Finite Group Databasebeta

← #27151390

Subgroups — #27151390

Order 12416.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#13095501C211yesmaximal, characteristic, complemented
2C2×#15690818C211yesmaximal, characteristic
3#20261311C211yesmaximal, characteristic
4#20261309C211yesmaximal, characteristic
5#20260992C214yesmaximal, complemented
6#20260975C214yesmaximal, complemented
7#20260771C211yesmaximal, characteristic, complemented
8#20260745C211yesmaximal, characteristic, complemented
9#419×C97C211yesmaximal, characteristic, complemented
10C2×#13095511C2211yescharacteristic
11C22×Dic194C2211yescharacteristic
12C2×#13095501—24no—
13C2×#13095501C2212yes—
14C2×#13095501C2212yes—
15#15690818—24no—
16C2×#13095494C2212yes—
17C22×#11715895C2211yescharacteristic
18C2×C8×D97C2212yes—
19#15690810—21no—
20#15690808—21no—
21#15690799—21no—
22#15690798—21no—
23#15690794—22no—
24#15690793—22no—
25#15690790C2214yes—
26#15690786C2214yes—
27#15690769C2211yescharacteristic
28#15690743C2212yes—
29#15690741C2212yes—
30#15690735C2214yes—
31C22×Q8×C97C2211yescharacteristic
32C2×M16×C97C2211yescharacteristic
33#117×C97C2211yescharacteristic
34#103×C97C2214yes—
35#13095511—22no—
36#13095511—22no—
37C2×Dic194—22no—
38C2×Dic194—22no—
39C2×Dic194—21no—
40C2×Dic194D411yescharacteristic
41C2×Dic194C2312yes—
42C2×D388D411yescharacteristic
43C2×C22⋊D97D412yes—
44C22×Dic97—21no—
45C2×C4×D97C2312yes—
46#13095501—24no—
47#13095501—24no—
48#13095494—24no—
49C2×#11715895D414yes—
50C2×#11715895C2311yescharacteristic
51C2×#11715895C2311yescharacteristic
52C8×D97—24no—
53C4⋊Dic97—24no—
54C4⋊Dic97C2312yes—
55#13095490—24no—
56C22⋊Dic97—21no—
57#13095488—24no—
58#13095488—24no—
59C4×Dic97—21no—
60C2×Q8×C97—21no—
61C2×Q8×C97—22no—
62C2×Q8×C97—22no—
63C2×Q8×C97C2312yes—
64C22×C4×C97—21no—
65C22×C4×C97C2311yescharacteristic
66M16×C97—22no—
67C2×C8×C97C2312yes—
68C4⋊C4×C97C2312yes—
69C22⋊C4×C97—21no—
70C42×C97—21no—
71C23×C97C2×D411yescharacteristic
72C22×D97C2×D412yes—
73Q8×C97—22no—
74Q8×C97—22no—
75Q8×C97—42no—
76C2×C4×C97—22no—
77C2×C4×C97—22no—
78C2×C4×C97—42no—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D412yes—
81C2×C4×C97#4112yes—
82C2×C4×C97C2411yescharacteristic
83C22⋊D97—24no—
84C2×Dic97—22no—
85C2×Dic97—22no—
86C2×Dic97—42no—
87C2×Dic97C2×D412yes—
88D388—22no—
89C4×D97—22no—
90C4×D97—22no—
91Dic194—22no—
92Dic194—22no—
93Dic194—22no—
94Dic194—42no—
95C8×C97—22no—
96#11715895C2×D412yes—
97#11715895C2×D412yes—
98C22×C97—22no—
99C22×C97#12112yes—
100C22×C97C22×D411yescharacteristic
101C2×D97—22no—
102C2×D97—24no—
103C4×C97—42no—
104C4×C97—42no—
105C4×C97#12112yes—
106C4×C97C22×D411yescharacteristic
107C4×C97C2×#4111yescharacteristic
108Dic97—22no—
109Dic97—42no—
110Dic97—42no—
111C2×C97—22no—
