Finite Group Databasebeta

← #30617798

Subgroups — #30617798

Order 15616.

1442 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840786C212yesmaximal, complemented
2#21461944C212yesmaximal, complemented
3#21461943C212yesmaximal, complemented
4#21461942C212yesmaximal, complemented
5#21461940C212yesmaximal, complemented
6#21461643C211yesmaximal, characteristic, complemented
7#21461643C211yesmaximal, characteristic, complemented
8#21461641C212yesmaximal, complemented
9C2×#16840268C211yesmaximal, characteristic
10#16840786—28no—
11C2×#14519844C2212yescomplemented
12C2×C4⋊#92420C2212yescomplemented
13C2×#14519834C2211yescharacteristic, complemented
14C2×#14519834C2212yescomplemented
15C2×#14519834C2211yescharacteristic, complemented
16C2×C22⋊#92420C2212yes—
17C2×C22⋊#92420C2212yes—
18C2×C22⋊#92420C2212yes—
19C2×C22⋊#92420C2212yes—
20C2×C22⋊#92420C2212yes—
21C2×C22⋊#92420C2212yes—
22C2×C22⋊#92420C2211yescharacteristic, complemented
23C2×C22⋊#92420C2212yescomplemented
24C2×C22⋊#92420C2211yescharacteristic, complemented
25C2×#14519832C2212yescomplemented
26C2×C4×#92420C2212yescomplemented
27C22×#11758224C2211yescharacteristic
28C24×Dic61C2211yescharacteristic
29#16840268—28no—
30C2×#14519761C411yescharacteristic
31C2×#14519758C411yescharacteristic
32C2×#14519754C412yes—
33C2×#14519749C412yes—
34C2×#14519746C411yescharacteristic
35C2×D4×Dic61C2212yes—
36C2×#14519737C2212yes—
37C2×#14519733C2211yescharacteristic
38C2×#120×C61C411yescharacteristic
39C22×#92420C2312yes—
40C22×#92420C2312yes—
41C22×#92420C2312yes—
42C22×#92420C2312yes—
43#14519844—24no—
44C4⋊#92420—24no—
45#14519834—41no—
46#14519834—42no—
47#14519834—41no—
48C22⋊#92420—24no—
49C22⋊#92420—24no—
50C22⋊#92420—42no—
51C22⋊#92420—42no—
52C22⋊#92420—42no—
53C22⋊#92420—42no—
54C22⋊#92420—41no—
55C22⋊#92420—42no—
56C22⋊#92420—41no—
57#14519832—24no—
58C4×#92420—24no—
59C2×#11758224—24no—
60C2×#11758224—42no—
61C2×#11758224—41no—
62C2×#11758224—41no—
63C22×Dic122C2×C411yescharacteristic
64C22×C22⋊D61C2×C412yes—
65C22×C22⋊D61C2×C411yescharacteristic
66C22×C22⋊D61C2×C411yescharacteristic
67C23×Dic61—24no—
68C23×Dic61—41no—
69C23×Dic61—41no—
70C23×Dic61—42no—
71C23×Dic61C2×C412yes—
72C23×Dic61C2×C411yescharacteristic
73C23×Dic61C2312yes—
74C23×Dic61C2311yescharacteristic
75C23×Dic61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77#14519761—24no—
78#14519758—24no—
79#14519754—28no—
80#14519749—28no—
81#14519746—24no—
82D4×Dic61—24no—
83D4×Dic61—24no—
84#14519737—24no—
85#14519737—24no—
86#14519733D418yes—
87C2×C4⋊Dic61C2×C412yes—
88C2×#11758196C2×C412yes—
89C2×#11758196C2×C412yes—
90C2×C22⋊Dic61C2×C412yes—
91C2×C22⋊Dic61C2×C412yes—
92C2×C22⋊Dic61C2×C411yescharacteristic
93C2×#11758198C2×C412yes—
94C2×C4×Dic61C2312yes—
95C25×C61C2×C411yescharacteristic
96C22×D4×C61C2×C412yes—
97C22×D4×C61C2×C411yescharacteristic
98#120×C61—24no—
99C2×C22⋊C4×C61C2×C412yes—
100C2×C22⋊C4×C61C2×C411yescharacteristic
101C24×C61—22no—
102C24×C61—24no—
103C24×C61—41no—
104C24×C61—41no—
105C24×C61—42no—
106C24×C61C22×C411yescharacteristic
107C24×C61C22×C411yescharacteristic
108C24×C61C22×C412yes—
109C23×D61C22×C411yescharacteristic
110C2×D4×C61—22no—
111C2×D4×C61—24no—
112C2×D4×C61—24no—
113C2×D4×C61—41no—
