Finite Group Databasebeta

← #30613601

Subgroups — #30613601

Order 15616.

650 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462751C211yesmaximal, characteristic, complemented
2#21462745C211yesmaximal, characteristic, complemented
3#21461462C212yesmaximal, complemented
4#21461432C212yesmaximal, complemented
5#21461420C212yesmaximal, complemented
6SD16⋊#92420C212yesmaximal, complemented
7#21461376C211yesmaximal, characteristic, complemented
8#21461154C211yesmaximal, characteristic, complemented
9#21461154C211yesmaximal, characteristic, complemented
10#21461028C211yesmaximal, characteristic, complemented
11#135×Dic61C211yesmaximal, characteristic
12#16840813C2212yes—
13Q8×#92420C2212yes—
14#16840797—22no—
15#16840797C2212yes—
16#16840793—22no—
17D4×#92420C2212yes—
18#16840782—21no—
19#16840779—21no—
20#16840779C2211yescharacteristic
21#16840765—21no—
22#16840765C2211yescharacteristic
23#16840765C2211yescharacteristic
24C2×#14519844—21no—
25C4×#11758217C2211yescharacteristic
26#16840626C2212yes—
27#16840625C2212yes—
28Q8⋊#92420C2212yes—
29Q8⋊#92420C2212yes—
30D4⋊#92420C2212yes—
31D4⋊#92420C2212yes—
32#92420⋊C8C2212yes—
33#16840603C2212yes—
34#41×Dic61C2211yescharacteristic
35#41×Dic61C2211yescharacteristic
36Q16×Dic61C411yescharacteristic
37Q16×Dic61C2211yescharacteristic
38SD16×Dic61C411yescharacteristic
39SD16×Dic61C411yescharacteristic
40SD16×Dic61C2211yescharacteristic
41SD16×Dic61C2211yescharacteristic
42D8×Dic61C411yescharacteristic
43D8×Dic61C2211yescharacteristic
44#16839800C411yescharacteristic
45#16839758C411yescharacteristic
46#16839758C411yescharacteristic
47C2×C8×Dic61C2211yescharacteristic
48C2×#135×C61C411yescharacteristic
49C2×#11758217—21no—
50C2×#11758217—21no—
51C2×#11758217—21no—
52C2×#11758217—22no—
53C22×#92420—22no—
54#14519844—22no—
55#14519844—21no—
56#14519844—21no—
57#14519844D411yescharacteristic
58#14519844D411yescharacteristic
59C4⋊#92420—22no—
60C4⋊#92420C2312yes—
61C4⋊#92420C2312yes—
62C22⋊#92420—22no—
63C22⋊#92420—22no—
64C22⋊#92420D412yes—
65#14519832—22no—
66#14519832—21no—
67C4×#92420—22no—
68C4×#92420C2312yes—
69C4×#92420C2312yes—
70C22×#92418C2×C411yescharacteristic
71Q8×Dic61C2×C411yescharacteristic
72Q8×Dic61C2×C411yescharacteristic
73Q8×Dic61C2311yescharacteristic
74Q8×Dic61C2311yescharacteristic
75D4×Dic61—21no—
76D4×Dic61—21no—
77D4×Dic61—21no—
78D4×Dic61—21no—
79D4×Dic61C2×C411yescharacteristic
80D4×Dic61C2×C411yescharacteristic
81D4×Dic61C2311yescharacteristic
82D4×Dic61C2311yescharacteristic
83#14519736—21no—
84#14519736—21no—
85#14519736C2×C411yescharacteristic
86#14519736C2×C411yescharacteristic
87C2×C4×Dic61—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519711C2×C411yescharacteristic
91#14519711C2×C411yescharacteristic
92#14519709C2×C411yescharacteristic
93#14519709C2×C411yescharacteristic
94#14519701C2×C411yescharacteristic
95#14519701C2×C411yescharacteristic
96#14519701C2×C411yescharacteristic
97#14519701C2×C411yescharacteristic
