Finite Group Databasebeta

← #30610980

Subgroups — #30610980

Order 15616.

1084 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462497C211yesmaximal, characteristic
2C4×#14519856C211yesmaximal, characteristic
3#21461217C214yesmaximal
4#21461213C212yesmaximal, complemented
5#21461211C212yesmaximal, complemented
6#21461201C214yesmaximal, complemented
7#533×D61C211yesmaximal, characteristic
8C2×#14519856—21no—
9#16840785—21no—
10#16840784C2212yes—
11#16840781—21no—
12#16840780C2212yes—
13#16840765—21no—
14#16840734C2212yes—
15C4×#11758217—21no—
16C4×C4⋊#6923—21no—
17C4×C4⋊#6923C2212yes—
18C4×C22⋊#6923—21no—
19C4×C22⋊#6923C2212yes—
20#16840698C2212yes—
21C42×#6923—21no—
22#16840616C2218yes—
23#16840615C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318—21no—
26#16840317C2211yescharacteristic
27#16840317C2211yescharacteristic
28#16840316—21no—
29#16840316C2212yes—
30#16840315—21no—
31#16840314—21no—
32#16840314C412yescomplemented
33C4⋊Q8×D61C411yescharacteristic, complemented
34C4⋊Q8×D61C411yescharacteristic, complemented
35#125×D61—21no—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61C2212yes—
38C4×Q8×D61—21no—
39C4×Q8×D61C412yescomplemented
40#117×D61—21no—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C2×#11758217—22no—
44#14519856—28no—
45C2×C4⋊#6923—22no—
46C2×C22⋊#6923—22no—
47C2×C4×#6923—22no—
48C2×C4×#6923—22no—
49C2×C4×#6923—22no—
50#14519844—22no—
51C4⋊#92420—22no—
52#14519834—22no—
53#14519834C2314yes—
54#14519832—22no—
55C4×#92420—22no—
56(C2×#6923)⋊C4—22no—
57(C2×#6923)⋊C4C2314yes—
58C2×Q8×D61—21no—
59C2×Q8×D61C2×C412yes—
60C22×C4×D61D411yescharacteristic
61C22×C4×D61D411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519777C2×C411yescharacteristic
65#14519776—21no—
66#14519776—21no—
67#14519776—21no—
68#14519776C2×C412yes—
69#14519775—21no—
70#14519775C2×C412yes—
71#14519769—21no—
72#14519768—21no—
73#14519768—21no—
74#14519767—22no—
75#14519767—22no—
76#14519757—21no—
77#14519757C2312yes—
78#14519756—21no—
79#14519756C2×C412yes—
80#14519755—22no—
81#14519755—22no—
82#14519754—22no—
83#14519754—22no—
84#14519753—21no—
85#14519753C2312yes—
86Q8×Dic61—21no—
87Q8×Dic61C2×C412yes—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122C2×C412yes—
92#14519742—22no—
93#14519742—22no—
94#14519742—21no—
95#14519742C2×C412yes—
96#14519736—21no—
97#14519735—21no—
98#14519735—21no—
99#14519734—22no—
100#14519733—22no—
101C4⋊C4×D61—21no—
102C4⋊C4×D61—21no—
103C4⋊C4×D61—21no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61C2×C412yes—
107C22⋊C4×D61—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
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×C61—21no—
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61C2×C412yes—
125C4×Q8×C61—21no—
126C4×Q8×C61C2×C412yes—
127#117×C61—21no—
128C2×C42×C61C2×C411yescharacteristic
129C23×D61C2×D411yescharacteristic
130C22×#6923—24no—
131C2×Q8×C61—21no—
132C2×Q8×C61C22×C412yes—
133C22×C4×C61C22⋊C411yescharacteristic
134C22×C4×C61C22⋊C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×Dic61C22⋊C411yescharacteristic
