Finite Group Databasebeta

← C2×#10593742 #11859272

Subgroups — C2×#10593742 #11859272

Order 1152.

978 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 978 conjugacy classes of subgroups — too many to draw. Only its 186 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×#3792C211yesmaximal, characteristic, complemented
2C22×#3788C211yesmaximal, characteristic, complemented
3C2×#64090C211yesmaximal, characteristic
4C2×C22⋊#622C211yesmaximal, characteristic, complemented
5C2×#63779C211yesmaximal, characteristic, complemented
6C2×#63778C211yesmaximal, characteristic, complemented
7C2×C4×#622C211yesmaximal, characteristic, complemented
8#10593742C218yesmaximal, direct product factor
9C22×#638C411yescharacteristic
10C23×C3⋊Dic3C2211yescharacteristic
11C22×C4×C3⋊D3C2211yescharacteristic
12C2×#3792—24no—
13C2×#3792C2214yescomplemented
14C22×#622C2211yescharacteristic
15C22×#622C2211yescharacteristic
16C22×#622C2211yescharacteristic
17C22×#622C2211yescharacteristic
18C2×#3788—24no—
19C2×#3788C2214yescomplemented
20C2×C32×C22⋊C4C411yescharacteristic
21C2×#3752C411yescharacteristic
22C2×#3750C411yescharacteristic
23C2×C4×C3⋊Dic3C2211yescharacteristic
24#64090C414yes—
25#64090C2214yes—
26C22⋊#622C2214yescomplemented
27#63779C2214yescomplemented
28#63778C2214yescomplemented
29C4×#622C2214yescomplemented
30C2×#67⋊C4—62no—
31C24×C32C2×C411yescharacteristic
32C23×C3⋊D3C2×C411yescharacteristic
33C22×C32×C4C2×C411yescharacteristic
34C22×C32×C4C2×C411yescharacteristic
35C2×#638—24no—
36C2×#638C2×C414yes—
37C22×C3⋊Dic3—22no—
38C22×C3⋊Dic3—22no—
39C22×C3⋊Dic3C2×C412yes—
40C22×C3⋊Dic3C2×C411yescharacteristic
41C22×C3⋊Dic3C2×C411yescharacteristic
42C22×C3⋊Dic3C2312yes—
43C22×C3⋊Dic3C2311yescharacteristic
44C2×C4×C3⋊D3—22no—
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3C2×C412yes—
47C2×C4×C3⋊D3C2312yes—
48#3792—28no—
49C2×#622—22no—
50C2×#622—22no—
51C2×#622—22no—
52C2×#622D414yescomplemented
53C2×#622C2312yes—
54C2×#622C2312yes—
55C2×#622C2312yes—
56C2×#622C2312yes—
57#3788—28no—
58C32×C22⋊C4C2×C414yes—
59#3752C2×C414yes—
60#3750C2×C414yes—
61C4×C3⋊Dic3C2×C412yes—
62C4×C3⋊Dic3C2312yes—
63C2×#436—91nomaximal, Sylow
64C22×#67—62no—
65C23×Dic3—62no—
66C22×C4×D3—62no—
67C2×C3×C22⋊C4—22no—
68C2×(C2×D3)⋊C4—62no—
69C2×Dic3⋊C4—62no—
70C2×C4×Dic3—62no—
71#67⋊C4—68no—
72#67⋊C4—68no—
73C23×C32—22no—
74C23×C32—22no—
75C23×C32—22no—
76C23×C32C22×C412yes—
77C23×C32C22×C411yescharacteristic
78C22×C3⋊D3—22no—
79C22×C3⋊D3—22no—
80C22×C3⋊D3—22no—
81C22×C3⋊D3C22×C412yes—
82C2×C32×C4—22no—
83C2×C32×C4—22no—
84C2×C32×C4C22×C412yes—
85C2×C32×C4C22×C412yes—
86#638—28no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—24no—
