Finite Group Databasebeta

← #29647199

Subgroups — #29647199

Order 15616.

1194 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459114C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic, complemented
3C4×C4⋊C4×D61C211yesmaximal, characteristic, complemented
4#21444341C214yesmaximal, complemented
5#21444320C214yesmaximal, complemented
6#21444319C211yesmaximal, characteristic, complemented
7#21444317C211yesmaximal, characteristic, complemented
8#21444314C211yesmaximal, characteristic, complemented
9C61×#1502C211yesmaximal, characteristic, complemented
10#16840319C2211yescharacteristic
11#16840317C411yescharacteristic, complemented
12#16840317C411yescharacteristic, complemented
13#16840310C411yescharacteristic, complemented
14#16840310C411yescharacteristic, complemented
15#16840294—21no—
16#16840294C412yescomplemented
17#16840293—21no—
18#16840293C2212yes—
19#16840291—21no—
20#16840291—21no—
21#16840272—21no—
22#16840272C2212yes—
23#16840269—21no—
24#41×Dic61—21no—
25C4×#11758224—21no—
26C4×#11758224C412yescomplemented
27C2×C4⋊C4×D61—21no—
28C2×C42×D61—21no—
29C2×C42×D61C2211yescharacteristic
30#16839818—21no—
31#16839816C2212yes—
32#16839815C2211yescharacteristic
33#16839813—21no—
34#16839784—21no—
35#16839724C2212yes—
36C4×#11758189—21no—
37C4⋊C4×Dic61—21no—
38C4×#11758196—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40#16839613C2212yes—
41#95×D61C2212yes—
42#16839500C2214yes—
43#16839498C2214yes—
44#16839496C2214yes—
45C61×#532C2211yescharacteristic
46C61×#431C2211yescharacteristic
47C4×C4⋊C4×C61C2211yescharacteristic
48C61×#327C2214yes—
49C2×#11758224—21no—
50C2×#11758224C2×C412yes—
51C22×C4×D61—22no—
52C22×C4×D61D411yescharacteristic
53C22×C4×D61D411yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C2×#11758189—22no—
56#14519777C2×C411yescharacteristic, complemented
57#14519777C2×C411yescharacteristic, complemented
58#14519770—22no—
59#14519770—22no—
60#14519769—21no—
61#14519768—21no—
62#14519768C2×C412yescomplemented
63#14519764—21no—
64#14519764C2×C412yescomplemented
65#14519759—22no—
66#14519759—22no—
67#14519758—22no—
68#14519757—21no—
69#14519757C2×C412yes—
70#14519756—21no—
71#14519756C2312yes—
72#14519753—21no—
73#14519753—21no—
74#14519748—21no—
75#14519748—21no—
76#14519748—21no—
77#14519748C2×C412yes—
78#14519745—22no—
79Q8×Dic61—21no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122C2×C412yescomplemented
83#14519742—21no—
84#14519742C2×C412yescomplemented
85D4×Dic61—22no—
86D4×Dic61—22no—
87D4×Dic61—21no—
88D4×Dic61—21no—
89D4×Dic61—21no—
90D4×Dic61C2×C412yescomplemented
91C4×C22⋊D61—22no—
92C4×C22⋊D61—22no—
93#14519737—22no—
94#14519737—22no—
95#14519736—21no—
96#14519736—22no—
97#14519735—21no—
98#14519735C2×C412yes—
99#14519733—22no—
100#14519733—22no—
101C2×C4⋊Dic61—21no—
102C2×#11758196—22no—
103C4⋊C4×D61—28no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—22no—
107C2×C4×Dic61D411yescharacteristic
108C2×C4×Dic61D411yescharacteristic
109C2×C4×Dic61C2311yescharacteristic
110C42×D61—44no—
111C42×D61C2×C411yescharacteristic
112C42×D61C2×C411yescharacteristic
113C42×D61C2×C411yescharacteristic
114C42×D61C2×C411yescharacteristic
115C42×D61D411yescharacteristic
116C42×D61D411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C4⋊#92418—21no—
120#14519706—22no—
121C22⋊#92418C2312yes—
122#14519689—22no—
123#14519687—22no—
124C4×#92418—21no—
125#14519684—22no—
126#14519684C2312yes—
127C2×#41×C61—21no—
128C4⋊Q8×C61C2×C411yescharacteristic, complemented
129#127×C61C2×C411yescharacteristic, complemented
130#124×C61—21no—
131#121×C61—21no—
132#121×C61C2312yes—
133C4×D4×C61—21no—
134C4×D4×C61C2×C412yescomplemented
135C2×C4⋊C4×C61—21no—
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138C4⋊C8×C61—21no—
