Finite Group Databasebeta

← #30612495

Subgroups — #30612495

Order 15616.

660 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462738C211yesmaximal, characteristic, complemented
2#21462734C211yesmaximal, characteristic, complemented
3#21461444C212yesmaximal, complemented
4#21461440C212yesmaximal, complemented
5#21461428C212yesmaximal, complemented
6#21461424C212yesmaximal, complemented
7C2×#16840624C211yesmaximal, characteristic, complemented
8#21461114C211yesmaximal, characteristic, complemented
9#21461114C211yesmaximal, characteristic, complemented
10#21461032C211yesmaximal, characteristic, complemented
11#135×Dic61C211yesmaximal, characteristic
12#16840813C2212yes—
13Q8×#92420C2212yes—
14#16840797C2212yes—
15#16840793—22no—
16D4×#92420—22no—
17D4×#92420C2212yes—
18#16840783—21no—
19#16840778—21no—
20#16840778C2211yescharacteristic
21#16840778C2211yescharacteristic
22#16840765—21no—
23C2×C4⋊#92420—21no—
24C2×C4⋊#92420C2211yescharacteristic
25C2×C4⋊#92420C2211yescharacteristic
26#16840624C2212yes—
27#16840624C2212yes—
28Q8⋊#92420C2212yes—
29Q8⋊#92420C2212yes—
30D4⋊#92420C2212yes—
31D4⋊#92420C2212yes—
32#92420⋊C8C2212yes—
33#92420⋊C8C2212yes—
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—22no—
50C22×#92420—21no—
51C22×#92420—21no—
52C22×#92420—22no—
53#14519844—22no—
54C4⋊#92420—22no—
55C4⋊#92420—21no—
56C4⋊#92420—21no—
57C4⋊#92420—21no—
58C4⋊#92420D411yescharacteristic
59C4⋊#92420D411yescharacteristic
60C4⋊#92420C2312yes—
61C4⋊#92420C2312yes—
62C22⋊#92420—21no—
63C22⋊#92420—22no—
64C22⋊#92420—22no—
65C22⋊#92420D412yes—
66#14519832—22no—
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—22no—
117#41×C61M1614yescomplemented
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—82no—
134C2×#92420—22no—
135C2×#92420—22no—
136C2×#92420—22no—
137C2×#92420—82no—
138C2×#92420—81no—
139C2×#92420—81no—
140C2×#92420C2×D412yes—
141C2×#92420C2×D412yes—
142Q16×C61—22no—
143SD16×C61—22no—
144SD16×C61—22no—
145D8×C61—22no—
146C2×C8×C61—22no—
147C2×C8×C61C22×C411yescharacteristic
148C2×C8×C61C22×C411yescharacteristic
149C22⋊Dic61—21no—
150C22⋊Dic61—21no—
151C22⋊Dic61—21no—
152C22⋊Dic61—21no—
153C22⋊Dic61—21no—
154C22⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—21no—
157C4⋊Dic61—21no—
158C4⋊Dic61—21no—
159C4⋊Dic61C22×C411yescharacteristic
160C4⋊Dic61C22×C411yescharacteristic
161C4⋊Dic61C22×C411yescharacteristic
162C4⋊Dic61C22×C411yescharacteristic
163C4×Dic61—21no—
164C4×Dic61—21no—
165C4×Dic61—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169C4×Dic61C22×C411yescharacteristic
170C4×Dic61C2×D411yescharacteristic
171C4×Dic61#4111yescharacteristic
172C4×Dic61C2411yescharacteristic
173C2×#92418—22no—
174C2×#92418C22×C411yescharacteristic
175C2×#92418C22×C411yescharacteristic
176C23×C61—21no—
177C23×C61—21no—
178C23×C61C4×D411yescharacteristic
179Q8×C61—21no—
180Q8×C61—21no—
181Q8×C61—21no—
182Q8×C61C2×M1612yes—
183D4×C61—24no—
184D4×C61—21no—
185D4×C61—21no—
186D4×C61—21no—
187D4×C61—42no—
188D4×C61C2×M1612yes—
189C2×C4×C61—21no—
190C2×C4×C61—24no—
191C2×C4×C61—21no—
192C2×C4×C61—42no—
193C2×C4×C61C4×D411yescharacteristic
194C2×C4×C61C2×M1612yes—
195C2×C4×C61#13112yes—
196C2×C4×C61C23×C411yescharacteristic
