Finite Group Databasebeta

← #27142237

Subgroups — #27142237

Order 12416.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#20261533C211yesmaximal, characteristic
2#20260752C211yesmaximal, characteristic, complemented
3C2×#15690738C211yesmaximal, characteristic, complemented
4#20260708C211yesmaximal, characteristic, complemented
5#20260505C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C4×#41×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841—21no—
10#15690840C412yescomplemented
11#15690836—21no—
12#15690835C412yescomplemented
13#15690830—22no—
14C2×#13095495C2211yescharacteristic
15C2×#13095495C2211yescharacteristic
16#15690769C2211yescharacteristic
17#15690738—21no—
18#15690738—21no—
19#15690737—22no—
20#15690724—22no—
21C2×#41×C97C2211yescharacteristic
22C4×Q8×C97—21no—
23C4×D4×C97—21no—
24C4×D4×C97—22no—
25#117×C97—21no—
26#117×C97C2211yescharacteristic
27C2×C42×C97—21no—
28C2×C42×C97C2211yescharacteristic
29#13095511—21no—
30#13095511—21no—
31#13095511—21no—
32#13095511—21no—
33C2×Dic194C2×C411yescharacteristic, complemented
34C2×D388C2×C411yescharacteristic, complemented
35C2×C22⋊D97—21no—
36C2×C4×D97—21no—
37#13095501—22no—
38#13095499—22no—
39#13095498—22no—
40#13095496—22no—
41#13095495—21no—
42#13095495—21no—
43#13095495—21no—
44#13095495C2×C412yescomplemented
45C2×#11715895—21no—
46C2×#11715895—21no—
47C4⋊Dic97—21no—
48C22⋊Dic97—21no—
49C4×Dic97—21no—
50#41×C97—21no—
51#41×C97—21no—
52#41×C97C2×C412yes—
53#41×C97D412yes—
54C2×Q8×C97D411yescharacteristic
55C2×D4×C97—21no—
56C2×D4×C97D411yescharacteristic
57C22×C4×C97—21no—
58C22×C4×C97—22no—
59C22×C4×C97—21no—
60C22×C4×C97C2311yescharacteristic
61C4⋊C4×C97—22no—
62C4⋊C4×C97—21no—
63C22⋊C4×C97—22no—
64C22⋊C4×C97D412yes—
65C42×C97—22no—
66C42×C97—21no—
67C42×C97—21no—
68C42×C97D411yescharacteristic
69C42×C97D411yescharacteristic
70C23×C97—21no—
71C23×C97C2×D411yescharacteristic
72C22×D97—21no—
73Q8×C97—21no—
74Q8×C97C22⋊C412yes—
75D4×C97—22no—
76D4×C97—22no—
77D4×C97—21no—
78D4×C97C22⋊C412yes—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—22no—
84C2×C4×C97—21no—
85C2×C4×C97—42no—
86C2×C4×C97—44no—
87C2×C4×C97C22×C411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97C2×D411yescharacteristic
90C2×C4×C97C2×D411yescharacteristic
91C2×C4×C97#4111yescharacteristic
92C2×C4×C97#4111yescharacteristic
93C22⋊D97—41no—
94C22⋊D97—41no—
95C2×Dic97—21no—
96C2×Dic97—41no—
97D388—21no—
98D388—21no—
99C4×D97—41no—
100C4×D97—41no—
101Dic194—21no—
102Dic194—21no—
103#11715895—22no—
104#11715895—41no—
105C22×C97—22no—
106C22×C97—41no—
107C22×C97—42no—
108C22×C97C4×D411yescharacteristic
109C22×C97#12011yescharacteristic
110C22×C97#12111yescharacteristic
111C2×D97—41no—
112C2×D97—81no—
113C4×C97—22no—
114C4×C97—22no—
115C4×C97—42no—
116C4×C97—41no—
117C4×C97C2×C22⋊C411yescharacteristic
118C4×C97C4×D411yescharacteristic
119C4×C97#12111yescharacteristic
120C4×C97#12311yescharacteristic
121Dic97—41no—
122Dic97—81no—
123C2×C97—21no—
124C2×C97—21no—
125C2×C97—21no—
126C2×C97—42no—
127C2×C97#38611yescharacteristic
128D97—81no—
129#1906—971nomaximal, Sylow
