Finite Group Databasebeta

← C2×#10593283 #11859298

Subgroups — C2×#10593283 #11859298

Order 1152.

850 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 167 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×C4×GO+(4,𝔽2)C211yesmaximal, characteristic
2C2×#63791C212yesmaximal, complemented
3C2×C8×#157C211yesmaximal, characteristic, complemented
4C2×#63732C211yesmaximal, characteristic, complemented
5C2×D32⋊C8C212yesmaximal, complemented
6#10593283C218yesmaximal, direct product factor
7C22×GO+(4,𝔽2)C411yescharacteristic
8C2×C4×D32C2212yes—
9C2×C4×#157C2211yescharacteristic
10C2×#3787C2211yescharacteristic
11C2×#3773C411yescharacteristic
12C2×D32⋊C4C412yes—
13C4×GO+(4,𝔽2)C414yes—
14C4×GO+(4,𝔽2)C2214yes—
15C2×C8×C3⋊D3C2211yescharacteristic
16C2×C3×ΓL(1,𝔽25)—22no—
17C2×#3711—22no—
18C2×#3710C2212yes—
19#63791—24no—
20#63791C2218yescomplemented
21C8×#157C2214yescomplemented
22#63732C2214yescomplemented
23D32⋊C8C2218yescomplemented
24C2×D3×M16—62no—
25C22×D32C2×C412yes—
26C22×#157C2×C411yescharacteristic
27C22×#157C2×C411yescharacteristic
28C2×GO+(4,𝔽2)—24no—
29C2×GO+(4,𝔽2)C2×C414yes—
30C2×C4×C3⋊D3C2311yescharacteristic
31C2×C3×C4×D3—22no—
32C2×#624C2×C412yes—
33C2×D3×Dic3—22no—
34C4×D32—22no—
35C4×D32—22no—
36C4×D32C2×C414yes—
37C4×D32C2314yes—
38C2×#622—21no—
39C4×#157C2×C412yes—
40C4×#157C2312yes—
41#3787—21no—
42#3787C2312yes—
43#3773C2×C414yes—
44D32⋊C4C2×C418yes—
45C2×C32×C8D411yescharacteristic, complemented
46C2×C3⋊(C3⋊C8)D411yescharacteristic, complemented
47C8×C3⋊D3D412yescomplemented
48C8×C3⋊D3C2312yes—
49C2×C3×C3⋊C8—22no—
50C3×ΓL(1,𝔽25)—28no—
51#3711—28no—
52#3710—22no—
53#3710C2314yes—
54C2×#436—91nomaximal, Sylow
55C22×C4×D3—62no—
56C2×C3×M16—62no—
57C2×#208—62no—
58D3×M16—68no—
59D3×M16—68no—
60C2×ΓL(1,𝔽25)—22no—
61C2×ΓL(1,𝔽25)—62no—
62C2×D3×C8—62no—
63C2×D3×C8—62no—
64C22×C3⋊D3C22×C411yescharacteristic
65C22×C3×D3—22no—
66C2×D32—22no—
67C2×D32—24no—
68C2×D32—24no—
69C2×D32C22×C414yes—
70C2×#157—22no—
71C2×#157M1614yescomplemented
72C2×#157C22×C412yes—
73C2×#157C22×C412yes—
74GO+(4,𝔽2)—28no—
75C2×C32×C4C2×D411yescharacteristic
76C2×C3⋊Dic3#4111yescharacteristic
77C4×C3⋊D3C22×C411yescharacteristic
78C4×C3⋊D3C2×D411yescharacteristic
79C4×C3⋊D3#4111yescharacteristic
80C4×C3⋊D3C2411yescharacteristic
81C2×C3×Dic3—22no—
82C3×C4×D3—24no—
83C3×C4×D3—24no—
84#624—22no—
85#624C22×C414yes—
86D3×Dic3—24no—
87D3×Dic3—24no—
