Finite Group Databasebeta

← #30611480

Subgroups — #30611480

Order 15616.

1036 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462498C211yesmaximal, characteristic
2C2×C4×#11758213C211yesmaximal, characteristic
3#21461291C214yesmaximal, complemented
4#21461282C214yesmaximal, complemented
5#21461277C212yesmaximal, complemented
6#21461277C212yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8C22×#11758213—21no—
9#16840785C2212yes—
10#16840782C2212yes—
11#16840780—21no—
12#16840779—21no—
13#16840767—21no—
14#16840733—21no—
15C4×#11758213—21no—
16C4×#11758213—21no—
17C4×#11758213C2212yes—
18C4×#11758213C2212yes—
19C4×#11758213C2212yes—
20C2×#14519832C2212yes—
21C2×C4×#92420—21no—
22#16840619C2218yescomplemented
23#16840619C2218yescomplemented
24#16840319C411yescharacteristic
25#16840318C412yes—
26#16840317C2211yescharacteristic
27#16840317C2211yescharacteristic
28#16840316—21no—
29#16840316—21no—
30#16840315C2212yes—
31#16840314—21no—
32#16840314—21no—
33C4⋊Q8×D61C411yescharacteristic
34C4⋊Q8×D61C411yescharacteristic
35#125×D61C2212yes—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61—21no—
38C4×Q8×D61—21no—
39C4×Q8×D61—21no—
40#117×D61C412yes—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic
43C2×#11758213—28no—
44C2×#11758213—22no—
45C2×#11758213—22no—
46C2×#11758213—22no—
47C22×#92420—22no—
48#14519844—22no—
49#14519844—22no—
50#14519834—22no—
51C22⋊#92420—22no—
52#14519832—22no—
53#14519832C2314yes—
54#14519832C2314yes—
55C4×#92420—22no—
56C4×#92420—22no—
57C4×#92420—22no—
58C2×Q8×D61—21no—
59C2×Q8×D61—21no—
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61D411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519776—21no—
66#14519776—21no—
67#14519776C2×C412yes—
68#14519776C2×C412yes—
69#14519775—21no—
70#14519775—21no—
71#14519769C2×C412yes—
72#14519768C2312yes—
73#14519768C2312yes—
74#14519767—22no—
75#14519767—22no—
76#14519757—21no—
77#14519757—21no—
78#14519756—21no—
79#14519756—21no—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519754—22no—
84#14519753—21no—
85#14519753—21no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88C4×Dic122—21no—
89C4×Dic122—21no—
90C4×Dic122—22no—
91C4×Dic122—22no—
92#14519742—22no—
93#14519742—22no—
94#14519742—21no—
95#14519742—21no—
96#14519736C2×C412yes—
97#14519735C2×C412yes—
98#14519735C2×C412yes—
99#14519734—22no—
100#14519733—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61C2×C412yes—
106C4⋊C4×D61C2×C412yes—
107C22⋊C4×D61—22no—
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61C2311yescharacteristic
111C42×D61—22no—
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61D411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61C2311yescharacteristic
119C42×D61C2311yescharacteristic
120C4⋊Q8×C61C2×C411yescharacteristic
121C4⋊Q8×C61C2×C411yescharacteristic
122#125×C61C2×C412yes—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127#117×C61C2×C412yes—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C22⋊C411yescharacteristic
