Finite Group Databasebeta

← #27142101

Subgroups — #27142101

Order 12416.

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

normal subgroups only

This group has 310 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
2#20261501C211yesmaximal, characteristic, complemented
3#20260752C211yesmaximal, characteristic, complemented
4#20260732C211yesmaximal, characteristic, complemented
5#20260506C211yesmaximal, characteristic, complemented
6#20260500C211yesmaximal, characteristic, complemented
7#408×C97C211yesmaximal, characteristic, complemented
8C2×#13095511C2211yescharacteristic
9#15690841C412yescomplemented
10#15690840—21no—
11#15690836C412yescomplemented
12#15690835—21no—
13#15690830—22no—
14#15690818—22no—
15#15690816—22no—
16C2×#13095495C2211yescharacteristic
17C2×#13095495C2211yescharacteristic
18M16×D97—22no—
19#15690769C2211yescharacteristic
20#15690737—22no—
21#15690735—21no—
22#15690732—21no—
23C2×#41×C97C2211yescharacteristic
24C2×M16×C97C2211yescharacteristic
25C22×C8×C97C2211yescharacteristic
26C22⋊C8×C97—22no—
27#13095511—21no—
28#13095511—21no—
29#13095511C2×C412yes—
30#13095511D412yes—
31C2×Dic194D411yescharacteristic
32C2×D388D411yescharacteristic
33C2×C22⋊D97—21no—
34C2×C4×D97—21no—
35#13095501—22no—
36#13095499—22no—
37#13095498—22no—
38#13095496—22no—
39#13095495—21no—
40#13095495—21no—
41#13095495C2×C412yes—
42#13095495D412yes—
43#13095494—24no—
44C2×#11715895—21no—
45C2×#11715895—21no—
46C8×D97—24no—
47C4⋊Dic97—21no—
48C22⋊Dic97—21no—
49C4×Dic97—21no—
50#41×C97—21no—
51#41×C97—21no—
52#41×C97—21no—
53#41×C97—21no—
54C2×Q8×C97C2×C411yescharacteristic
55C2×D4×C97—21no—
56C2×D4×C97C2×C411yescharacteristic
57C22×C4×C97—21no—
58C22×C4×C97C2311yescharacteristic
59M16×C97—21no—
60M16×C97—21no—
61C2×C8×C97—21no—
62C2×C8×C97—21no—
63C2×C8×C97—21no—
64C2×C8×C97D412yes—
65C23×C97—21no—
66C23×C97#4111yescharacteristic
67C22×D97—21no—
68Q8×C97—21no—
69Q8×C97—21no—
70D4×C97—21no—
71D4×C97—21no—
72D4×C97—41no—
73D4×C97—41no—
74C2×C4×C97—21no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97C22×C411yescharacteristic
78C2×C4×C97C2×D411yescharacteristic
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97#4111yescharacteristic
83C22⋊D97—22no—
84C22⋊D97—22no—
85C2×Dic97—21no—
86C2×Dic97—41no—
87D388—21no—
88D388C22⋊C412yes—
89C4×D97—22no—
90C4×D97—22no—
91Dic194—21no—
92Dic194C22⋊C412yes—
93C8×C97—22no—
94C8×C97—41no—
95#11715895—22no—
96#11715895—22no—
97C22×C97—41no—
98C22×C97—42no—
99C22×C97—41no—
100C22×C97C4×D411yescharacteristic
101C22×C97#12111yescharacteristic
102C22×C97#12311yescharacteristic
103C2×D97—22no—
104C2×D97—81no—
105C4×C97—41no—
106C4×C97C2×C22⋊C411yescharacteristic
107C4×C97C4×D411yescharacteristic
108C4×C97#12011yescharacteristic
109C4×C97#12111yescharacteristic
110Dic97—22no—
111Dic97—81no—
112C2×C97—21no—
113C2×C97—21no—
114C2×C97—21no—
115C2×C97—81no—
116C2×C97#38611yescharacteristic
117D97—42no—
118#1881—971nomaximal, Sylow
