Finite Group Databasebeta

← #30610734

Subgroups — #30610734

Order 15616.

1044 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic
2C2×C4×#11758217C211yesmaximal, characteristic
3#21461167C214yesmaximal, complemented
4#21461163C212yesmaximal, complemented
5#21461161C212yesmaximal, complemented
6#21461141C214yesmaximal, complemented
7C2×#16840314C211yesmaximal, characteristic
8C22×#11758217—21no—
9#16840783C2212yes—
10#16840782—21no—
11#16840779—21no—
12#16840778C2212yes—
13#16840765C2212yes—
14#16840732—21no—
15C4×#11758217—21no—
16C4×#11758217—21no—
17C4×#11758217—21no—
18C4×#11758217C2212yes—
19C4×#11758217C2212yes—
20C2×#14519832—21no—
21C2×C4×#92420C2212yes—
22#16840612C2218yescomplemented
23Q8⋊#92420C2218yescomplemented
24#16840314—21no—
25#16840314—21no—
26#16840314—21no—
27#16840314—21no—
28#16840314C412yes—
29#16840314C412yes—
30#16840314C2212yes—
31#16840314C2212yes—
32C2×#14519776C411yescharacteristic
33C2×#14519775C411yescharacteristic
34C2×#14519767—21no—
35C2×#14519756C411yescharacteristic
36C2×#14519754—21no—
37C2×Q8×Dic61C2211yescharacteristic
38C2×#14519742—21no—
39C2×#14519742C2211yescharacteristic
40C2×#14519733—21no—
41C22×C4×Dic61C2211yescharacteristic
42C2×C22⋊Q8×C61C411yescharacteristic
43C2×#11758217—28no—
44C2×#11758217—22no—
45C2×#11758217—22no—
46C2×#11758217—22no—
47C22×#92420—22no—
48#14519844—22no—
49C4⋊#92420C2314yes—
50C22⋊#92420—22no—
51C22⋊#92420—22no—
52#14519832—22no—
53#14519832—22no—
54#14519832—22no—
55C4×#92420—22no—
56C4×#92420—22no—
57C4×#92420C2314yes—
58C22×Dic122—21no—
59C22×Dic122C2×C411yescharacteristic
60C23×Dic61D411yescharacteristic
61C23×Dic61D411yescharacteristic
62#14519776—21no—
63#14519776—21no—
64#14519776C2×C412yes—
65#14519776C2×C412yes—
66#14519775—21no—
67#14519775—21no—
68#14519775C2×C412yes—
69#14519775C2×C412yes—
70#14519767—22no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519756—21no—
75#14519756—21no—
76#14519756C2×C412yes—
77#14519756C2×C412yes—
78#14519754—22no—
79#14519754—22no—
80#14519754—22no—
81#14519754—22no—
82Q8×Dic61—21no—
83Q8×Dic61—21no—
84Q8×Dic61C2×C412yes—
85Q8×Dic61C2312yes—
86#14519742—22no—
87#14519742—22no—
88#14519742—22no—
89#14519742—22no—
90#14519742—21no—
91#14519742—21no—
92#14519742C2×C412yes—
93#14519742C2312yes—
94#14519733—22no—
95#14519733—22no—
96#14519733—22no—
97#14519733—22no—
98C2×C4⋊Dic61—21no—
99C2×C4⋊Dic61C2×C411yescharacteristic
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×#11758196—21no—
105C2×C22⋊Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D411yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118C2×C4×Dic61C2311yescharacteristic
119C2×C4×Dic61C2311yescharacteristic
120C22×Q8×C61C2×C411yescharacteristic
121C23×C4×C61C2×C411yescharacteristic
122C22⋊Q8×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61C2×C412yes—
125C22⋊Q8×C61C2×C412yes—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2×C411yescharacteristic
128C2×C22⋊C4×C61—21no—
129C24×C61C22⋊C411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—42no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—42no—
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22×C411yescharacteristic
140C22×C4×C61C22×C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×Dic61—24no—
143C22×Dic61—24no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61C22⋊C411yescharacteristic
147C22×Dic61C22⋊C411yescharacteristic
148C22×Dic61C22⋊C411yescharacteristic
149C22×Dic61C22⋊C411yescharacteristic
150C22×Dic61C22⋊C411yescharacteristic
151C22×Dic61C22⋊C411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122—42no—
162C2×Dic122—44no—
163C2×Dic122C22×C412yes—
164#11758217—28no—
165#11758217—28no—
166#11758217—44no—
167C2×#92420—28no—
