Finite Group Databasebeta

← #30599752

Subgroups — #30599752

Order 15616.

428 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462365C211yesmaximal, characteristic
2#21462002C211yesmaximal, characteristic
3#21461757C212yesmaximal
4#21460696C212yesmaximal
5#330×D61C211yesmaximal, characteristic
6C2×#14519846—22no—
7C2×#14519845—22no—
8#16840721C2211yescharacteristic
9#16840713—22no—
10#16840711—22no—
11#16840711—22no—
12#16840708C2211yescharacteristic
13#16840706C2212yes—
14#16840635—22no—
15#125×D61C2211yescharacteristic
16#97×D61C2212yes—
17#16839504C411yescharacteristic, complemented
18#16839503C412yescomplemented
19C61×#330C411yescharacteristic, complemented
20C2×D61⋊C8—21no—
21C2×C4⋊#6923—22no—
22C2×C4⋊#6923—21no—
23C2×C4⋊#6923—22no—
24C2×C4⋊#6923D412yes—
25C2×C4⋊#6923D412yes—
26C2×C4×#6923—22no—
27#14519846—44no—
28#14519845—44no—
29#14519832—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—22no—
32(C2×#6923)⋊C4—22no—
33(C2×#6923)⋊C4—22no—
34C2×C8×D61—22no—
35#14519769C2×C411yescharacteristic
36#14519768Q812yes—
37#14519767—22no—
38C4⋊C4×D61—22no—
39C4⋊C4×D61D412yes—
40C42×D61C2311yescharacteristic
41#14519688C2×C412yes—
42#14519687—22no—
43#125×C61C2×C411yescharacteristic
44#97×C61C2×C412yes—
45C22×#6923—22no—
46C22×#6923—22no—
47C22×#6923—41no—
48C2×C4×D61—22no—
49C2×C4×D61C2×D412yes—
50C2×C4×D61C2×D411yescharacteristic
51C2×#92420—41no—
52C4⋊#6923—42no—
53C4⋊#6923—42no—
54C4⋊#6923—42no—
55C4⋊#6923—42no—
56C4⋊#6923—42no—
57C4⋊#6923—82no—
58C4×#6923—82no—
59D61⋊C8—22no—
60D61⋊C8—41no—
61C2×C8×C61—22no—
62C4⋊C4×C61—22no—
63C4⋊C4×C61C4⋊C412yes—
64C42×C61C22×C411yescharacteristic
65C4⋊Dic61—22no—
66C4⋊Dic61C4⋊C412yes—
67#11758196—22no—
68#11758196—22no—
69#11758196—22no—
70C4×Dic61C2×Q811yescharacteristic
71C4×Dic61#4112yes—
72C2×#92418—22no—
73C8×D61—24no—
74C8×D61—24no—
75C22×D61#12011yescharacteristic
76C2×#6923—42no—
77C2×#6923—42no—
78C2×#6923—42no—
79C2×#6923—42no—
80C2×#6923—42no—
81C2×#6923—42no—
82C2×#6923—82no—
83C2×#6923—81no—
84C2×C4×C61—22no—
85C2×C4×C61C2×C4⋊C411yescharacteristic
86C2×C4×C61C4×D412yes—
87C2×Dic61—22no—
88C2×Dic61C22⋊Q812yes—
89C2×Dic61#12311yescharacteristic
90C4×D61—22no—
91C4×D61—21no—
92C4×D61—22no—
93C4×D61—21no—
94C4×D61—42no—
95C4×D61—42no—
96#92420—42no—
97C8×C61—24no—
98#92418—24no—
99#8913—611nomaximal, Sylow
100C22×C61#38511yescharacteristic
101C2×D61#45312yes—
102C2×D61#45411yescharacteristic
103C2×D61#45611yescharacteristic
104C2×D61#45612yes—
105#6923—82no—
106#6923—82no—
107#6923—161no—
108C4×C61—21no—
109C4×C61—22no—
110C4×C61—42no—
111Dic61—21no—
112Dic61—22no—
113Dic61—42no—
114#2382—611no—
115#1992—611no—
116#1847—612no—
117C2×#330—611no—