119C97#188111yescharacteristic, complemented, Sylow
120#575—971no—
121#568—971no—
122#421—971no—
123#416—971no—
124#408D9711yescharacteristic, complemented
125#344—971no—
126#341—971no—
127C2×#41—971no—
128C2×#41C2×D9711yescharacteristic
129#137—972no—
130#137—1941no—
131AGL(1,ℤ/8)—972no—
132AGL(1,ℤ/8)—1941no—
133#135—1942no—
134#131—1942no—
135#131—1942no—
136C2×M16—971no—
137C2×M16—971no—
138C2×M16—1942no—
139C2×M16C2×D9711yescharacteristic
140C22×C8C2×D9711yescharacteristic
141#117—971no—
142#104—1942no—
143#103—1941no—
144#102—1941no—
145C22⋊C8—22no—
146#41—21no—
147#41—21no—
148#41—21no—
149#41—21no—
150#41—972no—
151#41—972no—
152#41—1941no—
153#41—1941no—
154C2×Q8—971no—
155C2×Q8C4×D9711yescharacteristic
156C2×D4—21no—
157C2×D4—971no—
158C2×D4—1941no—
159C2×D4C4×D9711yescharacteristic
160C22×C4—21no—
161C22×C4—1941no—
162C22×C4C22×D9711yescharacteristic
163Q16—1942no—
164SD16—1942no—
165SD16—1942no—
166D8—1942no—
167M16—21no—
168M16—21no—
169M16—972no—
170M16—972no—
171M16—1944no—
172M16—1941no—
173M16—1941no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—21no—
177C2×C8—1944no—
178C2×C8—1941no—
179C2×C8—1941no—
180C2×C8C22⋊D9712yes—
181C4⋊C4—1941no—
182C22⋊C4—1941no—
183C42—1941no—
184C23—21no—
185C23—1941no—
186C23#1309551111yescharacteristic
187Q8—21no—
188Q8—21no—
189Q8—972no—
190Q8—1941no—
191D4—21no—
192D4—21no—
193D4—41no—
194D4—41no—
195D4—972no—
196D4—1942no—
197D4—1942no—
198D4—1941no—
199C2×C4—21no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—3881no—
203C2×C4—1941no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4C2×C4×D9711yescharacteristic
207C2×C4D4×D9711yescharacteristic
208C2×C4D4×D9711yescharacteristic
209C2×C4D4×D9711yescharacteristic
210C2×C4C2×C22⋊D9711yescharacteristic
211C2×C4#1309551011yescharacteristic
212C8—22no—
213C8—41no—
214C8—1942no—
215C8—1942no—
216C22—41no—
217C22—42no—
218C22—41no—
219C22—7761no—
220C22—1942no—
221C22C4×C22⋊D9711yescharacteristic
222C22#1569078311yescharacteristic
223C22#1569079211yescharacteristic
224C4—41no—
225C4—7761no—
226C4—1942no—
227C4C22⋊C4×D9711yescharacteristic
228C4#1569077011yescharacteristic
229C4#1569077811yescharacteristic
230C4#1569078111yescharacteristic
231C2—21no—
232C2—21no—
233C2—21no—
234C2—81no—
235C2—3882no—
236C2#2026061111yescharacteristic
subgroupG/H
centerC4#15690778
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#408×C97C2
socleC2×C97#386
prime radical#20261533C2
cosocleC22×C4×C97C23
hypercenter#408D97
hypocenterC97#1881
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #20261501C2semidirect
3. #20260752C2semidirect
4. #20260732C2semidirect
5. #20260506C2semidirect
6. #20260500C2semidirect
7. #408×C97C2semidirect
9. #15690841C4semidirect
11. #15690836C4semidirect
119. C97#1881semidirect
124. #408D97semidirect

Maximal subgroups

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

subgroupindex
#202615332
#202615012
#202607522
#202607322
#202605062
#202605002
#408×C972
#188197