Finite Group Databasebeta

← #29836736

Subgroups — #29836736

Order 15616.

664 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14519723C211yesmaximal, characteristic, complemented
2C2×Q32×D61C211yesmaximal, characteristic, complemented
3#21457568C211yesmaximal, characteristic
4Q16×C22⋊D61C211yesmaximal, characteristic
5#21448070C211yesmaximal, characteristic, complemented
6#21447909C211yesmaximal, characteristic, complemented
7#21447908C211yesmaximal, characteristic, complemented
8#21447868C211yesmaximal, characteristic, complemented
9#21447867C211yesmaximal, characteristic, complemented
10#21447755C211yesmaximal, characteristic, complemented
11#21447754C211yesmaximal, characteristic, complemented
12#21447751C211yesmaximal, characteristic, complemented
13#21447675C211yesmaximal, characteristic, complemented
14#21447636C211yesmaximal, characteristic
15C61×#2188C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C2×Q16×D61C2211yescharacteristic
18C2×Q16×D61C2211yescharacteristic
19Q8×C22⋊D61—21no—
20#16840425—21no—
21C2×Dic488C2211yescharacteristic
22C2×Dic488C2211yescharacteristic
23C2×#14519723—24no—
24C2×#14519723C2211yescharacteristic
25C2×#14519723C2211yescharacteristic
26C2×#14519723C2211yescharacteristic
27C2×#14519723C2211yescharacteristic
28C2×#14519723C2211yescharacteristic
29Q32×D61—24no—
30C22×#11758185C2211yescharacteristic
31C2×C16×D61C2211yescharacteristic
32#16840140C2211yescharacteristic
33#16840140C2211yescharacteristic
34#16840004C2211yescharacteristic
35#16840003C2211yescharacteristic
36#16840000C2211yescharacteristic
37#16839971—21no—
38#16839970—21no—
39#16839919—21no—
40#16839918—21no—
41#16839917—21no—
42#16839916—21no—
43Q16×Dic61C2211yescharacteristic
44#16839856C2211yescharacteristic
45C8×C22⋊D61C2211yescharacteristic
46#16839587C2211yescharacteristic
47#16839580C2211yescharacteristic
48#16839567C2211yescharacteristic
49#16839567C2211yescharacteristic
50#16839566C2211yescharacteristic
51#16839565C2211yescharacteristic
52#16839541C2211yescharacteristic
53C16×Dic61C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×Q32×C61C2211yescharacteristic
56C2×Q32×C61C2211yescharacteristic
57C61×#467C2211yescharacteristic
58C61×#358C2211yescharacteristic
59C61×#358C2211yescharacteristic
60C61×C22⋊C16C2211yescharacteristic
61C22×Dic122—21no—
62C2×Q8×D61—21no—
63C2×Q8×D61—21no—
64C2×Dic244—22no—
65C2×Dic244—21no—
66C2×Dic244—41no—
67C2×Dic244C2311yescharacteristic
68C2×Dic244C2311yescharacteristic
69C2×Dic244C2311yescharacteristic
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74Q16×D61—22no—
75Q16×D61—22no—
76Q16×D61—41no—
77Q16×D61—41no—
78C2×C8×D61C2311yescharacteristic
79#14519776—41no—
80#14519775—21no—
81#14519775—21no—
82#14519775—41no—
83#14519757—21no—
84#14519756—21no—
85#14519756—41no—
86#14519755—41no—
87#14519754—41no—
88#14519753—21no—
89#14519753—41no—
90Q8×Dic61—21no—
91#14519742—21no—
92C4×C22⋊D61—41no—
93C4×C22⋊D61D411yescharacteristic
94#14519737—41no—
95Dic488—22no—
96Dic488—41no—
97#14519723—22no—
98#14519723—22no—
99#14519723—22no—
100#14519723—22no—
101#14519723D414yescomplemented
102C2×#11758185—22no—
103C2×#11758185C2311yescharacteristic
104C2×#11758185C2311yescharacteristic
105C16×D61—22no—
106#14519711C2311yescharacteristic