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C22×C4×C61C22⋊C411yescharacteristic
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×Dic61C22⋊C411yescharacteristic
136C22×Dic61C2×D411yescharacteristic
137C22×Dic61C2×D411yescharacteristic
138Q8×D61—44no—
139Q8×D61—44no—
140C2×C4×D61—24no—
141C2×C4×D61—24no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—42no—
145C2×C4×D61C22⋊C411yescharacteristic
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22⋊C411yescharacteristic
151C2×C4×D61C22×C411yescharacteristic
152C2×C4×D61C22×C411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×#92420—28no—
162C2×#92420—24no—
163C2×#92420—24no—
164#11758213—28no—
165#11758213—28no—
166#11758213—44no—
167C4⋊C4×C61—22no—
168C4⋊C4×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61C22×C412yes—
172C4⋊C4×C61C22×C412yes—
173C22⋊C4×C61—22no—
174C42×C61—22no—
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22×C411yescharacteristic
178C42×C61C22×C411yescharacteristic
179C22⋊Dic61—22no—
180#11758198—22no—
181#11758198—22no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—22no—
185C4⋊Dic61—22no—
186C4⋊Dic61C22×C412yes—
187C4⋊Dic61C22×C412yes—
188#11758196—22no—
189#11758196—22no—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196C4×Dic61—22no—
197C4×Dic61—22no—
198C4×Dic61—22no—
199C4×Dic61C22⋊C411yescharacteristic
200C4×Dic61C22⋊C411yescharacteristic
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22×C411yescharacteristic
204C4×Dic61C2×D411yescharacteristic
205C4×Dic61C2×D411yescharacteristic
206C4×Dic61C2×D411yescharacteristic
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2411yescharacteristic
211C23×C61C2×C22⋊C411yescharacteristic
212C22×D61—22no—
213C22×D61—22no—
214C22×D61—41no—
215C22×D61C2×C22⋊C411yescharacteristic
216C22×D61C2×C22⋊C411yescharacteristic
217Q8×C61—42no—
218Q8×C61—42no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61C2×C22⋊C411yescharacteristic
225C2×C4×C61C2×C22⋊C411yescharacteristic
226C2×C4×C61C2×C22⋊C411yescharacteristic
227C2×C4×C61C2×C22⋊C411yescharacteristic
228C2×C4×C61C2×C22⋊C411yescharacteristic
229C2×C4×C61C23×C411yescharacteristic
230C2×Dic61—22no—
231C2×Dic61—22no—
232C2×Dic61—22no—
233C2×Dic61—41no—
234C2×Dic61#10114yes—
235C2×Dic61C2×C22⋊C411yescharacteristic
236C2×Dic61C2×C22⋊C411yescharacteristic
237C2×Dic61C2×C22⋊C411yescharacteristic
238C2×Dic61C2×C22⋊C411yescharacteristic
239C2×Dic61C22×D411yescharacteristic
240C2×Dic61C22×D411yescharacteristic
241C4×D61—22no—
242C4×D61—22no—
243C4×D61—22no—
244C4×D61—22no—
245C4×D61—22no—
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—41no—
251C4×D61—41no—
252C4×D61—44no—
253C4×D61—44no—
254C4×D61#10414yes—
255Dic122—44no—
256Dic122—42no—
257Dic122—42no—
258Dic122—44no—
259#92420—28no—
260#92420—44no—
261#17648—611nomaximal, Sylow
262C22×C61—21no—
263C22×C61—21no—
264C22×C61C2×#10112yes—
265C22×C61C22×C22⋊C411yescharacteristic
266C2×D61—21no—
267C2×D61—21no—
268C2×D61—21no—
269C2×D61—21no—
270C2×D61—21no—
271C2×D61—44no—
