Finite Group Databasebeta

← #29685386

Subgroups — #29685386

Order 15616.

1100 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
1C2×#16840314C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic
3C2×M16×Dic61C211yesmaximal, characteristic
4#21446806C212yesmaximal, complemented
5#21446661C212yesmaximal, complemented
6#21446288C211yesmaximal, characteristic
7#21446287C211yesmaximal, characteristic
8#21446285C211yesmaximal, characteristic
9#101×Dic61C212yesmaximal, complemented
10#21445614C212yesmaximal, complemented
11C61×#1901C211yesmaximal, characteristic
12C22×#11758194C2211yescharacteristic
13#16840314—21no—
14#16840314—21no—
15#16840314—21no—
16#16840314—21no—
17#16840314—21no—
18#16840314—21no—
19#16840314—21no—
20#16840314—21no—
21C2×#14519776C2211yescharacteristic
22C2×#14519775C2211yescharacteristic
23C2×#14519767C412yes—
24C2×#14519756C2211yescharacteristic
25C2×#14519754C412yes—
26C2×Q8×Dic61C2211yescharacteristic
27C2×#14519742C412yes—
28C2×#14519742C2211yescharacteristic
29C2×#14519733C412yes—
30C22×C4×Dic61C2211yescharacteristic
31#16839816—21no—
32#16839816—21no—
33#16839812—21no—
34#16839812—21no—
35#16839785C2212yes—
36C2×#14519696C2212yes—
37C2×#14519695C2212yes—
38C2×#14519695C2212yes—
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×Dic122C2×C412yes—
56C22×Dic122C2311yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C23×Dic61C2×C411yescharacteristic
59C2×#11758194—21no—
60C2×#11758194—24no—
61C2×#11758194C2312yes—
62C2×#11758194C2312yes—
63C22×#92418—21no—
64#14519776—21no—
65#14519776—21no—
66#14519776—21no—
67#14519776—21no—
68#14519775—21no—
69#14519775—21no—
70#14519775—21no—
71#14519775—21no—
72#14519767—22no—
73#14519767—22no—
74#14519767—22no—
75#14519767—22no—
76#14519756—21no—
77#14519756—21no—
78#14519756—21no—
79#14519756—21no—
80#14519754—22no—
81#14519754—22no—
82#14519754—22no—
83#14519754—22no—
84Q8×Dic61—21no—
85Q8×Dic61—21no—
86Q8×Dic61—21no—
87Q8×Dic61—21no—
88#14519742—22no—
89#14519742—22no—
90#14519742—22no—
91#14519742—22no—
92#14519742—21no—
93#14519742—21no—
94#14519742—21no—
95#14519742—21no—
96#14519733—22no—
97#14519733—22no—
98#14519733—22no—
99#14519733—22no—
100C2×C4⋊Dic61C2×C412yes—
101C2×C4⋊Dic61C2311yescharacteristic
102C2×#11758196C2×C412yes—
103C2×#11758196C2×C412yes—
104C2×#11758196C2×C412yes—
105C2×#11758196C2×C412yes—
106C2×#11758196C2×C412yes—
107C2×C22⋊Dic61C2×C412yes—
108C2×C4×Dic61C2×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#14519695—24no—
126#14519695—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
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—22no—
156C2×Q8×C61—22no—
157C22×C4×C61—22no—
158C22×C4×C61—22no—
159C22×C4×C61C22×C412yes—
160C22×C4×C61C22×C411yescharacteristic
161C22×C4×C61C22×C412yes—
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61C2×D411yescharacteristic
165C22×C4×C61C2×D411yescharacteristic
166C22×C4×C61C2411yescharacteristic
167C22×Dic61—24no—
168C22×Dic61—42no—
169C22×Dic61C22⋊C411yescharacteristic
170C22×Dic61C22⋊C411yescharacteristic
171C22×Dic61C22⋊C411yescharacteristic
172C22×Dic61C22⋊C411yescharacteristic
173C22×Dic61C22⋊C411yescharacteristic
174C22×Dic61C22⋊C411yescharacteristic
175C22×Dic61C22⋊C411yescharacteristic
176C22×Dic61C22⋊C411yescharacteristic
177C22×Dic61C22×C412yes—
178C22×Dic61C22×C411yescharacteristic
179C22×Dic61C22×C411yescharacteristic