88#622—22no—
89C32×C8C2×D412yescomplemented
90C3⋊(C3⋊C8)C2×D412yescomplemented
91C3×C3⋊C8—24no—
92C22×M16—92no—
93C2×C4×D4—91no—
94#436—98no—
95C2×C4⋊C8—91no—
96C2×C22⋊C8—92no—
97C2×C4×C8—91no—
98C23×D3—62no—
99C22×C3×C4—62no—
100C22×Dic3—62no—
101C2×C4×D3—22no—
102C2×C4×D3—64no—
103C2×C4×D3—64no—
104C2×C4×D3—62no—
105C2×C4×D3—62no—
106C2×C4×D3—64no—
107C2×C4×D3—64no—
108C2×C4×D3—62no—
109C3×M16—68no—
110C2×C3×C8—22no—
111C2×C3×C8—62no—
112#208—68no—
113C2×C3⋊C8—22no—
114C2×C3⋊C8—62no—
115ΓL(1,𝔽25)—28no—
116ΓL(1,𝔽25)—68no—
117D3×C8—64no—
118D3×C8—64no—
119D3×C8—64no—
120D3×C8—64no—
121C22×C32C4×D411yescharacteristic
122C2×C3⋊D3C4×D411yescharacteristic
123C2×C3⋊D3C2×M1612yes—
124C2×C3⋊D3#13112yes—
125C2×C3⋊D3C23×C411yescharacteristic
126C2×C3×D3—24no—
127C2×C3×D3—44no—
128D32—24no—
129D32—24no—
130D32—44no—
131#157—22no—
132#157C2×M1614yescomplemented
133C32×C4C4×D411yescharacteristic
134C32×C4C22×D411yescharacteristic
135C3⋊Dic3C4×D411yescharacteristic
136C3⋊Dic3C2×#4111yescharacteristic
137C3×Dic3—24no—
138C22×D4—91no—
139C23×C4—92no—
140C2×M16—62no—
141C2×M16—98no—
142C2×M16—182no—
143C2×M16—184no—
144C22×C8—92no—
145C22×C8—91no—
146C22×C8—91no—
147C4×D4—94no—
148C4×D4—94no—
149C2×C4⋊C4—91no—
150C2×C22⋊C4—92no—
151C2×C42—91no—
152C4⋊C8—94no—
153C22⋊C8—98no—
154C4×C8—94no—
155C23×C3—62no—
156C22×D3—22no—
157C22×D3—64no—
158C22×D3—62no—
159C22×D3—64no—
160C22×D3—124no—
161C22×D3—124no—
162C2×C3×C4—22no—
163C2×C3×C4—64no—
164C2×C3×C4—64no—
165C2×C3×C4—62no—
166C2×Dic3—22no—
167C2×Dic3—62no—
168C2×Dic3—64no—
169C2×Dic3—64no—
170C4×D3—24no—
171C4×D3—24no—
172C4×D3—64no—
173C4×D3—62no—
174C4×D3—62no—
175C4×D3—62no—
176C4×D3—62no—
177C4×D3—64no—
178C4×D3—64no—
179C4×D3—64no—
180C3×C8—24no—
181C3×C8—64no—
182C3⋊C8—24no—
183C3⋊C8—64no—
184C2×C32#43612yescomplemented
185C2×C32C2×C4×D411yescharacteristic
186C3⋊D3#43612yescomplemented
187C3⋊D3C22×M1611yescharacteristic
188C3⋊D3C2×#13111yescharacteristic
189C3×D3—44no—
190C24—92no—
191C2×D4—94no—
192C2×D4—184no—
193C22×C4—62no—
194C22×C4—94no—
195C22×C4—92no—
196C22×C4—94no—
197C22×C4—91no—
198C22×C4—91no—
199C22×C4—91no—
200C22×C4—182no—
201C22×C4—182no—
202C22×C4—182no—
203M16—68no—
204M16—188no—
205C2×C8—62no—
206C2×C8—94no—
207C2×C8—91no—
208C2×C8—92no—
209C2×C8—92no—
210C2×C8—92no—
211C2×C8—182no—
212C2×C8—181no—
213C2×C8—181no—
214C2×C8GO+(4,𝔽2)11yescharacteristic, complemented
