Finite Group Databasebeta

← #27152266

Subgroups — #27152266

Order 12416.

340 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 149 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#20261366C211yesmaximal, characteristic
2#20261363C211yesmaximal, characteristic
3#20261143C212yesmaximal, complemented
4#20261142C212yesmaximal, complemented
5#20261136C212yesmaximal, complemented
6#20261132C212yesmaximal, complemented
7#20260777C211yesmaximal, characteristic
8#20260771C211yesmaximal, characteristic
9C2×#15690741C211yesmaximal, characteristic
10#20260693C211yesmaximal, characteristic
11#428×C97C211yesmaximal, characteristic
12C22×#11715895C2211yescharacteristic
13#15690810—21no—
14#15690809—21no—
15#15690809C2212yes—
16#15690809C2212yes—
17#15690802—21no—
18#15690801—21no—
19#15690801C2212yes—
20#15690801C2212yes—
21#15690789—22no—
22#15690787—22no—
23#15690769C2211yescharacteristic
24#15690769C2211yescharacteristic
25C2×C4⋊Dic97C2211yescharacteristic
26#15690744C2212yes—
27#15690743C2211yescharacteristic
28#15690743C2211yescharacteristic
29#15690742C2212yes—
30#15690741C2211yescharacteristic
31#15690741C2212yes—
32#15690741C2211yescharacteristic
33#15690741C2212yes—
34#15690720C2212yes—
35C8×Dic97C2212yes—
36C2×M16×C97C2211yescharacteristic
37#117×C97C2211yescharacteristic
38C2×C4⋊C4×C97C2211yescharacteristic
39#107×C97C2212yes—
40#106×C97C2212yes—
41C2×Dic194—22no—
42C2×Dic194—22no—
43C22×Dic97—21no—
44C2×#11715895Q814yescomplemented
45C2×#11715895C2311yescharacteristic
46C2×#11715895C2311yescharacteristic
47C4⋊Dic97—21no—
48C4⋊Dic97—22no—
49C4⋊Dic97—22no—
50C4⋊Dic97D411yescharacteristic
51C4⋊Dic97D411yescharacteristic
52C4⋊Dic97C2312yes—
53C4⋊Dic97C2311yescharacteristic
54C4⋊Dic97C2311yescharacteristic
55#13095490—22no—
56#13095490—22no—
57#13095490—22no—
58#13095490—22no—
59C22⋊Dic97—21no—
60C22⋊Dic97D412yes—
61C4×Dic97—21no—
62C4×Dic97C2312yes—
63C22×C4×C97—21no—
64C22×C4×C97C2311yescharacteristic
65M16×C97Q814yescomplemented
66C2×C8×C97C2312yes—
67C4⋊C4×C97—21no—
68C4⋊C4×C97C2311yescharacteristic
69C4⋊C4×C97C2312yes—
70C4⋊C4×C97C2311yescharacteristic
71C22⋊C4×C97—21no—
72C42×C97—21no—
73C23×C97C2×D411yescharacteristic
74C2×C4×C97—21no—
75C2×C4×C97—21no—
76C2×C4×C97—22no—
77C2×C4×C97—81no—
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×Q812yes—
80C2×C4×C97C2×Q812yes—
81C2×C4×C97C2411yescharacteristic
82C2×Dic97—21no—
83C2×Dic97—21no—
84C2×Dic97—22no—
85C2×Dic97—81no—
86C2×Dic97C2×D412yes—
87C2×Dic97C2×D412yes—
88Dic194—82no—
89Dic194—82no—
90C8×C97C2×Q814yes—
91#11715895C2×Q812yes—
92#11715895C2×Q812yes—
93C22×C97—21no—
94C22×C97—21no—
95C22×C97C4⋊Q812yes—
96C22×C97C22×D411yescharacteristic
97C4×C97—41no—
98C4×C97—42no—
99C4×C97—41no—
100C4×C97C4⋊Q812yes—
101C4×C97C22×Q811yescharacteristic
102C4×C97C22×Q811yescharacteristic
103Dic97—22no—
104Dic97—22no—
105Dic97—41no—
106Dic97—42no—
107Dic97—41no—
