Finite Group Databasebeta

← #11784225

Subgroups — #11784225

Order 1152.

398 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 98 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
1C2×#63767C211yesmaximal, characteristic
2C2×C8×#157C211yesmaximal, characteristic
3#10593733C211yesmaximal, characteristic
4#10593726C211yesmaximal, characteristic
5C3⋊D3×C4⋊C8C211yesmaximal, characteristic
6#10590181C211yesmaximal, characteristic
7#10590181C211yesmaximal, characteristic
8C2×C4×#157C2211yescharacteristic
9C2×C4×#157C2211yescharacteristic
10C2×#3787C2211yescharacteristic
11C2×#3787C2211yescharacteristic
12C2×C8×C3⋊D3C2211yescharacteristic
13C2×C8×C3⋊D3C2211yescharacteristic
14C42×C3⋊D3C2211yescharacteristic
15#63781C412yes—
16#63781C412yes—
17#63779C412yescomplemented
18C4⋊#622C412yescomplemented
19#63767—22no—
20C8×#157—22no—
21C32×C4⋊C8C411yescharacteristic, complemented
22#63641C411yescharacteristic, complemented
23#63641C411yescharacteristic, complemented
24C4⋊(C3⋊(C3⋊C8))C411yescharacteristic, complemented
25D3×C4⋊C8—62no—
26C22×#157C2×C412yes—
27C22×#157C2×C412yes—
28C2×C4×C3⋊D3C2×C411yescharacteristic
29C2×C4×C3⋊D3C2×C411yescharacteristic
30C2×C4×C3⋊D3C2311yescharacteristic
31C2×#622C2×C412yes—
32C2×#622C2×C412yes—
33C4×#157—21no—
34C4×#157—21no—
35C4×#157D412yes—
36C4×#157Q812yes—
37#3787—21no—
38#3787—21no—
39#3787—21no—
40#3787—21no—
41C2×C32×C8C2×C411yescharacteristic
42C2×C32×C8C2×C411yescharacteristic
43C32×C42C2×C411yescharacteristic
44C4×C3⋊Dic3C2×C411yescharacteristic
45C4×C3⋊Dic3C2×C411yescharacteristic
46C4×C3⋊Dic3C2×C411yescharacteristic
47C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
48C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
49C8×C3⋊D3—21no—
50C8×C3⋊D3—21no—
51C8×C3⋊D3—21no—
52C8×C3⋊D3—21no—
53#1768—91nomaximal, Sylow
54C2×D3×C8—62no—
55C2×D3×C8—62no—
56C42×D3—62no—
57C3×C4⋊C8—22no—
58Dic3⋊C8—62no—
59Dic3⋊C8—62no—
60C4⋊(C3⋊C8)—62no—
61C22×C3⋊D3C22×C411yescharacteristic
62C2×#157—22no—
63C2×#157—22no—
64C2×#157—22no—
65C2×#157—22no—
66C2×#157—22no—
67C2×#157C4⋊C414yes—
68C2×C32×C4C4211yescharacteristic
69C2×C32×C4C4211yescharacteristic
70C2×C32×C4C22×C411yescharacteristic
71C2×C3⋊Dic3C4211yescharacteristic
72C2×C3⋊Dic3C4211yescharacteristic
73C2×C3⋊Dic3C22×C411yescharacteristic
74C4×C3⋊D3—22no—
75C4×C3⋊D3—22no—
76C4×C3⋊D3C2×D411yescharacteristic
77C4×C3⋊D3C2×Q811yescharacteristic
78C4×C3⋊D3#4111yescharacteristic
79C4×C3⋊D3#4111yescharacteristic
80#622—22no—
81#622—22no—
82C32×C8—21no—
83C32×C8—21no—
84C3⋊(C3⋊C8)—21no—
85C3⋊(C3⋊C8)—21no—
86C2×C4⋊C8—91no—
87C2×C4⋊C8—91no—
88C2×#97—91no—
89C2×C4×C8—91no—
90#376—91no—
91#336—91no—
92#336—91no—
93C2×C4×D3—62no—
94C2×C4×D3—62no—
95C2×C4×D3—62no—
96C2×C3×C8—22no—
97C2×C3×C8—22no—
98C3×C42—22no—
99C4×Dic3—62no—
100C4×Dic3—62no—
101C4×Dic3—62no—
102C2×C3⋊C8—62no—
103C2×C3⋊C8—62no—
104D3×C8—122no—
105D3×C8—122no—
106D3×C8—122no—
107D3×C8—122no—
108C22×C32C2×C4211yescharacteristic
109C2×C3⋊D3C2×C4⋊C411yescharacteristic
110C2×C3⋊D3#11711yescharacteristic
111C2×C3⋊D3#13112yes—
112C2×C3⋊D3#13112yes—
113#157—24no—
114#157—42no—
115C32×C4—21no—
116C32×C4—21no—
