Finite Group Databasebeta

← C2×#5569⋊C3 #11923175

Subgroups — C2×#5569⋊C3 #11923175

Order 1152.

601 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 601 conjugacy classes of subgroups — too many to draw. Only its 111 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×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
3C2×Dic3×SL(2,𝔽3)C211yesmaximal, characteristic
4#5569⋊C3C214yesmaximal, direct product factor
5C2×#5569C311yesmaximal, characteristic, complemented
6C2×#557⋊C3—31nomaximal
7C22×C3×SL(2,𝔽3)C2211yescharacteristic
8C2×C3×Dic3⋊C4—41nomaximal
9C3×C4×SL(2,𝔽3)C2212yes—
10Dic3×SL(2,𝔽3)C2212yes—
11Dic3×SL(2,𝔽3)C2212yes—
12C2×C3×C4×Q8C2×C311yescharacteristic
13C2×Q8×Dic3C2×C311yescharacteristic
14C2×Q8×Dic3C2×C311yescharacteristic
15#5569—34no—
16#5569C2×C314yescomplemented
17C2×#890—31no—
18C2×#888—31no—
19C2×#870—31no—
20C2×C4×Dic6—31no—
21#557⋊C3—34no—
22C2×C4×SL(2,𝔽3)—21no—
23C2×C4×SL(2,𝔽3)—31no—
24C2×C4×SL(2,𝔽3)—31no—
25C2×C4×SL(2,𝔽3)D311yescharacteristic
26C22×C32×C4—41no—
27C22×C3×Dic3—41no—
28C22×C3×Dic3—41no—
29C2×C3×SL(2,𝔽3)Q812yescomplemented
30C2×C3×SL(2,𝔽3)C2311yescharacteristic
31C3×Dic3⋊C4—44no—
32C2×#557—31noSylow
33C22×C3×Q8C22×C311yescharacteristic
34C22×Dic6—31no—
35C22×SL(2,𝔽3)—21no—
36C22×SL(2,𝔽3)C2×D311yescharacteristic
37C3×C4×Q8—32no—
38C3×C4×Q8C22×C312yes—
39C2×C3×C4⋊C4—31no—
40C2×C3×C4⋊C4—121no—
41C2×C3×C42—31no—
42Q8×Dic3—32no—
43Q8×Dic3—32no—
44Q8×Dic3C22×C312yes—
45Q8×Dic3C22×C312yes—
46C2×C4⋊Dic3—31no—
47C2×C4⋊Dic3—31no—
48C2×C4⋊Dic3—31no—
49C2×Dic3⋊C4—32no—
50C2×Dic3⋊C4—31no—
51C2×Dic3⋊C4A411yescharacteristic
52C2×C4×Dic3—31no—
53C2×C4×Dic3—31no—
54#890—34no—
55#890—34no—
56#888—34no—
57#888—34no—
58#870—34no—
59#870—34no—
60C4×Dic6—34no—
61C4×Dic6—34no—
62C4×SL(2,𝔽3)—22no—
63C4×SL(2,𝔽3)—32no—
64C4×SL(2,𝔽3)—32no—
65C4×SL(2,𝔽3)C2×D312yes—
66C23×C32—41no—
67C2×C32×C4—42no—
68C2×C32×C4—82no—
69C2×C3×Dic3—44no—
70C2×C3×Dic3—42no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3—82no—
73C3×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
74#557—34no—
75#557—94no—
76C2×C4⋊Q8—91no—
77C2×#125—91no—
78C2×#125—91no—
79C2×C4×Q8—31no—
80C2×C4×Q8—31no—
81C2×C4×Q8—91no—
82C2×C4×Q8C3×D311yescharacteristic
83C2×C3×Q8—31no—
84C2×C3×Q8—32no—
85C2×C3×Q8C3×Q812yes—
86C2×C3×Q8C23×C311yescharacteristic
87C22×C3×C4—31no—
88C22×C3×C4—31no—
89C22×C3×C4—41no—
90C22×C3×C4—81no—
91C22×C3×C4—121no—
92C22×C3×C4—121no—
93C22×C3×C4C2×A411yescharacteristic
94C22×Dic3—31no—
95C22×Dic3—31no—
96C22×Dic3C2×A411yescharacteristic
97C22×Dic3C2×A411yescharacteristic
98C2×Dic6—34no—
99C2×Dic6—64no—
100C2×SL(2,𝔽3)—22no—
101C2×SL(2,𝔽3)—21no—
102C2×SL(2,𝔽3)Dic612yescomplemented
103C2×SL(2,𝔽3)C22×D311yescharacteristic
104C3×C4⋊C4—32no—
105C3×C4⋊C4—32no—
106C3×C4⋊C4—124no—
107C3×C42—32no—
108C3×C42—32no—
109C4⋊Dic3—32no—
110C4⋊Dic3—32no—
111C4⋊Dic3—34no—
112C4⋊Dic3—32no—
113C4⋊Dic3—32no—
114Dic3⋊C4—38no—
115Dic3⋊C4—34no—
116Dic3⋊C4C2×A414yes—
117C4×Dic3—32no—
118C4×Dic3—32no—
119C4×Dic3—32no—
120C4×Dic3—32no—
121C22×C32—42no—
122C22×C32—42no—
123C22×C32—42no—
124C22×C32—41no—
125C32×C4—82no—
126C3×Dic3—44no—
127C3×Dic3—82no—
128C22×Q8—91no—