168C2×#92420—24no—
169C2×#92420—24no—
170C4⋊C4×C61—22no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C22×C412yes—
174C22⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊Dic61—22no—
177C22⋊Dic61—22no—
178C4⋊Dic61—21no—
179C4⋊Dic61—22no—
180C4⋊Dic61—22no—
181C4⋊Dic61C22×C412yes—
182#11758196—22no—
183#11758196—22no—
184#11758196—22no—
185#11758196—22no—
186#11758196—22no—
187#11758196—22no—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192C4×Dic61—22no—
193C4×Dic61—22no—
194C4×Dic61—22no—
195C4×Dic61—22no—
196C4×Dic61C22⋊C411yescharacteristic
197C4×Dic61C22⋊C411yescharacteristic
198C4×Dic61C22⋊C411yescharacteristic
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22×C411yescharacteristic
203C4×Dic61C22×C411yescharacteristic
204C4×Dic61C22×C411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2411yescharacteristic
212C23×C61—22no—
213C23×C61—22no—
214C23×C61—41no—
215C23×C61C2×C22⋊C411yescharacteristic
216C23×C61C2×C22⋊C411yescharacteristic
217C23×C61C2×C22⋊C411yescharacteristic
218Q8×C61—24no—
219Q8×C61—42no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—42no—
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C23×C411yescharacteristic
234C2×Dic61—22no—
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—42no—
247C2×Dic61—42no—
248C2×Dic61#10414yes—
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61#13714yes—
256C2×Dic61C22×D411yescharacteristic
257C2×Dic61C22×D411yescharacteristic
258Dic122—24no—
259Dic122—42no—
260Dic122—44no—
261Dic122—44no—
262#92420—28no—
263#92420—44no—
264#17501—611nomaximal, Sylow
265C22×C61—21no—
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61—21no—
270C22×C61—21no—
271C22×C61—21no—
272C22×C61—44no—
273C22×C61#41912yes—
274C22×C61C2×#10412yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C4×C61—21no—
277C4×C61—21no—
278C4×C61—21no—
279C4×C61—42no—
280C4×C61—42no—
281C4×C61#56812yes—
282Dic61—21no—
283Dic61—21no—
284Dic61—21no—
285Dic61—21no—
286Dic61—42no—
287Dic61—42no—
288Dic61#41912yes—
289Dic61C2×#10412yes—
290Dic61C2×#10412yes—
291Dic61C2×#13712yes—
292C2×#533—611no—
293#2920—611no—
294C2×C4×M16—611no—
295#1493—614no—
296#1491—612no—
297#1491—612no—
298#1485—614no—
299C2×C61—22no—
300C2×C61—22no—
301C2×C61—41no—
302C2×C61#287911yescharacteristic
303C2×C61C2×#41911yescharacteristic
304C2×C61C22×#10411yescharacteristic
305C22×M16—1221no—
306#533—612no—
307#533—612no—
308#533—612no—
309#533—612no—
310#533—1221no—
311#533—1221no—
312#533—1221no—
313#533—1221no—
314C2×C4⋊Q8—611no—
315C2×C4⋊Q8—611no—
316C2×#125—1221no—
317C2×C22⋊Q8—611no—
318C2×C22⋊Q8—1221no—
319C2×C22⋊Q8#692311yescharacteristic
320C2×C4×Q8—611no—
321C2×C4×Q8—611no—
322C2×C4×Q8—1221no—
323C2×#117—1221no—
324C22×C42—611no—
325#432—612no—
326#432—1221no—
327#431—612no—
328#431—1221no—
329#423—612no—
330#407—1221no—
331C4×M16—612no—
332C4×M16—612no—
333C4×M16—1221no—
334C4×M16—1221no—
335C4×M16—1221no—
336C2×#97—1221no—
337C2×C4×C8—612no—
338Q8⋊C8—618no—
339Q8⋊C8—618no—
340C61#1750111yescharacteristic, complemented, Sylow
341C22×Q8—611no—
342C22×Q8—1221no—
343C22×Q8C2×#692311yescharacteristic
344C23×C4—611no—
345C23×C4—611no—
346C23×C4C2×#692311yescharacteristic
347C2×M16—1222no—
348C2×M16—1222no—
349C2×M16—1228no—
350C2×M16—1222no—
351C22×C8—1222no—
352C4⋊Q8—612no—
353C4⋊Q8—612no—
354C4⋊Q8—612no—
355C4⋊Q8—612no—
356C4⋊Q8—1221no—
357C4⋊Q8—1221no—
358C4⋊Q8—1221no—
359C4⋊Q8—1221no—
360#125—1222no—
361#125—1222no—
362#125—1222no—
363#125—1222no—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—612no—
367C22⋊Q8—612no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C22⋊Q8C2×#692312yes—
375C22⋊Q8C2×#692312yes—
376C4×Q8—612no—
377C4×Q8—612no—
378C4×Q8—612no—
