Finite Group Databasebeta

← #29685397

Subgroups — #29685397

Order 15616.

1132 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840298C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic, complemented
3C2×M16×Dic61C211yesmaximal, characteristic, complemented
4#21446810C212yesmaximal, complemented
5#21446667C212yesmaximal, complemented
6#21446288C211yesmaximal, characteristic, complemented
7#21446282C211yesmaximal, characteristic
8#21446280C211yesmaximal, characteristic
9#101×Dic61C212yesmaximal, complemented
10#21445610C212yesmaximal, complemented
11C61×#1901C211yesmaximal, characteristic, complemented
12C22×#11758194C2211yescharacteristic
13#16840298—21no—
14#16840298—21no—
15#16840298—21no—
16#16840298—21no—
17#16840298—21no—
18#16840298—21no—
19#16840298—21no—
20#16840298—21no—
21C2×#14519772C412yes—
22C2×#14519768C2211yescharacteristic
23C2×#14519766C2211yescharacteristic
24C2×#14519761C412yes—
25C2×#14519751C2211yescharacteristic
26C2×Q8×Dic61C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×#14519737C412yes—
29C2×#14519735C412yes—
30C22×C4×Dic61C2211yescharacteristic
31#16839816—21no—
32#16839816—21no—
33#16839812—21no—
34#16839812—21no—
35#16839785C2212yes—
36C2×#14519696C2212yes—
37C2×#14519693C2212yes—
38C2×#14519693C2212yes—
39#16839726C2211yescharacteristic
40M16×Dic61—21no—
41M16×Dic61—21no—
42M16×Dic61C2212yes—
43M16×Dic61C2212yes—
44M16×Dic61C2212yes—
45C2×#14519687—21no—
46C2×C8×Dic61—21no—
47#16839531C2212yes—
48#16839530C2212yes—
49#16839530C2212yes—
50C22×M16×C61C2211yescharacteristic
51C2×C22⋊Q8×C61C2211yescharacteristic
52C2×#101×C61C2212yes—
53C61×#407C2211yescharacteristic
54C61×#343C2212yes—
55C22×D244C2311yescharacteristic
56C22×C22⋊D61C2×C412yes—
57C23×Dic61C2×C411yescharacteristic
58C23×Dic61C2×C411yescharacteristic
59C22×C4×D61C2×C412yes—
60C2×#11758194—21no—
61C2×#11758194—24no—
62C2×#11758194C2312yes—
63C2×#11758194C2312yes—
64C22×#92418—21no—
65#14519772—22no—
66#14519772—22no—
67#14519772—22no—
68#14519772—22no—
69#14519768—21no—
70#14519768—21no—
71#14519768—21no—
72#14519768—21no—
73#14519766—21no—
74#14519766—21no—
75#14519766—21no—
76#14519766—21no—
77#14519761—22no—
78#14519761—22no—
79#14519761—22no—
80#14519761—22no—
81#14519751—21no—
82#14519751—21no—
83#14519751—21no—
84#14519751—21no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88Q8×Dic61—21no—
89#14519739—21no—
90#14519739—21no—
91#14519739—21no—
92#14519739—21no—
93#14519737—22no—
94#14519737—22no—
95#14519737—22no—
96#14519737—22no—
97#14519735—22no—
98#14519735—22no—
99#14519735—22no—
100#14519735—22no—
101C2×C4⋊Dic61C2×C412yes—
102C2×C4⋊Dic61C2×C412yes—
103C2×C4⋊Dic61C2311yescharacteristic
104C2×#11758196C2×C412yes—
105C2×#11758198C2×C412yes—
106C2×#11758198C2×C412yes—
107C2×#11758198C2×C412yes—
108C2×#11758198C2×C412yes—
109C2×C4×Dic61C2×C412yes—
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D411yescharacteristic
117C2×C4×Dic61D411yescharacteristic
118C2×C4×Dic61D411yescharacteristic
119C2×C4×Dic61D411yescharacteristic