137C22×Dic61C22⋊C411yescharacteristic
138C22×Dic61C2×D411yescharacteristic
139Q8×D61—44no—
140Q8×D61—44no—
141C2×C4×D61—24no—
142C2×C4×D61—24no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—42no—
146C2×C4×D61C22⋊C411yescharacteristic
147C2×C4×D61C22⋊C411yescharacteristic
148C2×C4×D61C22⋊C411yescharacteristic
149C2×C4×D61C22⋊C411yescharacteristic
150C2×C4×D61C22×C411yescharacteristic
151C2×C4×D61C2×D411yescharacteristic
152C2×C4×D61C2×D411yescharacteristic
153C2×C4×D61C2×D411yescharacteristic
154C2×C4×D61C2×D411yescharacteristic
155C2×C4×D61C2×D411yescharacteristic
156C2×C4×D61C2×D411yescharacteristic
157C2×C4×D61C2411yescharacteristic
158C2×Dic122—21no—
159C2×Dic122—22no—
160C2×Dic122—22no—
161C2×Dic122C22×C412yes—
162C22⋊#6923—28no—
163#11758217—44no—
164C2×#92420—24no—
165C4⋊#6923—28no—
166C4×#6923—28no—
167C4×#6923—44no—
168C4×#6923—44no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61C22×C412yes—
175C22⋊C4×C61—22no—
176C42×C61—22no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22×C411yescharacteristic
180C42×C61C22×C411yescharacteristic
181C22⋊Dic61—22no—
182#11758198—22no—
183#11758198—22no—
184C4⋊Dic61—21no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61C22×C412yes—
190#11758196—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198C4×Dic61—22no—
199C4×Dic61—22no—
200C4×Dic61—22no—
201C4×Dic61C22⋊C411yescharacteristic
202C4×Dic61C22⋊C411yescharacteristic
203C4×Dic61C22⋊C411yescharacteristic
204C4×Dic61C22⋊C411yescharacteristic
205C4×Dic61C22⋊C411yescharacteristic
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22×C411yescharacteristic
208C4×Dic61C22×C411yescharacteristic
209C4×Dic61C2×D411yescharacteristic
210C4×Dic61C2×D411yescharacteristic
211C4×Dic61C2×D411yescharacteristic
212C4×Dic61C2×D411yescharacteristic
213C23×C61C2×C22⋊C411yescharacteristic
214C22×D61—22no—
215C22×D61—22no—
216C22×D61—41no—
217C22×D61C2×C22⋊C411yescharacteristic
218C22×D61C22×D411yescharacteristic
219C2×#6923—28no—
220C2×#6923—44no—
221C2×#6923—44no—
222Q8×C61—42no—
223Q8×C61—42no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—41no—
229C2×C4×C61C2×C22⋊C411yescharacteristic
230C2×C4×C61C2×C22⋊C411yescharacteristic
231C2×C4×C61C2×C22⋊C411yescharacteristic
232C2×C4×C61C2×C22⋊C411yescharacteristic
233C2×C4×C61C2×C22⋊C411yescharacteristic
234C2×C4×C61C23×C411yescharacteristic
235C2×Dic61—22no—
236C2×Dic61—22no—
237C2×Dic61—22no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61C2×C22⋊C411yescharacteristic
241C2×Dic61C2×C22⋊C411yescharacteristic
242C2×Dic61C2×C22⋊C411yescharacteristic
243C2×Dic61C2×C22⋊C411yescharacteristic
244C2×Dic61C2×C22⋊C411yescharacteristic
245C2×Dic61C22×D411yescharacteristic
246C4×D61—22no—
247C4×D61—22no—
248C4×D61—22no—
249C4×D61—22no—
250C4×D61—22no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—41no—
255C4×D61—41no—
256C4×D61—44no—
257C4×D61—44no—
258C4×D61#10414yes—
259C4×D61#13714yes—
260Dic122—42no—
261Dic122—44no—
262Dic122—44no—
263Dic122—42no—
264#92420—44no—
265#17545—611nomaximal, Sylow
266C22×C61—21no—
267C22×C61—21no—
268C22×C61—21no—
