Finite Group Databasebeta

← C2×#10593623 #11922866

Subgroups — C2×#10593623 #11922866

Order 1152.

568 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 97 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×C3×C4×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×C3⋊#859C211yesmaximal, characteristic
3C2×C3⋊#859C211yesmaximal, characteristic
4#10593623C214yesmaximal, direct product factor
5C2×#5389—33nomaximal
6C2×#5022—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×#3750—41nomaximal
9C3×C4×SL(2,𝔽3)C2212yes—
10C3⋊#859C2212yes—
11C3⋊#859C2212yes—
12C2×C3×C4×Q8D311yescharacteristic
13C2×#870—31no—
14C2×C4×SL(2,𝔽3)D313yes—
15C2×#859—33no—
16C2×#859—33no—
17#5389—312no—
18C2×#835—31no—
19C2×#835—31no—
20#5022—34no—
21#5022—34no—
22C2×#808—31no—
23C2×#807—31no—
24C2×C4⋊(C3⋊C8)—31no—
25C22×C32×C4—41no—
26C22×C3⋊Dic3—41no—
27C22×C3⋊Dic3—41no—
28C2×C3×SL(2,𝔽3)Q812yes—
29C2×C3×SL(2,𝔽3)C2311yescharacteristic
30#3750—44no—
31C2×#479—91noSylow
32C22×C3×Q8C2×D311yescharacteristic
33C22×SL(2,𝔽3)C2×D313yes—
34C3×C4×Q8—32no—
35C3×C4×Q8C2×D312yes—
36C2×C3×C4⋊C4—31no—
37C2×C3×C42—31no—
38C2×C4⋊Dic3—31no—
39C2×C4⋊Dic3—31no—
40C2×Dic3⋊C4—61no—
41C2×Dic3⋊C4—61no—
42C2×Dic3⋊C4—123no—
43C22×C3⋊C8—31no—
44C22×C3⋊C8—31no—
45#870—34no—
46#870—62no—
47C4×SL(2,𝔽3)C2×D316yes—
48#859—36no—
49#859—36no—
50#835—32no—
51#835—32no—
52#835—32no—
53#835—32no—
54#808—34no—
55#807—34no—
56C4⋊(C3⋊C8)—34no—
57C23×C32—41no—
58C2×C32×C4—42no—
59C2×C32×C4—82no—
60C2×C3⋊Dic3—42no—
61C2×C3⋊Dic3—42no—
62C2×C3⋊Dic3—44no—
63C2×C3⋊Dic3—82no—
64C3×SL(2,𝔽3)C2×Q811yescharacteristic
65C2×#125—91no—
66C2×C4×Q8C3⋊D311yescharacteristic
67#479—94no—
68#479—94no—
69C2×#107—91no—
70C2×#106—91no—
71C2×C4⋊C8—91no—
72C2×#103—91no—
73C2×#103—91no—
74C2×C3×Q8—31no—
75C2×C3×Q8—32no—
76C2×C3×Q8Dic612yes—
77C2×C3×Q8C22×D311yescharacteristic
78C22×C3×C4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—43no—
81C22×C3×C4S411yescharacteristic
82C22×Dic3—61no—
83C22×Dic3—61no—
84C22×Dic3—123no—
85C22×Dic3—123no—
86C2×SL(2,𝔽3)Dic616yes—
87C2×SL(2,𝔽3)C22×D313yes—
88C3×C4⋊C4—32no—
89C3×C4⋊C4—61no—
90C3×C42—32no—
91C3×C42—32no—
92C4⋊Dic3—32no—
93C4⋊Dic3—32no—
94C4⋊Dic3—61no—
95C4⋊Dic3—61no—
96Dic3⋊C4—64no—
97Dic3⋊C4—64no—
98Dic3⋊C4—1212no—
99C2×C3⋊C8—32no—
100C2×C3⋊C8—32no—
101C2×C3⋊C8—62no—
102C2×C3⋊C8—62no—
103C22×C32—42no—
104C22×C32—42no—
105C22×C32—42no—
106C22×C32—41no—
107C32×C4—82no—
108C3⋊Dic3—44no—
109C3⋊Dic3—82no—
110C22×Q8C2×C3⋊D311yescharacteristic
111C22×C8—91no—
112C22×C8—91no—
113#125—94no—
114#125—182no—
115C4×Q8—32no—
116C4×Q8C2×C3⋊D312yes—
117C2×C4⋊C4—31no—
118C2×C4⋊C4—91no—
119C2×C4⋊C4—91no—
120C2×C4⋊C4—181no—
121C2×C4⋊C4—181no—
122C2×C42—31no—
123#107—94no—
124#106—94no—
125C4⋊C8—94no—
126#103—92no—
