Finite Group Databasebeta

← #30614298

Subgroups — #30614298

Order 15616.

1126 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840772C211yesmaximal, characteristic
2C2×#16840725C211yesmaximal, characteristic
3#21462257C212yesmaximal
4#21461635C214yesmaximal, complemented
5#106×#6923C214yesmaximal, complemented
6#21461506C212yesmaximal
7C2×#106×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2211yescharacteristic
9C2×#14519856C2212yes—
10C2×#14519856C2211yescharacteristic
11#16840772C418yescomplemented
12C2×#14519846C2212yes—
13C2×#14519845C2212yes—
14#16840725C418yescomplemented
15C2×#14519831C2212yes—
16C2×C8×#6923C2212yes—
17C4⋊C4×#6923—24no—
18C4⋊C4×#6923C2218yes—
19C4×C22⋊#6923—24no—
20#16840635C2218yes—
21C22×C8×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2212yes—
23C2×#14519709C411yescharacteristic, complemented
24C2×#14519708C412yescomplemented
25#106×D61C412yescomplemented
26#106×D61C412yescomplemented
27#106×D61C2212yes—
28#106×D61C2212yes—
29C2×#106×C61C411yescharacteristic, complemented
30C2×#11758213—21no—
31D61⋊M16—22no—
32D61⋊M16C2×C414yes—
33C2×#11758217—21no—
34C2×D61⋊C8C2312yes—
35#14519856—24no—
36#14519856C2×C418yes—
37#14519856C2×C414yes—
38#14519856Q814yes—
39C2×C4⋊#6923D412yes—
40C2×C4⋊#6923C2314yes—
41C2×C22⋊#6923—22no—
42C2×C22⋊#6923D412yes—
43C2×C4×#6923—22no—
44C2×C4×#6923—28no—
45C2×C4×#6923C2312yes—
46#14519846C2×C418yescomplemented
47#14519845C2×C418yescomplemented
48#14519831C2×C418yescomplemented
49C8×#6923C2×C418yescomplemented
50(C2×#6923)⋊C4—28no—
51C22×C4×D61—22no—
52C22×C4×D61C2311yescharacteristic
53C22×#92418C2×C411yescharacteristic
54C2×C8×D61C2×C411yescharacteristic
55C2×C8×D61C2×C411yescharacteristic
56C2×C8×D61C2×C414yes—
57C2×C8×D61C2×C414yes—
58C2×C8×D61C2311yescharacteristic
59C2×C8×D61C2311yescharacteristic
60C2×C4⋊Dic61C2×C412yes—
61C2×#11758196—22no—
62C4⋊C4×D61—22no—
63C4⋊C4×D61—22no—
64C4⋊C4×D61C2×C414yes—
65C4⋊C4×D61C2314yes—
66#14519709C2×C412yes—
67#14519709C2×C412yes—
68#14519708C2×C414yes—
69#14519708C2×C414yes—
70C22×C8×C61C2×C411yescharacteristic
71C2×C4⋊C4×C61C2×C412yes—
72#106×C61C2×C412yes—
73#106×C61C2×C412yes—
74C23×D61C2×D411yescharacteristic
75C22×#6923—24no—
76C22×#6923C2×D414yes—
77C22×C4×C61—22no—
78C22×C4×C61C22×C411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61#4111yescharacteristic
81C2×C4×D61—24no—
82C2×C4×D61—24no—
83C2×C4×D61—84no—
84C2×C4×D61C22×C411yescharacteristic
85C2×C4×D61C22×C412yes—
86C2×C4×D61C22×C412yes—
87C2×C4×D61C2×D411yescharacteristic
88C2×C4×D61C2×Q812yes—
89C2×C4×D61#4111yescharacteristic
90C2×C4×D61#4112yes—
91C2×C4×D61C2411yescharacteristic
92C22⋊#6923—28no—
93C22⋊#6923C4⋊C418yes—
94#11758217—24no—
95C2×#92420—22no—
96C4⋊#6923—24no—
97C4⋊#6923C4⋊C418yes—
98C4⋊#6923C22×C418yes—
99C4×#6923—28no—
100C4×#6923—88no—
101C4×#6923C22×C414yes—
102C4×#6923C2×Q814yes—
103#11758213—24no—
104D61⋊C8—22no—
105D61⋊C8C22×C414yes—
106C2×C8×C61C4214yescomplemented
107C2×C8×C61C22×C411yescharacteristic
108C2×C8×C61C22×C411yescharacteristic
109C4⋊C4×C61—22no—
110C4⋊C4×C61C22×C414yes—
111C4⋊Dic61—22no—
112C4⋊Dic61C22×C414yes—
113#11758196—24no—
114#11758196—24no—
115C2×#92418C4214yescomplemented