139C22⋊C8×C61C2312yes—
140C4×C8×C61—21no—
141#95×C61C2312yes—
142C23×D61C2×D411yescharacteristic
143#41×C61—42no—
144#41×C61—42no—
145C2×Q8×C61—21no—
146C2×D4×C61—22no—
147C2×D4×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61C22×C412yes—
150C22×C4×C61—22no—
151C22×C4×C61—22no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C2×C22⋊D61—22no—
156C2×C22⋊D61—22no—
157C22×Dic61—22no—
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163#11758224—44no—
164#11758224—44no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×C4×D61—24no—
168C2×C4×D61—28no—
169C2×C4×D61—42no—
170C2×C4×D61—44no—
171C2×C4×D61—44no—
172C2×C4×D61C22⋊C411yescharacteristic
173C2×C4×D61C22⋊C411yescharacteristic
174C2×C4×D61C22⋊C411yescharacteristic
175C2×C4×D61C22⋊C411yescharacteristic
176C2×C4×D61C22⋊C411yescharacteristic
177C2×C4×D61C22⋊C411yescharacteristic
178C2×C4×D61C22⋊C411yescharacteristic
179C2×C4×D61C22⋊C411yescharacteristic
180C2×C4×D61C22×C411yescharacteristic
181C2×C4×D61C22×C411yescharacteristic
182C2×C4×D61C2×D411yescharacteristic
183C2×C4×D61C2×D411yescharacteristic
184C2×Dic122—21no—
185C2×Dic122C22×C412yes—
186C2×C8×C61—22no—
187C4⋊C4×C61—24no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C22×C412yes—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C42×C61—42no—
193C42×C61C22×C411yescharacteristic
194C42×C61C22×C411yescharacteristic
195C42×C61C2×D411yescharacteristic
196C42×C61C2×D411yescharacteristic
197C22⋊Dic61—22no—
198C22⋊Dic61—22no—
199C22⋊Dic61—22no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202#11758198—22no—
203#11758198—22no—
204C4⋊Dic61—24no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61C22×C412yes—
211#11758196—28no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216C4×Dic61—22no—
217C4×Dic61—22no—
218C4×Dic61—22no—
219C4×Dic61—44no—
220C4×Dic61—42no—
221C4×Dic61C22⋊C411yescharacteristic
222C4×Dic61C22⋊C411yescharacteristic
223C4×Dic61C22⋊C411yescharacteristic
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22×C411yescharacteristic
230C4×Dic61C22×C411yescharacteristic
231C4×Dic61C2×D411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C2×#92418—22no—
234#11758189—44no—
235C23×C61—22no—
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—22no—
240C22×D61—22no—
241C22×D61C2×C22⋊C411yescharacteristic
242C22×D61C2×C22⋊C411yescharacteristic
243Q8×C61—42no—
244D4×C61—42no—
245D4×C61—42no—
246D4×C61—42no—
247D4×C61—42no—
248D4×C61—42no—
249C2×C4×C61—22no—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—41no—
253C2×C4×C61—42no—
254C2×C4×C61—42no—
255C2×C4×C61—44no—
256C2×C4×C61C2×C22⋊C411yescharacteristic
257C2×C4×C61C2×C22⋊C411yescharacteristic
258C2×C4×C61C2×C22⋊C411yescharacteristic
259C2×C4×C61C2×C22⋊C411yescharacteristic
260C2×C4×C61AGL(1,ℤ/8)12yes—
261C2×C4×C61#13712yes—
262C2×C4×C61C23×C411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C22⋊D61—44no—
265C22⋊D61—44no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—24no—
270C2×Dic61—22no—
271C2×Dic61—22no—
272C2×Dic61—41no—
273C2×Dic61—44no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—44no—
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—22no—
291C4×D61—22no—
292C4×D61—41no—
293C4×D61—41no—
294C4×D61—416no—
295C4×D61#10414yes—
296Dic122—42no—
297Dic122—42no—
298C8×C61—42no—
299#92418—42no—
300#17546—611nomaximal, Sylow
301C22×C61—21no—
302C22×C61—21no—
303C22×C61—42no—
304C22×C61—42no—
305C22×C61—42no—
306C22×C61C22×C22⋊C411yescharacteristic
307C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C2×D61—24no—
310C2×D61—24no—
311C2×D61—21no—
312C2×D61—21no—
313C2×D61—21no—
314C2×D61—21no—
315C2×D61#41012yes—
316C2×D61C2×#10412yes—
317C4×C61—21no—
318C4×C61—21no—