98#14519698C2×C411yescharacteristic
99#14519698C2×C411yescharacteristic
100#14519698C2×C411yescharacteristic
101#14519698C2×C411yescharacteristic
102C8×Dic61C2×C411yescharacteristic
103C8×Dic61C2×C411yescharacteristic
104C8×Dic61C2311yescharacteristic
105C8×Dic61C2311yescharacteristic
106C2×#41×C61C2×C411yescharacteristic
107C2×#41×C61C2×C411yescharacteristic
108#135×C61—24no—
109C2×Q16×C61C2×C411yescharacteristic
110C2×SD16×C61C2×C411yescharacteristic
111C2×SD16×C61C2×C411yescharacteristic
112C2×D8×C61C2×C411yescharacteristic
113C22×C8×C61C2×C411yescharacteristic
114#41×C61—22no—
115#41×C61—22no—
116#41×C61—41no—
117#41×C61—41no—
118C2×Q8×C61C22×C411yescharacteristic
119C2×Q8×C61C22×C411yescharacteristic
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61C22×C411yescharacteristic
123C2×D4×C61C22×C411yescharacteristic
124C22×C4×C61—21no—
125C22×C4×C61—21no—
126C22×C4×C61C22×C411yescharacteristic
127C22×Dic61—21no—
128C22×Dic61—21no—
129C22×Dic61—21no—
130C22×Dic61—21no—
131C22×Dic61C2×D411yescharacteristic
132C22×Dic61#4111yescharacteristic
133#11758217—42no—
134#11758217—82no—
135#11758217—81no—
136#11758217—81no—
137C2×#92420—22no—
138C2×#92420—22no—
139C2×#92420—22no—
140C2×#92420—82no—
141C2×#92420C2×D412yes—
142C2×#92420C2×D412yes—
143Q16×C61—22no—
144SD16×C61—22no—
145SD16×C61M1614yescomplemented
146D8×C61—22no—
147C2×C8×C61—22no—
148C2×C8×C61C22×C411yescharacteristic
149C2×C8×C61C22×C411yescharacteristic
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152C22⋊Dic61—21no—
153C22⋊Dic61—21no—
154C22⋊Dic61—21no—
155C22⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61—21no—
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162C4⋊Dic61C22×C411yescharacteristic
163C4⋊Dic61C22×C411yescharacteristic
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61—21no—
170C4×Dic61C22×C411yescharacteristic
171C4×Dic61C2×D411yescharacteristic
172C4×Dic61#4111yescharacteristic
173C4×Dic61C2411yescharacteristic
174C2×#92418—41no—
175C2×#92418C22×C411yescharacteristic
176C2×#92418C22×C411yescharacteristic
177C23×C61—21no—
178C23×C61—21no—
179C23×C61C4×D411yescharacteristic
180Q8×C61—21no—
181Q8×C61—21no—
182Q8×C61—21no—
183Q8×C61C2×M1612yes—
184D4×C61—21no—
185D4×C61—21no—
186D4×C61—21no—
187D4×C61—42no—
188D4×C61—42no—
189D4×C61C2×M1612yes—
190C2×C4×C61—21no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—21no—
194C2×C4×C61—42no—
195C2×C4×C61—42no—
196C2×C4×C61C4×D411yescharacteristic
197C2×C4×C61C23×C411yescharacteristic
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—22no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—21no—
204C2×Dic61—21no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—41no—
209C2×Dic61—42no—
210C2×Dic61—42no—
211C2×Dic61C4×D411yescharacteristic
212C2×Dic61C4×D411yescharacteristic
213C2×Dic61C22×D411yescharacteristic
214C2×Dic61C2×#4111yescharacteristic
215#92420—22no—
216#92420—22no—
217#92420—42no—
218#92420—42no—
219#92420—42no—
220C8×C61—21no—
