Finite Group Databasebeta

← #27153655

Subgroups — #27153655

Order 12416.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#20261458C211yesmaximal, characteristic, complemented
2#20261389C211yesmaximal, characteristic
3#20261086C211yesmaximal, characteristic, complemented
4#20261079C212yesmaximal, complemented
5#20261015C211yesmaximal, characteristic, complemented
6#20260975C212yesmaximal, complemented
7#20260945C211yesmaximal, characteristic, complemented
8#20260894C212yesmaximal, complemented
9C8×D388C211yesmaximal, characteristic, complemented
10#20260728C212yesmaximal, complemented
11C4×SD16×C97C211yesmaximal, characteristic, complemented
12C2×#13095510C2212yes—
13C2×#13095501C2212yes—
14C2×C8×D97C2212yes—
15#15690810C2211yescharacteristic
16#15690810C2211yescharacteristic
17#15690805—22no—
18#15690798—21no—
19#15690794—22no—
20#15690790C2212yes—
21#15690786C2212yes—
22#15690784—21no—
23C4×Dic194—21no—
24C4×D388C2211yescharacteristic
25#15690768C2212yes—
26C2×C4⋊Dic97—21no—
27#15690743C2212yes—
28C4⋊#11715895C2211yescharacteristic
29#15690735C2212yes—
30#15690735C2212yes—
31#15690733C2212yes—
32#15690731C2212yes—
33#15690722C2212yes—
34C2×SD16×C97C2211yescharacteristic
35C4×Q8×C97C2211yescharacteristic
36C4×D4×C97C2211yescharacteristic
37#106×C97C2211yescharacteristic
38#103×C97C2211yescharacteristic
39#102×C97C2211yescharacteristic
40C4×C8×C97C2211yescharacteristic
41#13095510—24no—
42#13095510—42no—
43C2×Dic194—21no—
44C2×Dic194C2312yes—
45C2×D388D411yescharacteristic
46C2×C22⋊D97—22no—
47C22×Dic97—22no—
48C2×C4×D97C2312yes—
49#13095501—42no—
50C2×#11715895C2312yes—
51C8×D97—42no—
52C4⋊Dic97—21no—
53C4⋊Dic97—21no—
54C4⋊Dic97—21no—
55C4⋊Dic97—22no—
56C4⋊Dic97D411yescharacteristic
57C4⋊Dic97C2312yes—
58#13095490—22no—
59#13095488—22no—
60#13095488—22no—
61#13095488—22no—
62#13095488D412yes—
63C4×Dic97—22no—
64C2×Q8×C97C2311yescharacteristic
65C2×D4×C97C2311yescharacteristic
66C22×C4×C97—21no—
67SD16×C97—22no—
68C2×C8×C97C2311yescharacteristic
69C2×C8×C97C2311yescharacteristic
70C4⋊C4×C97—21no—
71C4⋊C4×C97C2311yescharacteristic
72C4⋊C4×C97C2311yescharacteristic
73C22⋊C4×C97—21no—
74C42×C97—21no—
75C42×C97C2311yescharacteristic
76C23×C97—21no—
77C22×D97C2×D412yes—
78Q8×C97—21no—
79Q8×C97—21no—
80D4×C97—21no—
81D4×C97#4112yes—
82C2×C4×C97—21no—
83C2×C4×C97—21no—
84C2×C4×C97—21no—
85C2×C4×C97—42no—
86C2×C4×C97C2×D411yescharacteristic
87C2×C4×C97C2×D411yescharacteristic
88C2×C4×C97C2411yescharacteristic
89C22⋊D97—44no—
90C2×Dic97—22no—
91C2×Dic97—22no—
92C2×Dic97—44no—
93C2×Dic97C2×D412yes—
94D388—22no—
95C4×D97—22no—
96C4×D97—22no—
97Dic194—22no—
98Dic194—22no—
99Dic194—42no—
100C8×C97—21no—
101C8×C97—21no—
102#11715895—22no—
103C22×C97—22no—
104C22×C97—41no—
105C22×C97—41no—
106C22×C97C22×D411yescharacteristic
107C2×D97—22no—
108C2×D97—22no—
109C2×D97—22no—
110C4×C97—21no—
111C4×C97—41no—
112C4×C97—41no—
113C4×C97—41no—
114C4×C97#13512yes—
115C4×C97C2×#4111yescharacteristic
116C4×C97H(5,𝔽2)11yescharacteristic
117Dic97—22no—
118Dic97—42no—
119Dic97—42no—
120C2×C97—42no—
121C2×C97#54611yescharacteristic
122C2×C97C2×#13511yescharacteristic
123C2×C97#57811yescharacteristic
124D97—42no—