89C2×C3⋊Dic3—22no—
90C2×C3⋊Dic3—22no—
91C2×C3⋊Dic3M1614yescomplemented
92C2×C3⋊Dic3C22×C412yes—
93C2×C3⋊Dic3C22×C411yescharacteristic
94C2×C3⋊Dic3C22×C412yes—
95C2×C3⋊Dic3C2×D412yes—
96C2×C3⋊Dic3#4112yes—
97C2×C3⋊Dic3C2411yescharacteristic
98C4×C3⋊D3—24no—
99C4×C3⋊D3—22no—
100C4×C3⋊D3—22no—
101#622—22no—
102#622—22no—
103#622—22no—
104#622C2×D414yescomplemented
105C22×M16—91no—
106C22×M16—91no—
107C2×C4×D4—91no—
108#436—98no—
109C2×C4⋊C8—91no—
110C2×C22⋊C8—91no—
111C2×C22⋊C8—91no—
112C2×C4×C8—91no—
113C24×C3—22no—
114C23×D3—62no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C2×#67—68no—
118C2×#67—616no—
119C22×Dic3—62no—
120C22×Dic3—62no—
121C22×Dic3—64no—
122C22×Dic3—64no—
123C22×Dic3—62no—
124C22×Dic3—124no—
125C22×Dic3—124no—
126C2×C4×D3—64no—
127C2×C4×D3—64no—
128C2×C4×D3—124no—
129C2×C4×D3—124no—
130C3×C22⋊C4—28no—
131(C2×D3)⋊C4—68no—
132Dic3⋊C4—68no—
133C4×Dic3—64no—
134C4×Dic3—64no—
135C22×C32—22no—
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—44no—
139C22×C32C4×D412yes—
140C22×C32C2×M1612yes—
141C22×C32#13112yes—
142C22×C32C23×C411yescharacteristic
143C2×C3⋊D3—22no—
144C2×C3⋊D3—22no—
145C2×C3⋊D3—22no—
146C2×C3⋊D3—44no—
147C32×C4—22no—
148C32×C4—22no—
149C3⋊Dic3—22no—
150C3⋊Dic3C4×D412yes—
151C3⋊Dic3C2×M1614yescomplemented
152C3⋊Dic3C22×D411yescharacteristic
153C3⋊Dic3C2×#4111yescharacteristic
154C22×D4—91no—
155C23×C4—91no—
156C23×C4—91no—
157C2×M16—94no—
158C2×M16—94no—
159C2×M16—184no—
160C2×M16—184no—
161C22×C8—91no—
162C22×C8—91no—
163C22×C8—91no—
164C22×C8—91no—
165C4×D4—94no—
166C4×D4—94no—
167C2×C4⋊C4—91no—
168C2×C22⋊C4—91no—
169C2×C22⋊C4#15711yescharacteristic
170C2×C42—91no—
171C4⋊C8—94no—
172C22⋊C8—94no—
173C22⋊C8—94no—
174C4×C8—94no—
175C23×C3—28no—
176C23×C3—24no—
177C23×C3—22no—
178C23×C3—44no—
179C23×C3—44no—
180C22×D3—64no—
181C22×D3—68no—
182C22×D3—124no—
183C22×D3—124no—
184C2×C3×C4—24no—
185C2×C3×C4—24no—
186C2×C3×C4—44no—
187C2×C3×C4—44no—
188#67—632no—
189C2×Dic3—68no—
190C2×Dic3—64no—
191C2×Dic3—62no—
192C2×Dic3—64no—
193C2×Dic3—64no—
194C2×Dic3—62no—
195C2×Dic3—64no—
196C2×Dic3—124no—
197C2×Dic3—128no—
198C2×Dic3—124no—
199C2×Dic3—124no—
200C4×D3—124no—
201C4×D3—128no—
202C4×D3—124no—
203C2×C32—42no—
204C2×C32#43614yescomplemented
205C2×C32C2×C4×D411yescharacteristic
206C2×C32C22×M1611yescharacteristic
207C2×C32C2×#13111yescharacteristic
208C3⋊D3—42no—
209C24—91no—
210C24C2×#15711yescharacteristic
211C2×D4—94no—
212C2×D4—184no—
213C22×C4—91no—
214C22×C4—91no—
215C22×C4—92no—
216C22×C4—92no—
217C22×C4—91no—
218C22×C4—92no—
219C22×C4—92no—
220C22×C4—182no—
221C22×C4—182no—
222C22×C4—182no—
223C22×C4—182no—
224C22×C4C2×#15711yescharacteristic
