Finite Group Databasebeta

← #29712006

Subgroups — #29712006

Order 15616.

1222 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839917C211yesmaximal, characteristic, complemented
2C2×#16839900C211yesmaximal, characteristic
3#21448403C211yesmaximal, characteristic, complemented
4#21446957C212yesmaximal, complemented
5#21446913C212yesmaximal, complemented
6#21446160C212yesmaximal, complemented
7#21446153C212yesmaximal, complemented
8#21446142C211yesmaximal, characteristic
9#21446132C211yesmaximal, characteristic, complemented
10#21445890C211yesmaximal, characteristic, complemented
11C2×C61×#542C211yesmaximal, characteristic
12C23×Dic122C2211yescharacteristic
13C22×#11758202C2212yes—
14C22×#11758200C2212yes—
15C2×#14519736C2211yescharacteristic
16#16839917—24no—
17#16839900—24no—
18C2×#14519701C2212yes—
19C2×#14519700C2212yes—
20C2×#14519700C2212yes—
21C2×#14519698C2212yes—
22C2×C22⋊#92418C2211yescharacteristic
23C2×C22⋊#92418C2211yescharacteristic
24#16839668C2212yes—
25#16839668C2212yes—
26#16839667—21no—
27#16839667—21no—
28#16839623—21no—
29#16839623—21no—
30#16839524C2212yes—
31#16839524C2212yes—
32#16839517C2212yes—
33#16839517C2212yes—
34C61×#542—24no—
35C61×#542—24no—
36C2×C4⋊Q8×C61C2211yescharacteristic
37C2×#126×C61—21no—
38C2×#124×C61—21no—
39C2×#124×C61C2211yescharacteristic
40C2×#123×C61—21no—
41C2×C22⋊Q8×C61—21no—
42C2×C22⋊Q8×C61C2211yescharacteristic
43C2×#121×C61C2211yescharacteristic
44C2×C4×Q8×C61C2211yescharacteristic
45C2×C4×D4×C61C2211yescharacteristic
46C2×#117×C61C2211yescharacteristic
47C22×Dic122—21no—
48C22×Dic122—24no—
49C22×Dic122—22no—
50C22×Dic122—22no—
51C22×Dic122—42no—
52C22×Dic122C2312yes—
53C23×Dic61—21no—
54C2×#11758202—24no—
55C2×#11758202—24no—
56C2×#11758202—42no—
57C2×#11758200—24no—
58C2×#11758200—24no—
59C2×#11758200—42no—
60C22×#92418C2312yes—
61C22×#92418C2312yes—
62#14519736—42no—
63C2×C4⋊Dic61C2312yes—
64C2×C22⋊Dic61—21no—
65C2×C4×Dic61—21no—
66#14519701—42no—
67#14519700—24no—
68#14519700—24no—
69#14519698—42no—
70C22⋊#92418—22no—
71C22⋊#92418—22no—
72#14519684—22no—
73#14519684—22no—
74#14519684—22no—
75#14519684—22no—
76C22×Q8×C61—21no—
77C22×Q8×C61C2311yescharacteristic
78C22×D4×C61—21no—
79C22×D4×C61C2311yescharacteristic
80C23×C4×C61—21no—
81C23×C4×C61C2311yescharacteristic
82C4⋊Q8×C61—22no—
83C4⋊Q8×C61—22no—
84#126×C61—28no—
85#124×C61—28no—
86#124×C61—22no—
87#124×C61—22no—
88#123×C61—28no—
89C22⋊Q8×C61—28no—
90C22⋊Q8×C61—22no—
91C22⋊Q8×C61D414yes—
92#121×C61—22no—
93#121×C61D414yes—
94C4×Q8×C61—22no—
95C4×Q8×C61—22no—
96C4×D4×C61—22no—
97C4×D4×C61—22no—
98#117×C61—22no—
99#117×C61—22no—
100C2×C4⋊C4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61—21no—
103C2×C4⋊C4×C61D411yescharacteristic
104C2×C4⋊C4×C61D411yescharacteristic
105C2×C4⋊C4×C61C2311yescharacteristic
106C2×C4⋊C4×C61C2311yescharacteristic
107C2×C22⋊C4×C61—21no—
108C2×C22⋊C4×C61—21no—
109C2×C22⋊C4×C61—21no—
110C2×C22⋊C4×C61—21no—
111C2×C22⋊C4×C61—21no—
112C2×C22⋊C4×C61D411yescharacteristic
113C2×C22⋊C4×C61D411yescharacteristic