114C2×D4×C61—41no—
115C2×D4×C61—42no—
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C412yes—
118C2×C22⋊D61—22no—
119C2×C22⋊D61—24no—
120C2×C22⋊D61—22no—
121C2×C22⋊D61—24no—
122C2×C22⋊D61—42no—
123C2×C22⋊D61—41no—
124C2×C22⋊D61—41no—
125C2×C22⋊D61—41no—
126C2×C22⋊D61—41no—
127C22×Dic61—22no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—22no—
131C22×Dic61—24no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—22no—
135C22×Dic61—22no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61—22no—
139C22×Dic61—22no—
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—42no—
143C22×Dic61—42no—
144C22×Dic61—41no—
145C22×Dic61—41no—
146C22×Dic61—48no—
147C22×Dic61—48no—
148C22×Dic61C22×C412yes—
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C411yescharacteristic
151C22×Dic61C22×C412yes—
152C22×Dic61C2×D414yes—
153C22×Dic61C2411yescharacteristic
154#11758224—48no—
155#11758224—48no—
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×Dic122—22no—
161C2×Dic122—41no—
162C2×Dic122—41no—
163C2×#92420—22no—
164C2×#92420—22no—
165C2×#92420—22no—
166C2×#92420—22no—
167C2×#92420—22no—
168C2×#92420—22no—
169C2×#92420—22no—
170C2×#92420—22no—
171C2×#92420—22no—
172C2×#92420—22no—
173C2×#92420—22no—
174C2×#92420—22no—
175C22⋊C4×C61—24no—
176C22⋊C4×C61C22⋊C414yes—
177C22⋊Dic61—24no—
178C22⋊Dic61—24no—
179C22⋊Dic61C22⋊C414yes—
180#11758198—24no—
181C4⋊Dic61—24no—
182#11758196—24no—
183#11758196C22⋊C418yes—
184C4×Dic61C2×D414yes—
185C4×Dic61C2×D414yes—
186C23×C61—22no—
187C23×C61—21no—
188C23×C61—21no—
189C23×C61—22no—
190C23×C61—22no—
191C23×C61—22no—
192C23×C61—22no—
193C23×C61—22no—
194C23×C61—22no—
195C23×C61—22no—
196C23×C61—21no—
197C23×C61—21no—
198C23×C61—21no—
199C23×C61—22no—
200C23×C61—22no—
201C23×C61—48no—
202C23×C61—44no—
203C23×C61—44no—
204C23×C61—48no—
205C23×C61—44no—
206C23×C61C2×C22⋊C412yes—
207C23×C61#13114yes—
208C23×C61C23×C411yescharacteristic
209C22×D61—21no—
210C22×D61—21no—
211C22×D61—21no—
212C22×D61—22no—
213C22×D61—22no—
214D4×C61—48no—
215D4×C61—44no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61C2×C22⋊C414yes—
222C22⋊D61—48no—
223C22⋊D61—44no—
224C22⋊D61—44no—
225C2×Dic61—21no—
226C2×Dic61—21no—
227C2×Dic61—22no—
228C2×Dic61—22no—
229C2×Dic61—24no—
230C2×Dic61—24no—
231C2×Dic61—22no—
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—48no—
238C2×Dic61—44no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244C2×Dic61—42no—
245C2×Dic61C2×C22⋊C414yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C22×D412yes—
248C2×Dic61H(5,𝔽2)12yes—
249C2×Dic61H(5,𝔽2)11yescharacteristic
250C2×Dic61H(5,𝔽2)11yescharacteristic
251C4×D61—42no—
252C4×D61—42no—
253Dic122—44no—
254#92420—42no—
255#92420—42no—
256#92420—42no—
257#92420—42no—
258#21792—611nomaximal, Sylow
259C22×C61—22no—
260C22×C61—24no—
261C22×C61—24no—
262C22×C61—24no—
263C22×C61—24no—
264C22×C61—416no—
265C22×C61—42no—
266C22×C61—42no—
267C22×C61—42no—
268C22×C61—42no—
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—42no—
272C22×C61—41no—
273C22×C61—42no—
274C22×C61—42no—
275C22×C61—41no—
276C22×C61—41no—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61#51811yescharacteristic
279C22×C61#51811yescharacteristic
280C22×C61C2×#13112yes—
281C22×C61#56812yes—
