Finite Group Databasebeta

← #29647201

Subgroups — #29647201

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459115C211yesmaximal, characteristic
2#21454452C211yesmaximal, characteristic, complemented
3#21444342C214yesmaximal, complemented
4#21444321C214yesmaximal, complemented
5#21444317C212yesmaximal, complemented
6#21444313C212yesmaximal, complemented
7C2×C4×C4⋊C4×C61C211yesmaximal, characteristic, complemented
8#16840319C411yescharacteristic, complemented
9#16840319C411yescharacteristic, complemented
10#16840319C2211yescharacteristic
11#16840312C411yescharacteristic, complemented
12#16840312C411yescharacteristic, complemented
13#16840312C2211yescharacteristic
14#16840296—21no—
15#16840296—21no—
16#16840296—21no—
17#16840296—21no—
18#16840296C412yescomplemented
19#16840296C2212yes—
20#16840273—21no—
21#16840273—21no—
22#16840273C2212yes—
23C4×#11758222—21no—
24C4×#11758222—21no—
25C4×#11758222C412yescomplemented
26#16839819—21no—
27#16839819C2212yes—
28#16839815—21no—
29#16839815C2212yes—
30#16839786—21no—
31#16839726C2212yes—
32C4×#11758194—21no—
33#16839498C2218yes—
34#16839495C2218yes—
35C22×C4⋊C4×C61—21no—
36C22×C42×C61—21no—
37C22×C42×C61C2211yescharacteristic
38C4×C4⋊C4×C61—21no—
39C4×C4⋊C4×C61—21no—
40C4×C4⋊C4×C61C2212yes—
41C4×C4⋊C4×C61C2212yes—
42C2×#95×C61C2212yes—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222C2×C412yes—
46C2×#11758194—22no—
47#14519777C2×C411yescharacteristic, complemented
48#14519777C2×C411yescharacteristic, complemented
49#14519777C2×C411yescharacteristic, complemented
50#14519774C2×C411yescharacteristic, complemented
51#14519771—22no—
52#14519771—22no—
53#14519769—21no—
54#14519769—21no—
55#14519769C2×C412yescomplemented
56#14519765—21no—
57#14519765—21no—
58#14519765C2×C412yescomplemented
59#14519760—22no—
60#14519760—22no—
61#14519760—22no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756—21no—
66#14519756C2×C412yes—
67#14519756C2312yes—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751—21no—
72#14519751C2×C412yes—
73#14519751C2312yes—
74#14519746—22no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122—21no—
79C4×Dic122C2×C412yescomplemented
80C4×Dic122C2×C412yescomplemented
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244—21no—
85C4×D244C2×C412yescomplemented
86C4×D244C2×C412yescomplemented
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519736—21no—
94#14519736—21no—
95#14519736C2×C412yes—
96#14519734—22no—
97#14519734—22no—
98#14519734—22no—
99C42×D61—22no—
100C4⋊#92418—22no—
101C4⋊#92418—22no—
102C22⋊#92418—22no—
103C22⋊#92418C2314yes—
104#14519688—22no—
105C4×#92418—22no—
106C23×C4×C61—22no—
107C23×C4×C61D411yescharacteristic
108C23×C4×C61D411yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C2×C4⋊C4×C61—28no—
111C2×C4⋊C4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C42×C61—22no—
114C2×C42×C61—22no—
115C2×C42×C61—44no—
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61C2×C411yescharacteristic
119C2×C42×C61C2×C411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61D411yescharacteristic
125C2×C42×C61D411yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#95×C61—22no—
129#95×C61C2314yes—
130C24×C61C2×D411yescharacteristic
131C22×C4×C61—24no—
132C22×C4×C61—22no—
133C22×C4×C61—22no—
134C22×C4×C61—28no—
135C22×C4×C61—24no—
136C22×C4×C61—42no—
137C22×C4×C61—44no—
138C22×C4×C61—44no—
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22⋊C411yescharacteristic
