Finite Group Databasebeta

← #30600285

Subgroups — #30600285

Order 15616.

602 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462681C211yesmaximal, characteristic, complemented
2#21461118C212yesmaximal, complemented
3#21461059C211yesmaximal, characteristic, complemented
4#21460643C212yesmaximal, complemented
5#21448545C211yesmaximal, characteristic
6#16840797—24no—
7#16840778C2212yes—
8#16840778C2212yes—
9#16840732—22no—
10D4⋊#92420—24no—
11#16840605—22no—
12C2×D4×Dic61C2211yescharacteristic
13C2×D4×Dic61C2211yescharacteristic
14#16839665C411yescharacteristic
15#16839665C411yescharacteristic
16#16839642—22no—
17#16839642—22no—
18#16839628C411yescharacteristic
19#16839608C2211yescharacteristic
20C2×#127×C61C411yescharacteristic
21C2×#11758217—24no—
22C4⋊#92420—22no—
23C4⋊#92420—22no—
24C22⋊#92420—24no—
25C22⋊#92420—22no—
26C22⋊#92420—22no—
27C4×#92420—22no—
28C4×#92420D414yescomplemented
29C23×Dic61—21no—
30C23×Dic61D412yes—
31D4×Dic61—22no—
32D4×Dic61—22no—
33D4×Dic61—22no—
34D4×Dic61—22no—
35C2×C4⋊Dic61C2×C411yescharacteristic
36C2×C4⋊Dic61C2×C411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61—22no—
39C2×C22⋊Dic61—22no—
40C2×C22⋊Dic61D412yes—
41C2×C4×Dic61C2×C411yescharacteristic
42C2×C4×Dic61C2311yescharacteristic
43#14519684—22no—
44#14519684—21no—
45#14519684—21no—
46C22×D4×C61—22no—
47C22×D4×C61C2×C411yescharacteristic
48C22×D4×C61C2×C411yescharacteristic
49#127×C61—22no—
50#127×C61—22no—
51C2×C42×C61C2×C411yescharacteristic
52C24×C61—21no—
53C24×C61C22⋊C412yes—
54C2×D4×C61—28no—
55C2×D4×C61—22no—
56C2×D4×C61—22no—
57C2×D4×C61—21no—
58C2×D4×C61—21no—
59C2×D4×C61—21no—
60C2×D4×C61—22no—
61C2×D4×C61—22no—
62C2×D4×C61—82no—
63C2×D4×C61—82no—
64C2×D4×C61M1612yescomplemented
65C22×C4×C61—21no—
66C22×C4×C61C22×C411yescharacteristic
67C22×Dic61—22no—
68C22×Dic61—22no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C22×Dic61—42no—
72C22×Dic61—82no—
73C22×Dic61—82no—
74C22×Dic61C22⋊C412yes—
75C22×Dic61C2×D411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77#11758217—84no—
78C2×#92420—24no—
79C2×#92420—42no—
80C42×C61—21no—
81C42×C61M1612yescomplemented
82C22⋊Dic61—24no—
83C22⋊Dic61—41no—
84C22⋊Dic61—41no—
85C22⋊Dic61—82no—
86C22⋊Dic61—82no—
87C4⋊Dic61—22no—
88C4⋊Dic61—22no—
89C4×Dic61—22no—
90C4×Dic61C2×D412yes—
91C4×Dic61#4112yes—
92C23×C61—22no—
93C23×C61—22no—
94C23×C61—22no—
95C23×C61—22no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61—41no—
101C23×C61—41no—
102C23×C61C2×C22⋊C411yescharacteristic
103D4×C61—24no—
104D4×C61—42no—
105D4×C61—42no—
106D4×C61—42no—
107D4×C61—88no—
108C2×C4×C61—22no—
109C2×C4×C61—41no—
110C2×C4×C61—41no—
111C2×C4×C61C4×D412yes—
112C2×C4×C61C2×M1611yescharacteristic
113C2×C4×C61C2×M1611yescharacteristic
114C2×C4×C61#13112yes—
115C2×Dic61—22no—
116C2×Dic61—41no—
117C2×Dic61—41no—
118C2×Dic61—44no—
119C2×Dic61—42no—
120C2×Dic61—41no—
121C2×Dic61—41no—
122C2×Dic61—42no—
123C2×Dic61—41no—
124C2×Dic61—41no—
125C2×Dic61—82no—
126C2×Dic61—84no—
127C2×Dic61#12011yescharacteristic
128C2×Dic61#12112yes—
129C2×Dic61#12311yescharacteristic
130#92420—44no—
131#92420—82no—
132#9105—611nomaximal, Sylow
133C22×C61—21no—
134C22×C61—21no—
135C22×C61—44no—
136C22×C61—44no—
137C22×C61—42no—
138C22×C61—42no—
139C22×C61—42no—
140C22×C61—42no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—81no—
144C22×C61—81no—
145C22×C61#38611yescharacteristic
146C22×C61#40711yescharacteristic
147C22×C61#40811yescharacteristic
148C4×C61—42no—
149C4×C61#43512yes—
150C4×C61#43612yes—
151Dic61—22no—
152Dic61—21no—
153Dic61—21no—
154Dic61—42no—
155Dic61—81no—
156Dic61—81no—
157#2975—611no—
158#2328—611no—
159#1478—612no—
160#1460—611no—
161#1274—612no—