114C2×C42×C61—21no—
115C2×C42×C61C2311yescharacteristic
116C2×C42×C61C2311yescharacteristic
117C24×C61—21no—
118C24×C61C2×D411yescharacteristic
119C2×Q8×C61—24no—
120C2×Q8×C61—21no—
121C2×Q8×C61—22no—
122C2×Q8×C61—22no—
123C2×Q8×C61—82no—
124C2×Q8×C61C2×D412yes—
125C2×D4×C61—24no—
126C2×D4×C61—21no—
127C2×D4×C61—22no—
128C2×D4×C61—22no—
129C2×D4×C61—82no—
130C2×D4×C61C2×D412yes—
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—24no—
137C22×C4×C61—82no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D412yes—
144C22×C4×C61C2×D412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—24no—
147C22×Dic61—22no—
148C22×Dic61—22no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C2×Dic122—22no—
152C2×Dic122—22no—
153C2×Dic122—22no—
154C2×Dic122—22no—
155C2×Dic122—24no—
156C2×Dic122—22no—
157C2×Dic122—48no—
158C2×Dic122—48no—
159C2×Dic122—42no—
160C2×Dic122—42no—
161C2×Dic122#4114yes—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—24no—
164C4⋊C4×C61—24no—
165C4⋊C4×C61—22no—
166C4⋊C4×C61—22no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61C2×D412yes—
170C4⋊C4×C61C2×D412yes—
171C22⋊C4×C61—24no—
172C22⋊C4×C61—24no—
173C22⋊C4×C61—24no—
174C22⋊C4×C61—24no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C22⋊C4×C61—24no—
178C42×C61—24no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C22⋊Dic61—42no—
184#11758202—48no—
185#11758200—48no—
186C4⋊Dic61—22no—
187C4⋊Dic61—22no—
188C4×Dic61—42no—
189C2×#92418—22no—
190C2×#92418—22no—
191C2×#92418—22no—
192C2×#92418—22no—
193C2×#92418—22no—
194C2×#92418—22no—
195C23×C61—21no—
196C23×C61—21no—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—22no—
200C23×C61—41no—
201C23×C61—41no—
202C23×C61—42no—
203C23×C61—42no—
204C23×C61#12012yes—
205C23×C61C22×D411yescharacteristic
206Q8×C61—24no—
207Q8×C61—42no—
208Q8×C61—84no—
209D4×C61—24no—
210D4×C61—42no—
211D4×C61—84no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—24no—
215C2×C4×C61—24no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—42no—
230C2×C4×C61—42no—
231C2×C4×C61—84no—
232C2×C4×C61#12012yes—
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C2×#4112yes—
236C2×C4×C61H(5,𝔽2)12yes—
237C2×Dic61—24no—
238C2×Dic61—48no—
239C2×Dic61—42no—
240C2×Dic61—42no—
241C2×Dic61—42no—
242C2×Dic61—42no—
243C2×Dic61—42no—
244Dic122—24no—
245Dic122—24no—
246Dic122—48no—
247Dic122—42no—
248Dic122—42no—
249#92418—42no—
250#92418—42no—
251#23946—611nomaximal, Sylow
252C22×C61—22no—
253C22×C61—22no—
254C22×C61—22no—
255C22×C61—42no—
256C22×C61—42no—
257C22×C61—42no—
258C22×C61—42no—
259C22×C61—42no—
260C22×C61—82no—
261C22×C61—82no—
262C22×C61C2×#12011yescharacteristic
263C22×C61#52511yescharacteristic
264C22×C61#53511yescharacteristic
265C22×C61#57712yes—
266C22×C61#57812yes—
267C4×C61—22no—
268C4×C61—42no—
269C4×C61—41no—
270C4×C61—41no—
271C4×C61—42no—
272C4×C61—42no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61#54612yes—
276C4×C61#54612yes—
277Dic61—44no—
278Dic61—82no—
279C2×#542D6111yescharacteristic