144C22×C4×C61C22⋊C411yescharacteristic
145C22×C4×C61C22⋊C411yescharacteristic
146C22×C4×C61C22⋊C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2411yescharacteristic
157C2×C22⋊D61—22no—
158C2×C22⋊D61—22no—
159C2×C22⋊D61—22no—
160#11758222—44no—
161#11758222—42no—
162#11758222—42no—
163#11758222—44no—
164C2×D244—21no—
165C2×D244—21no—
166C2×D244C22×C412yes—
167C2×C4×D61—22no—
168C2×C4×D61—22no—
169C2×C4×D61—22no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C4⋊C4×C61—28no—
174C4⋊C4×C61—28no—
175C42×C61—44no—
176C42×C61—44no—
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22⋊C411yescharacteristic
183C42×C61C22⋊C411yescharacteristic
184C42×C61C22⋊C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C22×C411yescharacteristic
187C42×C61C22×C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C42×C61C2×D411yescharacteristic
191C42×C61C2×D411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C22⋊Dic61—22no—
194C22⋊Dic61—22no—
195C22⋊Dic61—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200#11758198—22no—
201#11758198—22no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217#11758194—44no—
218C2×#92418—24no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—22no—
222C23×C61—22no—
223C23×C61C2×C22⋊C411yescharacteristic
224C23×C61C2×C22⋊C411yescharacteristic
225C23×C61C22×D411yescharacteristic
226C22×D61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—22no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—28no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—416no—
246C2×C4×C61—44no—
247C2×C4×C61—44no—
248C2×C4×C61#10414yes—
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×C22⋊C411yescharacteristic
255C2×C4×C61C2×C22⋊C411yescharacteristic
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61AGL(1,ℤ/8)12yes—
260C2×C4×C61#13712yes—
261C2×C4×C61C23×C411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C22⋊D61—42no—
264C22⋊D61—42no—
265C22⋊D61—44no—
266C22⋊D61—44no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270D244—42no—
271D244—42no—
272D244—42no—
273C4×D61—44no—
274C4×D61—44no—
275C4×D61—44no—
276Dic122—42no—
277Dic122—42no—
278Dic122—42no—
279#92418—44no—
280#17546—611nomaximal, Sylow
281C22×C61—24no—
282C22×C61—24no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—21no—
287C22×C61—21no—
288C22×C61—21no—
289C22×C61#41012yes—
290C22×C61C2×#10412yes—
291C22×C61C22×C22⋊C411yescharacteristic
292C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
293C22×C61C2×#13711yescharacteristic
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—21no—
303C4×C61—21no—
304C4×C61—21no—
305C4×C61—48no—
306C4×C61#41712yes—
307C4×C61#41712yes—
308C4×C61C2×#10412yes—
309C4×C61C2×#10412yes—
310Dic61—42no—
311Dic61—42no—
312Dic61—42no—
313#3444—611no—
314#2920—611no—
315C2×C4×C4⋊C4D6111yescharacteristic, complemented
316#1509—614no—
317#1504—614no—
318#1503—612no—
319#1502—612no—
320C2×C61—21no—
321C2×C61—21no—
322C2×C61—22no—
323C2×C61—22no—
324C2×C61C2×#41011yescharacteristic
325C2×C61C2×#41711yescharacteristic
326C2×C61C22×#10411yescharacteristic
327D61—82no—
328#533—611no—
329#533—611no—
330#533—611no—
331#532—611no—
332#532—611no—
333#532—611no—
334#529—612no—
335#529—612no—
336#529—1221no—
337#529—1221no—
338#529—1221no—
339#529—1221no—
340#525—612no—
341#525—1221no—
342#525—1221no—
343C4×#41—612no—
344C4×#41—1221no—
345C4×#41—1221no—
346C22×C4⋊C4—21no—