118#1295—612no—
119C2×C61#185711yescharacteristic
120C2×C61#185912yes—
121D61#165611yescharacteristic
122D61#166012yes—
123D61#166311yescharacteristic
124C2×#125—611no—
125C2×#107—1222no—
126C2×#106—1222no—
127C2×#97—612no—
128C2×#97—611no—
129#389—1222no—
130#387—1222no—
131#387—1222no—
132#384—611no—
133#382—612no—
134#340—1222no—
135#330—611no—
136#330—612no—
137#330#692311yescharacteristic, complemented
138C61#891311yescharacteristic, complemented, Sylow
139C22×C8—1222no—
140C22×C8—1221no—
141#125—612no—
142#125—611no—
143#125—1222no—
144#125C2×#692311yescharacteristic
145C2×C4⋊C4—612no—
146C2×C4⋊C4—612no—
147C2×C4⋊C4—612no—
148C2×C4⋊C4—1222no—
149C2×C4⋊C4—1222no—
150C2×C4⋊C4—1221no—
151C2×C4⋊C4—1222no—
152C2×C42—611no—
153C2×C42—1222no—
154#107—2444no—
155#106—2444no—
156#97—612no—
157#97—1222no—
158#97—1222no—
159#97C2×#692312yes—
160#95—1222no—
161#95—1222no—
162#95—1222no—
163#95—1222no—
164C22×C4—611no—
165C22×C4—612no—
166C22×C4—1222no—
167C22×C4—1222no—
168C22×C4—1222no—
169C22×C4—2441no—
170C2×C8—22no—
171C2×C8—1224no—
172C2×C8—1224no—
173C2×C8—1222no—
174C2×C8—1222no—
175C2×C8—2441no—
176C2×C8—2441no—
177C4⋊C4—22no—
178C4⋊C4—612no—
179C4⋊C4—1222no—
180C4⋊C4—1222no—
181C4⋊C4—1222no—
182C4⋊C4—1222no—
183C4⋊C4—4882no—
184C4⋊C4—2442no—
185C4⋊C4—2442no—
186C4⋊C4—2442no—
187C4⋊C4—2442no—
188C4⋊C4—2442no—
189C4⋊C4C4⋊#692312yes—
190C42—611no—
191C42—612no—
192C42—4882no—
193C42C22×#692311yescharacteristic
194C23—611no—
195C2×C4—22no—
196C2×C4—612no—
197C2×C4—611no—
198C2×C4—1222no—
199C2×C4—1221no—
200C2×C4—1221no—
201C2×C4—1222no—
202C2×C4—1222no—
203C2×C4—4881no—
204C2×C4—4882no—
205C2×C4—2442no—
206C2×C4—2442no—
207C2×C4—2442no—
208C2×C4—2442no—
209C2×C4—2442no—
210C2×C4—2442no—
211C2×C4—2442no—
212C2×C4—2442no—
213C2×C4C2×C4⋊#692311yescharacteristic
214C2×C4D4×#692312yes—
215C8—24no—
216C8—1224no—
217C8—2442no—
218C22—611no—
219C22—611no—
220C22—612no—
221C22—612no—
222C22#1684070911yescharacteristic
223C4—21no—
224C4—22no—
225C4—42no—
226C4—1221no—
227C4—1222no—
228C4—9761no—
229C4—4882no—
230C4—4882no—
231C4—2442no—
232C2—611no—
233C2—612no—
234C2—611no—
235C2#2146177511yescharacteristic
236C2#2146177812yes—
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#330C4
socleC22×C61#385
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#330#6923
hypocenterC61#8913
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
17. #16839504C4semidirect
18. #16839503C4semidirect
19. C61×#330C4semidirect
137. #330#6923semidirect
138. C61#8913semidirect

Maximal subgroups

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

subgroupindex
#214623652
#214620022
#214617572
#214606962
#330×D612
#891361