319C4×C61—21no—
320C4×C61—44no—
321C4×C61—42no—
322C4×C61#41712yes—
323C4×C61#41712yes—
324C4×C61C2×#10412yes—
325Dic61—21no—
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—44no—
331Dic61—42no—
332Dic61—42no—
333Dic61C2×#10412yes—
334#3444—611no—
335#2920—611no—
336C2×C4×C4⋊C4—611no—
337#1509—614no—
338#1504—614no—
339#1503—611no—
340#1503—611no—
341#1502—611no—
342#1502D6111yescharacteristic, complemented
343C2×C61—21no—
344C2×C61—21no—
345C2×C61—82no—
346C2×C61C2×#41011yescharacteristic
347C2×C61C2×#41711yescharacteristic
348C2×C61C22×#10411yescharacteristic
349D61—22no—
350D61—22no—
351#533—611no—
352#533—611no—
353#533—611no—
354#532—611no—
355#532—611no—
356#532C2×D6111yescharacteristic
357#529—612no—
358#529—612no—
359#529—1221no—
360#529—1221no—
361#529—1221no—
362#529—1221no—
363#525—612no—
364#525—1221no—
365#525—1221no—
366C4×#41—612no—
367C4×#41—1221no—
368C4×#41—1221no—
369C22×C4⋊C4—1221no—
370C22×C42—611no—
371C22×C42—1221no—
372#432—612no—
373#432—1221no—
374#431—611no—
375#431—1221no—
376#431C2×D6111yescharacteristic
377#423—1221no—
378#407—612no—
379C4×M16—1221no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—612no—
382C4×C4⋊C4—1221no—
383C4×C4⋊C4—1221no—
384C4×C4⋊C4C2×D6111yescharacteristic
385C2×#95—612no—
386#328—614no—
387#328—614no—
388#327—614no—
389#327C2×D6114yes—
390C61#1754611yescharacteristic, complemented, Sylow
391C2×#41—21no—
392C2×#41—612no—
393C2×#41—1221no—
394C23×C4—611no—
395C23×C4—611no—
396C23×C4—611no—
397C23×C4—1222no—
398C2×M16—1222no—
399C4⋊Q8—611no—
400C4⋊Q8—611no—
401C4⋊Q8C4×D6111yescharacteristic, complemented
402#127C4×D6111yescharacteristic, complemented
403#126—1222no—
404#126—1222no—
405#125—612no—
406#125—1221no—
407#125—1221no—
408#124—21no—
409#124—612no—
410#124—1221no—
411#123—1222no—
412#123—1222no—
413#123—1222no—
414C22⋊Q8—612no—
415C22⋊Q8—612no—
416C22⋊Q8—1221no—
417C22⋊Q8—1221no—
418C22⋊Q8—1221no—
419C22⋊Q8—1221no—
420#121—21no—
421#121—612no—
422#121—1221no—
423#121—1221no—
424#121—1221no—
425#121C22×D6112yes—
426#120—1222no—
427C4×Q8—612no—
428C4×Q8—612no—
429C4×Q8—1221no—
430C4×Q8—1221no—
431C4×Q8—1221no—
432C4×Q8—1221no—
433C4×D4—21no—
434C4×D4—612no—
435C4×D4—1221no—
436C4×D4—1221no—
437C4×D4—1221no—
438C4×D4—1222no—
439C4×D4—1222no—
440C4×D4—1222no—
441C4×D4—1222no—
442C4×D4—1222no—
443C4×D4—1222no—
444C4×D4C4×D6112yescomplemented
445#117—612no—
446#117—1222no—
447#117—1222no—
448#117—1222no—
449#117—1221no—
450#117—1221no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—1228no—
453C2×C4⋊C4—1222no—
454C2×C4⋊C4—1221no—
455C2×C42—21no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—611no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—1222no—
468C2×C42—1222no—
469C2×C42—1221no—
470C2×C42—2444no—
471C2×C42C22×D6111yescharacteristic
472C4⋊C8—21no—
473C4⋊C8—1222no—
474C4⋊C8—1221no—
475C22⋊C8—612no—
476C22⋊C8—1222no—
477C22⋊C8C22×D6112yes—
478#97—1222no—
479C4×C8—21no—
480C4×C8—1221no—
481#95—612no—
482#95—1222no—
483#95C22×D6112yes—
484C24—611no—
485#41—42no—
486#41—42no—
487#41—2444no—
488#41—2444no—
489C2×Q8—21no—
490C2×Q8—612no—
491C2×Q8—1221no—
492C2×D4—22no—
493C2×D4—21no—
494C2×D4—21no—
495C2×D4—1222no—
496C2×D4—1222no—
497C2×D4C2×C4×D6112yes—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—611no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1224no—
516C22×C4—1228no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1222no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—2442no—
523C22×C4—2444no—
524C22×C4—2444no—
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528M16—2444no—
529C2×C8—22no—
530C2×C8—1222no—
531C4⋊C4—24no—
532C4⋊C4—21no—
533C4⋊C4—612no—
534C4⋊C4—1222no—
535C4⋊C4—1222no—
536C4⋊C4—1222no—
