Finite Group Databasebeta

← #27154381

Subgroups — #27154381

Order 12416.

350 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 123 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#20261445C211yesmaximal, characteristic, complemented
2#20261089C212yesmaximal, complemented
3#20261085C211yesmaximal, characteristic, complemented
4#20261042C212yesmaximal, complemented
5#20261025C211yesmaximal, characteristic, complemented
6#20260947C212yesmaximal
7#20260937C211yesmaximal, characteristic, complemented
8Q8×#11715895C211yesmaximal, characteristic, complemented
9C4×#13095498C212yesmaximal, complemented
10C4×#13095496C211yesmaximal, characteristic
11#556×C97C211yesmaximal, characteristic, complemented
12C2×#13095498C2212yes—
13C2×#13095496C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690802C2212yes—
16#15690788—22no—
17#15690786—21no—
18C4×D388C2211yescharacteristic
19C4×D388C2212yes—
20C4⋊C4×D97—21no—
21#15690743C2211yescharacteristic
22#15690741C2212yes—
23C4⋊#11715895C2211yescharacteristic
24C4⋊#11715895C2212yes—
25#15690734C2212yes—
26#15690734C2212yes—
27#15690734C2212yes—
28#15690731C2212yes—
29#15690731C2211yescharacteristic
30#15690730C2211yescharacteristic
31C4×#11715895C2211yescharacteristic
32C4×#11715895C2212yes—
33C4⋊Q8×C97C2211yescharacteristic
34#126×C97—22no—
35#124×C97C2212yes—
36C22⋊Q8×C97—21no—
37C4×Q8×C97C2212yes—
38C4×Q8×C97C2211yescharacteristic
39C4×D4×C97C2211yescharacteristic
40#117×C97—21no—
41C2×Dic194—21no—
42C2×D388C2311yescharacteristic
43C2×C4×D97—21no—
44C2×C4×D97—22no—
45#13095498—24no—
46#13095496—22no—
47C2×#11715895C2311yescharacteristic
48C2×#11715895C2312yes—
49C2×#11715895C2311yescharacteristic
50C4⋊Dic97—21no—
51C4⋊Dic97C2311yescharacteristic
52C4⋊Dic97C2312yes—
53#13095490—22no—
54#13095490—22no—
55#13095488—22no—
56#13095488—21no—
57C2×Q8×C97D411yescharacteristic
58C2×Q8×C97C2312yes—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C4⋊C4×C97—21no—
62C4⋊C4×C97—22no—
63C4⋊C4×C97—21no—
64C4⋊C4×C97D411yescharacteristic
65C4⋊C4×C97D412yes—
66C4⋊C4×C97C2311yescharacteristic
67C22⋊C4×C97—22no—
68C22⋊C4×C97—21no—
69C22⋊C4×C97—22no—
70C42×C97—22no—
71C42×C97C2312yes—
72C42×C97C2311yescharacteristic
73C23×C97—21no—
74C22×D97—21no—
75Q8×C97—22no—
76Q8×C97—22no—
77Q8×C97—22no—
78D4×C97—21no—
79D4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—22no—
82C2×C4×C97—42no—
83C2×C4×C97C2×D412yes—
84C2×C4×C97C2×D411yescharacteristic
85C2×C4×C97C2×D412yes—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2411yescharacteristic
88C2×Dic97—21no—
89C2×Dic97—22no—
90D388—21no—
91D388#4112yes—
92C4×D97—42no—
93C4×D97—82no—
94Dic194—81no—
95#11715895—21no—
96#11715895—22no—
97#11715895—21no—
98C22×C97—41no—
99C22×C97—41no—
100C22×C97—41no—
101C22×C97C22×D411yescharacteristic
102C2×D97—22no—
103C2×D97—41no—
104C2×D97—41no—
105C4×C97—22no—
106C4×C97—21no—
107C4×C97—22no—
108C4×C97—42no—
109C4×C97—41no—
110C4×C97#13512yes—
111C4×C97C2×#4111yescharacteristic
112C4×C972−1+411yescharacteristic
113Dic97—41no—
114Dic97—42no—
115C2×C97—81no—
116C2×C97#54811yescharacteristic
