no—
431#117—41no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22⋊D6111yescharacteristic
439C2×C4⋊C4C22⋊D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4C22⋊D6111yescharacteristic
446C2×C22⋊C4C22⋊D6111yescharacteristic
447C2×C42—21no—
448C2×C42—1221no—
449C2×C42—1221no—
450C2×C42C22×D6111yescharacteristic
451C2×C42C22×D6111yescharacteristic
452#107—2441no—
453#106—2441no—
454#103—614no—
455#103—614no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459#103—2441no—
460#103—2441no—
461#103—2441no—
462C22⋊C8—614no—
463C22⋊C8—1222no—
464#97—2441no—
465C4×C8—2441no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24C2×C22⋊D6111yescharacteristic
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—44no—
492C2×Q8—44no—
493C2×Q8—42no—
494C2×Q8—42no—
495C2×Q8—44no—
496C2×Q8—44no—
497C2×Q8—612no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—1221no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8C2×C4×D6112yes—
504C2×Q8C2×C4×D6112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—24no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—42no—
515C22×C4—42no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4C2×C4×D6112yes—
521C22×C4#1175822212yes—
522C22×C4C2×C22⋊D6111yescharacteristic
523C22×C4C2×C22⋊D6111yescharacteristic
524C22×C4C2×C22⋊D6111yescharacteristic
525C22×C4C2×C22⋊D6111yescharacteristic
526C22×C4C2×C22⋊D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528Q16—2444no—
529Q16—2444no—
530C2×C8—612no—
531C2×C8—612no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—41no—
547C4⋊C4—41no—
548C4⋊C4—612no—
549C4⋊C4—1221no—
550C4⋊C4—1224no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—4881no—
556C4⋊C4—2442no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—42no—
560C22⋊C4—1224no—
561C22⋊C4—4881no—
562C42—21no—
563C42—21no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—4881no—
568C42—2442no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23C4×C22⋊D6112yes—
576C23C22×C22⋊D6111yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—22no—
580Q8—22no—
581Q8—48no—
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—1222no—
587Q8—1222no—
588Q8—2441no—
589Q8—2441no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—48no—
611C2×C4—41no—
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—1222no—
622C2×C4—1222no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4C4×C22⋊D6112yes—
634C2×C4C22×C4×D6111yescharacteristic
635C2×C4C2×#1175822211yescharacteristic
636C2×C4#1451982211yescharacteristic
637C2×C4#1451982211yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982211yescharacteristic
640C8—1222no—
641C8—1222no—
642C8—2441no—
643C8—2441no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22—41no—
651C22C2×C4×C22⋊D6111yescharacteristic
652C22#1684032111yescharacteristic
653C22#1684032911yescharacteristic
654C22#1684057011yescharacteristic
655C22#1684057711yescharacteristic
656C22#1684057711yescharacteristic
657C22#1684057711yescharacteristic
658C4—22no—
659C4—42no—
660C4—42no—
661C4—41no—
662C4—41no—
663C4—41no—
664C4—41no—
665C4—81no—
666C4—81no—
667C4—4881no—
668C4—4881no—
669C4—2442no—
670C4—2442no—
671C4#1684023511yescharacteristic
672C4#1684023511yescharacteristic
673C4#1684035011yescharacteristic
674C4#1684035011yescharacteristic
675C2—42no—
676C2#2144847711yescharacteristic
677C2#2145466211yescharacteristic
678C2#2145468511yescharacteristic
679C2#2145670111yescharacteristic
680C2#2145671011yescharacteristic
681C2#2145690211yescharacteristic
682C2#2145692811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2310C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#2310D61
hypocenterC61#22054
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
3. #21446689C2semidirect
4. #21446679C2semidirect
5. #21446512C2semidirect
6. #21446498C2semidirect
7. #21446457C2semidirect
8. #21446450C2semidirect
9. #21446163C2semidirect
10. #21446163C2semidirect
20. #16839918C4semidirect
401. C61#22054semidirect

Maximal subgroups

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

subgroupindex
C2×#168399182
#214485012
#214466892
#214466792
#214465122
#214464982
#214464572
#214464502
#214461632
#214461312
#214460182
#214457542
#214455852
C61×#23102
#2205461