225C22×C4C2×#15711yescharacteristic
226M16—188no—
227M16—188no—
228C2×C8—92no—
229C2×C8—94no—
230C2×C8—92no—
231C2×C8—92no—
232C2×C8—92no—
233C2×C8—182no—
234C2×C8—182no—
235C2×C8—182no—
236C4⋊C4—94no—
237C22⋊C4—94no—
238C22⋊C4C2×#15714yes—
239C42—92no—
240C42—92no—
241C22×C3—28no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—24no—
245C22×C3—22no—
246C22×C3—44no—
247C22×C3—44no—
248C22×C3—416no—
249C2×D3—68no—
250C2×D3—124no—
251C2×D3—124no—
252C2×D3—1216no—
253C3×C4—44no—
254C3×C4—44no—
255Dic3—64no—
256Dic3—62no—
257Dic3—62no—
258Dic3—68no—
259Dic3—124no—
260C32C2×#43611yescharacteristic, complemented, Sylow
261C23—22no—
262C23—22no—
263C23—22no—
264C23—92no—
265C23—182no—
266C23—182no—
267C23—182no—
268C23C22×#15712yes—
269C23C22×#15711yescharacteristic
270D4—188no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—94no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4—92no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—92no—
280C2×C4—182no—
281C2×C4—182no—
282C2×C4—184no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4—184no—
286C2×C4—182no—
287C2×C4C22×#15712yes—
288C2×C4C22×#15712yes—
289C8—94no—
290C8—182no—
291C8—182no—
292C8—182no—
293C2×C3—22no—
294C2×C3—22no—
295C2×C3—22no—
296C2×C3—28no—
297C2×C3—48no—
298D3—128no—
299C22—22no—
300C22—22no—
301C22—22no—
302C22—44no—
303C22—182no—
304C22—182no—
305C22—182no—
306C22—364no—
307C22#6428312yes—
308C22#6428712yes—
309C22D4×#15712yes—
310C22C23×#15711yescharacteristic
311C4—22no—
312C4—22no—
313C4—94no—
314C4—91no—
315C4—91no—
316C4—92no—
317C4—182no—
318C3—22no—
319C2—42no—
320C2—362no—
321C2#1059374214yesdirect product factor
322C2C2×#6428311yescharacteristic
323C2C2×#6428711yescharacteristic
324C2C2×D4×#15711yescharacteristic
subgroupG/H
centerC23C22×#157
commutatorC22×C32C23×C4
FrattiniC22C23×#157
FittingC2×C32×C22⋊C4C4
socleC23×C32C22×C4
prime radicalC22×#638C4
cosocleC2×C3⋊Dic3C24
hypercenterC2×C22⋊C4#157
hypocenterC32C2×#436
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C22×#3792C2semidirect
2. C22×#3788C2semidirect
4. C2×C22⋊#622C2semidirect
5. C2×#63779C2semidirect
6. C2×#63778C2semidirect
7. C2×C4×#622C2semidirect
8. #10593742C2direct (N × H)
13. C2×#3792C22semidirect
19. C2×#3788C22semidirect
26. C22⋊#622C22semidirect
27. #63779C22semidirect
28. #63778C22semidirect
29. C4×#622C22semidirect
52. C2×#622D4semidirect
91. C2×C3⋊Dic3M16semidirect
104. #622C2×D4semidirect
151. C3⋊Dic3C2×M16semidirect
204. C2×C32#436semidirect
260. C32C2×#436semidirect
321. C2#10593742direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#37922
C22×#37882
C2×#640902
C2×C22⋊#6222
C2×#637792
C2×#637782
C2×C4×#6222
#105937422
C2×#4369