Finite Group Databasebeta

← #29698179

Subgroups — #29698179

Order 15616.

970 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839835C211yesmaximal, characteristic
2C2×#16839803C211yesmaximal, characteristic, complemented
3C2×#16839797C211yesmaximal, characteristic, complemented
4#21448770C211yesmaximal, characteristic
5#21448745C211yesmaximal, characteristic
6#21446585C211yesmaximal, characteristic, complemented
7#21446575C211yesmaximal, characteristic, complemented
8#21446017C211yesmaximal, characteristic, complemented
9#21446017C211yesmaximal, characteristic, complemented
10#21446015C211yesmaximal, characteristic, complemented
11#21446005C211yesmaximal, characteristic, complemented
12#21445849C211yesmaximal, characteristic, complemented
13#21445845C211yesmaximal, characteristic, complemented
14#21445716C211yesmaximal, characteristic, complemented
15C61×#1847C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519735C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839835C418yescomplemented
24#16839803—24no—
25#16839797—24no—
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519710C2211yescharacteristic
30C2×#14519709C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519689C2211yescharacteristic
34C2×#14519689C2211yescharacteristic
35#16839663—21no—
36#16839652—21no—
37#16839652—21no—
38#16839651—21no—
39#16839645—21no—
40#16839640—21no—
41#16839619C2211yescharacteristic
42#16839619C2211yescharacteristic
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4⋊C4×Dic61—21no—
46C4×C4⋊Dic61C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839609—21no—
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#106×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C61×#384C2211yescharacteristic
61C61×#382C2211yescharacteristic
62C22×D244D411yescharacteristic
63C22×C4×D61—21no—
64C22×C4×D61C2311yescharacteristic
65C22×C4×D61C2311yescharacteristic
66C2×#11758189—22no—
67C2×#11758189—22no—
68C2×#11758189C2×C414yes—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71C2×C8×D61—22no—
72C2×C8×D61—22no—
73C2×C8×D61C2×C414yescomplemented
74C4×D244C2×C414yes—
75C4×D244Q814yes—
76#14519735—22no—
77#14519735—22no—
78#14519735—22no—
79#14519735—41no—
80#14519735—41no—
81#14519735—41no—
82C2×C4⋊Dic61D411yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C4⋊C4×D61—22no—
89C4⋊C4×D61—41no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198D411yescharacteristic
98C2×#11758198D411yescharacteristic
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104#14519711—22no—
105#14519710—22no—
106#14519710—22no—
107#14519710—22no—
108#14519709D414yescomplemented
109#14519708—22no—
110C4⋊#92418C2×C414yescomplemented
111#14519689C2×C414yes—
112#14519689C2×C414yescomplemented
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61—21no—
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#107×C61—22no—
129#107×C61—41no—
130#106×C61—22no—
131#106×C61—41no—
132#97×C61C2×C414yescomplemented
133#95×C61—21no—
134#95×C61—21no—
135C23×D61C2×D411yescharacteristic
136C23×D61C2×D411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C2×D244—41no—
151C2×D244—41no—
152C2×D244C4⋊C414yes—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—42no—
162C2×C4×D61—42no—
163C2×C4×D61—81no—
164C2×C4×D61C22×C412yes—
165C2×C4×D61C22×C412yes—
166C2×C4×D61C2×Q812yes—
167C2×C4×D61C2×Q812yes—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C22×C412yes—
171C2×C8×C61C22×C412yes—
172C2×C8×C61C2×D412yes—
173C2×C8×C61#4112yes—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—81no—
184C42×C61C22×C412yes—
185C42×C61C2×Q812yes—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—81no—
191#11758198—81no—
192#11758198—81no—
193#11758198C4⋊C414yes—
194#11758198C4⋊C414yes—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61C4⋊C414yes—
