Finite Group Databasebeta

← #29670216

Subgroups — #29670216

Order 15616.

616 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458736C211yesmaximal, characteristic
2Q8×#11758201C211yesmaximal, characteristic
3Q16×D244C211yesmaximal, characteristic
4#21445498C211yesmaximal, characteristic, complemented
5#21445327C211yesmaximal, characteristic, complemented
6#21445267C211yesmaximal, characteristic
7#21445211C211yesmaximal, characteristic, complemented
8#21445189C211yesmaximal, characteristic, complemented
9#21445066C211yesmaximal, characteristic
10#21445061C211yesmaximal, characteristic, complemented
11#21445040C211yesmaximal, characteristic, complemented
12#21444982C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic
15C61×#3363C211yesmaximal, characteristic, complemented
16C2×Q16×D61—21no—
17Q8×D244C2211yescharacteristic
18Q8×D244C2211yescharacteristic
19#16840172—21no—
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840158—21no—
23#16840158C2211yescharacteristic
24#16840142C2211yescharacteristic
25#16840112C2211yescharacteristic
26#16840059—21no—
27#16840057—21no—
28#16840049—21no—
29#16840049C2211yescharacteristic
30#16840047—21no—
31#16840047C2211yescharacteristic
32#16840047C2211yescharacteristic
33#16840039C2211yescharacteristic
34#16840004—21no—
35#16839972C2211yescharacteristic
36#16839972C2211yescharacteristic
37#16839919—21no—
38#16839919—21no—
39#16839893C2211yescharacteristic
40Q8×#92418C2211yescharacteristic
41#16839860—21no—
42C4×#11758201—21no—
43C4×#11758201C2211yescharacteristic
44C8×D244C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839509C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57C61×#557C2211yescharacteristic
58C61×#487—21no—
59C61×#487C2211yescharacteristic
60C61×#477—21no—
61C61×#477C2211yescharacteristic
62C61×#462—21no—
63C61×#462C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×Q16×C61—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×Q8×D61—21no—
69C2×Dic244—21no—
70C2×Dic244—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73Q16×D61—44no—
74C2×#11758201D411yescharacteristic
75C2×C8×D61—21no—
76#14519777—21no—
77#14519777—21no—
78#14519777C2311yescharacteristic
79#14519777C2311yescharacteristic
80#14519757—21no—
81#14519757—21no—
82#14519757—41no—
83#14519757—41no—
84C4×D244—21no—
85C4×D244—21no—
86C4×D244C2311yescharacteristic
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—41no—
91#14519708—21no—
92#14519708—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96#14519708D411yescharacteristic
97C4⋊#92418—21no—
98C4⋊#92418—21no—
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102#14519701—21no—
103#14519701—21no—
104#14519701—41no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—21no—
108#14519697—21no—
109#14519697—21no—
110#14519697D411yescharacteristic
111#14519689—21no—
112#14519688—21no—
113C4×#92418—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×Q16×C61—21no—
117C2×Q16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—21no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124#125×C61—21no—
125#125×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×Q8×C61C2311yescharacteristic
131C4×Q8×C61C2311yescharacteristic
132C4×Q8×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61D411yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137#103×C61—21no—
138#103×C61—21no—
139#103×C61—21no—
140#103×C61—21no—
141#103×C61D411yescharacteristic
142#103×C61D411yescharacteristic
143C4×C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151Q8×D61—42no—
152Q8×D61—42no—
153Q8×D61—81no—
154Q8×D61—81no—
155C2×D244C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—41no—
158C2×C4×D61—41no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—41no—
162C2×Dic122—41no—
163Q16×C61—22no—
164Q16×C61—42no—
165C2×C8×C61—21no—
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61—41no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C42×C61—21no—
