Finite Group Databasebeta

← #27150217

Subgroups — #27150217

Order 12416.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15690857C211yesmaximal, characteristic
2#20261501C212yesmaximal, complemented
3#20260719C211yesmaximal, characteristic, complemented
4#20260716C214yesmaximal, complemented
5C2×#15690728C212yesmaximal, complemented
6#101×D97C214yesmaximal, complemented
7#413×C97C211yesmaximal, characteristic, complemented
8#15690857—24no—
9#15690857—24no—
10C2×#13095511C412yes—
11C2×#13095511C2211yescharacteristic
12C2×#13095510C414yes—
13C2×#13095510C2212yes—
14C22×Dic194C412yes—
15C22×Dic194C2211yescharacteristic
16C2×Q8×D97C2212yes—
17#15690818—24no—
18#15690816—24no—
19C2×#13095495C2212yes—
20M16×D97C2218yes—
21#15690729C2212yes—
22#15690728C2218yes—
23#15690727C2212yes—
24C2×#41×C97C2211yescharacteristic
25C2×M16×C97C2212yes—
26#101×C97C2212yes—
27#100×C97C2212yes—
28#13095511—24no—
29#13095511—24no—
30#13095511D414yes—
31#13095511D414yes—
32#13095510—216no—
33#13095510—28no—
34C2×Dic194—22no—
35C2×Dic194—22no—
36C2×Dic194—24no—
37C2×Dic194—24no—
38C2×Dic194C2×C414yes—
39C2×Dic194C2×C412yes—
40C2×Dic194C2×C411yescharacteristic
41C2×Dic194C2312yes—
42Q8×D97—28no—
43C2×D388C2×C411yescharacteristic
44C2×C22⋊D97C2×C414yes—
45C2×C22⋊D97C2×C412yes—
46C22×Dic97C2×C414yes—
47C22×Dic97C2×C412yes—
48C2×C4×D97C2×C414yes—
49C2×C4×D97C2312yes—
50#13095495—22no—
51#13095495C2314yes—
52#13095494—28no—
53C2×#11715895—22no—
54C8×D97—28no—
55#41×C97—22no—
56#41×C97—22no—
57C2×Q8×C97C2311yescharacteristic
58C2×D4×C97C2×C412yes—
59C2×D4×C97C2311yescharacteristic
60C22×C4×C97C2×C412yes—
61C22×C4×C97C2311yescharacteristic
62M16×C97—22no—
63M16×C97C2314yes—
64C2×C8×C97—22no—
65C23×C97C22×C412yes—
66C23×C97C22×C411yescharacteristic
67C22×D97C22×C412yes—
68Q8×C97—22no—
69D4×C97—24no—
70D4×C97—22no—
71C2×C4×C97—22no—
72C2×C4×C97—22no—
73C2×C4×C97C22×C412yes—
74C2×C4×C97C22×C411yescharacteristic
75C2×C4×C97C2×D412yes—
76C2×C4×C97C2×D412yes—
77C2×C4×C97C2411yescharacteristic
78C22⋊D97—28no—
79C22⋊D97C22⋊C418yes—
80C2×Dic97—24no—
81C2×Dic97—28no—
82C2×Dic97C22×C414yes—
83C2×Dic97C22×C412yes—
84D388C22⋊C412yes—
85D388C22⋊C412yes—
86C4×D97—28no—
87C4×D97C2×D418yes—
88Dic194—22no—
89Dic194—22no—
90Dic194—24no—
91Dic194—28no—
92Dic194C22⋊C412yes—
93Dic194C22⋊C412yes—
94C8×C97—24no—
95#11715895—24no—
96C22×C97—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97C2×C22⋊C412yes—
100C22×C97C23×C411yescharacteristic
101C2×D97—42no—
102C2×D97C2×C22⋊C414yes—
103C4×C97—22no—
104C4×C97C2×C22⋊C411yescharacteristic
105C4×C97C2×C22⋊C411yescharacteristic
106C4×C97C22×D412yes—
107Dic97—24no—
108Dic97C2×C22⋊C414yes—
109C2×C97—21no—
110C2×C97—22no—
111C2×C97—42no—
112C2×C97C22×C22⋊C411yescharacteristic
113D97—24no—
114#2889—971nomaximal, Sylow