347C22×C42—21no—
348C22×C42C2×D6111yescharacteristic
349#432—612no—
350#432—1221no—
351#431—612no—
352#431—1221no—
353#423—1221no—
354#407—612no—
355C4×M16—1221no—
356C4×C4⋊C4—21no—
357C4×C4⋊C4—21no—
358C4×C4⋊C4C2×D6112yes—
359C4×C4⋊C4C2×D6112yes—
360C2×#95C2×D6112yes—
361#328—618no—
362#327—618no—
363C61#1754611yescharacteristic, complemented, Sylow
364C2×#41—612no—
365C2×#41—1221no—
366C2×#41—1221no—
367C23×C4—22no—
368C23×C4C22⋊D6111yescharacteristic
369C23×C4C22⋊D6111yescharacteristic
370C23×C4C22×D6111yescharacteristic
371C2×M16—1222no—
372C4⋊Q8—611no—
373C4⋊Q8—611no—
374C4⋊Q8—611no—
375#127—611no—
376#126—1222no—
377#126—1222no—
378#125—612no—
379#125—1221no—
380#125—1221no—
381#124—612no—
382#124—1221no—
383#124—1221no—
384#123—1222no—
385#123—1222no—
386#123—1222no—
387C22⋊Q8—612no—
388C22⋊Q8—612no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391C22⋊Q8—1221no—
392C22⋊Q8—1221no—
393#121—612no—
394#121—612no—
395#121—1221no—
396#121—1221no—
397#121—1221no—
398#121—1221no—
399#120—1222no—
400C4×Q8—612no—
401C4×Q8—612no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1221no—
406C4×D4—612no—
407C4×D4—612no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4—1221no—
416C4×D4—1221no—
417C4×D4—1221no—
418#117—612no—
419#117—1221no—
420#117—1221no—
421#117—1222no—
422#117—1222no—
423#117—1222no—
424C2×C4⋊C4—28no—
425C2×C4⋊C4—22no—
426C2×C4⋊C4—22no—
427C2×C42—22no—
428C2×C42—22no—
429C2×C42—44no—
430C2×C42—1222no—
431C2×C42C4×D6111yescharacteristic
432C2×C42C4×D6111yescharacteristic
433C2×C42C4×D6111yescharacteristic
434C2×C42C4×D6111yescharacteristic
435C2×C42D24411yescharacteristic
436C2×C42D24411yescharacteristic
437C2×C42D24411yescharacteristic
438C2×C42D24411yescharacteristic
439C2×C42C22⋊D6111yescharacteristic
440C2×C42C22⋊D6111yescharacteristic
441C2×C42C22×D6111yescharacteristic
442C2×C42C22×D6111yescharacteristic
443C4⋊C8—1222no—
444C4⋊C8—1222no—
445C22⋊C8—614no—
446C22⋊C8—1222no—
447#97—1222no—
448C4×C8—1222no—
449#95—22no—
450#95C22×D6114yes—
451C24C2×C22⋊D6111yescharacteristic
452#41—2444no—
453#41—2442no—
454#41—2444no—
455#41—2442no—
456C2×Q8—612no—
457C2×Q8—1221no—
458C2×Q8—1221no—
459C2×D4—612no—
460C2×D4—1222no—
461C2×D4—1222no—
462C2×D4—1222no—
463C2×D4—1221no—
464C2×D4—1221no—
465C22×C4—22no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—28no—
469C22×C4—24no—
470C22×C4—42no—
471C22×C4—44no—
472C22×C4—44no—
473C22×C4—1222no—
474C22×C4—1222no—
475C22×C4—1222no—
476C22×C4#1175819811yescharacteristic
477C22×C4#1175819811yescharacteristic
478C22×C4#1175819811yescharacteristic
479C22×C4#1175819811yescharacteristic
480C22×C4#1175819811yescharacteristic
481C22×C4#1175819811yescharacteristic
482C22×C4#1175819811yescharacteristic
483C22×C4#1175819811yescharacteristic
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C2×C22⋊D6111yescharacteristic
492C22×C4C2×C22⋊D6111yescharacteristic
493C22×C4C23×D6111yescharacteristic
494M16—2444no—
495C2×C8—1224no—
496C4⋊C4—28no—
497C4⋊C4—28no—
498C4⋊C4—612no—
499C4⋊C4—612no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1222no—
503C4⋊C4—1222no—
504C4⋊C4—1222no—
505C4⋊C4—1222no—
506C4⋊C4—1221no—
507C4⋊C4—1221no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C22⋊C4—1222no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C42—44no—
520C42—44no—
521C42—1222no—
522C42—1222no—
523C42—1222no—
524C42#1175819811yescharacteristic
525C42#1175819811yescharacteristic