272C2×D61C2×#10412yes—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—21no—
277C4×C61—42no—
278C4×C61—42no—
279C4×C61C2×#10412yes—
280C4×C61#42112yes—
281Dic61—21no—
282Dic61—21no—
283Dic61—21no—
284Dic61—42no—
285Dic61—42no—
286Dic61C2×#10112yes—
287Dic61C2×#10112yes—
288Dic61C2×#10412yes—
289C2×#533—611no—
290#2920—611no—
291C2×C4×M16—611no—
292#1531—614no—
293#1528—614no—
294#1526—612no—
295#1526—612no—
296C2×C61—22no—
297C2×C61C22×#10111yescharacteristic
298C2×C61C22×#10411yescharacteristic
299C2×C61C2×#42111yescharacteristic
300D61—22no—
301D61—41no—
302C22×M16—1221no—
303#533—611no—
304#533—611no—
305#533—612no—
306#533—612no—
307#533—611no—
308#533—1221no—
309#533—1221no—
310#533—1221no—
311#533—1221no—
312#533#692311yescharacteristic
313C2×C4⋊Q8—611no—
314C2×C4⋊Q8—611no—
315C2×#125—612no—
316C2×C22⋊Q8—1221no—
317C2×C22⋊Q8—1221no—
318C2×C4×Q8—1221no—
319C2×C4×Q8—1221no—
320C2×#117—612no—
321C22×C42—611no—
322#432—612no—
323#432—612no—
324#431—1221no—
325#431—1221no—
326#423—1221no—
327#407—1221no—
328C4×M16—612no—
329C4×M16—612no—
330C4×M16—612no—
331C4×M16—1221no—
332C4×M16—1221no—
333C2×#97—612no—
334C2×C4×C8—1221no—
335#330—618no—
336#330—618no—
337C61#1764811yescharacteristic, complemented, Sylow
338C22×Q8—1221no—
339C22×Q8—1221no—
340C23×C4—611no—
341C23×C4—611no—
342C23×C4—611no—
343C2×M16—1222no—
344C2×M16—1222no—
345C2×M16—1228no—
346C2×M16—1222no—
347C22×C8—1222no—
348C4⋊Q8—612no—
349C4⋊Q8—612no—
350C4⋊Q8—611no—
351C4⋊Q8—611no—
352C4⋊Q8—1221no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8C2×#692311yescharacteristic
357C4⋊Q8C2×#692311yescharacteristic
358#125—612no—
359#125—612no—
360#125—612no—
361#125—1222no—
362#125—1222no—
363#125C2×#692312yes—
364C22⋊Q8—21no—
365C22⋊Q8—21no—
366C22⋊Q8—1222no—
367C22⋊Q8—1222no—
368C22⋊Q8—1222no—
369C22⋊Q8—1222no—
370C22⋊Q8—1221no—
371C22⋊Q8—1221no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—1222no—
379C4×Q8—1222no—
380C4×Q8—1221no—
381C4×Q8—1221no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1221no—
385C4×Q8—1221no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388#117—612no—
389#117—612no—
390#117—612no—
391#117—1222no—
392#117—1222no—
393#117C2×#692312yes—
394C2×C4⋊C4—612no—
395C2×C4⋊C4—612no—
396C2×C4⋊C4—1222no—
397C2×C4⋊C4—1222no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1221no—
400C2×C22⋊C4—1222no—
401C2×C42—611no—
402C2×C42—611no—
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—1222no—
413C2×C42C2×#692311yescharacteristic
414C4⋊C8—1222no—
415C4⋊C8—1222no—
416C22⋊C8—1222no—
417C22⋊C8—1222no—
418#97—614no—
419#97—614no—
420#97—1222no—
421C4×C8—1222no—
422C4×C8—1222no—
423C4×C8—1222no—
424C24—611no—
425C2×Q8—21no—
426C2×Q8—21no—
427C2×Q8—1222no—
428C2×Q8—1222no—
429C2×Q8—1221no—
430C2×Q8—1221no—
431C2×Q8—2444no—
432C2×Q8—2444no—
433C22×C4—611no—
434C22×C4—611no—
435C22×C4—611no—
436C22×C4—611no—
437C22×C4—611no—
438C22×C4—611no—
439C22×C4—611no—
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—1224no—