379C4×Q8—612no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1222no—
385C4×Q8—1222no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388#117—1222no—
389#117—1222no—
390#117—1222no—
391#117—1222no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—611no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C4⋊C4C2×#692311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1221no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—611no—
406C2×C42—611no—
407C2×C42—611no—
408C2×C42—611no—
409C2×C42—611no—
410C2×C42—611no—
411C2×C42—611no—
412C2×C42—611no—
413C2×C42—611no—
414C2×C42—611no—
415C2×C42—1221no—
416C2×C42—1221no—
417C4⋊C8—614no—
418C4⋊C8—1222no—
419C22⋊C8—1222no—
420C22⋊C8—1222no—
421#97—1222no—
422#97—1222no—
423#97—1222no—
424C4×C8—614no—
425C4×C8—1222no—
426C4×C8—1222no—
427C24C22⋊#692311yescharacteristic
428C2×Q8—21no—
429C2×Q8—42no—
430C2×Q8—612no—
431C2×Q8—1222no—
432C2×Q8—1222no—
433C2×Q8—1221no—
434C2×Q8—2442no—
435C2×Q8—2444no—
436C2×Q8C22×#692312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—42no—
440C22×C4—611no—
441C22×C4—611no—
442C22×C4—611no—
443C22×C4—611no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—1224no—
453C22×C4—1224no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4C22⋊#692311yescharacteristic
457C22×C4C22⋊#692311yescharacteristic
458C22×C4C22⋊#692311yescharacteristic
459C22×C4C22×#692311yescharacteristic
460C22×C4C22×#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462M16—1228no—
463M16—1228no—
464M16—2444no—
465C2×C8—1224no—
466C2×C8—1228no—
467C2×C8—1224no—
468C4⋊C4—22no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—612no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1222no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1221no—
483C4⋊C4—1222no—
484C4⋊C4—1222no—
485C4⋊C4C22×#692312yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—611no—
500C42—611no—
501C42—611no—
502C42—611no—
503C42—611no—
504C42—611no—
505C42—611no—
506C42—1222no—
507C42—1222no—
508C42—1222no—
509C42—1222no—
510C23—22no—
511C23—22no—
512C23—41no—
513C23C2×C22⋊#692311yescharacteristic
514C23C2×C22⋊#692311yescharacteristic
515C23C2×C22⋊#692311yescharacteristic
516Q8—24no—
517Q8—42no—
518Q8—1224no—
519Q8—2442no—
520Q8—2444no—
521Q8—2444no—
522C2×C4—22no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—614no—
533C2×C4—614no—
534C2×C4—611no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—611no—
540C2×C4—611no—
541C2×C4—611no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—2442no—
555C2×C4—2442no—
556C2×C4C2×C22⋊#692311yescharacteristic
557C2×C4C2×C22⋊#692311yescharacteristic
558C2×C4C2×C22⋊#692311yescharacteristic
559C2×C4C23×#692311yescharacteristic
560C8—1228no—
561C8—2444no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—44no—
570C22#1684075512yes—
571C22D61⋊#10412yes—
572C22C22×C22⋊#692311yescharacteristic
573C4—21no—
574C4—21no—
575C4—21no—
576C4—42no—
577C4—42no—
578C4—612no—
579C4—612no—
580C4—612no—
581C4—612no—
582C4—1221no—
583C4—1221no—
584C4—1221no—
585C4—1221no—
586C4—2442no—
587C4—2442no—
588C4#1684084012yes—
589C2—22no—
590C2—22no—
591C2—41no—
592C2#2146240311yescharacteristic
593C2C2×#1684075511yescharacteristic
594C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22⋊C4
cosocleC4×Dic61C24
hypercenterC2×C22⋊Q8#6923
hypocenterC61#17501
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21461167C2semidirect
4. #21461163C2semidirect
5. #21461161C2semidirect
6. #21461141C2semidirect
22. #16840612C22semidirect
23. Q8⋊#92420C22semidirect
340. C61#17501semidirect

Maximal subgroups

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

subgroupindex
#214624952
C2×C4×#117582172
#214611672
#214611632
#214611612
#214611412
C2×#168403142
#1750161