269C22×C61C22×C22⋊C411yescharacteristic
270C2×D61—21no—
271C2×D61—21no—
272C2×D61—21no—
273C2×D61—21no—
274C2×D61—44no—
275C2×D61C2×#10412yes—
276C2×D61C2×#13712yes—
277#6923—48no—
278C4×C61—21no—
279C4×C61—21no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—42no—
283C4×C61—42no—
284C4×C61#41912yes—
285C4×C61C2×#10412yes—
286Dic61—21no—
287Dic61—21no—
288Dic61—21no—
289Dic61—42no—
290Dic61—42no—
291Dic61#41012yes—
292Dic61#41912yes—
293Dic61C2×#10412yes—
294C2×#533—611no—
295#2920—611no—
296C4×#117—611no—
297#1508—614no—
298#1506—612no—
299#1506—612no—
300#1503—614no—
301C2×C61—41no—
302C2×C61C2×#41011yescharacteristic
303C2×C61C2×#41911yescharacteristic
304C2×C61C22×#10411yescharacteristic
305D61—22no—
306D61—22no—
307#533—611no—
308#533—611no—
309#533—612no—
310#533—612no—
311#533—611no—
312#533—1221no—
313#533—1221no—
314#533—1221no—
315#533—1221no—
316#533#692311yescharacteristic, complemented
317C2×C4⋊Q8—611no—
318C2×C4⋊Q8—611no—
319C2×#125—1221no—
320C2×C22⋊Q8—612no—
321C2×C22⋊Q8—1221no—
322C2×C4×Q8—612no—
323C2×C4×Q8—1221no—
324C2×#117—1221no—
325C2×#117—1221no—
326C22×C42—611no—
327#432—612no—
328#432—1221no—
329#431—612no—
330#431—1221no—
331#423—1221no—
332#407—612no—
333C4×M16—1221no—
334C4×C4⋊C4—612no—
335C4×C4⋊C4—1221no—
336C4×C22⋊C4—612no—
337C4×C22⋊C4—1221no—
338#376—612no—
339C43—1221no—
340#328—618no—
341#328—618no—
342C61#1754511yescharacteristic, complemented, Sylow
343C22×Q8—612no—
344C22×Q8—1221no—
345C23×C4—611no—
346C23×C4—611no—
347C23×C4—611no—
348C2×M16—1222no—
349C4⋊Q8—612no—
350C4⋊Q8—612no—
351C4⋊Q8—611no—
352C4⋊Q8—611no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—1221no—
356C4⋊Q8—1221no—
357C4⋊Q8C2×#692311yescharacteristic
358C4⋊Q8C2×#692311yescharacteristic
359#125—21no—
360#125—1222no—
361#125—1222no—
362#125—1221no—
363#125—1221no—
364#125—1221no—
365C22⋊Q8—21no—
366C22⋊Q8—612no—
367C22⋊Q8—612no—
368C22⋊Q8—612no—
369C22⋊Q8—1222no—
370C22⋊Q8—1222no—
371C22⋊Q8—1222no—
372C22⋊Q8—1222no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376C22⋊Q8C2×#692312yes—
377C4×Q8—21no—
378C4×Q8—612no—
379C4×Q8—612no—
380C4×Q8—612no—
381C4×Q8—1222no—
382C4×Q8—1222no—
383C4×Q8—1222no—
384C4×Q8—1221no—
385C4×Q8—1222no—
386C4×Q8—1221no—
387C4×Q8—1221no—
388C4×Q8C2×#692312yes—
389#117—21no—
390#117—1222no—
391#117—1222no—
392#117—1221no—
393#117—1221no—
394#117—1228no—
395#117—1221no—
396C2×C4⋊C4—612no—
397C2×C4⋊C4—1221no—
398C2×C4⋊C4—1221no—
399C2×C4⋊C4—1222no—
400C2×C4⋊C4—1222no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1222no—
403C2×C22⋊C4—1222no—
404C2×C22⋊C4—1222no—
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—611no—
416C2×C42—1222no—
417C2×C42—1222no—
418C2×C42—1222no—
419C2×C42—1222no—
420C2×C42C2×#692311yescharacteristic
421C4⋊C8—1222no—
422C4⋊C8—1222no—
423C22⋊C8—614no—
424C22⋊C8—1222no—
425#97—1222no—
426C4×C8—1222no—
427#95—614no—
428#95—1222no—
429C24—611no—
430C2×Q8—21no—
431C2×Q8—612no—
432C2×Q8—1222no—
433C2×Q8—1222no—
434C2×Q8—1221no—
435C2×Q8—2444no—
436C2×Q8—2444no—
437C2×Q8C22×#692312yes—