108C2×C97—22no—
109C2×C97C2×C4⋊Q811yescharacteristic
110C2×C97#57711yescharacteristic
111C2×C97#57711yescharacteristic
112#3163—971nomaximal, Sylow
113C97#316311yescharacteristic, complemented, Sylow
114#544—971no—
115#544—971no—
116#501—972no—
117#501—972no—
118#499—972no—
119#498—972no—
120#428—971no—
121#428D9711yescharacteristic
122#427—971no—
123C2×#106—971no—
124#405—971no—
125C2×M16C2×D9711yescharacteristic
126C22×C8—971no—
127C4⋊Q8—972no—
128C4⋊Q8—972no—
129C4⋊Q8—1941no—
130C4⋊Q8—1941no—
131#125—972no—
132#125—972no—
133#125—1941no—
134#125—1941no—
135C22⋊Q8—1942no—
136C22⋊Q8—1942no—
137#117—971no—
138#117—971no—
139#117C2×D9711yescharacteristic
140C2×C4⋊C4—971no—
141C2×C4⋊C4C2×D9711yescharacteristic
142#107—972no—
143#107—971no—
144#107—971no—
145#107C2×D9712yes—
146#106—972no—
147#106—972no—
148#106—972no—
149#106—971no—
150#106—971no—
151#106C2×D9712yes—
152#97—972no—
153C4×C8—972no—
154C2×Q8—1942no—
155C2×Q8—1942no—
156C22×C4—21no—
157C22×C4—1941no—
158C22×C4C22×D9711yescharacteristic
159M16Dic19414yescomplemented
160C2×C8—974no—
161C2×C8—971no—
162C2×C8—971no—
163C2×C8C22×D9712yes—
164C4⋊C4—21no—
165C4⋊C4—971no—
166C4⋊C4—971no—
167C4⋊C4—972no—
168C4⋊C4—971no—
169C4⋊C4—971no—
170C4⋊C4—1942no—
171C4⋊C4—1942no—
172C4⋊C4—1942no—
173C4⋊C4—1942no—
174C4⋊C4—1942no—
175C4⋊C4—1942no—
176C4⋊C4—1941no—
177C4⋊C4C22×D9712yes—
178C4⋊C4C22×D9711yescharacteristic
179C4⋊C4C22×D9711yescharacteristic
180C22⋊C4—21no—
181C22⋊C4—972no—
182C22⋊C4—1941no—
183C42—21no—
184C42—972no—
185C42—1941no—
186C23D4×D9711yescharacteristic
187Q8—7762no—
188Q8—7762no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—22no—
192C2×C4—81no—
193C2×C4—7761no—
194C2×C4—972no—
195C2×C4—972no—
196C2×C4—1941no—
197C2×C4—1941no—
198C2×C4—1942no—
199C2×C4D4×D9711yescharacteristic
200C2×C4Q8×D9712yes—
201C2×C4C2×Dic19412yes—
202C2×C4C23×D9711yescharacteristic
203C8—972no—
204C8—972no—
205C8C2×Dic19414yes—
206C22—21no—
207C22—21no—
208C22#1569080912yes—
209C22C2×D4×D9711yescharacteristic
210C4—41no—
211C4—42no—
212C4—41no—
213C4—3882no—
214C4—3881no—
215C4—3881no—
216C4—1942no—
217C4—1942no—
218C4#1569080912yes—
219C4C2×Q8×D9711yescharacteristic
220C4C22×Dic19411yescharacteristic
221C2—22no—
222C2C2×#1569080911yescharacteristic
223C2#2026154411yescharacteristic
224C2#2026154511yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radicalC23×C97C2×D4
hypercenter#428D97
hypocenterC97#3163
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
3. #20261143C2semidirect
4. #20261142C2semidirect
5. #20261136C2semidirect
6. #20261132C2semidirect
44. C2×#11715895Q8semidirect
65. M16×C97Q8semidirect
113. C97#3163semidirect
159. M16Dic194semidirect

Maximal subgroups

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

subgroupindex
#202613662
#202613632
#202611432
#202611422
#202611362
#202611322
#202607772
#202607712
C2×#156907412
#202606932
#428×C972
#316397