107#14519706D411yescharacteristic
108#14519702—21no—
109#14519701—21no—
110#14519701—21no—
111#14519700—21no—
112#14519700—21no—
113C22⋊#92418D411yescharacteristic
114#14519689D411yescharacteristic
115C8×Dic61C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×Q16×C61—22no—
118C2×Q16×C61—21no—
119C2×Q16×C61—41no—
120C2×Q16×C61C2311yescharacteristic
121C2×Q16×C61C2311yescharacteristic
122C2×Q16×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22⋊Q8×C61—21no—
125Q32×C61—22no—
126Q32×C61—41no—
127C2×C16×C61C2311yescharacteristic
128C2×C16×C61C2311yescharacteristic
129#107×C61C2311yescharacteristic
130#103×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—42no—
135C2×Q8×C61—41no—
136C2×Q8×C61—81no—
137C22×C4×C61—41no—
138C22×C4×C61C2×D411yescharacteristic
139C2×C22⋊D61D811yescharacteristic
140C22×Dic61—41no—
141Q8×D61—42no—
142Q8×D61—42no—
143Q8×D61—81no—
144Q8×D61—81no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61C2×D411yescharacteristic
148C2×Dic122—21no—
149C2×Dic122—21no—
150C2×Dic122—21no—
151C2×Dic122—42no—
152C2×Dic122—41no—
153C2×Dic122—41no—
154C2×Dic122—41no—
155C2×Dic122—81no—
156Q16×C61—21no—
157Q16×C61—21no—
158Q16×C61—21no—
159Q16×C61—21no—
160Q16×C61—21no—
161Q16×C61—42no—
162Q16×C61C2×D412yes—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2411yescharacteristic
167C4⋊C4×C61—21no—
168C4⋊C4×C61—41no—
169C22⋊C4×C61—41no—
170C22⋊Dic61—41no—
171C22⋊Dic61D811yescharacteristic
172#11758202—42no—
173#11758202—42no—
174#11758202—81no—
175#11758202—81no—
176#11758198—41no—
177#11758198—41no—
178#11758198D811yescharacteristic
179C4⋊Dic61—21no—
180C4⋊Dic61—41no—
181C4⋊Dic61—41no—
182#11758196—41no—
183#11758196—41no—
184#11758196—41no—
185#11758196—41no—
186#11758196—41no—
187#11758196—41no—
188#11758196—41no—
189#11758196D811yescharacteristic
190C4×Dic61—41no—
191C4×Dic61—41no—
192C4×Dic61C2×D411yescharacteristic
193C2×#92418C2×D411yescharacteristic
194C2×#92418C2×D411yescharacteristic
195Dic244—21no—
196Dic244—21no—
197Dic244—21no—
198Dic244—21no—
199Dic244—21no—
200Dic244—42no—
201Dic244C2×D412yes—
202C8×D61—21no—
203C8×D61—21no—
204C16×C61—21no—
205C16×C61—21no—
206#11758185—21no—
207#11758185C2×D412yes—
208C23×C61C2×D811yescharacteristic
209C22×D61C2×D811yescharacteristic
210Q8×C61—22no—
211Q8×C61—41no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—82no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—82no—
224C2×C4×C61C2×D811yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C22⋊D61Q3214yescomplemented
227C2×Dic61—41no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230C2×Dic61—41no—
231C2×Dic61—82no—
232C2×Dic61C2×D811yescharacteristic
233C2×Dic61C2×D811yescharacteristic
234C2×Dic61C2×D811yescharacteristic
235C4×D61—21no—
236C4×D61—21no—
237C4×D61—82no—
238C4×D61—82no—
239Dic122—22no—
240Dic122—41no—
241Dic122—41no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—82no—
246Dic122—82no—
247Dic122—82no—
248C8×C61—21no—
249C8×C61C22×D411yescharacteristic
250C8×C61H(5,𝔽2)11yescharacteristic
251#92418—21no—
252#92418—21no—
253D4×Q32—611nomaximal, Sylow
254C22×C61—21no—
255C22×C61—21no—
256C22×C61C2×Q3212yescomplemented
257C22×C61C22×D811yescharacteristic
258C2×D61—21no—
259C2×D61—21no—
260C2×D61C2×Q3212yescomplemented