117C32×C4C4×D411yescharacteristic
118C32×C4C4×Q811yescharacteristic
119C3⋊Dic3—21no—
120C3⋊Dic3—21no—
121C3⋊Dic3C4×D411yescharacteristic
122C3⋊Dic3C4×Q811yescharacteristic
123C22×C8—91no—
124C22×C8—91no—
125C22×C8—91no—
126C22×C8—91no—
127C2×C42—91no—
128C2×C42—91no—
129C2×C42—91no—
130C4⋊C8—92no—
131C4⋊C8—92no—
132C4⋊C8—91no—
133C4⋊C8—91no—
134C4⋊C8—91no—
135C4⋊C8#15711yescharacteristic, complemented
136#97—182no—
137C4×C8—182no—
138#95—92no—
139#95—92no—
140C22×D3—62no—
141C2×C3×C4—22no—
142C2×C3×C4—22no—
143C2×C3×C4—22no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—62no—
147C4×D3—62no—
148C4×D3—62no—
149C4×D3—62no—
150C4×D3—62no—
151C4×D3—68no—
152C4×D3—124no—
153C3×C8—42no—
154C3×C8—42no—
155C3⋊C8—122no—
156C3⋊C8—122no—
157C2×C32C4×C4⋊C411yescharacteristic
158C2×C32#40511yescharacteristic
159C2×C32#40511yescharacteristic
160C3⋊D3#42412yes—
161C3⋊D3#43312yes—
162C22×C4—91no—
163C22×C4—92no—
164C22×C4—91no—
165C22×C4—91no—
166C22×C4—92no—
167C2×C8—92no—
168C2×C8—92no—
169C2×C8—91no—
170C2×C8—91no—
171C2×C8—181no—
172C2×C8—181no—
173C2×C8—181no—
174C2×C8—181no—
175C2×C8—181no—
176C2×C8—181no—
177C2×C8—181no—
178C2×C8—181no—
179C2×C8C2×#15711yescharacteristic
180C2×C8C2×#15711yescharacteristic
181C42—92no—
182C42—92no—
183C42—91no—
184C42—91no—
185C42—91no—
186C42—181no—
187C42—181no—
188C42C2×#15711yescharacteristic
189C22×C3—22no—
190C2×D3—62no—
191C2×D3—62no—
192C2×D3—64no—
193C2×D3—64no—
194C3×C4—24no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—42no—
198Dic3—62no—
199Dic3—62no—
200Dic3—64no—
201Dic3—122no—
202C32#176811yescharacteristic, complemented, Sylow
203C23—91no—
204C2×C4—91no—
205C2×C4—91no—
206C2×C4—91no—
207C2×C4—91no—
208C2×C4—91no—
209C2×C4—94no—
210C2×C4—91no—
211C2×C4—91no—
212C2×C4—182no—
213C2×C4—182no—
214C2×C4—182no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—182no—
218C2×C4—182no—
219C2×C4C4×#15711yescharacteristic
220C2×C4C4×#15711yescharacteristic
221C2×C4C22×#15711yescharacteristic
222C8—21no—
223C8—21no—
224C8—182no—
225C8—182no—
226C8—181no—
227C8—181no—
228C2×C3—22no—
229C2×C3—22no—
230C2×C3—22no—
231D3—64no—
232D3—64no—
233C22—91no—
234C22—91no—
235C22—92no—
236C22—92no—
237C22C2×C4×#15711yescharacteristic
238C4—21no—
239C4—21no—
240C4—91no—
241C4—91no—
242C4—184no—
243C4—181no—
244C4—181no—
245C4—362no—
246C4D4×#15711yescharacteristic
247C4Q8×#15711yescharacteristic
248C3—22no—
249C2—92no—
250C2—92no—
251C2C4⋊C4×#15711yescharacteristic
252C2#1059376511yescharacteristic
253C2#1059377511yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×#157
commutator, socleC22×C32C2×C42
FittingC32×C4⋊C8C4
prime radicalC22×C3⋊D3C22×C4
cosocleC2×C4×C3⋊D3C23
hypercenterC4⋊C8#157
hypocenterC32#1768
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
17. #63779C4semidirect
18. C4⋊#622C4semidirect
21. C32×C4⋊C8C4semidirect
22. #63641C4semidirect
23. #63641C4semidirect
24. C4⋊(C3⋊(C3⋊C8))C4semidirect
135. C4⋊C8#157semidirect
202. C32#1768semidirect

Maximal subgroups

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

subgroupindex
C2×#637672
C2×C8×#1572
#105937332
#105937262
C3⋊D3×C4⋊C82
#105901812
#17689