112C2×C97C2×#12111yescharacteristic
113C2×C97#57812yes—
114D97—42no—
115#3059—971nomaximal, Sylow
116C97#305911yescharacteristic, complemented, Sylow
117C22×Q16—971no—
118C2×#131—971no—
119#536—971no—
120#536—971no—
121#470—974no—
122#467—974no—
123#427—971no—
124#419—971no—
125#419D9711yescharacteristic, complemented
126C2×#41—971no—
127C22×Q8—971no—
128C22×Q8C2×D9711yescharacteristic
129C2×Q16—972no—
130C2×Q16—972no—
131C2×Q16—1944no—
132#131—1944no—
133C2×M16—972no—
134C2×M16C2×D9711yescharacteristic
135C22×C8—971no—
136C22×C8—972no—
137C4⋊Q8—1941no—
138C4⋊Q8—1941no—
139#124—1941no—
140#124—1941no—
141#123—1942no—
142#123—1942no—
143C22⋊Q8—974no—
144C22⋊Q8—974no—
145#117—971no—
146#117C2×D9711yescharacteristic
147#107—972no—
148#106—972no—
149#103—974no—
150#103C2×D9714yes—
151#41—1942no—
152#41—1942no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—971no—
157C2×Q8—972no—
158C2×Q8—1942no—
159C2×Q8—1942no—
160C2×Q8—1941no—
161C2×Q8C22×D9712yes—
162C2×D4—972no—
163C2×D4—971no—
164C22×C4—21no—
165C22×C4—972no—
166C22×C4—1941no—
167C22×C4C22×D9711yescharacteristic
168Q16—1944no—
169Q16—1944no—
170M16—22no—
171M16—1944no—
172C2×C8—974no—
173C2×C8—971no—
174C2×C8—971no—
175C2×C8—1944no—
176C2×C8C22×D9712yes—
177C4⋊C4—972no—
178C4⋊C4—1944no—
179C4⋊C4—1944no—
180C4⋊C4C22×D9712yes—
181C22⋊C4—21no—
182C22⋊C4—1944no—
183C22⋊C4—1944no—
184C22⋊C4—1941no—
185C42—21no—
186C42—1941no—
187C23—972no—
188C23D4×D9711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—3882no—
193Q8—1942no—
194Q8—1942no—
195Q8—1942no—
196D4—1944no—
197D4—1942no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—972no—
202C2×C4—3882no—
203C2×C4—1942no—
204C2×C4—1942no—
205C2×C4—1942no—
206C2×C4—1942no—
207C2×C4D4×D9711yescharacteristic
208C2×C4D4×D9712yes—
209C2×C4#1309551112yes—
210C2×C4C23×D9711yescharacteristic
211C8—22no—
212C8—972no—
213C8—972no—
214C22—22no—
215C22—1942no—
216C22—1944no—
217C22#1569078312yes—
218C22C2×D4×D9711yescharacteristic
219C4—42no—
220C4—42no—
221C4—3882no—
222C4—3882no—
223C4—1942no—
224C4#1569078312yes—
225C4C2×D4×D9711yescharacteristic
226C4C2×#1309551111yescharacteristic
227C2—22no—
228C2—3882no—
229C2C2×#1569078311yescharacteristic
230C2#2026155112yes—
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#419×C97C2
socleC22×C97C22×D4
prime radicalC2×#13095511C22
hypercenter#419D97
hypocenterC97#3059
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
5. #20260992C2semidirect
6. #20260975C2semidirect
7. #20260771C2semidirect
8. #20260745C2semidirect
9. #419×C97C2semidirect
116. C97#3059semidirect
125. #419D97semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×#156908182
#202613112
#202613092
#202609922
#202609752
#202607712
#202607452
#419×C972
#305997