Finite Group Databasebeta

← #29689574

Subgroups — #29689574

Order 15616.

1100 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840315C211yesmaximal, characteristic
2#21454448C211yesmaximal, characteristic
3C4×#14519733C211yesmaximal, characteristic
4#21446794C214yesmaximal, complemented
5#21446761C214yesmaximal, complemented
6#21445919C211yesmaximal, characteristic
7#21445918C211yesmaximal, characteristic
8#21445916C211yesmaximal, characteristic
9C61×#1826C211yesmaximal, characteristic
10#16840315—21no—
11#16840315—21no—
12#16840315—21no—
13#16840315—21no—
14#16840315C412yescomplemented
15#16840315C412yescomplemented
16#16840315C2212yes—
17#16840315C2212yes—
18C2×#14519776C411yescharacteristic, complemented
19C2×#14519776C411yescharacteristic, complemented
20C2×#14519768C411yescharacteristic, complemented
21C2×#14519768C411yescharacteristic, complemented
22C2×#14519754—21no—
23C2×#14519754—21no—
24C2×C4×Dic122—21no—
25C2×C4×Dic122—21no—
26C2×#14519736C2211yescharacteristic
27C2×#14519733—21no—
28C22×C4×Dic61C2211yescharacteristic
29#16839816—21no—
30#16839816—21no—
31#16839812—21no—
32#16839812—21no—
33#16839785C2212yes—
34#16839726C2211yescharacteristic
35M16×Dic61C2212yes—
36C4×#11758196C2212yes—
37#16839613—21no—
38C22⋊C4×Dic61—21no—
39C4×C22⋊Dic61C2211yescharacteristic
40#16839607—21no—
41C42×Dic61C2212yes—
42#16839523C2214yes—
43#16839522C2214yes—
44#16839521C2214yes—
45C2×#117×C61C2211yescharacteristic
46C61×#407C2211yescharacteristic
47C4×C22⋊C4×C61C2211yescharacteristic
48C61×#339C2214yes—
49C22×Dic122—21no—
50C22×Dic122—21no—
51C23×Dic61D411yescharacteristic
52C23×Dic61D411yescharacteristic
53C2×#11758194C2312yes—
54#14519776—21no—
55#14519776—21no—
56#14519776—21no—
57#14519776—21no—
58#14519776C2×C412yes—
59#14519776C2×C412yes—
60#14519776C2×C412yes—
61#14519776C2×C412yes—
62#14519768—21no—
63#14519768—21no—
64#14519768—21no—
65#14519768—21no—
66#14519768C2×C412yes—
67#14519768C2×C412yes—
68#14519768C2×C412yes—
69#14519768C2×C412yes—
70#14519754—22no—
71#14519754—22no—
72#14519754—22no—
73#14519754—22no—
74#14519754—22no—
75#14519754—22no—
76#14519754—22no—
77#14519754—22no—
78C4×Dic122—22no—
79C4×Dic122—22no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85C4×Dic122—22no—
86#14519736—21no—
87#14519736—21no—
88#14519736C2×C412yes—
89#14519736C2312yes—
90#14519733—28no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61—21no—
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×C4⋊Dic61C2×C411yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—22no—
100C2×C22⋊Dic61—21no—
101C2×C22⋊Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—22no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61C2×C411yescharacteristic
106C2×C4×Dic61C2×C411yescharacteristic
107C2×C4×Dic61C2×C411yescharacteristic
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61D411yescharacteristic
110C2×C4×Dic61D411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61C2312yes—
116C2×C4×Dic61C2311yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118#14519706—22no—
119#14519706—22no—
120C22⋊#92418—21no—
121C22⋊#92418—21no—
122#14519687—22no—
123C8×Dic61—22no—
124#14519684—21no—
125#14519684—22no—
126C23×C4×C61C2311yescharacteristic
127C2×M16×C61C2312yes—
128#117×C61—21no—
129#117×C61—21no—
130#117×C61C2×C412yes—
131#117×C61C2312yes—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C4⋊C4×C61C2×C411yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C2×C42×C61—21no—
137C2×C42×C61C2312yes—
138C22⋊C8×C61—21no—
139C22⋊C8×C61—21no—
140#95×C61—21no—
141C24×C61C2×D411yescharacteristic
142C22×C4×C61—24no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—22no—
146C22×C4×C61C22×C411yescharacteristic
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—24no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156C22×Dic61—42no—
157C22×Dic61C22⋊C411yescharacteristic
158C22×Dic61C22⋊C411yescharacteristic
159C22×Dic61C22⋊C411yescharacteristic
160C22×Dic61C22⋊C411yescharacteristic
161C22×Dic61C22⋊C411yescharacteristic
162C22×Dic61C22⋊C411yescharacteristic
163C22×Dic61C22⋊C411yescharacteristic
164C22×Dic61C22⋊C411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D411yescharacteristic
169C2×Dic122—22no—