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—611no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1224no—
454C22×C4—1224no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1224no—
458C22×C4—2442no—
459C22×C4C22⋊#692311yescharacteristic
460C22×C4C22⋊#692311yescharacteristic
461C22×C4C22×#692311yescharacteristic
462M16—2444no—
463C2×C8—1224no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—22no—
468C4⋊C4—22no—
469C4⋊C4—612no—
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—1228no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4C22×#692312yes—
485C22⋊C4—22no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1228no—
489C22⋊C4—1222no—
490C42—22no—
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—1222no—
504C42—1222no—
505C42—1228no—
506C42—1222no—
507C42—2444no—
508C42—2444no—
509C42C22⋊#692311yescharacteristic
510C42C22⋊#692311yescharacteristic
511C42C22×#692311yescharacteristic
512C42C22×#692311yescharacteristic
513C23—611no—
514C23—611no—
515C23—1222no—
516C23—1222no—
517C23—2441no—
518C23C2×C22⋊#692311yescharacteristic
519Q8—42no—
520Q8—42no—
521Q8—2442no—
522Q8—2444no—
523Q8—2444no—
524Q8—2442no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—41no—
530C2×C4—611no—
531C2×C4—611no—
532C2×C4—611no—
533C2×C4—614no—
534C2×C4—614no—
535C2×C4—611no—
536C2×C4—611no—
537C2×C4—611no—
538C2×C4—1222no—
539C2×C4—1222no—
540C2×C4—1222no—
541C2×C4—1222no—
542C2×C4—1222no—
543C2×C4—1222no—
544C2×C4—1222no—
545C2×C4—1222no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1222no—
549C2×C4—1228no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2444no—
555C2×C4—2444no—
556C2×C4—2444no—
557C2×C4—2444no—
558C2×C4C2×C22⋊#692311yescharacteristic
559C2×C4C2×C22⋊#692311yescharacteristic
560C2×C4C2×C22⋊#692311yescharacteristic
561C2×C4C2×C22⋊#692311yescharacteristic
562C2×C4C2×C22⋊#692311yescharacteristic
563C2×C4C23×#692311yescharacteristic
564C8—2444no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—612no—
569C22—612no—
570C22—1221no—
571C22—1221no—
572C22—1221no—
573C22—1221no—
574C22—2444no—
575C22C22×C22⋊#692311yescharacteristic
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—42no—
581C4—42no—
582C4—612no—
583C4—612no—
584C4—612no—
585C4—1221no—
586C4—1221no—
587C4—1221no—
588C4—2448no—
589C4—2442no—
590C4—2442no—
591C4#1684075512yes—
592C4D61⋊#10412yes—
593C2—41no—
594C2—1222no—
595C2—1222no—
596C2D61⋊#41011yescharacteristic
597C2C2×#1684075511yescharacteristic
598C2C2×D61⋊#10411yescharacteristic
subgroupG/H
centerC2×C4C2×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#17545
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
4. #21461213C2semidirect
5. #21461211C2semidirect
6. #21461201C2semidirect
24. #16840319C4semidirect
32. #16840314C4semidirect
33. C4⋊Q8×D61C4semidirect
34. C4⋊Q8×D61C4semidirect
39. C4×Q8×D61C4semidirect
42. C61×#533C4semidirect
316. #533#6923semidirect
342. C61#17545semidirect

Maximal subgroups

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

subgroupindex
#214624972
C4×#145198562
#214612172
#214612132
#214612112
#214612012
#533×D612
#1754561