537C4⋊C4—1222no—
538C4⋊C4—1228no—
539C4⋊C4—1224no—
540C4⋊C4—1222no—
541C4⋊C4—1222no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4C2×C4×D6112yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—1222no—
549C22⋊C4—1222no—
550C22⋊C4—1222no—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1222no—
554C22⋊C4—1222no—
555C42—42no—
556C42—611no—
557C42—611no—
558C42—611no—
559C42—611no—
560C42—611no—
561C42—611no—
562C42—611no—
563C42—611no—
564C42—611no—
565C42—611no—
566C42—611no—
567C42—611no—
568C42—1222no—
569C42—1222no—
570C42—1222no—
571C42—2444no—
572C42—2442no—
573C42C2×C4×D6111yescharacteristic
574C42C2×C4×D6111yescharacteristic
575C42D4×D6111yescharacteristic
576C42D4×D6111yescharacteristic
577C23—22no—
578C23—611no—
579C23—611no—
580C23—1222no—
581C23—1222no—
582C23—1221no—
583C23—1221no—
584C23C2×D4×D6111yescharacteristic
585Q8—42no—
586Q8—2442no—
587Q8—2442no—
588D4—42no—
589D4—42no—
590D4—42no—
591D4—42no—
592D4—42no—
593D4—2444no—
594D4—2444no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—41no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—44no—
602C2×C4—611no—
603C2×C4—614no—
604C2×C4—611no—
605C2×C4—611no—
606C2×C4—611no—
607C2×C4—611no—
608C2×C4—611no—
609C2×C4—1222no—
610C2×C4—1222no—
611C2×C4—1222no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1224no—
622C2×C4—1222no—
623C2×C4—1222no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—24416no—
627C2×C4—2444no—
628C2×C4—2441no—
629C2×C4—2444no—
630C2×C4—2442no—
631C2×C4—2442no—
632C2×C4C22⋊C4×D6111yescharacteristic
633C2×C4C22⋊C4×D6111yescharacteristic
634C2×C4C22⋊C4×D6111yescharacteristic
635C2×C4C22⋊C4×D6111yescharacteristic
636C2×C4#1451980112yes—
637C2×C4#1451980612yes—
638C2×C4C22×C4×D6111yescharacteristic
639C2×C4C2×D4×D6111yescharacteristic
640C8—42no—
641C8—2442no—
642C22—21no—
643C22—21no—
644C22—42no—
645C22—42no—
646C22—42no—
647C22—612no—
648C22—612no—
649C22—1224no—
650C22—1224no—
651C22—1221no—
652C22—1221no—
653C22—1221no—
654C22—1221no—
655C22C2×C22⋊C4×D6111yescharacteristic
656C22C2×#1451980111yescharacteristic
657C22C2×#1451980611yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—44no—
662C4—42no—
663C4—612no—
664C4—1221no—
665C4—1221no—
666C4—1221no—
667C4—1221no—
668C4—1221no—
669C4—2444no—
670C4—2442no—
671C4—2442no—
672C4#1683976012yes—
673C4#1683976012yes—
674C4#104×D6112yes—
675C2—21no—
676C2—21no—
677C2—82no—
678C2—1222no—
679C2—1222no—
680C2#410×D6111yescharacteristic
681C2C2×#1683976011yescharacteristic
682C2C2×#104×D6111yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1502C2
socleC22×C61C22×C22⋊C4
prime radical#21459114C2
cosocleC22×C4×C61C24
hypercenter#1502D61
hypocenterC61#17546
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
2. #21454447C2semidirect
3. C4×C4⋊C4×D61C2semidirect
4. #21444341C2semidirect
5. #21444320C2semidirect
6. #21444319C2semidirect
7. #21444317C2semidirect
8. #21444314C2semidirect
9. C61×#1502C2semidirect
11. #16840317C4semidirect
12. #16840317C4semidirect
13. #16840310C4semidirect
14. #16840310C4semidirect
16. #16840294C4semidirect
26. C4×#11758224C4semidirect
56. #14519777C2×C4semidirect
57. #14519777C2×C4semidirect
62. #14519768C2×C4semidirect
64. #14519764C2×C4semidirect
82. C4×Dic122C2×C4semidirect
84. #14519742C2×C4semidirect
90. D4×Dic61C2×C4semidirect
128. C4⋊Q8×C61C2×C4semidirect
129. #127×C61C2×C4semidirect
134. C4×D4×C61C2×C4semidirect
342. #1502D61semidirect
390. C61#17546semidirect
401. C4⋊Q8C4×D61semidirect
402. #127C4×D61semidirect
444. C4×D4C4×D61semidirect

Maximal subgroups

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

subgroupindex
#214591142
#214544472
C4×C4⋊C4×D612
#214443412
#214443202
#214443192
#214443172
#214443142
C61×#15022
#1754661