449C22×C4—1224no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—2442no—
453C22×C4C22⋊#692311yescharacteristic
454C22×C4C22⋊#692311yescharacteristic
455C22×C4C22×#692311yescharacteristic
456M16—1228no—
457M16—1228no—
458M16—2444no—
459C2×C8—1224no—
460C2×C8—1228no—
461C2×C8—1224no—
462C4⋊C4—22no—
463C4⋊C4—22no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—612no—
467C4⋊C4—612no—
468C4⋊C4—1222no—
469C4⋊C4—1222no—
470C4⋊C4—1222no—
471C4⋊C4—1222no—
472C4⋊C4—1222no—
473C4⋊C4—1222no—
474C4⋊C4—1222no—
475C4⋊C4—1222no—
476C4⋊C4—1222no—
477C4⋊C4—1222no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4C22×#692312yes—
481C4⋊C4C22×#692312yes—
482C22⋊C4—22no—
483C22⋊C4—1222no—
484C22⋊C4—1222no—
485C22⋊C4—1222no—
486C42—22no—
487C42—611no—
488C42—611no—
489C42—611no—
490C42—611no—
491C42—611no—
492C42—611no—
493C42—611no—
494C42—611no—
495C42—611no—
496C42—611no—
497C42—611no—
498C42—611no—
499C42—1222no—
500C42—1222no—
501C42—1222no—
502C42C22⋊#692311yescharacteristic
503C42C22⋊#692311yescharacteristic
504C42C22×#692311yescharacteristic
505C42C22×#692311yescharacteristic
506C23—611no—
507C23—611no—
508C23—1222no—
509C23—1222no—
510C23—2441no—
511C23C2×C22⋊#692311yescharacteristic
512Q8—42no—
513Q8—42no—
514Q8—2444no—
515Q8—2444no—
516Q8—2442no—
517Q8—2442no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—22no—
523C2×C4—614no—
524C2×C4—611no—
525C2×C4—611no—
526C2×C4—611no—
527C2×C4—614no—
528C2×C4—611no—
529C2×C4—611no—
530C2×C4—611no—
531C2×C4—1222no—
532C2×C4—1222no—
533C2×C4—1222no—
534C2×C4—1222no—
535C2×C4—1222no—
536C2×C4—1222no—
537C2×C4—1222no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—2441no—
544C2×C4—2441no—
545C2×C4—2444no—
546C2×C4—2444no—
547C2×C4—2441no—
548C2×C4C2×C22⋊#692311yescharacteristic
549C2×C4C2×C22⋊#692311yescharacteristic
550C2×C4C2×C22⋊#692311yescharacteristic
551C2×C4C2×C22⋊#692311yescharacteristic
552C2×C4C2×C22⋊#692311yescharacteristic
553C2×C4C23×#692311yescharacteristic
554C8—1228no—
555C8—2444no—
556C22—21no—
557C22—21no—
558C22—612no—
559C22—1221no—
560C22—1221no—
561C22—1221no—
562C22—1221no—
563C22—1221no—
564C22—2444no—
565C22D61⋊#10112yes—
566C22C22×C22⋊#692311yescharacteristic
567C4—21no—
568C4—21no—
569C4—21no—
570C4—21no—
571C4—42no—
572C4—42no—
573C4—612no—
574C4—612no—
575C4—612no—
576C4—1221no—
577C4—1221no—
578C4—1221no—
579C4—2442no—
580C4—2442no—
581C4D61⋊#10412yes—
582C4#1684075912yes—
583C2—22no—
584C2—1222no—
585C2—2441no—
586C2C2×D61⋊#10111yescharacteristic
587C2C2×D61⋊#10411yescharacteristic
588C2C2×#1684075911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC4×Dic61C24
hypercenter#533#6923
hypocenterC61#17648
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461291C2semidirect
4. #21461282C2semidirect
5. #21461277C2semidirect
6. #21461277C2semidirect
22. #16840619C22semidirect
23. #16840619C22semidirect
337. C61#17648semidirect

Maximal subgroups

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

subgroupindex
#214624982
C2×C4×#117582132
#214612912
#214612822
#214612772
#533×D612
#1764861