Finite Group Databasebeta

← #11783110

Subgroups — #11783110

Order 1152.

714 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 714 conjugacy classes of subgroups — too many to draw. Only its 138 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×#64286C211yesmaximal, characteristic
2C22×#3787C211yesmaximal, characteristic
3C2×M16×C3⋊D3C211yesmaximal, characteristic
4#10590181C214yesmaximal, complemented
5C22×C4×C3⋊D3C2211yescharacteristic
6C2×C22⋊#157C412yes—
7C22×#622C412yescomplemented
8#64286—22no—
9#64286—22no—
10C2×C4⋊#157C412yes—
11C2×C4×#157C2212yes—
12C2×#3787—22no—
13C2×#3787—22no—
14C2×#3787C412yescomplemented
15C2×#3787C2212yes—
16C2×C32×M16C411yescharacteristic, complemented
17C2×#3748C411yescharacteristic, complemented
18M16×C3⋊D3—22no—
19M16×C3⋊D3—22no—
20C2×#3743C412yescomplemented
21C2×C8×C3⋊D3C2212yes—
22C2×D3×M16—62no—
23C23×C3⋊D3C2×C411yescharacteristic
24C22×#157C2×C414yes—
25C22×C32×C4C2×C411yescharacteristic
26C22×C3⋊Dic3C2×C411yescharacteristic
27C2×C4×C3⋊D3C2×C411yescharacteristic
28C2×C4×C3⋊D3C2×C411yescharacteristic
29C2×C4×C3⋊D3C2×C411yescharacteristic
30C2×C4×C3⋊D3D412yes—
31C2×C4×C3⋊D3D412yes—
32C2×C4×C3⋊D3D412yes—
33C2×C4×C3⋊D3Q812yes—
34C2×C4×C3⋊D3C2311yescharacteristic
35C22⋊#157—24no—
36C2×#622—24no—
37C2×#622C2×C414yes—
38C4⋊#157—24no—
39C4×#157—22no—
40C4×#157—22no—
41#3787—24no—
42#3787—22no—
43#3787—22no—
44C32×M16—22no—
45C2×C32×C8C2×C412yes—
46#3748—22no—
47C2×C3⋊(C3⋊C8)C2×C412yes—
48#3743—24no—
49C8×C3⋊D3—22no—
50C8×C3⋊D3—22no—
51#1729—91nomaximal, Sylow
52C22×C4×D3—62no—
53C2×C3×M16—22no—
54C2×#208—62no—
55D3×M16—68no—
56D3×M16—68no—
57C2×ΓL(1,𝔽25)—64no—
58C2×D3×C8—64no—
59C23×C32C4211yescharacteristic
60C22×C3⋊D3—24no—
61C22×C3⋊D3C4211yescharacteristic
62C22×C3⋊D3C22⋊C412yes—
63C22×C3⋊D3C4⋊C412yes—
64C22×C3⋊D3C22×C411yescharacteristic
65C2×#157—24no—
66C2×#157—28no—
67C2×C32×C4C4211yescharacteristic
68C2×C32×C4C22⋊C412yes—
69C2×C32×C4C4⋊C412yes—
70C2×C32×C4C22×C411yescharacteristic
71C2×C3⋊Dic3C4211yescharacteristic
72C2×C3⋊Dic3C22⋊C412yes—
73C2×C3⋊Dic3C4⋊C412yes—
74C2×C3⋊Dic3C22×C411yescharacteristic
75C4×C3⋊D3C22⋊C412yes—
76C4×C3⋊D3C22⋊C412yes—
77C4×C3⋊D3C22⋊C412yes—
78C4×C3⋊D3C4⋊C412yes—
79C4×C3⋊D3C4⋊C412yes—
80C4×C3⋊D3C4⋊C412yes—
81C4×C3⋊D3C2×D411yescharacteristic
82C4×C3⋊D3C2×D411yescharacteristic
83C4×C3⋊D3C2×D411yescharacteristic
84C4×C3⋊D3C2×Q811yescharacteristic
85#622—24no—
86C32×C8—22no—
87C3⋊(C3⋊C8)—22no—
88C22×M16—91no—
89C23×C8—91no—
90C2×#117—91no—
91#336—94no—
92C23×D3—62no—
93C22×C3×C4—22no—
94C22×Dic3—62no—
95C2×C4×D3—62no—
96C2×C4×D3—62no—
97C2×C4×D3—64no—
98C2×C4×D3—64no—
99C2×C4×D3—62no—
100C2×C4×D3—62no—
101C2×C4×D3—64no—
102C2×C4×D3—64no—
103C3×M16—28no—
104C2×C3×C8—24no—
105#208—68no—
106C2×C3⋊C8—64no—
107ΓL(1,𝔽25)—616no—
108D3×C8—68no—
109D3×C8—68no—
110C22×C32—22no—
111C22×C32#9512yes—
112C22×C32C2×C4211yescharacteristic
113C2×C3⋊D3—22no—
114C2×C3⋊D3—24no—
115C2×C3⋊D3—24no—
116C2×C3⋊D3#9512yes—
117C2×C3⋊D3C2×C22⋊C411yescharacteristic
118C2×C3⋊D3C2×C4⋊C411yescharacteristic
119C2×C3⋊D3#13114yes—
120#157—44no—
121C32×C4#9512yes—
122C32×C4C2×C22⋊C411yescharacteristic
123C32×C4C2×C4⋊C411yescharacteristic
124C3⋊Dic3#9512yes—
125C3⋊Dic3C2×C22⋊C411yescharacteristic