261C4×C61—21no—
262C4×C61—42no—
263C4×C61—81no—
264C4×C61—81no—
265C4×C61—81no—
266C4×C61D4211yescharacteristic
267C4×C61#57611yescharacteristic
268Dic61—21no—
269Dic61—21no—
270Dic61—42no—
271Dic61—81no—
272Dic61—81no—
273Dic61—81no—
274Dic61C2×Q3212yescomplemented
275C22×Q32—611no—
276C22×Q32—611no—
277D4×Q16—611no—
278D4×Q16—611no—
279#2247—611no—
280#2216—611no—
281#2216—611no—
282#2209—611no—
283#2209—611no—
284#2188—611no—
285#2188—611no—
286#2188—611no—
287#2188D6111yescharacteristic, complemented
288C4×Q32—611no—
289D4×C16—611no—
290C2×C61—22no—
291C2×C61D4×D811yescharacteristic
292C2×C61C22×Q3211yescharacteristic
293C2×C61#341711yescharacteristic
294D61—22no—
295C22×Q16—611no—
296C22×Q16—611no—
297C22×Q16—611no—
298C22×Q16C2×D6111yescharacteristic
299D4×Q8—1221no—
300D4×Q8—1221no—
301C2×Q32—611no—
302C2×Q32—611no—
303C2×Q32—611no—
304C2×Q32—611no—
305C2×Q32—611no—
306C2×Q32—611no—
307C2×Q32—611no—
308C2×Q32—1224no—
309C2×Q32—1224no—
310C2×Q32C2×D6111yescharacteristic
311C2×Q32C2×D6111yescharacteristic
312C22×C16—611no—
313C22×C16—611no—
314#494—611no—
315#494—611no—
316#467—611no—
317#467—611no—
318#467—611no—
319#467C2×D6111yescharacteristic
320#462—1221no—
321#462—1221no—
322#451—1221no—
323#451—1221no—
324#451—1221no—
325#451—1221no—
326C4×Q16—611no—
327C4×Q16—611no—
328C8×D4—611no—
329#366—611no—
330C4⋊C16—611no—
331#358—611no—
332#358—611no—
333#358—611no—
334#358—611no—
335#358C2×D6111yescharacteristic
336#358C2×D6111yescharacteristic
337C22⋊C16—611no—
338C22⋊C16C2×D6111yescharacteristic
339C4×C16—611no—
340C61D4×Q3211yescharacteristic, complemented, Sylow
341C22×Q8—21no—
342C22×Q8—1221no—
343C22×Q8—1221no—
344C22×Q8—1221no—
345C2×Q16—22no—
346C2×Q16—21no—
347C2×Q16—41no—
348C2×Q16—611no—
349C2×Q16—611no—
350C2×Q16—611no—
351C2×Q16—1222no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×Q16—1221no—
356C2×Q16—1222no—
357C2×Q16—1222no—
358C2×Q16—1221no—
359C2×Q16—2441no—
360C2×Q16—2441no—
361C2×Q16—2441no—
362C2×Q16C22×D6111yescharacteristic
363C2×Q16C22×D6111yescharacteristic
364C2×Q16C22×D6111yescharacteristic
365C22×C8—611no—
366C22×C8C22×D6111yescharacteristic
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8—2441no—
370C4⋊Q8—2441no—
371C22⋊Q8—21no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—2441no—
376C22⋊Q8—2441no—
377C22⋊Q8—2441no—
378C22⋊Q8—2441no—
379C4×Q8—1221no—
380C4×Q8—1221no—
381C4×D4—611no—
382C4×D4—2441no—
383C4×D4—2441no—
384Q32—22no—
385Q32—41no—
386Q32—614no—
387Q32—1222no—
388Q32—1222no—
389Q32—1222no—
390Q32—1222no—
391Q32—1222no—
392Q32—2441no—
393C2×C16—611no—
394C2×C16—611no—
395C2×C16—1222no—
396C2×C16—1222no—
397C2×C16C22×D6111yescharacteristic
398C2×C16C22×D6111yescharacteristic
399#107—611no—
400#107C22×D6111yescharacteristic
401C4⋊C8—611no—
402#103—21no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—1221no—
407#103—1221no—
408C22⋊C8—611no—
409C22⋊C8—611no—
410C4×C8—611no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—42no—
415C2×Q8—41no—
416C2×Q8—81no—
417C2×Q8—1221no—
418C2×Q8—1221no—
419C2×Q8—1221no—
420C2×Q8—4881no—
421C2×Q8—4881no—
422C2×Q8—4881no—
423C2×Q8—2442no—
424C2×Q8—2441no—
425C2×Q8—2441no—
426C2×Q8—2442no—
427C2×Q8—2442no—
428C2×Q8—2441no—
429C2×D4—611no—
430C22×C4—41no—