197C2×Dic61—21no—
198C2×Dic61—21no—
199C2×Dic61—21no—
200C2×Dic61—21no—
201C2×Dic61—21no—
202C2×Dic61—21no—
203C2×Dic61—41no—
204C2×Dic61—41no—
205C2×Dic61—41no—
206C2×Dic61—41no—
207C2×Dic61—41no—
208C2×Dic61—81no—
209C2×Dic61—81no—
210C2×Dic61C4×D411yescharacteristic
211C2×Dic61C4×D411yescharacteristic
212C2×Dic61C22×D411yescharacteristic
213C2×Dic61C2×#4111yescharacteristic
214#92420—22no—
215#92420—22no—
216#92420—42no—
217#92420—42no—
218#92420—42no—
219C8×C61—21no—
220C8×C61—21no—
221#92418—21no—
222#92418—21no—
223#17882—611nomaximal, Sylow
224C22×C61—21no—
225C22×C61—21no—
226C22×C61—22no—
227C22×C61—41no—
228C22×C61—41no—
229C22×C61—41no—
230C22×C61—41no—
231C22×C61—41no—
232C22×C61#43512yes—
233C22×C61C2×C4×D411yescharacteristic
234C4×C61—22no—
235C4×C61—41no—
236C4×C61#43512yes—
237C4×C61C22×M1611yescharacteristic
238C4×C61C2×#13111yescharacteristic
239Dic61—21no—
240Dic61—21no—
241Dic61—21no—
242Dic61—41no—
243Dic61—41no—
244Dic61—41no—
245Dic61—41no—
246Dic61#57711yescharacteristic
247Dic61#57711yescharacteristic
248#2989—611no—
249#2988—611no—
250C4×#135—611no—
251#1587—612no—
252#1586—612no—
253#1583—612no—
254#1582—612no—
255C2×#334—611no—
256#1477—611no—
257#1477—611no—
258#1452—611no—
259C2×C61—22no—
260C2×C61—42no—
261C2×C61—81no—
262C2×C61C2×#43511yescharacteristic
263C2×C61#295211yescharacteristic
264C2×C61#295711yescharacteristic
265C2×#135#692311yescharacteristic
266C4×#41—611no—
267C4×#41—611no—
268#446—612no—
269C8×Q8—612no—
270C4×Q16—611no—
271C4×Q16—611no—
272C4×SD16—611no—
273C4×SD16—611no—
274C4×SD16—611no—
275C4×SD16—611no—
276C4×D8—611no—
277C4×D8—611no—
278#436—612no—
279#435—1222no—
280C8×D4—612no—
281C8×D4—1222no—
282#432—1221no—
283#431—611no—
284#431—611no—
285#431—1221no—
286#427—611no—
287#423—1221no—
288C2×C4⋊C8—611no—
289C2×C4⋊C8—611no—
290C2×C4⋊C8—1221no—
291#416—611no—
292#416—611no—
293C2×C4×C8—611no—
294#334—612no—
295#334—612no—
296Q8⋊C8—612no—
297Q8⋊C8—612no—
298D4⋊C8—612no—
299D4⋊C8—612no—
300#322—612no—
301#322—612no—
302C61#1788211yescharacteristic, complemented, Sylow
303C2×#41C2×#692311yescharacteristic
304C2×#41C2×#692311yescharacteristic
305#135—24no—
306C2×Q16C2×#692311yescharacteristic
307C2×SD16C2×#692311yescharacteristic
308C2×SD16C2×#692311yescharacteristic
309C2×D8C2×#692311yescharacteristic
310C2×M16—1222no—
311C22×C8—611no—
312C22×C8—1222no—
313C22×C8—1221no—
314C22×C8—1221no—
315C22×C8C2×#692311yescharacteristic
316C4×Q8—611no—
317C4×Q8—611no—
318C4×Q8—611no—
319C4×Q8—611no—
320C4×D4—611no—
321C4×D4—611no—
322C4×D4—611no—
323C4×D4—611no—
324C4×D4—1221no—
325C4×D4—1221no—
326C4×D4—1221no—
327C4×D4—1221no—
328#117—611no—
329#117—611no—
330#117—1221no—
331#117—1221no—
332C2×C42—611no—
333C2×C42—1221no—
334C2×C42—1221no—
335#107—611no—
336#107—611no—
337#106—611no—
338#106—611no—
339C4⋊C8—611no—
340C4⋊C8—611no—
341C4⋊C8—612no—
342C4⋊C8—612no—
343C4⋊C8—1221no—
344C4⋊C8—1221no—
345C4⋊C8—1221no—
346C4⋊C8—1222no—
347C4⋊C8—1222no—
348#103—611no—
349#103—611no—
350#103—611no—
351#103—611no—
352#102—611no—
353#102—611no—
354#102—611no—
355#102—611no—
356C22⋊C8—612no—
357C22⋊C8—1222no—
358C22⋊C8—1222no—
359C22⋊C8—1221no—
360#97—1222no—
361C4×C8—612no—
362C4×C8—611no—
363C4×C8—611no—