120C2×C4×Dic61C2311yescharacteristic
121C2×C4×Dic61C2311yescharacteristic
122#14519706—22no—
123#14519706—22no—
124#14519696—24no—
125#14519693—24no—
126#14519693—24no—
127C22⋊#92418—21no—
128C22⋊#92418—21no—
129#14519687—22no—
130#14519687—22no—
131#14519687—22no—
132C8×Dic61—22no—
133C8×Dic61—22no—
134C8×Dic61—22no—
135C22×Q8×C61C2311yescharacteristic
136C23×C4×C61C2311yescharacteristic
137C2×M16×C61—21no—
138C2×M16×C61—24no—
139C2×M16×C61C2312yes—
140C2×M16×C61C2312yes—
141C22×C8×C61—21no—
142C22⋊Q8×C61—21no—
143C22⋊Q8×C61—21no—
144C22⋊Q8×C61—21no—
145C22⋊Q8×C61—21no—
146C2×C4⋊C4×C61C2×C412yes—
147C2×C4⋊C4×C61C2311yescharacteristic
148C2×C22⋊C4×C61C2×C412yes—
149#101×C61—24no—
150C22⋊C8×C61—21no—
151C22⋊C8×C61—21no—
152C24×C61C22×C411yescharacteristic
153C23×D61C22×C412yes—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—22no—
157C2×Q8×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61—22no—
160C22×C4×C61C22×C412yes—
161C22×C4×C61C22×C411yescharacteristic
162C22×C4×C61C22×C412yes—
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2×D411yescharacteristic
167C22×C4×C61C2411yescharacteristic
168C2×C22⋊D61—22no—
169C2×C22⋊D61—22no—
170C2×C22⋊D61—44no—
171C22×Dic61—42no—
172C22×Dic61—42no—
173C22×Dic61C22⋊C411yescharacteristic
174C22×Dic61C22⋊C411yescharacteristic
175C22×Dic61C22⋊C411yescharacteristic
176C22×Dic61C22⋊C411yescharacteristic
177C22×Dic61C22⋊C411yescharacteristic
178C22×Dic61C22⋊C411yescharacteristic
179C22×Dic61C22⋊C411yescharacteristic
180C22×Dic61C22⋊C411yescharacteristic
181C22×Dic61C22×C411yescharacteristic
182C22×Dic61C22×C412yes—
183C22×Dic61C22×C411yescharacteristic
184C22×Dic61C22×C411yescharacteristic
185C22×Dic61C22×C411yescharacteristic
186C2×D244—21no—
187C2×D244—21no—
188C2×D244—22no—
189C2×D244—22no—
190C2×C4×D61—22no—
191C2×C4×D61—22no—
192C2×C4×D61—44no—
193M16×C61—24no—
194M16×C61—22no—
195M16×C61—22no—
196M16×C61—22no—
197M16×C61#4114yes—
198C2×C8×C61—24no—
199C2×C8×C61—22no—
200C2×C8×C61—22no—
201C4⋊C4×C61—22no—
202C4⋊C4×C61—22no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C22⋊C4×C61—22no—
206C22⋊C4×C61—22no—
207#11758198—22no—
208#11758198—22no—
209#11758198—22no—
210#11758198—22no—
211#11758198—22no—
212#11758198—22no—
213#11758198—22no—
214#11758198—22no—
215C4⋊Dic61—21no—
216C4⋊Dic61—21no—
217C4⋊Dic61—22no—
218C4⋊Dic61—22no—
219C4⋊Dic61—22no—
220C4⋊Dic61—22no—
221#11758196—22no—
222#11758196—22no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22⋊C411yescharacteristic
232C4×Dic61C22⋊C411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D411yescharacteristic
239C4×Dic61C2×D411yescharacteristic
240C4×Dic61C2×D411yescharacteristic
241#11758194—24no—
242#11758194—22no—
243#11758194—22no—
244#11758194—22no—
245#11758194#4114yes—
246C2×#92418—24no—
247C2×#92418—22no—
248C2×#92418—22no—
249C23×C61—22no—
250C23×C61—22no—
251C23×C61—41no—
252C23×C61C2×C22⋊C411yescharacteristic
253C23×C61C2×C22⋊C411yescharacteristic
254C23×C61C23×C411yescharacteristic
255C22×D61—22no—
256C22×D61—24no—