116C2×#92418C22×C411yescharacteristic
117C2×#92418C22×C411yescharacteristic
118C8×D61C4214yescomplemented
119C8×D61C4214yescomplemented
120C8×D61C22×C412yes—
121C8×D61C22×C412yes—
122C8×D61C22×C412yes—
123C8×D61C22×C412yes—
124C23×C61C4×D411yescharacteristic
125C22×D61—24no—
126C22×D61C2×C4⋊C412yes—
127C22×D61C4×D412yes—
128C22×D61C4×D411yescharacteristic
129C22×D61C22×D411yescharacteristic
130C2×#6923—28no—
131C2×#6923—28no—
132C2×#6923—44no—
133C2×#6923—44no—
134C2×#6923C2×C4⋊C418yes—
135C2×C4×C61—24no—
136C2×C4×C61—82no—
137C2×C4×C61C2×C4212yes—
138C2×C4×C61C4×D411yescharacteristic
139C2×C4×C61C4×Q812yes—
140C2×C4×C61C23×C411yescharacteristic
141C2×Dic61—24no—
142C2×Dic61—82no—
143C2×Dic61#11712yes—
144C2×Dic61C4×D411yescharacteristic
145C2×Dic61C4×Q812yes—
146C2×Dic61C2×#4111yescharacteristic
147C4×D61—48no—
148C4×D61—84no—
149C4×D61C2×C4212yes—
150C4×D61C2×C4⋊C412yes—
151C4×D61#11712yes—
152C4×D61C4×D412yes—
153C4×D61C4×Q812yes—
154C4×D61C4×Q812yes—
155C4×D61C23×C411yescharacteristic
156C4×D61C23×C411yescharacteristic
157C4×D61C22×Q811yescharacteristic
158C4×D61C2×#4111yescharacteristic
159#92420—24no—
160C8×C61C2×C4212yes—
161C8×C61C2×C4212yes—
162#92418C2×C4212yes—
163#92418C2×C4212yes—
164#20454—611nomaximal, Sylow
165C22×C61—22no—
166C22×C61C4×C4⋊C412yes—
167C22×C61C2×C4×D411yescharacteristic
168C2×D61—22no—
169C2×D61—24no—
170C2×D61—24no—
171C2×D61C4×C4⋊C412yes—
172C2×D61C22×C4⋊C411yescharacteristic
173C2×D61C2×C4×D411yescharacteristic
174C2×D61#57714yes—
175#6923—28no—
176#6923—48no—
177C4×C61—44no—
178C4×C61C4×C4⋊C412yes—
179C4×C61C22×C4211yescharacteristic
180C4×C61C2×C4×Q811yescharacteristic
181Dic61—44no—
182Dic61C4×C4⋊C412yes—
183Dic61C2×#11711yescharacteristic
184Dic61C2×C4×Q811yescharacteristic
185C2×#427—611no—
186C22×#106—611no—
187C2×#405—611no—
188#2274—612no—
189#1776—614no—
190C4×#106—614no—
191#1736—612no—
192C2×C61—22no—
193C2×C61C2×C4×C4⋊C411yescharacteristic
194C2×C61#294912yes—
195D61—22no—
196D61#291212yes—
197D61#295212yes—
198C22×M16—611no—
199C23×C8—611no—
200C2×#117—611no—
201C2×#117—612no—
202C22×C4⋊C4—612no—
203#427—618no—
204C2×#107—612no—
205C2×#106—612no—
206C2×#106—611no—
207C2×#106—612no—
208C2×#106—612no—
209C2×#106—612no—
210C2×#106—612no—
211C2×#106—612no—
212C2×#106#692311yescharacteristic, complemented
213#405—618no—
214C2×#97—612no—
215C2×C4×C8—612no—
216C4×C4⋊C4—618no—
217C4×C4⋊C4—1224no—
218C4×C22⋊C4—1224no—
219#340—618no—
220C61#2045411yescharacteristic, complemented, Sylow
221C23×C4—611no—
222C23×C4—1222no—
223C2×M16—614no—
224C2×M16—1221no—
225C2×M16—1222no—
226C2×M16—1221no—
227C22×C8—611no—
228C22×C8—614no—
229C22×C8—614no—
230C22×C8—611no—
231C22×C8—611no—
232C22×C8—611no—
233C22×C8—611no—
234C22×C8—612no—
235C22×C8C2×#692311yescharacteristic
236#117—614no—
237#117—614no—
238#117—618no—
239#117—1224no—
240C2×C4⋊C4—612no—
241C2×C4⋊C4—614no—
242C2×C4⋊C4—612no—
243C2×C4⋊C4—614no—
244C2×C4⋊C4—614no—
245C2×C4⋊C4—1222no—
246C2×C4⋊C4—1222no—
247C2×C4⋊C4—1222no—
248C2×C4⋊C4C2×#692312yes—
249C2×C22⋊C4—612no—
250C2×C22⋊C4—1222no—
251C2×C42—612no—
252C2×C42—1222no—
253C2×C42—1228no—
254#107—618no—
255#106—618no—
256#106—612no—
257#106—612no—
258#106—614no—
259#106—614no—
260#106C2×#692312yes—