526C42#1175819811yescharacteristic
527C42#1175819811yescharacteristic
528C42#1175819811yescharacteristic
529C42#1175819811yescharacteristic
530C42#1175819811yescharacteristic
531C42#1175819811yescharacteristic
532C42C2×C4×D6111yescharacteristic
533C42C2×C4×D6111yescharacteristic
534C42C2×C4×D6111yescharacteristic
535C42C2×C4×D6111yescharacteristic
536C42C2×D24411yescharacteristic
537C42C2×D24411yescharacteristic
538C42C2×D24411yescharacteristic
539C42C2×D24411yescharacteristic
540C23—22no—
541C23—22no—
542C23—21no—
543C23—21no—
544C23—1222no—
545C23C2×#1175819811yescharacteristic
546C23C2×#1175819811yescharacteristic
547C23C22×C22⋊D6111yescharacteristic
548Q8—2442no—
549Q8—2442no—
550Q8—2442no—
551D4—2442no—
552D4—2444no—
553D4—2444no—
554D4—2442no—
555D4—2442no—
556D4—2442no—
557D4—2442no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—22no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—28no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—416no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—1222no—
582C2×C4—2444no—
583C2×C4—2444no—
584C2×C4—2444no—
585C2×C4#1451970514yes—
586C2×C4C2×#1175819811yescharacteristic
587C2×C4C2×#1175819811yescharacteristic
588C2×C4C2×#1175819811yescharacteristic
589C2×C4C2×#1175819811yescharacteristic
590C2×C4C2×#1175819811yescharacteristic
591C2×C4C2×#1175819811yescharacteristic
592C2×C4C2×#1175819811yescharacteristic
593C2×C4C2×#1175819811yescharacteristic
594C2×C4C2×#1175819811yescharacteristic
595C2×C4C2×#1175819811yescharacteristic
596C2×C4#1451980312yes—
597C2×C4#1451980812yes—
598C2×C4C22×C4×D6111yescharacteristic
599C2×C4C22×D24411yescharacteristic
600C8—2444no—
601C22—24no—
602C22—24no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22#1683973512yes—
613C22C2×#1451970512yes—
614C22C22×#1175819811yescharacteristic
615C22C2×#1451980311yescharacteristic
616C22C2×#1451980811yescharacteristic
617C4—21no—
618C4—21no—
619C4—21no—
620C4—21no—
621C4—21no—
622C4—21no—
623C4—21no—
624C4—21no—
625C4—48no—
626C4—2442no—
627C4—2442no—
628C4—2442no—
629C4#1683976112yes—
630C4#1683976112yes—
631C4C2×#1451970512yes—
632C4C2×#1451970512yes—
633C2—21no—
634C2—22no—
635C2—22no—
636C2—21no—
637C2—4882no—
638C2C2×#1683973511yescharacteristic
639C2C2×#1683976111yescharacteristic
640C2C22×#1451970511yescharacteristic
subgroupG/H
centerC2×C4C2×#11758198
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C4×C4⋊C4×C61C2
socleC22×C61C22×C22⋊C4
prime radical#21459115C2
cosocleC22×C4×C61C24
hypercenterC2×C4×C4⋊C4D61
hypocenterC61#17546
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #21454452C2semidirect
3. #21444342C2semidirect
4. #21444321C2semidirect
5. #21444317C2semidirect
6. #21444313C2semidirect
7. C2×C4×C4⋊C4×C61C2semidirect
8. #16840319C4semidirect
9. #16840319C4semidirect
11. #16840312C4semidirect
12. #16840312C4semidirect
18. #16840296C4semidirect
25. C4×#11758222C4semidirect
47. #14519777C2×C4semidirect
48. #14519777C2×C4semidirect
49. #14519777C2×C4semidirect
50. #14519774C2×C4semidirect
55. #14519769C2×C4semidirect
58. #14519765C2×C4semidirect
79. C4×Dic122C2×C4semidirect
80. C4×Dic122C2×C4semidirect
85. C4×D244C2×C4semidirect
86. C4×D244C2×C4semidirect
315. C2×C4×C4⋊C4D61semidirect
363. C61#17546semidirect

Maximal subgroups

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

subgroupindex
#214591152
#214544522
#214443422
#214443212
#214443172
#214443132
C2×C4×C4⋊C4×C612
#1754661