282C2×D61—41no—
283C2×D61—42no—
284C2×D61—42no—
285C2×D61—41no—
286C2×D61—41no—
287C4×C61—24no—
288C4×C61—41no—
289Dic61—22no—
290Dic61—24no—
291Dic61—41no—
292Dic61—42no—
293Dic61#53512yes—
294Dic61#53512yes—
295C2×#525—611no—
296C2×#433—612no—
297#1955—612no—
298#1955—612no—
299#1954—612no—
300#1954—612no—
301#1780—611no—
302#1780—611no—
303#1779—612no—
304C2×C61—42no—
305C2×C61—44no—
306C2×C61—81no—
307C2×C61#230911yescharacteristic
308C2×C61#288012yes—
309C2×C61#297612yes—
310C2×C61#297711yescharacteristic
311C2×C61#297811yescharacteristic
312D61—81no—
313C22×#41—611no—
314C24×C4—611no—
315#525—1228no—
316C2×#123—611no—
317C2×#123—611no—
318C2×C22⋊Q8—612no—
319C2×#121—612no—
320C2×#120—611no—
321C2×#120#692311yescharacteristic
322C2×C4×D4—612no—
323C2×C4×D4—612no—
324C2×#117—611no—
325#433—1228no—
326C2×C4⋊C8—612no—
327C2×C4⋊C8—612no—
328C2×C22⋊C8—612no—
329C2×C22⋊C8—612no—
330C2×C22⋊C8—612no—
331C2×C22⋊C8—612no—
332C2×C22⋊C8—612no—
333C2×C22⋊C8—612no—
334C2×C22⋊C8—611no—
335C2×C22⋊C8—612no—
336C2×C22⋊C8—611no—
337C2×C22⋊C8—611no—
338C2×C22⋊C8—612no—
339C2×C22⋊C8—611no—
340C2×#97—612no—
341C2×C4×C8—612no—
342C61#2179211yescharacteristic, complemented, Sylow
343C25C2×#692311yescharacteristic
344C2×#41—1224no—
345C2×#41—2442no—
346C2×#41—2441no—
347C2×#41—2441no—
348C22×Q8—611no—
349C22×D4—612no—
350C22×D4—611no—
351C22×D4—611no—
352C22×D4C2×#692312yes—
353C22×D4C2×#692311yescharacteristic
354C23×C4—611no—
355C23×C4—612no—
356C23×C4—611no—
357C23×C4—612no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4—1224no—
361C23×C4—2441no—
362C23×C4—2441no—
363C23×C4—2442no—
364C22×C8—612no—
365C22×C8—612no—
366C22×C8—612no—
367C22×C8—612no—
368#123—1224no—
369#123—1224no—
370C22⋊Q8—1228no—
371#121—1228no—
372#120—24no—
373#120—1224no—
374C4×D4—1224no—
375C4×D4—1224no—
376C4×D4—1224no—
377C4×D4—1224no—
378#117—618no—
379C2×C4⋊C4—612no—
380C2×C4⋊C4—612no—
381C2×C4⋊C4—612no—
382C2×C22⋊C4—611no—
383C2×C22⋊C4—612no—
384C2×C22⋊C4—612no—
385C2×C22⋊C4—612no—
386C2×C22⋊C4C2×#692312yes—
387C2×C22⋊C4C2×#692311yescharacteristic
388C2×C42—612no—
389C4⋊C8—1224no—
390C4⋊C8—1224no—
391C22⋊C8—1224no—
392C22⋊C8—1224no—
393C22⋊C8—2442no—
394C22⋊C8—2442no—
395C22⋊C8—2442no—
396C22⋊C8—2442no—
397C22⋊C8—2441no—
398C22⋊C8—2442no—
399C22⋊C8—2441no—
400C22⋊C8—2441no—
401C22⋊C8—2442no—
402C22⋊C8—2441no—
403#97—1224no—
404C4×C8—1224no—
405C24—22no—
406C24—24no—
407C24—41no—
408C24—41no—
409C24—42no—
410C24—611no—
411C24C22×#692311yescharacteristic
412C24C22×#692311yescharacteristic
413C24C22×#692312yes—
414#41—2448no—
415#41—2448no—
416C2×Q8—1222no—
417C2×Q8—2441no—
418C2×Q8—2441no—
419C2×D4—22no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—41no—
423C2×D4—41no—
424C2×D4—42no—
425C2×D4—1224no—
426C2×D4—1222no—
427C2×D4—1222no—
428C2×D4—1224no—
429C2×D4—2441no—
430C2×D4—2441no—
431C2×D4—2441no—
432C2×D4—2441no—
433C2×D4—2442no—
434C22×C4—611no—
435C22×C4—614no—
436C22×C4—612no—
437C22×C4—611no—
438C22×C4—612no—
439C22×C4—611no—
440C22×C4—1222no—
441C22×C4—1221no—
442C22×C4—1221no—
443C22×C4—1222no—
444C22×C4—1222no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1222no—
448C22×C4—1222no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—1222no—