130C97#190611yescharacteristic, complemented, Sylow
131#575—971no—
132C4×#41D9711yescharacteristic, complemented
133#421—971no—
134C2×#104—971no—
135#410—971no—
136#344—971no—
137#339—971no—
138C2×#41—971no—
139C2×#41C2×D9711yescharacteristic
140#137—972no—
141#137—1941no—
142AGL(1,ℤ/8)—972no—
143AGL(1,ℤ/8)—1941no—
144#135—1942no—
145C2×M16—971no—
146C2×M16—971no—
147C4×Q8—21no—
148C4×D4—21no—
149C4×D4—22no—
150#117—21no—
151#117—971no—
152#117C2×D9711yescharacteristic
153C2×C42—21no—
154C2×C42C2×D9711yescharacteristic
155#104—1942no—
156#104—1941no—
157#104—1941no—
158#99—1942no—
159#41—21no—
160#41—21no—
161#41—1941no—
162#41—1941no—
163#41—1941no—
164#41—1941no—
165#41C4×D9712yes—
166#41D38812yes—
167C2×Q8—971no—
168C2×Q8C22⋊D9711yescharacteristic
169C2×D4—21no—
170C2×D4—971no—
171C2×D4—1941no—
172C2×D4C22⋊D9711yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178Q16—1942no—
179SD16—1942no—
180SD16—1942no—
181D8—1942no—
182M16—972no—
183M16—1941no—
184M16—1941no—
185M16—1941no—
186C2×C8—1941no—
187C2×C8—1941no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—1941no—
191C22⋊C4—22no—
192C22⋊C4—1941no—
193C22⋊C4C22⋊D9712yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—1941no—
198C42C22⋊D9711yescharacteristic
199C42C22⋊D9711yescharacteristic
200C23—21no—
201C23—1941no—
202C23C2×C22⋊D9711yescharacteristic
203Q8—21no—
204Q8—1941no—
205Q8—1941no—
206Q8#1309548812yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—3881no—
211D4—3881no—
212D4—1941no—
213D4—1941no—
214D4#1309548812yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—3881no—
224C2×C4—3881no—
225C2×C4—3881no—
226C2×C4—1941no—
227C2×C4C2×C4×D9711yescharacteristic
228C2×C4C2×C22⋊D9711yescharacteristic
229C2×C4C2×C22⋊D9711yescharacteristic
230C2×C4C2×D38811yescharacteristic
231C2×C4#1309551111yescharacteristic
232C2×C4#1309551111yescharacteristic
233C8—3881no—
234C8—1942no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—7761no—
239C22—3881no—
240C22C4×C22⋊D9711yescharacteristic
241C22#1569077911yescharacteristic
242C22#1569078411yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—7761no—
248C4—3881no—
249C4C2×#1309548811yescharacteristic
250C4C4×C22⋊D9711yescharacteristic
251C4#1569078411yescharacteristic
252C4#1569079311yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—7761no—
258C2#2026061411yescharacteristic
subgroupG/H
centerC4#15690784
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC4×#41×C97C2
socleC2×C97#386
prime radical#20261533C2
cosocleC22×C4×C97C23
hypercenterC4×#41D97
hypocenterC97#1906
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #20260752C2semidirect
3. C2×#15690738C2semidirect
4. #20260708C2semidirect
5. #20260505C2semidirect
6. #20260495C2semidirect
7. C4×#41×C97C2semidirect
10. #15690840C4semidirect
12. #15690835C4semidirect
33. C2×Dic194C2×C4semidirect
34. C2×D388C2×C4semidirect
44. #13095495C2×C4semidirect
130. C97#1906semidirect
132. C4×#41D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
#202607522
C2×#156907382
#202607082
#202605052
#202604952
C4×#41×C972
#190697