257C22×D61—42no—
258C22×D61—42no—
259Q8×C61—41no—
260Q8×C61—41no—
261Q8×C61—42no—
262C2×C4×C61—22no—
263C2×C4×C61—21no—
264C2×C4×C61—21no—
265C2×C4×C61—21no—
266C2×C4×C61—21no—
267C2×C4×C61—22no—
268C2×C4×C61—22no—
269C2×C4×C61—22no—
270C2×C4×C61—22no—
271C2×C4×C61—24no—
272C2×C4×C61—42no—
273C2×C4×C61C2×C22⋊C411yescharacteristic
274C2×C4×C61C2×C22⋊C411yescharacteristic
275C2×C4×C61C22×D411yescharacteristic
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C2×#4112yes—
278C2×C4×C61C2×#4112yes—
279C22⋊D61—44no—
280C22⋊D61—44no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—22no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61—41no—
294C2×Dic61—42no—
295C2×Dic61C2×C22⋊C411yescharacteristic
296C2×Dic61C2×C22⋊C411yescharacteristic
297C2×Dic61C2×C22⋊C411yescharacteristic
298C2×Dic61C2×C22⋊C411yescharacteristic
299C2×Dic61C2×C22⋊C411yescharacteristic
300C2×Dic61C2×C22⋊C411yescharacteristic
301C2×Dic61C2×C22⋊C411yescharacteristic
302C2×Dic61C2×C22⋊C411yescharacteristic
303D244—41no—
304D244—41no—
305D244—42no—
306C4×D61—44no—
307C4×D61—44no—
308C8×C61—24no—
309C8×C61—42no—
310#92418—24no—
311#92418—42no—
312#21426—611nomaximal, Sylow
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—21no—
316C22×C61—21no—
317C22×C61—21no—
318C22×C61—21no—
319C22×C61—21no—
320C22×C61—44no—
321C22×C61#41312yes—
322C22×C61C22×C22⋊C411yescharacteristic
323C22×C61C4×#4112yes—
324C2×D61—42no—
325C2×D61—42no—
326C2×D61—42no—
327C2×D61—84no—
328C4×C61—21no—
329C4×C61—21no—
330C4×C61—42no—
331C4×C61—42no—
332C4×C61#52511yescharacteristic
333C4×C61#52511yescharacteristic
334C4×C61#53212yes—
335Dic61—21no—
336Dic61—21no—
337Dic61—21no—
338Dic61—21no—
339Dic61—21no—
340Dic61—21no—
341Dic61—21no—
342Dic61—42no—
343Dic61#41312yes—
344C2×#529—611no—
345#2920—611no—
346C2×C4×M16—611no—
347#1998—612no—
348#1967—612no—
349#1901—611no—
350#1901D6111yescharacteristic, complemented
351#1900—611no—
352#1900—611no—
353C4×#101—612no—
354C4×#100—612no—
355C2×C61—22no—
356C2×C61—22no—
357C2×C61—41no—
358C2×C61#229711yescharacteristic
359C2×C61C2×#41311yescharacteristic
360C2×C61C2×#41311yescharacteristic
361D61—82no—
362C22×M16—611no—
363C22×M16C2×D6111yescharacteristic
364#529—1221no—
365#529—1221no—
366#529—1221no—
367#529—1221no—
368#529—1221no—
369#529—1221no—
370#529—1221no—
371#529—1221no—
372C2×#126—612no—
373C2×#125—611no—
374C2×#124—611no—
375C2×#123—612no—
376C2×C22⋊Q8C2×D6111yescharacteristic
377C2×#121—611no—
378C2×C4×Q8—611no—
379C2×C4×D4—611no—
380C2×C4×D4—612no—
381C2×#117—612no—
382C22×C42—611no—
383#432—1221no—
384#432—1221no—
385#431—1221no—
386#431—1221no—
387#423—612no—
388C2×#101—612no—
389C2×#101C2×D6112yes—
390C2×#100—612no—
391C2×#100—612no—
392#407—611no—
393#407C2×D6111yescharacteristic
394C4×M16—612no—
395C4×M16—612no—
396C4×M16—612no—
397C4×M16—1221no—
398C4×M16—1221no—
399C2×#97—1221no—
400C2×C4×C8—1221no—
401#343—612no—
402#343—612no—
403#343—612no—
404#343C2×D6112yes—
405C61#2142611yescharacteristic, complemented, Sylow