180C22×Dic61C22×C411yescharacteristic
181C22×Dic61C22×C412yes—
182C22×Dic61C22×C411yescharacteristic
183C2×Dic122—21no—
184C2×Dic122—21no—
185C2×Dic122—22no—
186C2×Dic122—22no—
187C2×Dic122—22no—
188C2×Dic122—22no—
189C2×Dic122—44no—
190M16×C61—24no—
191M16×C61—22no—
192M16×C61—22no—
193M16×C61—22no—
194M16×C61#4114yes—
195C2×C8×C61—24no—
196C2×C8×C61—22no—
197C2×C8×C61—22no—
198C4⋊C4×C61—22no—
199C4⋊C4×C61—22no—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C22⋊C4×C61—22no—
203C22⋊C4×C61—22no—
204C22⋊Dic61—22no—
205C22⋊Dic61—22no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—22no—
216#11758196—22no—
217#11758196—22no—
218#11758196—22no—
219#11758196—22no—
220C4×Dic61—22no—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61—22no—
224C4×Dic61C22⋊C411yescharacteristic
225C4×Dic61C22⋊C411yescharacteristic
226C4×Dic61C22⋊C411yescharacteristic
227C4×Dic61C22⋊C411yescharacteristic
228C4×Dic61C22⋊C411yescharacteristic
229C4×Dic61C22⋊C411yescharacteristic
230C4×Dic61C22⋊C411yescharacteristic
231C4×Dic61C22⋊C411yescharacteristic
232C4×Dic61C2×D411yescharacteristic
233C4×Dic61C2×D411yescharacteristic
234C4×Dic61C2×D411yescharacteristic
235C4×Dic61C2×D411yescharacteristic
236C4×Dic61C2×D411yescharacteristic
237C4×Dic61C2×D411yescharacteristic
238C4×Dic61C2×D411yescharacteristic
239C4×Dic61C2×D411yescharacteristic
240#11758194—24no—
241#11758194—22no—
242#11758194—22no—
243#11758194—22no—
244#11758194#4114yes—
245C2×#92418—24no—
246C2×#92418—22no—
247C2×#92418—22no—
248C23×C61—22no—
249C23×C61—22no—
250C23×C61—41no—
251C23×C61C2×C22⋊C411yescharacteristic
252C23×C61C2×C22⋊C411yescharacteristic
253C23×C61C23×C411yescharacteristic
254Q8×C61—41no—
255Q8×C61—41no—
256Q8×C61—42no—
257C2×C4×C61—22no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—22no—
266C2×C4×C61—24no—
267C2×C4×C61—42no—
268C2×C4×C61C2×C22⋊C411yescharacteristic
269C2×C4×C61C2×C22⋊C411yescharacteristic
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C22×D411yescharacteristic
272C2×C4×C61C2×#4112yes—
273C2×C4×C61C2×#4112yes—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—22no—
277C2×Dic61—22no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—22no—
283C2×Dic61—22no—
284C2×Dic61—24no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—42no—
289C2×Dic61#10114yescomplemented
290C2×Dic61C2×C22⋊C411yescharacteristic
291C2×Dic61C2×C22⋊C411yescharacteristic
292C2×Dic61C2×C22⋊C411yescharacteristic
293C2×Dic61C2×C22⋊C411yescharacteristic
294C2×Dic61C2×C22⋊C411yescharacteristic
295C2×Dic61C2×C22⋊C411yescharacteristic
296C2×Dic61C2×C22⋊C411yescharacteristic
297C2×Dic61C2×C22⋊C411yescharacteristic
298Dic122—44no—
299Dic122—41no—
300Dic122—41no—
301Dic122—42no—
302Dic122—44no—
303C8×C61—24no—
304C8×C61—42no—
305#92418—24no—
306#92418—42no—
307#21425—611nomaximal, Sylow
308C22×C61—21no—
309C22×C61—21no—
310C22×C61—21no—
311C22×C61—21no—
312C22×C61—21no—
313C22×C61—21no—
314C22×C61—21no—
315C22×C61—44no—
316C22×C61C2×#10112yes—
317C22×C61C22×C22⋊C411yescharacteristic
318C22×C61C4×#4112yes—
319C4×C61—21no—
320C4×C61—21no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61#52511yescharacteristic
324C4×C61#52511yescharacteristic
325C4×C61#53212yes—
326Dic61—21no—
327Dic61—21no—
328Dic61—21no—
329Dic61—21no—
330Dic61—21no—
331Dic61—42no—
332Dic61—42no—
333Dic61C2×#10112yes—
334Dic61C2×#10112yes—
