Finite Group Databasebeta

← #27142240

Subgroups — #27142240

Order 12416.

332 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 79 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
2C4×#13095511C211yesmaximal, characteristic, complemented
3#20260751C211yesmaximal, characteristic, complemented
4#20260709C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260495C211yesmaximal, characteristic, complemented
7C2×#104×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841C412yescomplemented
10#15690840—21no—
11#15690836C412yescomplemented
12#15690835—21no—
13#15690830—22no—
14C2×#13095495C2211yescharacteristic
15C4×Dic194—21no—
16C4×D388—21no—
17C4×C22⋊D97—22no—
18#15690769C2211yescharacteristic
19#15690769C2211yescharacteristic
20#15690767—21no—
21C42×D97—21no—
22#15690736—22no—
23#15690725—22no—
24C2×#41×C97C2211yescharacteristic
25C2×M16×C97C2211yescharacteristic
26C2×C42×C97C2211yescharacteristic
27#104×C97—21no—
28#104×C97—21no—
29#13095511—21no—
30#13095511—21no—
31#13095511C2×C412yes—
32#13095511D412yes—
33C2×Dic194D411yescharacteristic
34C2×D388D411yescharacteristic
35C2×C22⋊D97—21no—
36C2×C4×D97—21no—
37C2×C4×D97—22no—
38#13095501—22no—
39#13095499—22no—
40#13095498—22no—
41#13095496—22no—
42#13095495—21no—
43#13095495C2×C412yescomplemented
44C2×#11715895—21no—
45C4⋊Dic97—21no—
46C4⋊Dic97—21no—
47#13095490—22no—
48C22⋊Dic97—21no—
49C22⋊Dic97D412yes—
50#13095488—22no—
51C4×Dic97—22no—
52C4×Dic97—21no—
53C4×Dic97—21no—
54#41×C97—21no—
55#41×C97—21no—
56#41×C97—21no—
57#41×C97—21no—
58C2×Q8×C97C2×C411yescharacteristic, complemented
59C2×D4×C97—21no—
60C2×D4×C97C2×C411yescharacteristic, complemented
61C22×C4×C97—21no—
62C22×C4×C97—21no—
63C22×C4×C97C2311yescharacteristic
64M16×C97—21no—
65M16×C97—21no—
66C2×C8×C97—21no—
67C42×C97—21no—
68C42×C97D411yescharacteristic
69C42×C97D411yescharacteristic
70C23×C97—21no—
71C23×C97C2×D411yescharacteristic
72C22×D97—21no—
73Q8×C97—21no—
74Q8×C97—21no—
75D4×C97—21no—
76D4×C97—21no—
77D4×C97—41no—
78D4×C97—41no—
79C2×C4×C97—21no—
80C2×C4×C97—21no—
81C2×C4×C97—21no—
82C2×C4×C97—42no—
83C2×C4×C97—41no—
84C2×C4×C97—41no—
85C2×C4×C97C22×C411yescharacteristic
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2×D411yescharacteristic
89C2×C4×C97#4111yescharacteristic
90C2×C4×C97#4111yescharacteristic
91C22⋊D97—22no—
92C22⋊D97—22no—
93C2×Dic97—21no—
94C2×Dic97—22no—
95C2×Dic97—41no—
96D388—21no—
97D388C22⋊C412yes—
98C4×D97—22no—
99C4×D97—22no—
100C4×D97—44no—
101Dic194—21no—
102Dic194C22⋊C412yes—
103C8×C97—41no—
104#11715895—22no—
105C22×C97—41no—
106C22×C97—41no—
107C22×C97—81no—
108C22×C97C4×D411yescharacteristic
109C22×C97#12011yescharacteristic
110C22×C97#12111yescharacteristic
111C2×D97—22no—
112C2×D97—42no—
113C4×C97—22no—
114C4×C97—41no—
115C4×C97—41no—
116C4×C97C2×C22⋊C411yescharacteristic
117C4×C97C4×D411yescharacteristic
118C4×C97#12111yescharacteristic
119C4×C97#12311yescharacteristic
120Dic97—22no—
121Dic97—42no—
122Dic97—81no—
123C2×C97—21no—
124C2×C97—21no—
125C2×C97—21no—
126C2×C97—81no—
127C2×C97#38611yescharacteristic
128D97—42no—
129#1906—971nomaximal, Sylow