221C8×C61C2×M1612yes—
222#92418—21no—
223#92418—21no—
224#18061—611nomaximal, Sylow
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—21no—
228C22×C61—21no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61—41no—
233C22×C61—41no—
234C22×C61C2×C4×D411yescharacteristic
235C4×C61—21no—
236C4×C61—22no—
237C4×C61—41no—
238C4×C61C22×M1611yescharacteristic
239C4×C61#56811yescharacteristic
240Dic61—21no—
241Dic61—21no—
242Dic61—41no—
243Dic61—41no—
244Dic61—41no—
245Dic61—41no—
246Dic61#44412yes—
247Dic61#57611yescharacteristic
248Dic61#57811yescharacteristic
249#2991—611no—
250#2990—611no—
251C4×#135—611no—
252#1592—612no—
253#1584—612no—
254#1581—612no—
255SD16⋊C8—612no—
256#1567—611no—
257#1489—611no—
258#1489—611no—
259#1451—611no—
260C2×C61—41no—
261C2×C61—42no—
262C2×C61—81no—
263C2×C61D4×M1611yescharacteristic
264C2×C61#295111yescharacteristic
265C2×C61C2×#44411yescharacteristic
266C2×#135#692311yescharacteristic
267C4×#41—611no—
268C4×#41—611no—
269#446—612no—
270C8×Q8—612no—
271C4×Q16—611no—
272C4×Q16—611no—
273C4×SD16—611no—
274C4×SD16—611no—
275C4×SD16—611no—
276C4×SD16—611no—
277C4×D8—611no—
278C4×D8—611no—
279#436—612no—
280#436—1222no—
281#435—1222no—
282C8×D4—612no—
283#432—1221no—
284#431—611no—
285#431—1221no—
286#427—611no—
287#423—611no—
288#423—611no—
289#423—1221no—
290C2×C4⋊C8—1221no—
291#416—611no—
292#416—611no—
293C4×M16—611no—
294C2×C4×C8—611no—
295#335—612no—
296#335—612no—
297Q8⋊C8—612no—
298Q8⋊C8—612no—
299D4⋊C8—612no—
300D4⋊C8—612no—
301#322—612no—
302#322—612no—
303C61#1806111yescharacteristic, complemented, Sylow
304C2×#41C2×#692311yescharacteristic
305C2×#41C2×#692311yescharacteristic
306#135—24no—
307C2×Q16C2×#692311yescharacteristic
308C2×SD16C2×#692311yescharacteristic
309C2×SD16C2×#692311yescharacteristic
310C2×D8C2×#692311yescharacteristic
311C2×M16—1222no—
312C2×M16—1221no—
313C2×M16—1221no—
314C2×M16—1221no—
315C22×C8—611no—
316C22×C8—1222no—
317C22×C8C2×#692311yescharacteristic
318C4×Q8—611no—
319C4×Q8—611no—
320C4×Q8—611no—
321C4×Q8—611no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—611no—
325C4×D4—611no—
326C4×D4—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330#117—611no—
331#117—611no—
332#117—1221no—
333#117—1221no—
334C2×C42—611no—
335C2×C42—1221no—
336C2×C42—1221no—
337#107—611no—
338#107—611no—
339#106—611no—
340#106—611no—
341C4⋊C8—611no—
342C4⋊C8—611no—
343C4⋊C8—612no—
344C4⋊C8—612no—
345C4⋊C8—1221no—
346C4⋊C8—1221no—
347C4⋊C8—1222no—
348C4⋊C8—1222no—
349#103—611no—
350#103—611no—
351#103—611no—
352#103—611no—
353#102—611no—
354#102—611no—
355#102—611no—
356#102—611no—
357C22⋊C8—612no—
358C22⋊C8—1222no—
359C22⋊C8—1222no—
360#97—1221no—
361#97—1222no—
362C4×C8—612no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8—611no—
367C4×C8—612no—
368C4×C8—1222no—
369#41—22no—
370#41—22no—
371#41—41no—
372#41—41no—
373C2×Q8C22×#692311yescharacteristic
374C2×Q8C22×#692311yescharacteristic
375C2×D4—21no—
376C2×D4—21no—