127#103—92no—
128#103—92no—
129#103—92no—
130C23×C3—43no—
131C23×C3C2×S411yescharacteristic
132C3×Q8—32no—
133C3×Q8—31no—
134C3×Q8—61no—
135C3×Q8C2×Dic611yescharacteristic
136C2×C3×C4—22no—
137C2×C3×C4—32no—
138C2×C3×C4—32no—
139C2×C3×C4—31no—
140C2×C3×C4—31no—
141C2×C3×C4—32no—
142C2×C3×C4—46no—
143C2×C3×C4—62no—
144C2×C3×C4—86no—
145C2×C3×C4C2×S412yes—
146C2×Dic3—62no—
147C2×Dic3—62no—
148C2×Dic3—126no—
149C2×Dic3—126no—
150C2×Dic3—1212no—
151C2×Dic3—122no—
152C2×Dic3—122no—
153C2×Dic3—246no—
154SL(2,𝔽3)C2×Dic613yes—
155C3⋊C8—62no—
156C3⋊C8—62no—
157C2×C32—44no—
158C2×C32—41no—
159C2×C32—41no—
160C2×C32—41no—
161C2×Q8—31no—
162C2×Q8—32no—
163C2×Q8#63412yes—
164C2×Q8C22×C3⋊D311yescharacteristic
165C22×C4—31no—
166C22×C4—31no—
167C22×C4—181no—
168C22×C4—181no—
169C22×C4C3⋊S411yescharacteristic
170C2×C8—92no—
171C2×C8—92no—
172C2×C8—182no—
173C2×C8—182no—
174C4⋊C4—32no—
175C4⋊C4—61no—
176C4⋊C4—92no—
177C4⋊C4—92no—
178C4⋊C4—181no—
179C4⋊C4—181no—
180C4⋊C4—184no—
181C4⋊C4—184no—
182C42—32no—
183C42—32no—
184C22×C3—46no—
185C22×C3—46no—
186C22×C3—46no—
187C22×C3—43no—
188C22×C3A4⋊Q812yes—
189C22×C3#98312yes—
190C22×C3#98512yes—
191C22×C3C22×S411yescharacteristic
192C3×C4—22no—
193C3×C4—31no—
194C3×C4—31no—
195C3×C4—32no—
196C3×C4—62no—
197C3×C4—86no—
198Dic3—1212no—
199Dic3—122no—
200Dic3—122no—
201Dic3—246no—
202C32—41noSylow
203C23C2×C3⋊S411yescharacteristic
204Q8—32no—
205Q8—31no—
206Q8—61no—
207Q8C2×#63411yescharacteristic
208C2×C4—22no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—32no—
214C2×C4—62no—
215C2×C4—182no—
216C2×C4—182no—
217C2×C4—362no—
218C2×C4—362no—
219C2×C4C2×C3⋊S412yes—
220C8—182no—
221C8—182no—
222C2×C3—412no—
223C2×C3—43no—
224C2×C3—43no—
225C2×C3—43no—
226C2×C3#538914yescomplemented
227C2×C3C2×A4⋊Q811yescharacteristic
228C2×C3C2×#98311yescharacteristic
229C2×C3C2×#98511yescharacteristic
230C22#6425912yes—
231C22#6426412yes—
232C22#6426612yes—
233C22C22×C3⋊S411yescharacteristic
234C4—22no—
235C4—31no—
236C4—31no—
237C4—32no—
238C4—62no—
239C4—362no—
240C4—362no—
241C3—43no—
242C3C2×#538911yescharacteristic, complemented
243C2#1059362314yesdirect product factor
244C2C2×#6425911yescharacteristic
245C2C2×#6426411yescharacteristic
246C2C2×#6426611yescharacteristic
subgroupG/H
centerC23C2×C3⋊S4
commutator, cosocleC2×C3×SL(2,𝔽3)C23
FrattiniC22C22×C3⋊S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4C3⋊S4
hypocenterC3×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C3×SL(2,𝔽3) ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #10593623C2direct (N × H)
226. C2×C3#5389semidirect
242. C3C2×#5389semidirect
243. C2#10593623direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×SL(2,𝔽3)2
C2×C3⋊#8592
#105936232
C2×#53893
C2×#50223
C2×#37504