Finite Group Databasebeta

← #27154385

Subgroups — #27154385

Order 12416.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#20261458C211yesmaximal, characteristic, complemented
2#20261092C212yesmaximal, complemented
3#20261087C211yesmaximal, characteristic, complemented
4#20261039C212yesmaximal, complemented
5#20261022C211yesmaximal, characteristic, complemented
6#20260950C212yesmaximal
7#20260940C211yesmaximal, characteristic, complemented
8Q8×#11715895C211yesmaximal, characteristic, complemented
9C4×#13095499C212yesmaximal, complemented
10C4×#13095496C211yesmaximal, characteristic
11#554×C97C211yesmaximal, characteristic, complemented
12C2×#13095499C2212yes—
13C2×#13095496C2211yescharacteristic
14#15690810C2211yescharacteristic
15#15690805—22no—
16#15690798C2212yes—
17#15690786—21no—
18C4×Dic194C2212yes—
19C4×D388C2211yescharacteristic
20#15690768—21no—
21#15690743C2211yescharacteristic
22#15690741C2212yes—
23C4⋊#11715895C2211yescharacteristic
24C4⋊#11715895C2212yes—
25#15690733C2212yes—
26#15690733C2212yes—
27#15690733C2212yes—
28#15690731C2211yescharacteristic
29#15690730C2212yes—
30#15690730C2211yescharacteristic
31C4×#11715895C2211yescharacteristic
32C4×#11715895C2212yes—
33C4⋊Q8×C97C2211yescharacteristic
34#125×C97C2212yes—
35C22⋊Q8×C97—21no—
36C22⋊Q8×C97—22no—
37C4×Q8×C97C2211yescharacteristic
38C4×D4×C97C2211yescharacteristic
39C4×D4×C97C2212yes—
40C2×C4⋊C4×C97—21no—
41C2×Dic194C2312yes—
42C2×D388C2311yescharacteristic
43C2×C4×D97—21no—
44#13095499—24no—
45#13095496—22no—
46C2×#11715895C2311yescharacteristic
47C2×#11715895C2312yes—
48C2×#11715895C2311yescharacteristic
49C4⋊Dic97—21no—
50C4⋊Dic97—21no—
51C4⋊Dic97C2311yescharacteristic
52#13095490—22no—
53#13095488—22no—
54#13095488—21no—
55#13095488—22no—
56C4×Dic97—22no—
57C2×Q8×C97—21no—
58C2×Q8×C97D411yescharacteristic
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97—21no—
61C22×C4×C97—22no—
62C4⋊C4×C97—22no—
63C4⋊C4×C97—21no—
64C4⋊C4×C97—22no—
65C4⋊C4×C97D411yescharacteristic
66C4⋊C4×C97D412yes—
67C4⋊C4×C97C2311yescharacteristic
68C4⋊C4×C97C2312yes—
69C22⋊C4×C97—21no—
70C22⋊C4×C97—22no—
71C42×C97C2311yescharacteristic
72C42×C97C2312yes—
73C23×C97—21no—
74C22×D97—21no—
75Q8×C97—22no—
76Q8×C97—81no—
77D4×C97—21no—
78D4×C97#4112yes—
79C2×C4×C97—21no—
80C2×C4×C97—22no—
81C2×C4×C97—42no—
82C2×C4×C97—82no—
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—21no—
92C4×D97—42no—
93Dic194—22no—
94Dic194—22no—
95#11715895—21no—
96#11715895—22no—
97#11715895—21no—
98C22×C97—22no—
99C22×C97—41no—
100C22×C97—41no—
101C22×C97C22×D411yescharacteristic
102C2×D97—41no—
103C2×D97—41no—
104C2×D97—41no—
105C4×C97—22no—
106C4×C97—22no—
107C4×C97—21no—
108C4×C97—41no—
109C4×C97—42no—
110C4×C97#13512yes—
111C4×C97C2×#4111yescharacteristic
112C4×C972−1+411yescharacteristic
113Dic97—42no—
114Dic97—41no—
115C2×C97—42no—
116C2×C97#54811yescharacteristic
117C2×C97C2×#13511yescharacteristic
118C2×C97#57711yescharacteristic
119D97—81no—
120#3360—971nomaximal, Sylow
121C97#336011yescharacteristic, complemented, Sylow
122#556—971no—
123#554D9711yescharacteristic, complemented
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—1942no—
138#125C2×D9712yes—
139#124—972no—
140C22⋊Q8—21no—
141C22⋊Q8—22no—
142C22⋊Q8—1941no—
143C4×Q8—972no—
144C4×Q8C2×D9711yescharacteristic
145C4×D4—971no—
146C4×D4C2×D9711yescharacteristic
147C4×D4C2×D9712yes—
148#117—1941no—
149C2×C4⋊C4—21no—
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—971no—
159#102—972no—
160C4×C8—971no—
161C4×C8—972no—
162C2×Q8—21no—
163C2×Q8—972no—
164C2×Q8C22⋊D9711yescharacteristic
165C2×D4—971no—
166C2×D4C22×D9711yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—1941no—
170SD16—1944no—
171D8—1942no—
172C2×C8—971no—
173C2×C8—972no—
174C2×C8—971no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—971no—
179C4⋊C4—1942no—
180C4⋊C4—1941no—
181C4⋊C4—1941no—
182C4⋊C4C22⋊D9711yescharacteristic
183C4⋊C4C22⋊D9712yes—
184C4⋊C4C22×D9712yes—
185C4⋊C4C22×D9711yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—1942no—
189C22⋊C4—1941no—
190C22⋊C4—1942no—
191C42—1942no—
192C42C22×D9711yescharacteristic
193C42C22×D9712yes—
194C23—21no—
195C23—1941no—
196Q8—22no—
197Q8—81no—
198Q8—1942no—
199Q8—1942no—
200D4—21no—
201D4—1941no—
202D4—1941no—
203D4#1309551212yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
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—22no—
220C22—41no—
221C22—41no—
222C22—3881no—
223C22—3881no—
224C22—3881no—
225C22C22×C22⋊D9711yescharacteristic
226C4—22no—
227C4—22no—
228C4—21no—
229C4—41no—
230C4—42no—
231C4—3882no—
232C4—3881no—
233C4#1569083012yes—
234C4C2×#1309551211yescharacteristic
235C4#1569085811yescharacteristic
236C2—42no—
237C2—7761no—
238C2#2026139611yescharacteristic
239C2C2×#1569083011yescharacteristic
240C2#2026154611yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#554×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095496C22
hypercenter#554D97
hypocenterC97#3360
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #20261458C2semidirect
2. #20261092C2semidirect
3. #20261087C2semidirect
4. #20261039C2semidirect
5. #20261022C2semidirect
7. #20260940C2semidirect
8. Q8×#11715895C2semidirect
9. C4×#13095499C2semidirect
11. #554×C97C2semidirect
121. C97#3360semidirect
123. #554D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202610922
#202610872
#202610392
#202610222
#202609502
#202609402
Q8×#117158952
C4×#130954992
C4×#130954962
#554×C972
#336097