377C2×D4C22×#692311yescharacteristic
378C2×D4C22×#692311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—611no—
382C22×C4—611no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4—1221no—
387C22×C4C22×#692311yescharacteristic
388Q16—22no—
389SD16—22no—
390SD16#1175821714yescomplemented
391D8—22no—
392M16—4882no—
393M16—4881no—
394M16—4881no—
395M16—2442no—
396C2×C8—22no—
397C2×C8—612no—
398C2×C8—612no—
399C2×C8—611no—
400C2×C8—611no—
401C2×C8—1222no—
402C2×C8—1222no—
403C2×C8—1222no—
404C2×C8—4882no—
405C2×C8—2441no—
406C2×C8C22×#692311yescharacteristic
407C2×C8C22×#692311yescharacteristic
408C4⋊C4—611no—
409C4⋊C4—611no—
410C4⋊C4—611no—
411C4⋊C4—611no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C4⋊C4—1221no—
415C4⋊C4—1221no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C22⋊C4—1221no—
421C22⋊C4—1221no—
422C42—611no—
423C42—611no—
424C42—611no—
425C42—611no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—1221no—
430C42—1221no—
431C42—1221no—
432C23—21no—
433C23—21no—
434C23D4×#692311yescharacteristic
435Q8—21no—
436Q8—21no—
437Q8—21no—
438Q8C2×#1175821712yes—
439D4—21no—
440D4—21no—
441D4—21no—
442D4—42no—
443D4—42no—
444D4C2×#1175821712yes—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—611no—
452C2×C4—611no—
453C2×C4—611no—
454C2×C4—611no—
455C2×C4—1221no—
456C2×C4—1222no—
457C2×C4—1221no—
458C2×C4—1221no—
459C2×C4—1221no—
460C2×C4—1221no—
461C2×C4—1221no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4—2441no—
465C2×C4—2441no—
466C2×C4—2442no—
467C2×C4—2442no—
468C2×C4D4×#692311yescharacteristic
469C2×C4C23×#692311yescharacteristic
470C8—21no—
471C8—1222no—
472C8—1222no—
473C8—1221no—
474C8—1221no—
475C8—2442no—
476C8—2442no—
477C8—2442no—
478C8C2×#1175821712yes—
479C22—21no—
480C22—22no—
481C22—21no—
482C22—21no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22C2×D4×#692311yescharacteristic
489C4—21no—
490C4—22no—
491C4—41no—
492C4—611no—
493C4—611no—
494C4—612no—
495C4—1221no—
496C4—1221no—
497C4—2441no—
498C4—2441no—
499C4—2441no—
500C4—2441no—
501C4C22×#1175821711yescharacteristic
502C4#1684084011yescharacteristic
503C2—41no—
504C2—42no—
505C2—81no—
506C2D4×#1175821711yescharacteristic
507C2#2146258111yescharacteristic
508C2D61⋊#44411yescharacteristic
subgroupG/H
centerC22C2×D4×#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC2×#135×C61C4
socleC22×C61C2×C4×D4
cosocleC4×Dic61C24
hypercenterC2×#135#6923
hypocenterC61#18061
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. #21462751C2semidirect
2. #21462745C2semidirect
3. #21461462C2semidirect
4. #21461432C2semidirect
5. #21461420C2semidirect
6. SD16⋊#92420C2semidirect
7. #21461376C2semidirect
8. #21461154C2semidirect
9. #21461154C2semidirect
10. #21461028C2semidirect
145. SD16×C61M16semidirect
303. C61#18061semidirect
390. SD16#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627512
#214627452
#214614622
#214614322
#214614202
SD16⋊#924202
#214613762
#214611542
#214610282
#135×Dic612
#1806161