280C2×#451—611no—
281C2×#448—611no—
282#2310—611no—
283#2024—612no—
284#2015—612no—
285#1878—612no—
286#1877—612no—
287#1875—611no—
288#1873—611no—
289#1818—611no—
290C2×C61—42no—
291C2×C61—82no—
292C2×C61#240411yescharacteristic
293C2×C61#300911yescharacteristic
294C2×C61#301111yescharacteristic
295C2×C61#312112yes—
296C2×C61#322512yes—
297C23×Q8—611no—
298C22×Q16—612no—
299C22×SD16—612no—
300#542—24no—
301#542—24no—
302C2×C4⋊Q8C2×D6111yescharacteristic
303C2×#126—21no—
304C2×#124—21no—
305C2×#124C2×D6111yescharacteristic
306C2×#123—21no—
307C2×C22⋊Q8—21no—
308C2×C22⋊Q8C2×D6111yescharacteristic
309C2×#121C2×D6111yescharacteristic
310C2×C4×Q8C2×D6111yescharacteristic
311C2×C4×D4C2×D6111yescharacteristic
312C2×#117—611no—
313C2×#117C2×D6111yescharacteristic
314#451—1224no—
315#448—1224no—
316C2×#103—612no—
317C2×#103—612no—
318C2×#103—612no—
319C2×#102—612no—
320C2×C22⋊C8—611no—
321C2×C22⋊C8—611no—
322#391—612no—
323#391—612no—
324#391—1221no—
325#391—1221no—
326#381—1221no—
327#381—1221no—
328#340—612no—
329#340—612no—
330#336—612no—
331#336—612no—
332C61#2394611yescharacteristic, complemented, Sylow
333C22×Q8—21no—
334C22×Q8—612no—
335C22×Q8—1224no—
336C22×Q8—1221no—
337C22×Q8—1222no—
338C22×Q8—1222no—
339C22×Q8—2442no—
340C22×Q8C22×D6111yescharacteristic
341C22×D4—21no—
342C22×D4C22×D6111yescharacteristic
343C23×C4—21no—
344C23×C4—1221no—
345C23×C4C22×D6111yescharacteristic
346C2×Q16—1224no—
347C2×Q16—1224no—
348C2×Q16—2442no—
349C2×SD16—1224no—
350C2×SD16—1224no—
351C2×SD16—2442no—
352C22×C8—612no—
353C22×C8—612no—
354C4⋊Q8—22no—
355C4⋊Q8—22no—
356#126—28no—
357#124—28no—
358#124—22no—
359#124—22no—
360#123—28no—
361C22⋊Q8—28no—
362C22⋊Q8—22no—
363C22⋊Q8C22⋊D6114yes—
364#121—22no—
365#121C22⋊D6114yes—
366C4×Q8—22no—
367C4×Q8—22no—
368C4×D4—22no—
369C4×D4—22no—
370#117—22no—
371#117—22no—
372#117—2442no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—612no—
377C2×C4⋊C4C22⋊D6111yescharacteristic
378C2×C4⋊C4C22⋊D6111yescharacteristic
379C2×C4⋊C4C22×D6111yescharacteristic
380C2×C4⋊C4C22×D6111yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—1221no—
387C2×C22⋊C4C22⋊D6111yescharacteristic
388C2×C22⋊C4C22⋊D6111yescharacteristic
389C2×C42—21no—
390C2×C42—1221no—
391C2×C42C22×D6111yescharacteristic
392C2×C42C22×D6111yescharacteristic
393#103—1224no—
394#103—1224no—
395#103—2442no—
396#102—2442no—
397C22⋊C8—1222no—
398C22⋊C8—1222no—
399#95—1222no—
400#95—1222no—
401#95—1222no—
402#95—1222no—
403C24—21no—
404C24C2×C22⋊D6111yescharacteristic
405C2×Q8—24no—
406C2×Q8—21no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—82no—
410C2×Q8—614no—
411C2×Q8—1224no—
412C2×Q8—1222no—
413C2×Q8—1222no—
414C2×Q8—1222no—
415C2×Q8—1222no—
416C2×Q8—1222no—
417C2×Q8—2448no—
418C2×Q8—2448no—
419C2×Q8—2442no—
420C2×Q8—2442no—
421C2×Q8C2×C22⋊D6112yes—
422C2×D4—24no—
423C2×D4—21no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—82no—
427C2×D4C2×C22⋊D6112yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—24no—
434C22×C4—82no—
435C22×C4—1224no—