202C4⋊Dic61C2×D412yes—
203C4⋊Dic61C2×D412yes—
204#11758196—42no—
205C4×Dic61—42no—
206C4×Dic61—42no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—81no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418C22×C412yes—
215C2×#92418C22×C412yes—
216#11758189—28no—
217C8×D61—28no—
218C23×C61C22×D411yescharacteristic
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61C2×C4⋊C412yes—
232C22×D61C2×C4⋊C412yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61C2×C4⋊C412yes—
250C2×C4×C61C2×C4⋊C412yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61C2×C4⋊C412yes—
274C2×Dic61C2×C4⋊C412yes—
275D244—44no—
276C4×D61—22no—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—22no—
282C4×D61—22no—
283C4×D61—22no—
284C4×D61—84no—
285C8×C61—22no—
286C8×C61C4×D414yes—
287#92418—22no—
288#92418—22no—
289#22660—611nomaximal, Sylow
290C22×C61C22×C4⋊C411yescharacteristic
291C22×C61D4211yescharacteristic
292C22×C61#54711yescharacteristic
293C22×C61#57511yescharacteristic
294C22×C61#57511yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57811yescharacteristic
297C2×D61—22no—
298C2×D61—22no—
299C2×D61—22no—
300C2×D61—22no—
301C2×D61—22no—
302C2×D61—22no—
303C2×D61—41no—
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—22no—
312C4×C61—22no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61—81no—
317C4×C61C2×C4×D411yescharacteristic
318C4×C61#52011yescharacteristic
319C4×C61D4×Q811yescharacteristic
320C4×C61#55411yescharacteristic
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327C2×#435—611no—
328C2×#428—611no—
329C2×#427—611no—
330#2354—611no—
331#2352—611no—
332#1949—611no—
333#1947—611no—
334#1854—611no—
335#1854—611no—
336#1854—611no—
337#1852—611no—
338#1847D6111yescharacteristic, complemented
339#1809—611no—
340#1808—611no—
341#1776—611no—
342C2×C61D4×C4⋊C411yescharacteristic
343C2×C61#291111yescharacteristic
344C2×C61#291311yescharacteristic
345C2×C61#326811yescharacteristic
346C2×C61#326911yescharacteristic
347C2×C61#334711yescharacteristic
348C2×C61#334911yescharacteristic
349D61—42no—
350D61—42no—
351C22×M16—611no—
352C23×C8—611no—
353C2×C4×D4—611no—
354C2×#117—611no—
355C2×#117—611no—
356C2×#117—611no—
357C22×C4⋊C4—611no—
358#435—618no—
359#428—1224no—
360#427—1224no—
361C2×#107—611no—
362C2×#107—611no—
363C2×#107—611no—
364C2×#107—611no—
365C2×#107C2×D6111yescharacteristic
366C2×#107C2×D6111yescharacteristic
367C2×#106—611no—
368C2×#106—611no—
369C2×#106C2×D6111yescharacteristic
370C2×#106C2×D6111yescharacteristic
371C2×C4⋊C8—611no—
372C2×C22⋊C8—611no—
373C2×C22⋊C8—611no—
374C2×#97C2×D6111yescharacteristic
375#390—1221no—
376#388—1221no—
377#388—1221no—
378#388—1221no—
379#387—1221no—
380#386—1221no—
381#384C2×D6111yescharacteristic
382#382C2×D6111yescharacteristic
383#380—611no—
384#380—611no—
385#380—611no—
386#380—611no—
387C4×C4⋊C4—611no—
388C4×C4⋊C4—611no—
389C4×C4⋊C4—1221no—
390C4×C22⋊C4—1221no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397C61#2266011yescharacteristic, complemented, Sylow
398C22×D4—611no—
399C23×C4—611no—
400C23×C4—611no—
401C23×C4—1221no—
402C2×M16—614no—
403C2×M16—1222no—
404C2×M16—1222no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8—614no—
408C22×C8—1222no—
409C22×C8—1222no—
410C22×C8C22×D6111yescharacteristic
411C22×C8C22×D6111yescharacteristic
412C4×D4—614no—
413C4×D4—614no—
414#117—1222no—
415#117—1222no—
416#117—1222no—
417#117—2441no—
418#117—2441no—
419#117—2441no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—2441no—
430C2×C4⋊C4C22×D6111yescharacteristic
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C22⋊C4—611no—
435C2×C22⋊C4—611no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C22⋊C4—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#107—22no—
450#107—41no—
451#107—1222no—
452#107—1222no—
453#107—1222no—
