Finite Group Databasebeta

← #29574957

Subgroups — #29574957

Order 15616.

458 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16840132C211yesmaximal, characteristic
2#21446791C212yesmaximal
3C2×#16839504C211yesmaximal, characteristic
4#21443700C212yesmaximal
5C61×#2382C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7#16840132—22no—
8#16840132C414yescomplemented
9C2×#14519700—22no—
10C2×#14519688C2212yes—
11C2×#14519688C2211yescharacteristic
12#16839524—22no—
13#16839524—22no—
14#16839504—22no—
15C2×#125×C61C2211yescharacteristic
16C61×#389—22no—
17C61×#387—22no—
18C61×#387—22no—
19C61×#382C2212yes—
20C22×Dic122—21no—
21C22×#92418—21no—
22C22×#92418—22no—
23#14519777—22no—
24#14519777—21no—
25#14519777C2×C412yes—
26C2×C4⋊Dic61—22no—
27#14519700—28no—
28#14519688—21no—
29#14519688—22no—
30#14519688—22no—
31#14519688C2×C412yes—
32#125×C61—22no—
33#125×C61C2×C412yes—
34#125×C61D412yes—
35C2×C4⋊C4×C61—22no—
36C2×C4⋊C4×C61—22no—
37C2×C4⋊C4×C61D412yes—
38C2×C4⋊C4×C61D412yes—
39C2×C4⋊C4×C61D412yes—
40C2×C42×C61—22no—
41C2×C42×C61C2311yescharacteristic
42#95×C61—22no—
43#95×C61—22no—
44#95×C61—22no—
45#95×C61—22no—
46C22×C4×C61—22no—
47C22×C4×C61—22no—
48C22×C4×C61—22no—
49C22×C4×C61C2×D411yescharacteristic
50C22×C4×C61C2×D412yes—
51C22×Dic61—41no—
52C2×Dic122—22no—
53C2×Dic122—42no—
54C2×Dic122—42no—
55C2×Dic122—41no—
56C4⋊C4×C61—24no—
57C4⋊C4×C61—24no—
58C4⋊C4×C61—22no—
59C4⋊C4×C61—42no—
60C4⋊C4×C61—42no—
61C4⋊C4×C61—82no—
62C4⋊C4×C61C22⋊C414yes—
63C42×C61—82no—
64C42×C61C22×C411yescharacteristic
65C42×C61C2×D411yescharacteristic
66C42×C61#4112yes—
67C4⋊Dic61—24no—
68C4⋊Dic61—42no—
69C2×#92418—24no—
70C2×#92418—22no—
71C2×#92418—41no—
72C2×#92418—41no—
73C2×#92418—42no—
74C2×#92418—42no—
75C23×C61#12011yescharacteristic
76C2×C4×C61—21no—
77C2×C4×C61—21no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—24no—
81C2×C4×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61—42no—
84C2×C4×C61—42no—
85C2×C4×C61—42no—
86C2×C4×C61—42no—
87C2×C4×C61—42no—
88C2×C4×C61—42no—
89C2×C4×C61C2×C22⋊C411yescharacteristic
90C2×C4×C61C4×D412yes—
91C2×C4×C61#12112yes—
92C2×C4×C61#12311yescharacteristic
93C2×Dic61—42no—
94C2×Dic61—81no—
95C2×Dic61—81no—
96Dic122—44no—
97Dic122—82no—
98#92418—44no—
99#92418—42no—
100#9004—611nomaximal, Sylow
101C22×C61#38611yescharacteristic
102C22×C61#45411yescharacteristic
103C22×C61#45411yescharacteristic
104C22×C61#45512yes—
105C22×C61#45612yes—
106C4×C61—22no—
107C4×C61—22no—
108C4×C61—21no—
109C4×C61—21no—
110C4×C61—44no—
111C4×C61—82no—
112C4×C61—82no—
113Dic61—82no—
114C2×#491—611no—
115#2382D6111yescharacteristic
116#1993—612no—
117C2×#330—611no—
118#1295—612no—
119C2×C61#166211yescharacteristic
120C2×C61#166311yescharacteristic
121C2×C61#166412yes—
122C2×C61#188212yes—
123C2×C61#188811yescharacteristic
124C2×C4⋊Q8—611no—
125C2×#125C2×D6111yescharacteristic
126#491—614no—
127#491—1222no—
128C2×#103—1222no—
129C2×#97—611no—
130C2×#97—612no—
131#389—22no—
132#387—22no—
133#387—22no—
134#382C2×D6112yes—
135#340—1222no—
136#340—1222no—
137#330—1222no—
138C61#900411yescharacteristic, complemented, Sylow
139C22×Q8—1221no—
140C22×C8—1222no—
141C22×C8—1221no—
142C4⋊Q8—612no—
143C4⋊Q8—1221no—
144C4⋊Q8—1222no—
145#125—22no—
146#125C4×D6112yes—
147#125D24412yes—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—1222no—
151C2×C4⋊C4C22⋊D6112yes—
152C2×C4⋊C4C22⋊D6112yes—
153C2×C4⋊C4C22⋊D6112yes—
154C2×C42—22no—
155C2×C42C22×D6111yescharacteristic
156#103—1228no—
157#97—612no—
158#97—1221no—
159#97—1222no—
160#97—1222no—
161#95—22no—
162#95—22no—
163#95—22no—
164#95—22no—
165C2×Q8—1222no—
166C2×Q8—2442no—
167C2×Q8—2442no—
168C2×Q8—2441no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—2441no—
173C22×C4C2×C22⋊D6111yescharacteristic
174C22×C4C2×C22⋊D6112yes—
175C2×C8—1224no—
176C2×C8—1222no—
177C2×C8—2442no—
178C2×C8—2441no—
179C2×C8—2441no—
180C2×C8—2442no—
181C4⋊C4—24no—
182C4⋊C4—24no—
183C4⋊C4—22no—
184C4⋊C4—42no—
185C4⋊C4—42no—
186C4⋊C4—82no—
187C4⋊C4—1224no—
188C4⋊C4—2442no—
189C4⋊C4#1175819814yes—
190C42—82no—
191C42C2×C4×D6111yescharacteristic
192C42C2×D24411yescharacteristic
193C42#1175822212yes—
194C23#1451974611yescharacteristic
195Q8—4882no—
196Q8—2444no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—24no—
202C2×C4—42no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—4881no—
211C2×C4—4881no—
212C2×C4—2442no—
213C2×C4C2×#1175819811yescharacteristic
214C2×C4C4×C22⋊D6112yes—
215C2×C4#1451975112yes—
216C2×C4#1451976011yescharacteristic
217C8—2444no—
218C8—2442no—
219C22#1683964111yescharacteristic
220C22#1683993511yescharacteristic
221C22#1683993511yescharacteristic
222C22#1683994012yes—
223C22#1683994512yes—
224C4—22no—
225C4—22no—
226C4—21no—
227C4—21no—
228C4—44no—
229C4—82no—
230C4—82no—
231C4—4882no—
232C2#2144514211yescharacteristic
233C2#2144514711yescharacteristic
234C2#2144515312yes—
235C2#2144618812yes—
236C2#2144622211yescharacteristic
subgroupG/H
centerC23#14519746
commutatorC42×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2382C2
socle, prime radicalC23×C61#120
cosocleC2×C42×C61C23
hypercenter#2382D61
hypocenterC61#9004
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2382

Complements

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

NH ≅ G/Ntype
8. #16840132C4semidirect
138. C61#9004semidirect

Maximal subgroups

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

subgroupindex
C2×#168401322
#214467912
C2×#168395042
#214437002
C61×#23822
#900461