162C2×C61—21no—
163C2×C61—21no—
164C2×C61—82no—
165C2×C61—82no—
166C2×C61#187011yescharacteristic
167C2×C61#191011yescharacteristic
168C2×C61#191111yescharacteristic
169C2×#127#692311yescharacteristic
170C2×C4×D4—611no—
171C2×C4×D4—611no—
172#436—1224no—
173#431—612no—
174#431—612no—
175#407—1222no—
176#390—611no—
177#390—611no—
178#386—1222no—
179#386—1222no—
180#382—611no—
181#376—611no—
182D4⋊C8—1224no—
183#323—1222no—
184C61#910511yescharacteristic, complemented, Sylow
185C22×D4—22no—
186C22×D4C2×#692311yescharacteristic
187C22×D4C2×#692311yescharacteristic
188C23×C4—612no—
189C23×C4—1221no—
190C2×M16—1224no—
191#127—22no—
192#127—22no—
193C4×D4—1222no—
194C4×D4—1222no—
195C4×D4—1222no—
196C4×D4—1222no—
197C2×C4⋊C4—611no—
198C2×C4⋊C4—611no—
199C2×C22⋊C4—612no—
200C2×C22⋊C4—1222no—
201C2×C22⋊C4—1222no—
202C2×C22⋊C4—1221no—
203C2×C42—611no—
204C2×C42—611no—
205C2×C42C2×#692311yescharacteristic
206C4⋊C8—1222no—
207C4⋊C8—1222no—
208C22⋊C8—1222no—
209C22⋊C8—1224no—
210C22⋊C8—1222no—
211C4×C8—614no—
212C4×C8—1222no—
213#95—1221no—
214#95—1221no—
215#95—1222no—
216C24—21no—
217C24C22⋊#692312yes—
218C2×D4—28no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—82no—
227C2×D4—82no—
228C2×D4#1175821712yescomplemented
229C22×C4—21no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—612no—
233C22×C4—1222no—
234C22×C4—1222no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—4882no—
238C22×C4—4882no—
239C22×C4—2442no—
240C22×C4C22×#692311yescharacteristic
241M16—4884no—
242C2×C8—1224no—
243C2×C8—2442no—
244C4⋊C4—1222no—
245C4⋊C4—1222no—
246C22⋊C4—1224no—
247C22⋊C4—4882no—
248C22⋊C4—4882no—
249C22⋊C4—2441no—
250C22⋊C4—2441no—
251C42—21no—
252C42—612no—
253C42—612no—
254C42—1222no—
255C42#1175821712yescomplemented
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—42no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—41no—
265C23—41no—
266C23C2×C22⋊#692311yescharacteristic
267D4—24no—
268D4—42no—
269D4—42no—
270D4—42no—
271D4—88no—
272C2×C4—22no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—611no—
276C2×C4—612no—
277C2×C4—611no—
278C2×C4—1222no—
279C2×C4—4884no—
280C2×C4—4882no—
281C2×C4—2441no—
282C2×C4—2441no—
283C2×C4—2444no—
284C2×C4—2441no—
285C2×C4—2441no—
286C2×C4—2442no—
287C2×C4—2441no—
288C2×C4—2441no—
289C2×C4—2442no—
290C2×C4D4×#692312yes—
291C2×C4C2×#1175821711yescharacteristic
292C2×C4C2×#1175821711yescharacteristic
293C2×C4#1451986412yes—
294C8—4882no—
295C8—2444no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—44no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—82no—
305C22—82no—
306C22—81no—
307C22—81no—
308C22#1684071011yescharacteristic
309C22#1684073211yescharacteristic
310C22#1684073511yescharacteristic
311C4—42no—
312C4—1222no—
313C4—1221no—
314C4—1221no—
315C4—4881no—
316C4—4881no—
317C4—2442no—
318C4#1684079312yes—
319C4#1684079712yes—
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#2146179611yescharacteristic
325C2#2146186211yescharacteristic
326C2#2146186511yescharacteristic
subgroupG/H
centerC22#16840710
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
Fitting, prime radicalC2×#127×C61C4
socleC22×C61#386
cosocleC2×C4×Dic61C23
hypercenterC2×#127#6923
hypocenterC61#9105
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21462681C2semidirect
2. #21461118C2semidirect
3. #21461059C2semidirect
4. #21460643C2semidirect
28. C4×#92420D4semidirect
64. C2×D4×C61M16semidirect
81. C42×C61M16semidirect
184. C61#9105semidirect
228. C2×D4#11758217semidirect
255. C42#11758217semidirect

Maximal subgroups

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

subgroupindex
#214626812
#214611182
#214610592
#214606432
#214485452
#910561