115C97#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—971no—
117#568—972no—
118#413—971no—
119#413—974no—
120#413D9711yescharacteristic, complemented
121C2×#101—972no—
122C2×#101—974no—
1232−1+4—1944no—
1242−1+4—1944no—
125C2×#41—974no—
126C2×#41—972no—
127C2×#41—972no—
128C2×#41—971no—
129C2×#41C2×D9711yescharacteristic
130C22×Q8—972no—
131C22×Q8—972no—
132C22×Q8—971no—
133#131—1944no—
134#131—1944no—
135C2×M16—972no—
136C2×M16—978no—
137C2×M16C2×D9712yes—
138#101—972no—
139#101—978no—
140#101C2×D9712yes—
141#100—972no—
142#100C2×D9712yes—
143#41—22no—
144#41—22no—
145#41—974no—
146#41—974no—
147#41—1944no—
148#41—1944no—
149#41—1948no—
150#41—19416no—
151C2×Q8—971no—
152C2×Q8—974no—
153C2×Q8—972no—
154C2×Q8—972no—
155C2×Q8—1944no—
156C2×Q8—1944no—
157C2×Q8—1948no—
158C2×Q8—1942no—
159C2×Q8—1942no—
160C2×Q8C22×D9711yescharacteristic
161C2×D4—972no—
162C2×D4—974no—
163C2×D4—971no—
164C2×D4C4×D9712yes—
165C2×D4C22×D9711yescharacteristic
166C22×C4—972no—
167C22×C4—974no—
168C22×C4—972no—
169C22×C4—974no—
170C22×C4C4×D9712yes—
171C22×C4C22×D9711yescharacteristic
172M16—22no—
173M16—974no—
174M16—1948no—
175M16—1942no—
176M16C22×D9714yes—
177C2×C8—22no—
178C2×C8—1948no—
179C2×C8—1942no—
180C23—972no—
181C23C2×C4×D9712yes—
182C23C2×C4×D9711yescharacteristic
183Q8—22no—
184Q8—972no—
185Q8—972no—
186Q8—1948no—
187Q8—1942no—
188Q8—1942no—
189Q8—1944no—
190D4—24no—
191D4—22no—
192D4—978no—
193D4—972no—
194D4—972no—
195D4—1948no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—972no—
199C2×C4—978no—
200C2×C4—974no—
201C2×C4—1948no—
202C2×C4—1948no—
203C2×C4—1944no—
204C2×C4C2×C4×D9712yes—
205C2×C4C2×C4×D9711yescharacteristic
206C2×C4D4×D9712yes—
207C2×C4D4×D9712yes—
208C2×C4C23×D9711yescharacteristic
209C8—24no—
210C8—1944no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—974no—
215C22—3882no—
216C22C22⋊C4×D9712yes—
217C22C22×C4×D9711yescharacteristic
218C4—22no—
219C4—974no—
220C4—1944no—
221C4C22⋊C4×D9711yescharacteristic
222C4C22⋊C4×D9711yescharacteristic
223C4C2×D4×D9712yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—1944no—
228C2C2×C22⋊C4×D9711yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D97
commutatorC22×C97C23×C4
FrattiniC2×C4C23×D97
Fitting#413×C97C2
socleC2×C97C22×C22⋊C4
prime radicalC2×#15690857C2
cosocleC2×C4×C97C24
hypercenter#413D97
hypocenterC97#2889
LCSG ▷ C22×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20261501C2semidirect
3. #20260719C2semidirect
4. #20260716C2semidirect
5. C2×#15690728C2semidirect
6. #101×D97C2semidirect
7. #413×C97C2semidirect
115. C97#2889semidirect
120. #413D97semidirect

Maximal subgroups

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

subgroupindex
C2×#156908572
#202615012
#202607192
#202607162
C2×#156907282
#101×D972
#413×C972
#288997