170C2×Dic122—22no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—44no—
174C2×Dic122—44no—
175M16×C61—24no—
176C2×C8×C61—22no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61C22×C412yes—
180C4⋊C4×C61C22×C412yes—
181C22⋊C4×C61—24no—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C42×C61—22no—
185C42×C61—22no—
186C42×C61—24no—
187C22⋊Dic61—24no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—22no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61—22no—
196C4⋊Dic61C22×C412yes—
197C4⋊Dic61C22×C412yes—
198#11758196—28no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207C4×Dic61—28no—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—24no—
211C4×Dic61—44no—
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22⋊C411yescharacteristic
219C4×Dic61C22⋊C411yescharacteristic
220C4×Dic61C22×C411yescharacteristic
221C4×Dic61C22×C411yescharacteristic
222C4×Dic61C22×C411yescharacteristic
223C4×Dic61C22×C411yescharacteristic
224C4×Dic61C2×D411yescharacteristic
225C4×Dic61C2×D411yescharacteristic
226C4×Dic61C2×D411yescharacteristic
227C4×Dic61C2×D411yescharacteristic
228#11758194—24no—
229C2×#92418—22no—
230C23×C61—22no—
231C23×C61—22no—
232C23×C61—41no—
233C23×C61C2×C22⋊C411yescharacteristic
234C23×C61C2×C22⋊C411yescharacteristic
235C23×C61C22×D411yescharacteristic
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—24no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—42no—
245C2×C4×C61—42no—
246C2×C4×C61—42no—
247C2×C4×C61—42no—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C612−1+414yes—
253C2×Dic61—22no—
254C2×Dic61—22no—
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—24no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—42no—
268C2×Dic61—42no—
269C2×Dic61—42no—
270C2×Dic61—42no—
271C2×Dic61#10414yes—
272C2×Dic61C2×C22⋊C411yescharacteristic
273C2×Dic61C2×C22⋊C411yescharacteristic
274C2×Dic61C2×C22⋊C411yescharacteristic
275C2×Dic61C2×C22⋊C411yescharacteristic
276C2×Dic61C2×C22⋊C411yescharacteristic
277C2×Dic61C2×C22⋊C411yescharacteristic
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280Dic122—44no—
281Dic122—44no—
282Dic122—44no—
283Dic122—44no—
284C8×C61—42no—
285#92418—42no—
286#21844—611nomaximal, Sylow
287C22×C61—21no—
288C22×C61—21no—
289C22×C61—21no—
290C22×C61—21no—
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—44no—
295C22×C61C2×#10412yes—
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61#53612yes—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—44no—
301C4×C61—42no—
302C4×C61—42no—
303C4×C61#51912yes—
304C4×C61#53612yes—
305Dic61—21no—
306Dic61—21no—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—44no—
311Dic61—42no—
312Dic61—42no—
313Dic61C2×#10412yes—
314Dic61C2×#10412yes—
315Dic61#42112yes—
316C2×#533—611no—
317#2920—611no—
318C4×#117—611no—
319#1994—614no—
320#1986—614no—
321#1826—611no—
322#1826—611no—
323#1826—611no—
324#1826D6111yescharacteristic
325C2×C61—22no—
326C2×C61—22no—
327C2×C61—41no—
328C2×C61#231011yescharacteristic
329C2×C61C22×#10411yescharacteristic
330C2×C61C2×#42111yescharacteristic
331#533—612no—
332#533—612no—
333#533—612no—
334#533—612no—
335#533—1221no—
336#533—1221no—
337#533—1221no—
338#533—1221no—
339C2×C4⋊Q8—611no—
340C2×C4⋊Q8—611no—
341C2×#125—611no—
342C2×#125—611no—
343C2×C22⋊Q8—1221no—
344C2×C22⋊Q8—1221no—
345C2×C4×Q8—1221no—
346C2×C4×Q8—1221no—
347C2×#117—611no—
348C2×#117—1221no—
349C2×#117C2×D6111yescharacteristic
350C22×C42—611no—
351#432—1221no—
352#432—1221no—
353#431—1221no—
354#431—1221no—
355#423—612no—
356#407—611no—
357#407C2×D6111yescharacteristic
358C4×M16—612no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—1221no—
361C4×C22⋊C4—611no—
362C4×C22⋊C4—1221no—
363C4×C22⋊C4C2×D6111yescharacteristic
364#376—1221no—
365C43—612no—
366#339—614no—
367#339—614no—
368#339—614no—
369#339C2×D6114yes—
370C61#2184411yescharacteristic, complemented, Sylow
371C22×Q8—1221no—
372C22×Q8—1221no—
373C23×C4—611no—
374C23×C4—611no—
375C23×C4C22×D6111yescharacteristic
376C2×M16—612no—
377C2×M16C22×D6112yes—
378C4⋊Q8—612no—
379C4⋊Q8—612no—
380C4⋊Q8—612no—
381C4⋊Q8—612no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384C4⋊Q8—1221no—
385C4⋊Q8—1221no—
386#125—612no—
387#125—612no—
388#125—612no—
389#125—612no—
390#125—1221no—