335Dic61#41312yes—
336C2×#533—611no—
337#2920—611no—
338C2×C4×M16—611no—
339#1997—612no—
340#1966—612no—
341#1901—611no—
342#1901—611no—
343#1901—611no—
344#1901D6111yescharacteristic
345C4×#101—612no—
346C4×#101—612no—
347C2×C61—22no—
348C2×C61—22no—
349C2×C61—41no—
350C2×C61#229711yescharacteristic
351C2×C61C22×#10111yescharacteristic
352C2×C61C2×#41311yescharacteristic
353C22×M16—611no—
354C22×M16C2×D6111yescharacteristic
355#533—1221no—
356#533—1221no—
357#533—1221no—
358#533—1221no—
359#533—1221no—
360#533—1221no—
361#533—1221no—
362#533—1221no—
363C2×C4⋊Q8—611no—
364C2×C4⋊Q8—611no—
365C2×#125—612no—
366C2×C22⋊Q8—611no—
367C2×C22⋊Q8—612no—
368C2×C22⋊Q8C2×D6111yescharacteristic
369C2×C4×Q8—611no—
370C2×C4×Q8—612no—
371C2×C4×Q8—611no—
372C2×#117—612no—
373C22×C42—611no—
374#432—1221no—
375#432—1221no—
376#431—1221no—
377#431—1221no—
378#423—612no—
379C2×#101—612no—
380C2×#101—612no—
381C2×#101—612no—
382C2×#101C2×D6112yes—
383#407—611no—
384#407C2×D6111yescharacteristic
385C4×M16—612no—
386C4×M16—612no—
387C4×M16—612no—
388C4×M16—1221no—
389C4×M16—1221no—
390C2×#97—1221no—
391C2×C4×C8—1221no—
392#343—612no—
393#343—612no—
394#343—612no—
395#343C2×D6112yes—
396C61#2142511yescharacteristic, complemented, Sylow
397C22×Q8—612no—
398C22×Q8—611no—
399C22×Q8C22×D6111yescharacteristic
400C23×C4—611no—
401C23×C4—611no—
402C23×C4C22×D6111yescharacteristic
403C2×M16—21no—
404C2×M16—24no—
405C2×M16—612no—
406C2×M16—612no—
407C2×M16—1224no—
408C2×M16—1221no—
409C2×M16C22×D6112yes—
410C2×M16C22×D6112yes—
411C22×C8—21no—
412C22×C8—1221no—
413C4⋊Q8—1221no—
414C4⋊Q8—1221no—
415C4⋊Q8—1221no—
416C4⋊Q8—1221no—
417C4⋊Q8—1221no—
418C4⋊Q8—1221no—
419C4⋊Q8—1221no—
420C4⋊Q8—1221no—
421#125—1222no—
422#125—1222no—
423#125—1222no—
424#125—1222no—
425C22⋊Q8—21no—
426C22⋊Q8—21no—
427C22⋊Q8—21no—
428C22⋊Q8—21no—
429C22⋊Q8—1222no—
430C22⋊Q8—1222no—
431C22⋊Q8—1222no—
432C22⋊Q8—1222no—
433C22⋊Q8—1221no—
434C22⋊Q8—1221no—
435C22⋊Q8—1221no—
436C22⋊Q8—1221no—
437C4×Q8—1221no—
438C4×Q8—1221no—
439C4×Q8—1221no—
440C4×Q8—1221no—
441C4×Q8—1222no—
442C4×Q8—1222no—
443C4×Q8—1222no—
444C4×Q8—1222no—
445C4×Q8—1221no—
446C4×Q8—1221no—
447C4×Q8—1221no—
448C4×Q8—1221no—
449#117—1222no—
450#117—1222no—
451#117—1222no—
452#117—1222no—
453C2×C4⋊C4—612no—
454C2×C4⋊C4—612no—
455C2×C4⋊C4—612no—
456C2×C4⋊C4—612no—
457C2×C4⋊C4—612no—
458C2×C4⋊C4—612no—
459C2×C4⋊C4—611no—
460C2×C4⋊C4C4×D6112yes—
461C2×C4⋊C4C22×D6111yescharacteristic
462C2×C22⋊C4—612no—
463C2×C22⋊C4C4×D6112yes—
464C2×C42—611no—
465C2×C42—611no—
466C2×C42—611no—
467C2×C42—611no—
468C2×C42—611no—
469C2×C42—611no—
470C2×C42—611no—
471C2×C42—611no—
472C2×C42—611no—
473C2×C42—611no—
474C2×C42—611no—
475C2×C42—611no—
476C2×C42—612no—
477C2×C42—612no—
478C4⋊C8—1222no—
479C4⋊C8—1222no—
480#101—24no—
481#101—1224no—
482#101—1224no—
483#101—1224no—
484C22⋊C8—21no—
485C22⋊C8—21no—
486C22⋊C8—1221no—
487C22⋊C8—1221no—
488#97—1222no—
489#97—1222no—
490#97—1222no—
491C4×C8—1222no—
492C4×C8—1222no—
493C4×C8—1222no—
494C24C2×C4×D6111yescharacteristic
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—22no—
498C2×Q8—22no—
499C2×Q8—1222no—
500C2×Q8—1222no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—1222no—
504C2×Q8—1222no—
505C2×Q8—2444no—
506C22×C4—22no—
507C22×C4—22no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—611no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—611no—
