Finite Group Databasebeta

← #27142596

Subgroups — #27142596

Order 12416.

312 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C22×#13095501C211yesmaximal, characteristic
2C2×#15690743C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic
4C2×#15690733C211yesmaximal, characteristic
5C2×C4×#11715895C211yesmaximal, characteristic
6#20260641C211yesmaximal, characteristic
7#391×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095501—22no—
10C2×#13095501—22no—
11C2×#13095501C414yescomplemented
12C22×#11715895C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14C2×C4⋊Dic97C2211yescharacteristic
15#15690743—22no—
16#15690734—22no—
17#15690733—22no—
18C4×#11715895—22no—
19#15690717—22no—
20C22×Q8×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×C42×C97C2211yescharacteristic
23#95×C97—22no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194C2×C412yes—
30C22×Dic97—21no—
31C22×Dic97—21no—
32#13095501—28no—
33C2×#11715895—21no—
34C2×#11715895—21no—
35C2×#11715895C2×C412yes—
36C2×#11715895D412yes—
37C2×#11715895D412yes—
38C2×#11715895D412yes—
39C4⋊Dic97—21no—
40C4⋊Dic97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97C2×C412yes—
47C22×C4×C97—21no—
48C22×C4×C97—21no—
49C22×C4×C97D412yes—
50C22×C4×C97C2311yescharacteristic
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C42×C97—21no—
54C42×C97—21no—
55C23×C97C2×D411yescharacteristic
56Q8×C97—22no—
57Q8×C97—22no—
58Q8×C97—41no—
59Q8×C97—41no—
60C2×C4×C97—22no—
61C2×C4×C97—22no—
62C2×C4×C97—22no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97—41no—
66C2×C4×C97—42no—
67C2×C4×C97Q1614yescomplemented
68C2×C4×C97C22×C411yescharacteristic
69C2×C4×C97C2×D411yescharacteristic
70C2×C4×C97C2×D411yescharacteristic
71C2×C4×C97C2×D411yescharacteristic
72C2×C4×C97#4111yescharacteristic
73C2×C4×C97#4111yescharacteristic
74C2×Dic97—22no—
75C2×Dic97—41no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—42no—
79Dic194—22no—
80Dic194—22no—
81Dic194—41no—
82Dic194—41no—
83#11715895—22no—
84#11715895C22⋊C414yes—
85C22×C97C4×D411yescharacteristic
86C22×C97#12111yescharacteristic
87C22×C97#12711yescharacteristic
88C22×C97C2×Q1612yes—
89C22×C97#13512yes—
90C4×C97—24no—
91C4×C97—42no—
92C4×C97—81no—
93C4×C97C2×C22⋊C411yescharacteristic
94C4×C97C4×D411yescharacteristic
95C4×C97#12111yescharacteristic
96C4×C97#12411yescharacteristic
97Dic97—42no—
98Dic97—81no—
99C2×C97#39011yescharacteristic
100C2×C97C4×Q1612yes—
101C2×C97#46712yes—
102C2×C97#49411yescharacteristic
103C2×C97#49511yescharacteristic
104#1969—971nomaximal, Sylow
105C97#196911yescharacteristic, complemented, Sylow
106C22×Q16—971no—
107C2×#107—971no—
108C2×#103—971no—
109C2×#103—971no—
110C2×C4×C8—971no—
111#391—971no—
112#391D9711yescharacteristic
113C22×Q8—971no—
114C22×Q8C2×D9711yescharacteristic
115C2×Q16—974no—
116C2×Q16—1942no—
117C2×Q16—1942no—
118C22×C8—971no—
119C22×C8—971no—
120C2×C4⋊C4—971no—
121C2×C4⋊C4C2×D9711yescharacteristic
122C2×C42C2×D9711yescharacteristic
123#107—1942no—
124#103—1942no—
125#103—1942no—
126C4×C8—1942no—
127#95—22no—
128#95—1942no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—972no—
135C2×Q8—1941no—
136C2×Q8—1941no—
137C2×Q8—1941no—
138C2×Q8—1941no—
139C2×Q8—1941no—
140C2×Q8C4×D9712yes—
141C22×C4—21no—
142C22×C4—21no—
143C22×C4—1941no—
144C22×C4—1941no—
145C22×C4C22⋊D9712yes—
146C22×C4C22×D9711yescharacteristic
147Q16—1948no—
148C2×C8—972no—
149C2×C8—972no—
150C2×C8—972no—
151C2×C8—972no—
152C2×C8—1941no—
153C2×C8—1941no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1941no—
157C4⋊C4—1941no—
158C42—21no—
159C42—21no—
160C23C2×C22⋊D9711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165Q8—3881no—
166Q8—3881no—
167Q8—1942no—
168Q8—1942no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—42no—
176C2×C4—3881no—
177C2×C4—3882no—
178C2×C4—3881no—
179C2×C4—3881no—
180C2×C4—1942no—
181C2×C4#1309550114yescomplemented
182C2×C4C2×C4×D9711yescharacteristic
183C2×C4D4×D9711yescharacteristic
184C2×C4D4×D9711yescharacteristic
185C2×C4D4×D9711yescharacteristic
186C2×C4#1309551011yescharacteristic
187C2×C4#1309551111yescharacteristic
188C8—974no—
189C8—1942no—
190C22C4×C22⋊D9711yescharacteristic
191C22#1569078211yescharacteristic
192C22#1569080611yescharacteristic
193C22C2×#1309550112yes—
194C22#1569083012yes—
195C4—24no—
196C4—42no—
197C4—81no—
198C4—7761no—
199C4—3882no—
200C4C22⋊C4×D9711yescharacteristic
201C4#1569077011yescharacteristic
202C4#1569078311yescharacteristic
203C4#1569079611yescharacteristic
204C2#2026063511yescharacteristic
205C2C4×#1309550112yes—
206C2#2026097412yes—
207C2#2026111411yescharacteristic
208C2#2026111711yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#391×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#391D97
hypocenterC97#1969
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
11. C2×#13095501C4semidirect
67. C2×C4×C97Q16semidirect
105. C97#1969semidirect
181. C2×C4#13095501semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×#156907432
C2×#156907342
C2×#156907332
C2×C4×#117158952
#202606412
#391×C972
#196997