117C2×C97C2×#13511yescharacteristic
118C2×C97#57711yescharacteristic
119D97—42no—
120#3360—971nomaximal, Sylow
121C97#336011yescharacteristic, complemented, Sylow
122#556D9711yescharacteristic, complemented
123#554—971no—
124#488—972no—
125#487—971no—
126#479—972no—
127#476—971no—
128#463—972no—
129#461—971no—
130C8×Q8—971no—
131C4×SD16—972no—
132C4×D8—971no—
133C2×SD16—972no—
134C2×D8—971no—
135C4⋊Q8—971no—
136C4⋊Q8C2×D9711yescharacteristic
137#126—22no—
138#125—972no—
139#124C2×D9712yes—
140C22⋊Q8—21no—
141C22⋊Q8—1941no—
142C22⋊Q8—1942no—
143C4×Q8C2×D9712yes—
144C4×Q8C2×D9711yescharacteristic
145C4×D4—971no—
146C4×D4—972no—
147C4×D4C2×D9711yescharacteristic
148#117—21no—
149C2×C4⋊C4—1941no—
150#107—971no—
151#106—972no—
152C4⋊C8—971no—
153C4⋊C8—972no—
154#103—972no—
155#103—972no—
156#103—972no—
157#102—971no—
158#102—972no—
159#102—971no—
160C4×C8—971no—
161C4×C8—972no—
162C2×Q8—1941no—
163C2×Q8C22⋊D9711yescharacteristic
164C2×Q8C22×D9712yes—
165C2×D4—971no—
166C2×D4C22×D9711yescharacteristic
167C22×C4—21no—
168C22×C4—1942no—
169C22×C4—1941no—
170SD16—1944no—
171D8—1942no—
172C2×C8—971no—
173C2×C8—972no—
174C2×C8—971no—
175C4⋊C4—21no—
176C4⋊C4—22no—
177C4⋊C4—21no—
178C4⋊C4—972no—
179C4⋊C4—971no—
180C4⋊C4—1942no—
181C4⋊C4—1942no—
182C4⋊C4—1941no—
183C4⋊C4C22⋊D9711yescharacteristic
184C4⋊C4C22⋊D9712yes—
185C4⋊C4C22×D9711yescharacteristic
186C22⋊C4—22no—
187C22⋊C4—21no—
188C22⋊C4—22no—
189C22⋊C4—1941no—
190C22⋊C4—1942no—
191C42—22no—
192C42C22×D9711yescharacteristic
193C42C22×D9712yes—
194C23—21no—
195C23—1941no—
196Q8—22no—
197Q8—22no—
198Q8—22no—
199Q8—7761no—
200D4—21no—
201D4—21no—
202D4—972no—
203D4—1941no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—7762no—
208C2×C4—3882no—
209C2×C4—1941no—
210C2×C4—1942no—
211C2×C4C2×C22⋊D9712yes—
212C2×C4C2×C22⋊D9712yes—
213C2×C4C2×C22⋊D9711yescharacteristic
214C2×C4C2×C22⋊D9711yescharacteristic
215C2×C4C23×D9711yescharacteristic
216C8—1941no—
217C8—1942no—
218C8—1941no—
219C22—41no—
220C22—41no—
221C22—41no—
222C22—3881no—
223C22—3881no—
224C22—1942no—
225C22C22×C22⋊D9711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—42no—
230C4—41no—
231C4—3881no—
232C4—3882no—
233C4#1569083012yes—
234C4#41×D9711yescharacteristic
235C4#1569085711yescharacteristic
236C2—81no—
237C2—3882no—
238C2#2026139211yescharacteristic
239C2C2×#1569083011yescharacteristic
240C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#556×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095496C22
hypercenter#556D97
hypocenterC97#3360
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20261445C2semidirect
2. #20261089C2semidirect
3. #20261085C2semidirect
4. #20261042C2semidirect
5. #20261025C2semidirect
7. #20260937C2semidirect
8. Q8×#11715895C2semidirect
9. C4×#13095498C2semidirect
11. #556×C97C2semidirect
121. C97#3360semidirect
122. #556D97semidirect

Maximal subgroups

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

subgroupindex
#202614452
#202610892
#202610852
#202610422
#202610252
#202609472
#202609372
Q8×#117158952
C4×#130954982
C4×#130954962
#556×C972
#336097