364C4×C8—611no—
365C4×C8—611no—
366C4×C8—612no—
367C4×C8—1222no—
368#41—22no—
369#41—22no—
370#41—22no—
371#41#1175821714yescomplemented
372C2×Q8C22×#692311yescharacteristic
373C2×Q8C22×#692311yescharacteristic
374C2×D4—21no—
375C2×D4—21no—
376C2×D4C22×#692311yescharacteristic
377C2×D4C22×#692311yescharacteristic
378C22×C4—21no—
379C22×C4—21no—
380C22×C4—611no—
381C22×C4—611no—
382C22×C4—1221no—
383C22×C4—1221no—
384C22×C4—1221no—
385C22×C4—1221no—
386C22×C4C22×#692311yescharacteristic
387Q16—22no—
388SD16—22no—
389SD16—22no—
390D8—22no—
391M16—4882no—
392C2×C8—22no—
393C2×C8—612no—
394C2×C8—612no—
395C2×C8—611no—
396C2×C8—611no—
397C2×C8—1222no—
398C2×C8—1222no—
399C2×C8—1222no—
400C2×C8—1222no—
401C2×C8—4882no—
402C2×C8—4881no—
403C2×C8—4881no—
404C2×C8C22×#692311yescharacteristic
405C2×C8C22×#692311yescharacteristic
406C4⋊C4—611no—
407C4⋊C4—611no—
408C4⋊C4—611no—
409C4⋊C4—611no—
410C4⋊C4—1221no—
411C4⋊C4—1221no—
412C4⋊C4—1221no—
413C4⋊C4—1221no—
414C22⋊C4—1221no—
415C22⋊C4—1221no—
416C22⋊C4—1221no—
417C22⋊C4—1221no—
418C22⋊C4—1221no—
419C22⋊C4—1221no—
420C42—611no—
421C42—611no—
422C42—611no—
423C42—611no—
424C42—1221no—
425C42—1221no—
426C42—1221no—
427C42—1221no—
428C42—1221no—
429C42—1221no—
430C23—21no—
431C23—21no—
432C23D4×#692311yescharacteristic
433Q8—21no—
434Q8—21no—
435Q8—21no—
436Q8C2×#1175821712yes—
437D4—24no—
438D4—21no—
439D4—21no—
440D4—21no—
441D4—42no—
442D4C2×#1175821712yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—42no—
447C2×C4—611no—
448C2×C4—611no—
449C2×C4—611no—
450C2×C4—611no—
451C2×C4—1221no—
452C2×C4—1221no—
453C2×C4—1221no—
454C2×C4—1221no—
455C2×C4—1221no—
456C2×C4—1221no—
457C2×C4—4881no—
458C2×C4—4881no—
459C2×C4—2441no—
460C2×C4—2441no—
461C2×C4—2441no—
462C2×C4—2441no—
463C2×C4—2441no—
464C2×C4D4×#692311yescharacteristic
465C2×C4C2×#1175821712yes—
466C2×C4#1451986412yes—
467C2×C4C23×#692311yescharacteristic
468C8—21no—
469C8—21no—
470C8—1222no—
471C8—1222no—
472C8—1221no—
473C8—1221no—
474C8—2442no—
475C8—2442no—
476C8—2442no—
477C22—21no—
478C22—21no—
479C22—22no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22#1684079312yes—
486C22C2×D4×#692311yescharacteristic
487C4—22no—
488C4—41no—
489C4—611no—
490C4—611no—
491C4—1221no—
492C4—1221no—
493C4—1221no—
494C4—2441no—
495C4—2441no—
496C4—2441no—
497C4—2441no—
498C4#1684079312yes—
499C4C22×#1175821711yescharacteristic
500C4C2×#1451986411yescharacteristic
501C2—22no—
502C2—42no—
503C2—81no—
504C2C2×#1684079311yescharacteristic
505C2#2146258211yescharacteristic
506C2#2146259611yescharacteristic
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#17882
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. #21462738C2semidirect
2. #21462734C2semidirect
3. #21461444C2semidirect
4. #21461440C2semidirect
5. #21461428C2semidirect
6. #21461424C2semidirect
7. C2×#16840624C2semidirect
8. #21461114C2semidirect
9. #21461114C2semidirect
10. #21461032C2semidirect
117. #41×C61M16semidirect
302. C61#17882semidirect
371. #41#11758217semidirect

Maximal subgroups

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

subgroupindex
#214627382
#214627342
#214614442
#214614402
#214614282
#214614242
C2×#168406242
#214611142
#214610322
#135×Dic612
#1788261