431C22×C4—611no—
432C22×C4—2441no—
433C22×C4—2441no—
434C22×C4—2441no—
435C22×C4D4×D6111yescharacteristic
436Q16—21no—
437Q16—21no—
438Q16—21no—
439Q16—21no—
440Q16—21no—
441Q16—42no—
442Q16—612no—
443Q16—1221no—
444Q16—1221no—
445Q16—1221no—
446Q16—1221no—
447Q16—1221no—
448Q16—4881no—
449Q16—4881no—
450Q16—2442no—
451Q16—2442no—
452Q16—2442no—
453Q16D4×D6112yes—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—611no—
457C2×C8—611no—
458C2×C8—1221no—
459C2×C8—1221no—
460C2×C8D4×D6111yescharacteristic
461C2×C8C23×D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—611no—
465C4⋊C4—1221no—
466C4⋊C4—2441no—
467C4⋊C4—2441no—
468C4⋊C4—2441no—
469C4⋊C4—2441no—
470C4⋊C4—2441no—
471C4⋊C4—2441no—
472C4⋊C4—2441no—
473C4⋊C4—2441no—
474C4⋊C4—2441no—
475C22⋊C4—41no—
476C22⋊C4—611no—
477C22⋊C4—611no—
478C22⋊C4—2441no—
479C22⋊C4—2441no—
480C22⋊C4—2441no—
481C42—611no—
482C42—2441no—
483C42—2441no—
484C16—21no—
485C16—21no—
486C16—612no—
487C16—1221no—
488C23—611no—
489C23D8×D6111yescharacteristic
490Q8—22no—
491Q8—41no—
492Q8—41no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—82no—
497Q8—1222no—
498Q8—4882no—
499Q8—4882no—
500Q8—4882no—
501Q8—2441no—
502Q8—2441no—
503Q8—2441no—
504Q8—2441no—
505Q8—2441no—
506D4—614no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—82no—
514C2×C4—611no—
515C2×C4—611no—
516C2×C4—611no—
517C2×C4—1221no—
518C2×C4—1221no—
519C2×C4—4882no—
520C2×C4—4882no—
521C2×C4—4882no—
522C2×C4—2441no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4D8×D6111yescharacteristic
527C2×C4C2×D4×D6111yescharacteristic
528C8—21no—
529C8—1221no—
530C8—1221no—
531C8C2×D4×D6111yescharacteristic
532C8#1451982211yescharacteristic
533C22—21no—
534C22—21no—
535C22—612no—
536C22—1221no—
537C22—1221no—
538C22Q32×D6112yescomplemented
539C22C2×D8×D6111yescharacteristic
540C4—21no—
541C4—42no—
542C4—81no—
543C4—81no—
544C4—81no—
545C4—612no—
546C4—1221no—
547C4—1221no—
548C4—4881no—
549C4—4881no—
550C4—4881no—
551C4—2442no—
552C4#1684039411yescharacteristic
553C4#1684056611yescharacteristic
554C2—22no—
555C2—1222no—
556C2#2145748911yescharacteristic
557C2C2×Q32×D6111yescharacteristic
558C2#2145897111yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2188C2
socleC22×C61C22×D8
prime radicalC2×C22⋊D61D8
hypercenter#2188D61
hypocenterC61D4×Q32
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2188

Complements

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

NH ≅ G/Ntype
1. C22×#14519723C2semidirect
2. C2×Q32×D61C2semidirect
5. #21448070C2semidirect
6. #21447909C2semidirect
7. #21447908C2semidirect
8. #21447868C2semidirect
9. #21447867C2semidirect
10. #21447755C2semidirect
11. #21447754C2semidirect
12. #21447751C2semidirect
13. #21447675C2semidirect
15. C61×#2188C2semidirect
101. #14519723D4semidirect
226. C22⋊D61Q32semidirect
256. C22×C61C2×Q32semidirect
260. C2×D61C2×Q32semidirect
274. Dic61C2×Q32semidirect
287. #2188D61semidirect
340. C61D4×Q32semidirect
538. C22Q32×D61semidirect

Maximal subgroups

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

subgroupindex
C22×#145197232
C2×Q32×D612
#214575682
Q16×C22⋊D612
#214480702
#214479092
#214479082
#214478682
#214478672
#214477552
#214477542
#214477512
#214476752
#214476362
C61×#21882
D4×Q3261