519C22×C4—611no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—1224no—
523C22×C4—2442no—
524C22×C4C2×C4×D6112yes—
525C22×C4C2×C4×D6111yescharacteristic
526C22×C4C2×C4×D6112yes—
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6111yescharacteristic
529C22×C4D4×D6111yescharacteristic
530C22×C4D4×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532M16—24no—
533M16—22no—
534M16—22no—
535M16—22no—
536M16—614no—
537M16—1224no—
538M16—1222no—
539M16—1222no—
540M16—1222no—
541M16#1175822414yes—
542C2×C8—24no—
543C2×C8—22no—
544C2×C8—22no—
545C2×C8—1224no—
546C2×C8—1222no—
547C2×C8—1222no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1222no—
555C4⋊C4—1222no—
556C4⋊C4—1222no—
557C4⋊C4—1222no—
558C4⋊C4—1222no—
559C4⋊C4—1222no—
560C4⋊C4—1222no—
561C4⋊C4—1222no—
562C4⋊C4—1221no—
563C4⋊C4—1221no—
564C4⋊C4—1222no—
565C4⋊C4—1222no—
566C22⋊C4—22no—
567C22⋊C4—22no—
568C22⋊C4—1222no—
569C22⋊C4—1222no—
570C42—611no—
571C42—611no—
572C42—611no—
573C42—611no—
574C42—611no—
575C42—611no—
576C42—611no—
577C42—611no—
578C42—611no—
579C42—611no—
580C42—611no—
581C42—611no—
582C42—611no—
583C42—611no—
584C42—611no—
585C42—611no—
586C42—1222no—
587C42—1222no—
588C42—1222no—
589C42—1222no—
590C23—22no—
591C23—22no—
592C23—41no—
593C23C22⋊C4×D6111yescharacteristic
594C23C22⋊C4×D6111yescharacteristic
595C23C22×C4×D6111yescharacteristic
596Q8—41no—
597Q8—41no—
598Q8—42no—
599Q8—2444no—
600Q8—2444no—
601Q8—2441no—
602Q8—2441no—
603Q8—2442no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—22no—
610C2×C4—22no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—24no—
614C2×C4—42no—
615C2×C4—611no—
616C2×C4—611no—
617C2×C4—611no—
618C2×C4—614no—
619C2×C4—611no—
620C2×C4—611no—
621C2×C4—611no—
622C2×C4—611no—
623C2×C4—611no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1222no—
632C2×C4—1222no—
633C2×C4—1222no—
634C2×C4—1224no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2442no—
639C2×C4C22⋊C4×D6111yescharacteristic
640C2×C4C22⋊C4×D6111yescharacteristic
641C2×C4C2×D4×D6111yescharacteristic
642C2×C4C2×D4×D6111yescharacteristic
643C2×C4#41×D6112yes—
644C2×C4C2×#1175822412yes—
645C8—24no—
646C8—42no—
647C8—1224no—
648C8—2442no—
649C22—21no—
650C22—21no—
651C22—21no—
652C22—21no—
653C22—21no—
654C22—21no—
655C22—21no—
656C22—44no—
657C22#101×D6112yes—
658C22C2×C22⋊C4×D6111yescharacteristic
659C22C4×#1175822412yes—
660C4—21no—
661C4—21no—
662C4—42no—
663C4—42no—
664C4—612no—
665C4—612no—
666C4—612no—
667C4—1221no—
668C4—1221no—
669C4—1221no—
670C4—1221no—
671C4—1221no—
672C4—2442no—
673C4—2442no—
674C4#1684026811yescharacteristic
675C4#1684027011yescharacteristic
676C4#1684030712yes—
677C2—22no—
678C2—22no—
679C2—41no—
680C2#2144830411yescharacteristic
681C2C2×#101×D6111yescharacteristic
682C2#413×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1901C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C22×C4
cosocleC22×C4×C61C24
hypercenter#1901D61
hypocenterC61#21425
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
4. #21446806C2semidirect
5. #21446661C2semidirect
9. #101×Dic61C2semidirect
10. #21445614C2semidirect
289. C2×Dic61#101semidirect
396. C61#21425semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
#214544482
C2×M16×Dic612
#214468062
#214466612
#214462882
#214462872
#214462852
#101×Dic612
#214456142
C61×#19012
#2142561