215C4⋊C4—94no—
216C22⋊C4—98no—
217C42—92no—
218C42—92no—
219C22×C3—22no—
220C22×C3—64no—
221C22×C3—124no—
222C2×D3—24no—
223C2×D3—44no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—64no—
227C2×D3—64no—
228C2×D3—124no—
229C2×D3—124no—
230C2×D3—128no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—64no—
234Dic3—24no—
235Dic3—62no—
236Dic3—62no—
237C32C2×#43611yescharacteristic, complemented, Sylow
238C23—62no—
239C23—94no—
240C23—91no—
241C23—182no—
242C23—184no—
243C23—184no—
244D4—188no—
245C2×C4—62no—
246C2×C4—64no—
247C2×C4—64no—
248C2×C4—94no—
249C2×C4—92no—
250C2×C4—94no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—91no—
255C2×C4—91no—
256C2×C4—92no—
257C2×C4—182no—
258C2×C4—184no—
259C2×C4—184no—
260C2×C4—182no—
261C2×C4C2×GO+(4,𝔽2)11yescharacteristic
262C8—64no—
263C8—92no—
264C8—182no—
265C8C2×GO+(4,𝔽2)12yescomplemented
266C2×C3—24no—
267C2×C3—22no—
268C2×C3—124no—
269D3—44no—
270D3—62no—
271D3—62no—
272D3—64no—
273C22—64no—
274C22—92no—
275C22—92no—
276C22—91no—
277C22—91no—
278C22—124no—
279C22—184no—
280C22—184no—
281C22—364no—
282C22C4×GO+(4,𝔽2)11yescharacteristic
283C4—64no—
284C4—94no—
285C4—91no—
286C4—91no—
287C4—182no—
288C4C4×GO+(4,𝔽2)11yescharacteristic
289C4C22×GO+(4,𝔽2)11yescharacteristic
290C3—22no—
291C2—91no—
292C2—91no—
293C2—92no—
294C2—124no—
295C2#1059328312yesdirect product factor
296C2C2×C4×GO+(4,𝔽2)11yescharacteristic
subgroupG/H
centerC2×C4C2×GO+(4,𝔽2)
commutatorC2×C3⋊D3C23×C4
FrattiniC4C22×GO+(4,𝔽2)
FittingC2×C32×C8D4
socleC22×C32C4×D4
prime radicalC22×GO+(4,𝔽2)C4
cosocleC4×C3⋊D3C24
hypercenterC2×C8GO+(4,𝔽2)
hypocenterC32C2×#436
LCSG ▷ C2×C3⋊D3 ▷ C32
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#63791C2semidirect
3. C2×C8×#157C2semidirect
4. C2×#63732C2semidirect
5. C2×D32⋊C8C2semidirect
6. #10593283C2direct (N × H)
20. #63791C22semidirect
21. C8×#157C22semidirect
22. #63732C22semidirect
23. D32⋊C8C22semidirect
45. C2×C32×C8D4semidirect
46. C2×C3⋊(C3⋊C8)D4semidirect
47. C8×C3⋊D3D4semidirect
71. C2×#157M16semidirect
89. C32×C8C2×D4semidirect
90. C3⋊(C3⋊C8)C2×D4semidirect
132. #157C2×M16semidirect
184. C2×C32#436semidirect
186. C3⋊D3#436semidirect
214. C2×C8GO+(4,𝔽2)semidirect
237. C32C2×#436semidirect
265. C8C2×GO+(4,𝔽2)semidirect
295. C2#10593283direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×GO+(4,𝔽2)2
C2×#637912
C2×C8×#1572
C2×#637322
C2×D32⋊C82
#105932832
C2×#4369