391#125—1221no—
392#125—1221no—
393#125—1221no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1222no—
399C22⋊Q8—1222no—
400C22⋊Q8—1222no—
401C22⋊Q8—1222no—
402C4×Q8—1222no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410#117—21no—
411#117—21no—
412#117—612no—
413#117—612no—
414#117—1228no—
415#117—1221no—
416#117—1221no—
417#117C4×D6112yes—
418#117C22×D6112yes—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4C4×D6111yescharacteristic
429C2×C4⋊C4C4×D6111yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—611no—
445C2×C42—611no—
446C2×C42—611no—
447C2×C42—612no—
448C2×C42—1222no—
449C2×C42—1221no—
450C2×C42—1222no—
451C2×C42C22×D6112yes—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454C22⋊C8—21no—
455C22⋊C8—21no—
456C22⋊C8—1221no—
457C22⋊C8—1221no—
458#97—1222no—
459C4×C8—1222no—
460#95—21no—
461#95—1222no—
462#95—1221no—
463C24D4×D6111yescharacteristic
464C2×Q8—1222no—
465C2×Q8—1222no—
466C2×Q8—1222no—
467C2×Q8—1222no—
468C2×Q8—2444no—
469C2×Q8—2444no—
470C22×C4—24no—
471C22×C4—22no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—1224no—
487C22×C4—1222no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—2442no—
491C22×C4C2×C4×D6111yescharacteristic
492C22×C4C2×C4×D6111yescharacteristic
493C22×C4D4×D6111yescharacteristic
494C22×C4D4×D6111yescharacteristic
495C22×C4D4×D6111yescharacteristic
496C22×C4C23×D6111yescharacteristic
497M16—24no—
498M16—1224no—
499C2×C8—22no—
500C2×C8—1222no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—612no—
504C4⋊C4—612no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1228no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4C2×C4×D6112yes—
521C4⋊C4C2×C4×D6112yes—
522C22⋊C4—24no—
523C22⋊C4—22no—
524C22⋊C4—22no—
525C22⋊C4—1224no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C42—22no—
529C42—22no—
530C42—24no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1228no—
548C42—1222no—
549C42—1222no—
550C42—1224no—
551C42—2444no—
552C23—22no—
553C23—22no—
554C23—41no—
555C23C22⋊C4×D6111yescharacteristic
556C23C22⋊C4×D6111yescharacteristic
557C23C2×D4×D6111yescharacteristic
558Q8—2444no—
559Q8—2444no—
560Q8—2444no—
561Q8—2444no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—22no—
567C2×C4—24no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—42no—
571C2×C4—42no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—611no—
575C2×C4—611no—
576C2×C4—611no—
577C2×C4—614no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—611no—
581C2×C4—611no—
582C2×C4—611no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1222no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—1222no—
591C2×C4—1224no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2442no—
598C2×C4—2442no—
599C2×C4—2442no—
600C2×C4—2442no—
601C2×C4C22⋊C4×D6111yescharacteristic
602C2×C4C22⋊C4×D6111yescharacteristic
603C2×C4C22×C4×D6111yescharacteristic
604C2×C4C2×D4×D6111yescharacteristic
605C2×C4#1451982414yes—
606C8—42no—
607C8—2442no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—44no—
616C22#104×D6112yes—
617C22C2×C22⋊C4×D6111yescharacteristic
618C22#1684033412yes—
619C4—21no—
620C4—21no—
621C4—44no—
622C4—42no—
623C4—42no—
624C4—612no—
625C4—612no—
626C4—612no—
627C4—1221no—
628C4—1221no—
629C4—1221no—
630C4—1221no—
631C4—1221no—
632C4—2444no—
633C4—2442no—
634C4—2442no—
635C4#1684023912yes—
636C4#1684033412yes—
637C2—22no—
638C2—22no—
639C2—41no—
640C2#2144840411yescharacteristic
641C2C2×#104×D6111yescharacteristic
642C2#421×D6111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1826C2
socleC22×C61C22×C22⋊C4
prime radicalC24×C61C2×D4
cosocleC22×C4×C61C24
hypercenter#1826D61
hypocenterC61#21844
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1826

Complements

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

NH ≅ G/Ntype
4. #21446794C2semidirect
5. #21446761C2semidirect
14. #16840315C4semidirect
15. #16840315C4semidirect
18. C2×#14519776C4semidirect
19. C2×#14519776C4semidirect
20. C2×#14519768C4semidirect
21. C2×#14519768C4semidirect
370. C61#21844semidirect

Maximal subgroups

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

subgroupindex
C2×#168403152
#214544482
C4×#145197332
#214467942
#214467612
#214459192
#214459182
#214459162
C61×#18262
#2184461