406C22×Q8C22×D6111yescharacteristic
407C22×D4—612no—
408C22×D4—611no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—612no—
412C23×C4C22×D6111yescharacteristic
413C2×M16—21no—
414C2×M16—24no—
415C2×M16—612no—
416C2×M16—612no—
417C2×M16—1224no—
418C2×M16—1221no—
419C2×M16C22×D6112yes—
420C2×M16C22×D6112yes—
421C22×C8—21no—
422C22×C8—1221no—
423#126—1222no—
424#126—1222no—
425#126—1222no—
426#126—1222no—
427#125—1221no—
428#125—1221no—
429#125—1221no—
430#125—1221no—
431#124—1221no—
432#124—1221no—
433#124—1221no—
434#124—1221no—
435#123—1222no—
436#123—1222no—
437#123—1222no—
438#123—1222no—
439C22⋊Q8—21no—
440C22⋊Q8—21no—
441C22⋊Q8—21no—
442C22⋊Q8—21no—
443#121—1221no—
444#121—1221no—
445#121—1221no—
446#121—1221no—
447C4×Q8—1221no—
448C4×Q8—1221no—
449C4×Q8—1221no—
450C4×Q8—1221no—
451C4×D4—1221no—
452C4×D4—1221no—
453C4×D4—1221no—
454C4×D4—1221no—
455C4×D4—1222no—
456C4×D4—1222no—
457C4×D4—1222no—
458C4×D4—1222no—
459#117—1222no—
460#117—1222no—
461#117—1222no—
462#117—1222no—
463C2×C4⋊C4—612no—
464C2×C4⋊C4—612no—
465C2×C4⋊C4—612no—
466C2×C4⋊C4—611no—
467C2×C4⋊C4C4×D6112yes—
468C2×C4⋊C4C22×D6111yescharacteristic
469C2×C22⋊C4—612no—
470C2×C22⋊C4—612no—
471C2×C22⋊C4—612no—
472C2×C22⋊C4—612no—
473C2×C22⋊C4C4×D6112yes—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—611no—
477C2×C42—611no—
478C2×C42—611no—
479C2×C42—611no—
480C2×C42—611no—
481C2×C42—611no—
482C2×C42—611no—
483C2×C42—611no—
484C2×C42—611no—
485C2×C42—611no—
486C2×C42—612no—
487C4⋊C8—1222no—
488C4⋊C8—1222no—
489#101—24no—
490#101—1224no—
491#100—1224no—
492#100—1224no—
493C22⋊C8—21no—
494C22⋊C8—21no—
495C22⋊C8—1221no—
496C22⋊C8—1221no—
497#97—1222no—
498#97—1222no—
499#97—1222no—
500C4×C8—1222no—
501C4×C8—1222no—
502C4×C8—1222no—
503C24—612no—
504C24C2×C4×D6111yescharacteristic
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—22no—
508C2×Q8—22no—
509C2×D4—1222no—
510C2×D4—1222no—
511C2×D4—1221no—
512C2×D4—1221no—
513C2×D4—1222no—
514C2×D4—1222no—
515C2×D4—2444no—
516C22×C4—22no—
517C22×C4—22no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—611no—
521C22×C4—611no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—611no—
526C22×C4—611no—
527C22×C4—611no—
528C22×C4—611no—
529C22×C4—611no—
530C22×C4—612no—
531C22×C4—1222no—
532C22×C4—1222no—
533C22×C4—2444no—
534C22×C4—2442no—
535C22×C4—2442no—
536C22×C4C2×C4×D6112yes—
537C22×C4C2×C4×D6111yescharacteristic
538C22×C4C2×C4×D6112yes—
539C22×C4D4×D6111yescharacteristic
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4D4×D6111yescharacteristic
543C22×C4C23×D6111yescharacteristic
544M16—24no—
545M16—22no—
546M16—22no—
547M16—22no—
548M16—614no—
549M16—1224no—
550M16—1222no—
551M16—1222no—
552M16—1222no—
553M16#1175822414yes—
554C2×C8—24no—
555C2×C8—22no—
556C2×C8—22no—
557C2×C8—1224no—
558C2×C8—1222no—
559C2×C8—1222no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—22no—
563C4⋊C4—22no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C4⋊C4—1222no—
567C4⋊C4—1222no—
568C4⋊C4—1222no—
569C4⋊C4—1222no—
570C4⋊C4—1221no—