459C22×C4—2442no—
460C22×C4—2442no—
461C22×C4—2441no—
462C22×C4—2441no—
463C22×C4—2448no—
464C22×C4—2448no—
465C22×C4C22×#692311yescharacteristic
466C22×C4C22×#692312yes—
467C2×C8—1222no—
468C2×C8—1222no—
469C2×C8—1222no—
470C2×C8—1222no—
471C2×C8—1222no—
472C2×C8—1222no—
473C2×C8—1222no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C4⋊C4—618no—
480C4⋊C4—1224no—
481C4⋊C4—1224no—
482C22⋊C4—24no—
483C22⋊C4—614no—
484C22⋊C4—1224no—
485C22⋊C4—1224no—
486C22⋊C4—1224no—
487C22⋊C4C22⋊#692314yes—
488C42—614no—
489C42—614no—
490C23—22no—
491C23—21no—
492C23—21no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—22no—
504C23—22no—
505C23—48no—
506C23—44no—
507C23—44no—
508C23—48no—
509C23—44no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1222no—
514C23—1222no—
515C23C2×C22⋊#692312yes—
516C23#1451986414yes—
517C23C23×#692311yescharacteristic
518Q8—2444no—
519D4—48no—
520D4—44no—
521D4—2448no—
522D4—2444no—
523D4—2444no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—612no—
530C2×C4—612no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—614no—
534C2×C4—612no—
535C2×C4—1224no—
536C2×C4—1224no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1221no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2442no—
548C2×C4—2442no—
549C2×C4—2442no—
550C2×C4—2442no—
551C2×C4—2442no—
552C2×C4—2442no—
553C2×C4—2442no—
554C2×C4—2442no—
555C2×C4—2448no—
556C2×C4—2444no—
557C2×C4C2×C22⋊#692314yes—
558C8—2442no—
559C8—2442no—
560C8—2442no—
561C8—2442no—
562C22—22no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—24no—
567C22—416no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—42no—
572C22—41no—
573C22—41no—
574C22—42no—
575C22—41no—
576C22—42no—
577C22—42no—
578C22—41no—
579C22—41no—
580C22—2441no—
581C22—2442no—
582C22—2442no—
583C22—2441no—
584C22—2441no—
585C22C22×C22⋊#692311yescharacteristic
586C22#1684082911yescharacteristic
587C22#1684082911yescharacteristic
588C22C2×#1451986412yes—
589C22#1684084012yes—
590C4—24no—
591C4—41no—
592C4—612no—
593C4—612no—
594C4—1222no—
595C4—1224no—
596C4—2442no—
597C4—2441no—
598C2—42no—
599C2—44no—
600C2—81no—
601C2—4881no—
602C2#2146229211yescharacteristic
603C2#2146240712yes—
604C2#2146268512yes—
605C2#2146269111yescharacteristic
606C2#2146269511yescharacteristic
subgroupG/H
center, FrattiniC23C23×#6923
commutator, socleC23×C61C23×C4
FittingC2×#120×C61C4
prime radicalC2×#16840268C2
cosocleC22×Dic61C24
hypercenterC2×#120#6923
hypocenterC61#21792
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#16840786C2semidirect
2. #21461944C2semidirect
3. #21461943C2semidirect
4. #21461942C2semidirect
5. #21461940C2semidirect
6. #21461643C2semidirect
7. #21461643C2semidirect
8. #21461641C2semidirect
11. C2×#14519844C22semidirect
12. C2×C4⋊#92420C22semidirect
13. C2×#14519834C22semidirect
14. C2×#14519834C22semidirect
15. C2×#14519834C22semidirect
22. C2×C22⋊#92420C22semidirect
23. C2×C22⋊#92420C22semidirect
24. C2×C22⋊#92420C22semidirect
25. C2×#14519832C22semidirect
26. C2×C4×#92420C22semidirect
342. C61#21792semidirect

Maximal subgroups

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

subgroupindex
C2×#168407862
#214619442
#214619432
#214619422
#214619402
#214616432
#214616412
C2×#168402682
#2179261