454#107—1222no—
455#106—22no—
456#106—41no—
457#106—614no—
458#106—1222no—
459C4⋊C8—614no—
460C22⋊C8—614no—
461C22⋊C8—614no—
462#97C4×D6114yescomplemented
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—611no—
472C24—611no—
473C2×D4—614no—
474C2×D4—2441no—
475C2×D4—2441no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—612no—
483C22×C4—612no—
484C22×C4—612no—
485C22×C4—612no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—2442no—
500C22×C4—2442no—
501C22×C4D4×D6111yescharacteristic
502C22×C4D4×D6111yescharacteristic
503C22×C4C23×D6111yescharacteristic
504M16—1228no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—612no—
508C2×C8—612no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1228no—
514C2×C8C2×C4×D6112yes—
515C2×C8C2×C4×D6112yes—
516C2×C8D4×D6112yes—
517C2×C8#1175822612yes—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—42no—
527C4⋊C4—81no—
528C4⋊C4—614no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—2442no—
538C22⋊C4—614no—
539C22⋊C4—614no—
540C22⋊C4—4881no—
541C22⋊C4—4881no—
542C22⋊C4—4881no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C22⋊C4—2442no—
547C42—4881no—
548C42—2442no—
549C42—2442no—
550C42—2442no—
551C42—2442no—
552C42C2×C4×D6112yes—
553C42Q8×D6112yes—
554C23—612no—
555C23—612no—
556C23—1221no—
557C23—1221no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23C2×D4×D6111yescharacteristic
569D4—2444no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—4884no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4C4⋊C4×D6112yes—
614C2×C4C4⋊C4×D6112yes—
615C2×C4C22×C4×D6111yescharacteristic
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C2×Q8×D6111yescharacteristic
618C2×C4C2×#1175822611yescharacteristic
619C2×C4#1451982211yescharacteristic
620C2×C4#1451982511yescharacteristic
621C8—22no—
622C8—1222no—
623C8—1222no—
624C8#1451973914yes—
625C22—1222no—
626C22—1222no—
627C22—1222no—
628C22—1222no—
629C22—1222no—
630C22—1222no—
631C22—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22C2×C4⋊C4×D6111yescharacteristic
640C22#1684039711yescharacteristic
641C22#1684041511yescharacteristic
642C22#1684055811yescharacteristic
643C22#1684056111yescharacteristic
644C22#1684056611yescharacteristic
645C22#1684057811yescharacteristic
646C4—22no—
647C4—22no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—81no—
652C4—1222no—
653C4—1222no—
654C4—4881no—
655C4—4881no—
656C4—2442no—
657C4—2442no—
658C4C2×#1451973911yescharacteristic
659C4#1684024311yescharacteristic
660C4#1684042711yescharacteristic
661C4#1684047311yescharacteristic
662C2—2442no—
663C2—2442no—
664C2#2144871911yescharacteristic
665C2#2145440611yescharacteristic
666C2#2145441811yescharacteristic
667C2#2145735811yescharacteristic
668C2#2145737311yescharacteristic
669C2#2145831811yescharacteristic
670C2#2145834811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1847C2
socleC23×C61C22×D4
prime radicalC22×D244D4
cosocleC22×C4×C61C24
hypercenter#1847D61
hypocenterC61#22660
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
2. C2×#16839803C2semidirect
3. C2×#16839797C2semidirect
6. #21446585C2semidirect
7. #21446575C2semidirect
8. #21446017C2semidirect
9. #21446017C2semidirect
10. #21446015C2semidirect
11. #21446005C2semidirect
12. #21445849C2semidirect
13. #21445845C2semidirect
14. #21445716C2semidirect
15. C61×#1847C2semidirect
23. #16839835C4semidirect
73. C2×C8×D61C2×C4semidirect
108. #14519709D4semidirect
110. C4⋊#92418C2×C4semidirect
112. #14519689C2×C4semidirect
132. #97×C61C2×C4semidirect
338. #1847D61semidirect
397. C61#22660semidirect
462. #97C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168398352
C2×#168398032
C2×#168397972
#214487702
#214487452
#214465852
#214465752
#214460172
#214460152
#214460052
#214458492
#214458452
#214457162
C61×#18472
#2266061