571C4⋊C4—1221no—
572C22⋊C4—22no—
573C22⋊C4—22no—
574C22⋊C4—1222no—
575C22⋊C4—1222no—
576C22⋊C4—1222no—
577C22⋊C4—1222no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—1222no—
581C22⋊C4—1222no—
582C42—611no—
583C42—611no—
584C42—611no—
585C42—611no—
586C42—611no—
587C42—611no—
588C42—611no—
589C42—611no—
590C42—611no—
591C42—611no—
592C42—611no—
593C42—611no—
594C42—611no—
595C42—611no—
596C42—611no—
597C42—611no—
598C42—1222no—
599C42—1222no—
600C23—22no—
601C23—22no—
602C23—41no—
603C23—1224no—
604C23—1222no—
605C23—2442no—
606C23—2442no—
607C23C22⋊C4×D6111yescharacteristic
608C23C22⋊C4×D6111yescharacteristic
609C23C22×C4×D6111yescharacteristic
610Q8—41no—
611Q8—41no—
612Q8—42no—
613D4—2444no—
614D4—2444no—
615D4—2441no—
616D4—2441no—
617D4—2442no—
618C2×C4—22no—
619C2×C4—21no—
620C2×C4—21no—
621C2×C4—21no—
622C2×C4—21no—
623C2×C4—22no—
624C2×C4—22no—
625C2×C4—22no—
626C2×C4—22no—
627C2×C4—24no—
628C2×C4—42no—
629C2×C4—611no—
630C2×C4—611no—
631C2×C4—611no—
632C2×C4—611no—
633C2×C4—611no—
634C2×C4—611no—
635C2×C4—611no—
636C2×C4—611no—
637C2×C4—1222no—
638C2×C4—1222no—
639C2×C4—1222no—
640C2×C4—1222no—
641C2×C4—1222no—
642C2×C4—1222no—
643C2×C4—2444no—
644C2×C4—2444no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4—2441no—
651C2×C4—2441no—
652C2×C4—2442no—
653C2×C4C22⋊C4×D6111yescharacteristic
654C2×C4C22⋊C4×D6111yescharacteristic
655C2×C4C2×D4×D6111yescharacteristic
656C2×C4C2×D4×D6111yescharacteristic
657C2×C4#41×D6112yes—
658C2×C4C2×#1175822412yes—
659C8—24no—
660C8—42no—
661C8—1224no—
662C8—2442no—
663C22—21no—
664C22—21no—
665C22—21no—
666C22—21no—
667C22—21no—
668C22—21no—
669C22—21no—
670C22—44no—
671C22—4884no—
672C22—2442no—
673C22—2442no—
674C22—2442no—
675C22#1683974512yes—
676C22C2×C22⋊C4×D6111yescharacteristic
677C22C4×#1175822412yes—
678C4—21no—
679C4—21no—
680C4—42no—
681C4—42no—
682C4—612no—
683C4—1221no—
684C4—1221no—
685C4—1221no—
686C4—1221no—
687C4—1221no—
688C4—1221no—
689C4—1221no—
690C4—2442no—
691C4#1684026811yescharacteristic
692C4#1684027011yescharacteristic
693C4#1684030712yes—
694C2—22no—
695C2—22no—
696C2—41no—
697C2—4882no—
698C2#2144830411yescharacteristic
699C2#413×D6111yescharacteristic
700C2C2×#1683974511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1901C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#14519751C22
cosocleC22×C4×C61C24
hypercenter#1901D61
hypocenterC61#21426
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1901

Complements

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

NH ≅ G/Ntype
2. #21454448C2semidirect
3. C2×M16×Dic61C2semidirect
4. #21446810C2semidirect
5. #21446667C2semidirect
6. #21446288C2semidirect
9. #101×Dic61C2semidirect
10. #21445610C2semidirect
11. C61×#1901C2semidirect
350. #1901D61semidirect
405. C61#21426semidirect

Maximal subgroups

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

subgroupindex
C2×#168402982
#214544482
C2×M16×Dic612
#214468102
#214466672
#214462882
#214462822
#214462802
#101×Dic612
#214456102
C61×#19012
#2142661