125#3302—971nomaximal, Sylow
126C97#330211yescharacteristic, complemented, Sylow
127#556—971no—
128#547—971no—
129#487—971no—
130#485—972no—
131#474—971no—
132#467—972no—
133#462—971no—
134#452—972no—
135C4×SD16D9711yescharacteristic, complemented
136C8×D4—971no—
137#416—972no—
138C2×#41—972no—
139C2×Q16—972no—
140C2×SD16C2×D9711yescharacteristic
141C22×C8—972no—
142C4⋊Q8—971no—
143C4⋊Q8—971no—
144#126—1942no—
145#124—1941no—
146#123—1942no—
147C22⋊Q8—972no—
148C22⋊Q8—972no—
149#121—1941no—
150C4×Q8—1941no—
151C4×Q8C2×D9711yescharacteristic
152C4×D4—971no—
153C4×D4C2×D9711yescharacteristic
154#117—972no—
155C2×C4⋊C4—1941no—
156#107—972no—
157#106C2×D9711yescharacteristic
158C4⋊C8—971no—
159#103—972no—
160#103—972no—
161#103—972no—
162#103C2×D9711yescharacteristic
163#102—972no—
164#102C2×D9711yescharacteristic
165C22⋊C8—972no—
166C4×C8C2×D9711yescharacteristic
167#41—3882no—
168#41—1944no—
169C2×Q8—972no—
170C2×Q8—1941no—
171C2×Q8C22×D9711yescharacteristic
172C2×D4—971no—
173C2×D4—1942no—
174C2×D4C22×D9711yescharacteristic
175C22×C4—21no—
176C22×C4—972no—
177C22×C4—1942no—
178Q16—3882no—
179SD16—22no—
180C2×C8—972no—
181C2×C8—3882no—
182C2×C8C22×D9711yescharacteristic
183C2×C8C22×D9711yescharacteristic
184C4⋊C4—21no—
185C4⋊C4—971no—
186C4⋊C4—972no—
187C4⋊C4—1942no—
188C4⋊C4—1942no—
189C4⋊C4—1941no—
190C4⋊C4—1941no—
191C4⋊C4—1941no—
192C4⋊C4C22×D9711yescharacteristic
193C4⋊C4C22×D9711yescharacteristic
194C22⋊C4—21no—
195C22⋊C4—972no—
196C22⋊C4—1942no—
197C22⋊C4—1942no—
198C22⋊C4—1942no—
199C42—21no—
200C42—1942no—
201C42C22×D9711yescharacteristic
202C23—21no—
203C23—972no—
204Q8—21no—
205Q8—21no—
206Q8—3882no—
207Q8—1942no—
208Q8—1942no—
209D4—21no—
210D4—3884no—
211D4—1942no—
212D4#1309551112yes—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—42no—
217C2×C4—972no—
218C2×C4—3884no—
219C2×C4—1942no—
220C2×C4—1942no—
221C2×C4—1942no—
222C2×C4—1942no—
223C2×C4D4×D9711yescharacteristic
224C2×C4D4×D9711yescharacteristic
225C2×C4C23×D9711yescharacteristic
226C8—21no—
227C8—21no—
228C8—1942no—
229C22—22no—
230C22—41no—
231C22—41no—
232C22—1942no—
233C22—1942no—
234C22—1942no—
235C22C2×D4×D9711yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—41no—
240C4—3882no—
241C4—3882no—
242C4—1942no—
243C4#1569083112yes—
244C4C2×#1309551111yescharacteristic
245C4#1569085611yescharacteristic
246C2—42no—
247C2—3882no—
248C2#2026138111yescharacteristic
249C2C2×#1569083111yescharacteristic
250C2#2026154911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC4×SD16×C97C2
socleC22×C97C22×D4
prime radical#20261389C2
hypercenterC4×SD16D97
hypocenterC97#3302
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #20261458C2semidirect
3. #20261086C2semidirect
4. #20261079C2semidirect
5. #20261015C2semidirect
6. #20260975C2semidirect
7. #20260945C2semidirect
8. #20260894C2semidirect
9. C8×D388C2semidirect
10. #20260728C2semidirect
11. C4×SD16×C97C2semidirect
126. C97#3302semidirect
135. C4×SD16D97semidirect

Maximal subgroups

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

subgroupindex
#202614582
#202613892
#202610862
#202610792
#202610152
#202609752
#202609452
#202608942
C8×D3882
#202607282
C4×SD16×C972
#330297