261#106C2×#692312yes—
262#97—618no—
263C4×C8—618no—
264#95—1228no—
265C24—611no—
266C22×C4—22no—
267C22×C4—611no—
268C22×C4—612no—
269C22×C4—612no—
270C22×C4—611no—
271C22×C4—611no—
272C22×C4—612no—
273C22×C4—612no—
274C22×C4—611no—
275C22×C4—611no—
276C22×C4—614no—
277C22×C4—1224no—
278C22×C4—1224no—
279C22×C4—1222no—
280C22×C4—1224no—
281C22×C4—4884no—
282C22×C4C22×#692311yescharacteristic
283M16—1224no—
284M16—1224no—
285C2×C8—614no—
286C2×C8—611no—
287C2×C8—611no—
288C2×C8—612no—
289C2×C8—612no—
290C2×C8—612no—
291C2×C8—612no—
292C2×C8—614no—
293C2×C8—614no—
294C2×C8—614no—
295C2×C8—1222no—
296C2×C8—1222no—
297C2×C8C4×#692314yescomplemented
298C2×C8C22×#692311yescharacteristic
299C2×C8C22×#692311yescharacteristic
300C4⋊C4—22no—
301C4⋊C4—618no—
302C4⋊C4—618no—
303C4⋊C4—614no—
304C4⋊C4—1224no—
305C4⋊C4—1224no—
306C4⋊C4—1224no—
307C4⋊C4—1222no—
308C4⋊C4C22×#692314yes—
309C22⋊C4—618no—
310C22⋊C4—1228no—
311C42—614no—
312C42—614no—
313C42—1228no—
314C42—4888no—
315C23—611no—
316C23—611no—
317C23—612no—
318C23—612no—
319C23—1224no—
320C23D4×#692311yescharacteristic
321C2×C4—24no—
322C2×C4—82no—
323C2×C4—611no—
324C2×C4—611no—
325C2×C4—612no—
326C2×C4—612no—
327C2×C4—612no—
328C2×C4—611no—
329C2×C4—611no—
330C2×C4—612no—
331C2×C4—611no—
332C2×C4—611no—
333C2×C4—612no—
334C2×C4—612no—
335C2×C4—612no—
336C2×C4—612no—
337C2×C4—618no—
338C2×C4—1228no—
339C2×C4—1228no—
340C2×C4—1224no—
341C2×C4—4884no—
342C2×C4—4882no—
343C2×C4—2448no—
344C2×C4—2444no—
345C2×C4—2444no—
346C2×C4C2×C4×#692312yes—
347C2×C4D4×#692311yescharacteristic
348C2×C4Q8×#692312yes—
349C2×C4C23×#692311yescharacteristic
350C8—612no—
351C8—612no—
352C8—1224no—
353C8C2×C4×#692312yes—
354C8C2×C4×#692312yes—
355C22—22no—
356C22—614no—
357C22—611no—
358C22—611no—
359C22—612no—
360C22—1224no—
361C22—1224no—
362C22—1222no—
363C22C4⋊C4×#692312yes—
364C22C2×D4×#692311yescharacteristic
365C4—44no—
366C4—611no—
367C4—611no—
368C4—612no—
369C4—1228no—
370C4—2448no—
371C4—2444no—
372C4C4⋊C4×#692312yes—
373C4C22×C4×#692311yescharacteristic
374C4C2×Q8×#692311yescharacteristic
375C2—22no—
376C2—612no—
377C2—612no—
378C2—1222no—
379C2C2×C4⋊C4×#692311yescharacteristic
380C2#2146257912yes—
subgroupG/H
centerC22C2×D4×#6923
commutatorC4×C61C22×C42
FrattiniC2×C4C23×#6923
FittingC2×#106×C61C4
socleC22×C61C2×C4×D4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenterC2×#106#6923
hypocenterC61#20454
LCSG ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
4. #21461635C2semidirect
5. #106×#6923C2semidirect
11. #16840772C4semidirect
14. #16840725C4semidirect
23. C2×#14519709C4semidirect
24. C2×#14519708C4semidirect
25. #106×D61C4semidirect
26. #106×D61C4semidirect
29. C2×#106×C61C4semidirect
46. #14519846C2×C4semidirect
47. #14519845C2×C4semidirect
48. #14519831C2×C4semidirect
49. C8×#6923C2×C4semidirect
106. C2×C8×C61C42semidirect
115. C2×#92418C42semidirect
118. C8×D61C42semidirect
119. C8×D61C42semidirect
212. C2×#106#6923semidirect
220. C61#20454semidirect
297. C2×C8C4×#6923semidirect

Maximal subgroups

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

subgroupindex
C2×#168407722
C2×#168407252
#214622572
#214616352
#106×#69232
#214615062
C2×#106×D612
#2045461