184C42×C61—21no—
185C42×C61—21no—
186C42×C61—41no—
187C42×C61—41no—
188C42×C61C2411yescharacteristic
189#11758202—42no—
190#11758202—42no—
191#11758201Q1614yescomplemented
192#11758198—21no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—41no—
200C4⋊Dic61—41no—
201C4⋊Dic61C2×D411yescharacteristic
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208Dic244—42no—
209Dic244—81no—
210C8×D61—42no—
211C22×D61—21no—
212Q8×C61—21no—
213Q8×C61—21no—
214Q8×C61—21no—
215Q8×C61—21no—
216Q8×C61—41no—
217Q8×C61—41no—
218Q8×C61—81no—
219Q8×C61—161no—
220Q8×C61C2×Q1612yescomplemented
221Q8×C61AGL(1,ℤ/8)12yes—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61C22×D411yescharacteristic
231C2×C4×C61C22×D411yescharacteristic
232C2×C4×C61H(5,𝔽2)11yescharacteristic
233C2×Dic61—21no—
234C2×Dic61—41no—
235C2×Dic61—41no—
236D244—21no—
237D244C2×Q1612yes—
238C4×D61—41no—
239C4×D61—41no—
240C4×D61—82no—
241C4×D61—82no—
242Dic122—41no—
243Dic122—41no—
244Dic122—41no—
245Dic122—41no—
246Dic122—82no—
247Dic122—82no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—41no—
251#92418—21no—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418C2×Q1612yescomplemented
257#19812—611nomaximal, Sylow
258C22×C61D4211yescharacteristic
259C2×D61—22no—
260C2×D61—41no—
261C2×D61—41no—
262C4×C61—22no—
263C4×C61—22no—
264C4×C61—21no—
265C4×C61—41no—
266C4×C61—41no—
267C4×C61—41no—
268C4×C61—41no—
269C4×C61—81no—
270C4×C61—81no—
271C4×C61C22×Q1611yescharacteristic
272C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
273C4×C61#57711yescharacteristic
274C4×C61#57811yescharacteristic
275Dic61—41no—
276Dic61—81no—
277Dic61—81no—
278#3386—611no—
279Q8×SD16—611no—
280#3363D6111yescharacteristic, complemented
281D4×Q16—611no—
282#1723—611no—
283#1692—611no—
284#1683—611no—
285#1674—611no—
286#1670—611no—
287#1649—611no—
288#1648—611no—
289#1645—611no—
290#1637—611no—
291#1584—611no—
292SD16⋊C8—611no—
293C2×C61#326511yescharacteristic
294C2×C61D4×Q1611yescharacteristic
295C2×C61#328911yescharacteristic
296D61—42no—
297C22×Q16—1221no—
298Q82C2×D6111yescharacteristic
299#557C2×D6111yescharacteristic
300D4×Q8—611no—
301D4×Q8—611no—
302#501—1221no—
303#500—611no—
304#500—611no—
305#498—611no—
306#498—1221no—
307#494—611no—
308#487—21no—
309#487—611no—
310#487C2×D6111yescharacteristic
311#478—1221no—
312#477—21no—
313#477—1221no—
314#477C2×D6111yescharacteristic
315#476—611no—
316#476—1221no—
317#475—611no—
318#475—611no—
319#475—1221no—
320#474—611no—
321#467—1221no—
322#462—21no—
323#462—611no—
324#462—611no—
325#462C2×D6111yescharacteristic
326#451—1221no—
327#451—1221no—
328#446—611no—
329C8×Q8—611no—
330C8×Q8C2×D6111yescharacteristic
331#440—1221no—
332C4×Q16—21no—
333C4×Q16C2×D6111yescharacteristic
334C4×SD16—611no—
335C4×SD16—1221no—
336C8×D4—611no—
337#335—611no—
338#335—611no—
339#333—611no—
340#332—611no—
341#332—611no—
342#332—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346D4⋊C8—611no—
347#322—611no—
348C61#1981211yescharacteristic, complemented, Sylow
349C22×Q8—1221no—
350C22×Q8—1221no—
351C2×Q16—21no—
352C2×Q16—1221no—
353C2×Q16—1221no—
354C2×Q16—1221no—
355C2×Q16—1221no—
356C2×Q16—2444no—
357C2×Q16C22⋊D6111yescharacteristic
358C2×SD16—611no—
359C22×C8—1221no—
360C4⋊Q8—21no—
361C4⋊Q8—21no—
362C4⋊Q8—21no—
363C4⋊Q8—41no—
364C4⋊Q8—611no—
365C4⋊Q8—611no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370#125—21no—
371#125—41no—
372C22⋊Q8—1221no—
373C22⋊Q8—1221no—
374C22⋊Q8—2441no—
375C22⋊Q8—2441no—
376C4×Q8—21no—
377C4×Q8—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22×D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×Q8C22×D6111yescharacteristic
383C4×D4—611no—
384C4×D4—1221no—
385C4×D4—1221no—
386#107—21no—
387#107—1221no—
388#107—1221no—
389#107—1221no—
390#107—2441no—
391#107C22⋊D6111yescharacteristic
392#106—611no—
393#106—1221no—