130C97#190611yescharacteristic, complemented, Sylow
131#575—971no—
132C4×#41—971no—
133#421—971no—
134C2×#104D9711yescharacteristic, complemented
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×M16C2×D9711yescharacteristic
147C4×Q8—1941no—
148C4×D4—1941no—
149C4×D4—1942no—
150#117—971no—
151#117—971no—
152#117—1941no—
153C2×C42—1941no—
154C2×C42C2×D9711yescharacteristic
155#104—21no—
156#104—21no—
157#104—1942no—
158#99—1942no—
159#41—21no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—972no—
164#41—972no—
165#41—1941no—
166#41—1941no—
167C2×Q8—971no—
168C2×Q8C4×D9711yescharacteristic, complemented
169C2×D4—21no—
170C2×D4—971no—
171C2×D4—1941no—
172C2×D4C4×D9711yescharacteristic, complemented
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—1942no—
176C22×C4—1941no—
177C22×C4C22×D9711yescharacteristic
178Q16—1942no—
179SD16—1942no—
180SD16—1942no—
181D8—1942no—
182M16—21no—
183M16—21no—
184M16—972no—
185M16—1941no—
186C2×C8—21no—
187C2×C8—1941no—
188C4⋊C4—1942no—
189C4⋊C4—1941no—
190C4⋊C4—1941no—
191C22⋊C4—972no—
192C22⋊C4—1942no—
193C22⋊C4—1941no—
194C42—21no—
195C42—1942no—
196C42—1941no—
197C42—1941no—
198C42C22⋊D9711yescharacteristic
199C42C22⋊D9711yescharacteristic
200C23—21no—
201C23—1941no—
202C23D4×D9711yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—972no—
206Q8—1941no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—972no—
212D4—1942no—
213D4—1942no—
214D4—1941no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—3884no—
222C2×C4—3881no—
223C2×C4—1942no—
224C2×C4—1941no—
225C2×C4—1942no—
226C2×C4—1942no—
227C2×C4C2×C4×D9711yescharacteristic
228C2×C4D4×D9711yescharacteristic
229C2×C4D4×D9711yescharacteristic
230C2×C4C2×C22⋊D9711yescharacteristic
231C2×C4#1309551011yescharacteristic
232C2×C4#1309551111yescharacteristic
233C8—41no—
234C8—1942no—
235C22—41no—
236C22—41no—
237C22—81no—
238C22—3882no—
239C22—1942no—
240C22#1569077011yescharacteristic
241C22#1569077811yescharacteristic
242C22#1569078311yescharacteristic
243C4—22no—
244C4—41no—
245C4—41no—
246C4—7761no—
247C4—3882no—
248C4—1942no—
249C4C22⋊C4×D9711yescharacteristic
250C4C4×C22⋊D9711yescharacteristic
251C4#1569078111yescharacteristic
252C4#1569079211yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—81no—
257C2—3882no—
258C2#2026061111yescharacteristic
subgroupG/H
centerC4#15690781
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#104×C97C2
socleC2×C97#386
prime radical#20261533C2
cosocleC22×C4×C97C23
hypercenterC2×#104D97
hypocenterC97#1906
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C4×#13095511C2semidirect
3. #20260751C2semidirect
4. #20260709C2semidirect
5. #20260506C2semidirect
6. #20260495C2semidirect
7. C2×#104×C97C2semidirect
9. #15690841C4semidirect
11. #15690836C4semidirect
43. #13095495C2×C4semidirect
58. C2×Q8×C97C2×C4semidirect
60. C2×D4×C97C2×C4semidirect
130. C97#1906semidirect
134. C2×#104D97semidirect
168. C2×Q8C4×D97semidirect
172. C2×D4C4×D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
C4×#130955112
#202607512
#202607092
#202605062
#202604952
C2×#104×C972
#190697