436C22×C4—1222no—
437C22×C4—1222no—
438C22×C4—2442no—
439C22×C4—2442no—
440C22×C4C2×C22⋊D6111yescharacteristic
441C22×C4C2×C22⋊D6111yescharacteristic
442C22×C4C2×C22⋊D6111yescharacteristic
443C22×C4C2×C22⋊D6111yescharacteristic
444C22×C4C2×C22⋊D6112yes—
445C22×C4C2×C22⋊D6112yes—
446C22×C4C2×C22⋊D6111yescharacteristic
447C22×C4C23×D6111yescharacteristic
448Q16—2448no—
449SD16—2448no—
450C2×C8—1222no—
451C2×C8—1222no—
452C2×C8—1222no—
453C2×C8—1222no—
454C2×C8—1222no—
455C2×C8—1222no—
456C4⋊C4—24no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—24no—
461C4⋊C4—22no—
462C4⋊C4—22no—
463C4⋊C4—1222no—
464C4⋊C4—1222no—
465C4⋊C4C2×C22⋊D6112yes—
466C4⋊C4C2×C22⋊D6112yes—
467C22⋊C4—24no—
468C22⋊C4—24no—
469C22⋊C4—24no—
470C22⋊C4—24no—
471C22⋊C4—22no—
472C22⋊C4—22no—
473C22⋊C4—24no—
474C22⋊C4—2442no—
475C42—24no—
476C42—21no—
477C42—21no—
478C42—21no—
479C42—21no—
480C42—2442no—
481C23—21no—
482C23—22no—
483C23—22no—
484C23—21no—
485C23—22no—
486C23—41no—
487C23—42no—
488C23—42no—
489C23—41no—
490C23#1451974612yes—
491C23C22×C22⋊D6111yescharacteristic
492Q8—24no—
493Q8—42no—
494Q8—84no—
495Q8—1224no—
496Q8—1224no—
497Q8—2448no—
498Q8—2442no—
499Q8—2442no—
500D4—24no—
501D4—42no—
502D4—84no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—22no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—41no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—84no—
523C2×C4—1224no—
524C2×C4—2442no—
525C2×C4—2442no—
526C2×C4—2442no—
527C2×C4—2448no—
528C2×C4—2442no—
529C2×C4—2442no—
530C2×C4#1451974612yes—
531C2×C4C22×C22⋊D6111yescharacteristic
532C2×C4C22×C22⋊D6111yescharacteristic
533C2×C4#41×D6112yes—
534C2×C4#1451982312yes—
535C8—2442no—
536C8—2442no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—82no—
546C22—82no—
547C22C2×#1451974611yescharacteristic
548C22#1684026911yescharacteristic
549C22#1684032611yescharacteristic
550C22#1684057312yes—
551C22#1684057712yes—
552C4—22no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—42no—
559C4—42no—
560C4—4882no—
561C4—2444no—
562C4#1684040012yes—
563C4#1684040012yes—
564C2—42no—
565C2—82no—
566C2#2144913411yescharacteristic
567C2#2145509611yescharacteristic
568C2#2145511411yescharacteristic
569C2#2145604212yes—
570C2#2145692212yes—
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#542C2
socleC23×C61C22×D4
prime radicalC2×#121×C61C22
hypercenterC2×#542D61
hypocenterC61#23946
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#542

Complements

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

NH ≅ G/Ntype
1. C2×#16839917C2semidirect
3. #21448403C2semidirect
4. #21446957C2semidirect
5. #21446913C2semidirect
6. #21446160C2semidirect
7. #21446153C2semidirect
9. #21446132C2semidirect
10. #21445890C2semidirect
332. C61#23946semidirect

Maximal subgroups

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

subgroupindex
C2×#168399172
C2×#168399002
#214484032
#214469572
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C2×C61×#5422
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