126C3⋊Dic3C2×C4⋊C411yescharacteristic
127C23×C4—91no—
128C2×M16—91no—
129C2×M16—92no—
130C2×M16—182no—
131C2×M16—182no—
132C2×M16#15711yescharacteristic, complemented
133C22×C8—92no—
134C22×C8—92no—
135C22×C8—92no—
136C22×C8—92no—
137C22×C8—182no—
138C22×C8—182no—
139#117—182no—
140#117—182no—
141C2×C4⋊C4—92no—
142C2×C22⋊C4—92no—
143C2×C42—92no—
144C23×C3—22no—
145C22×D3—62no—
146C22×D3—62no—
147C22×D3—64no—
148C22×D3—64no—
149C22×D3—128no—
150C2×C3×C4—22no—
151C2×C3×C4—22no—
152C2×C3×C4—24no—
153C2×C3×C4—24no—
154C2×Dic3—62no—
155C2×Dic3—64no—
156C2×Dic3—64no—
157C2×Dic3—62no—
158C4×D3—64no—
159C4×D3—64no—
160C4×D3—64no—
161C4×D3—64no—
162C4×D3—64no—
163C4×D3—64no—
164C4×D3—62no—
165C4×D3—62no—
166C4×D3—62no—
167C4×D3—62no—
168C3×C8—28no—
169C3⋊C8—68no—
170C2×C32—22no—
171C2×C32C2×#9511yescharacteristic
172C2×C32#40512yes—
173C3⋊D3—22no—
174C3⋊D3#40812yes—
175C3⋊D3#42412yes—
176C24—91no—
177C22×C4—91no—
178C22×C4—91no—
179C22×C4—91no—
180C22×C4—91no—
181C22×C4—91no—
182C22×C4—92no—
183C22×C4—92no—
184C22×C4—92no—
185C22×C4—92no—
186C22×C4—94no—
187C22×C4C2×#15711yescharacteristic
188M16—22no—
189M16—184no—
190M16—182no—
191C2×C8—94no—
192C2×C8—92no—
193C2×C8—182no—
194C2×C8—182no—
195C2×C8—182no—
196C2×C8—184no—
197C2×C8—182no—
198C2×C8—184no—
199C2×C8C2×#15712yes—
200C4⋊C4—184no—
201C22⋊C4—184no—
202C42—182no—
203C42—182no—
204C22×C3—22no—
205C22×C3—24no—
206C22×C3—44no—
207C2×D3—62no—
208C2×D3—62no—
209C2×D3—68no—
210C2×D3—64no—
211C2×D3—128no—
212C2×D3—128no—
213C2×D3—124no—
214C3×C4—22no—
215C3×C4—24no—
216C3×C4—22no—
217Dic3—64no—
218Dic3—62no—
219Dic3—62no—
220C32#172911yescharacteristic, complemented, Sylow
221C23—91no—
222C23—91no—
223C23—92no—
224C23—92no—
225C23—184no—
226C23C4×#15711yescharacteristic
227C2×C4—91no—
228C2×C4—91no—
229C2×C4—92no—
230C2×C4—92no—
231C2×C4—91no—
232C2×C4—91no—
233C2×C4—92no—
234C2×C4—92no—
235C2×C4—92no—
236C2×C4—92no—
237C2×C4—92no—
238C2×C4—91no—
239C2×C4—91no—
240C2×C4—92no—
241C2×C4—184no—
242C2×C4—188no—
243C2×C4C4×#15711yescharacteristic
244C2×C4C4⋊#15712yes—
245C2×C4C22⋊#15712yes—
246C2×C4C22×#15711yescharacteristic
247C8—22no—
248C8—184no—
249C8—182no—
250C2×C3—22no—
251C2×C3—24no—
252C2×C3—44no—
253D3—64no—
254D3—64no—
255D3—124no—
256C22—22no—
257C22—91no—
258C22—91no—
259C22—94no—
260C22—92no—
261C22—184no—
262C22—184no—
263C22—182no—
264C22#6378112yes—
265C22C2×C4×#15711yescharacteristic
266C4—91no—
267C4—91no—
268C4—92no—
269C4—364no—
270C4#6378112yes—
271C4C2×C4⋊#15711yescharacteristic
272C4C2×C22⋊#15711yescharacteristic
273C3—22no—
274C2—22no—
275C2—92no—
276C2—92no—
277C2—182no—
278C2#1059377512yes—
279C2C2×#6378111yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×#157
commutator, socleC22×C32C2×C42
FittingC2×C32×M16C4
prime radicalC23×C3⋊D3C2×C4
cosocleC2×C4×C3⋊D3C23
hypercenterC2×M16#157
hypocenterC32#1729
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. #10590181C2semidirect
7. C22×#622C4semidirect
14. C2×#3787C4semidirect
16. C2×C32×M16C4semidirect
17. C2×#3748C4semidirect
20. C2×#3743C4semidirect
132. C2×M16#157semidirect
220. C32#1729semidirect

Maximal subgroups

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

subgroupindex
C2×#642862
C22×#37872
C2×M16×C3⋊D32
#105901812
#17299