394#106—1221no—
395#106—1221no—
396#106—1221no—
397#106—1221no—
398C4⋊C8—21no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—1221no—
403C4⋊C8—1221no—
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—21no—
407#103—21no—
408#103—21no—
409#103—611no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—2441no—
415#103C22⋊D6111yescharacteristic
416#103C22⋊D6111yescharacteristic
417#102—611no—
418#102—1221no—
419#102—1221no—
420C22⋊C8—1221no—
421#97—1221no—
422C4×C8—21no—
423C4×C8—611no—
424C4×C8—611no—
425C4×C8—1221no—
426C4×C8C22×D6111yescharacteristic
427C2×Q8—21no—
428C2×Q8—21no—
429C2×Q8—41no—
430C2×Q8—1221no—
431C2×Q8—1221no—
432C2×Q8—4881no—
433C2×Q8—4881no—
434C2×Q8—2441no—
435C2×Q8—2441no—
436C2×Q8—2442no—
437C2×Q8—2442no—
438C2×Q8D4×D6111yescharacteristic
439C2×Q8C2×C22⋊D6111yescharacteristic
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—611no—
442C22×C4—1221no—
443C22×C4—2441no—
444C22×C4—2441no—
445Q16—22no—
446Q16—42no—
447Q16—4881no—
448Q16—2442no—
449Q16—2442no—
450Q16—2442no—
451SD16—614no—
452C2×C8—21no—
453C2×C8—611no—
454C2×C8—611no—
455C2×C8—1221no—
456C2×C8—1221no—
457C2×C8—1221no—
458C2×C8—1221no—
459C2×C8—2442no—
460C2×C8C2×C22⋊D6111yescharacteristic
461C2×C8C2×C22⋊D6111yescharacteristic
462C4⋊C4—21no—
463C4⋊C4—21no—
464C4⋊C4—21no—
465C4⋊C4—21no—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—41no—
470C4⋊C4—41no—
471C4⋊C4—41no—
472C4⋊C4—41no—
473C4⋊C4—41no—
474C4⋊C4—611no—
475C4⋊C4—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—2441no—
480C4⋊C4—2441no—
481C4⋊C4D4×D6111yescharacteristic
482C4⋊C4C2×C22⋊D6111yescharacteristic
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C22⋊C4—1221no—
485C22⋊C4—2441no—
486C22⋊C4—2441no—
487C42—21no—
488C42—21no—
489C42—21no—
490C42—41no—
491C42—41no—
492C42C23×D6111yescharacteristic
493C23—1221no—
494Q8—21no—
495Q8—21no—
496Q8—21no—
497Q8—21no—
498Q8—41no—
499Q8—41no—
500Q8—81no—
501Q8—161no—
502Q8—4882no—
503Q8—4882no—
504Q8—2441no—
505Q8—2441no—
506Q8—2441no—
507Q8—2441no—
508Q8Q16×D6112yescomplemented
509Q8#1451980212yes—
510D4—612no—
511D4—1221no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—1221no—
521C2×C4—4882no—
522C2×C4—4882no—
523C2×C4—2441no—
524C2×C4—2441no—
525C2×C4—2441no—
526C2×C4—2441no—
527C2×C4C2×D4×D6111yescharacteristic
528C2×C4C22×C22⋊D6111yescharacteristic
529C2×C4#1451982211yescharacteristic
530C8—21no—
531C8—21no—
532C8—41no—
533C8—612no—
534C8—1221no—
535C8—2441no—
536C8—2441no—
537C8—2441no—
538C8—2441no—
539C22—1222no—
540C22—2441no—
541C22—2441no—
542C22D4×C22⋊D6111yescharacteristic
543C4—22no—
544C4—22no—
545C4—21no—
546C4—41no—
547C4—41no—
548C4—41no—
549C4—41no—
550C4—81no—
551C4—81no—
552C4—4881no—
553C4—4881no—
554C4—2441no—
555C4C2×Q16×D6111yescharacteristic
556C4C2×#1451980211yescharacteristic
557C4#1684057011yescharacteristic
558C4#1684057511yescharacteristic
559C2—2442no—
560C2#2145731511yescharacteristic
561C2Q16×C22⋊D6111yescharacteristic
562C2#2145759211yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3363C2
socleC22×C61D42
prime radicalC2×D244C2×D4
hypercenter#3363D61
hypocenterC61#19812
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3363

Complements

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

NH ≅ G/Ntype
4. #21445498C2semidirect
5. #21445327C2semidirect
7. #21445211C2semidirect
8. #21445189C2semidirect
10. #21445061C2semidirect
11. #21445040C2semidirect
13. #21444670C2semidirect
15. C61×#3363C2semidirect
191. #11758201Q16semidirect
220. Q8×C61C2×Q16semidirect
256. #92418C2×Q16semidirect
280. #3363D61semidirect
348. C61#19812semidirect
508. Q8Q16×D61semidirect

Maximal subgroups

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

subgroupindex
#214587362
Q8×#117582012
Q16×D2442
#214454982
#214453272
#214452672
#214452112
#214451892
#214450662
#214450612
#214450402
#214449822
#214446702
#214446292
C61×#33632
#1981261