129C22×Q8C2×C3×D311yescharacteristic
130C4⋊Q8—94no—
131C4⋊Q8—94no—
132#125—94no—
133#125—94no—
134#125—94no—
135#125—94no—
136C4×Q8—32no—
137C4×Q8—32no—
138C4×Q8—32no—
139C4×Q8—94no—
140C4×Q8—94no—
141C4×Q8—92no—
142C4×Q8—92no—
143C4×Q8C2×C3×D312yes—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—91no—
147C2×C4⋊C4—91no—
148C2×C4⋊C4—91no—
149C2×C4⋊C4—92no—
150C2×C4⋊C4—91no—
151C2×C42—31no—
152C2×C42—91no—
153C2×C42—91no—
154C23×C3—41no—
155C23×C3—81no—
156C23×C3C22×A411yescharacteristic
157C3×Q8—32no—
158C3×Q8—31no—
159C3×Q8—32no—
160C3×Q8C2×C3×Q811yescharacteristic
161C2×C3×C4—22no—
162C2×C3×C4—32no—
163C2×C3×C4—32no—
164C2×C3×C4—31no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—42no—
168C2×C3×C4—62no—
169C2×C3×C4—82no—
170C2×C3×C4—84no—
171C2×C3×C4—82no—
172C2×C3×C4—122no—
173C2×C3×C4—124no—
174C2×C3×C4—122no—
175C2×C3×C4—242no—
176C2×C3×C4C22×A412yes—
177C2×Dic3—22no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—62no—
181C2×Dic3—62no—
182C2×Dic3#41⋊C314yes—
183C2×Dic3C22×A412yes—
184C2×Dic3C22×A412yes—
185Dic6—68no—
186SL(2,𝔽3)—21no—
187SL(2,𝔽3)C2×Dic611yescharacteristic, complemented
188C2×C32—41no—
189C2×C32—41no—
190C2×C32—41no—
191C2×C32—44no—
192C2×Q8—31no—
193C2×Q8—32no—
194C2×Q8—94no—
195C2×Q8—184no—
196C2×Q8C3×Dic612yes—
197C2×Q8C22×C3×D311yescharacteristic
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—31no—
202C22×C4—91no—
203C22×C4—91no—
204C22×C4D3×A411yescharacteristic
205C4⋊C4—32no—
206C4⋊C4—32no—
207C4⋊C4—34no—
208C4⋊C4—94no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4—92no—
213C4⋊C4—98no—
214C4⋊C4—94no—
215C42—32no—
216C42—32no—
217C42—92no—
218C42—92no—
219C42—92no—
220C42—92no—
221C22×C3—42no—
222C22×C3—42no—
223C22×C3—42no—
224C22×C3—41no—
225C22×C3—82no—
226C22×C3—82no—
227C22×C3—82no—
228C22×C3—81no—
229C22×C3Q8×A412yes—
230C22×C3C2×#41⋊C312yes—
231C22×C32−1+4⋊C312yes—
232C22×C3C23×A411yescharacteristic
233C3×C4—22no—
234C3×C4—31no—
235C3×C4—32no—
236C3×C4—31no—
237C3×C4—62no—
238C3×C4—82no—
239C3×C4—84no—
240C3×C4—124no—
241C3×C4—242no—
242Dic3—22no—
243Dic3—62no—
244Dic3—62no—
245Dic3C2×#41⋊C314yes—
246C32—41noSylow
247C23C2×D3×A411yescharacteristic
248Q8—32no—
249Q8—31no—
250Q8—32no—
251Q8—188no—
252Q8C2×C3×Dic611yescharacteristic
253C2×C4—22no—
254C2×C4—32no—
255C2×C4—32no—
256C2×C4—34no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—32no—
260C2×C4—32no—
261C2×C4—32no—
262C2×C4—62no—
263C2×C4—62no—
264C2×C4—92no—
265C2×C4—92no—
266C2×C4—182no—
267C2×C4—182no—
268C2×C4C2×D3×A412yes—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—44no—
272C2×C3—41no—
273C2×C3—81no—
274C2×C3—81no—
275C2×C3—84no—
276C2×C3—81no—
277C2×C3#557⋊C314yescomplemented
278C2×C3C2×Q8×A411yescharacteristic
279C2×C3C22×#41⋊C311yescharacteristic
280C2×C3C2×2−1+4⋊C311yescharacteristic
281C22A4×Dic612yes—
282C22#1007⋊C312yes—
283C22D3×#41⋊C312yes—
284C22C22×D3×A411yescharacteristic
285C4—22no—
286C4—34no—
287C4—31no—
288C4—32no—
289C4—31no—
290C4—62no—
291C4—62no—
292C4—182no—
293C4—182no—
294C3—41no—
295C3—81no—
296C3C2×#557⋊C311yescharacteristic, complemented
297C2#5569⋊C314yesdirect product factor
298C2C2×A4×Dic611yescharacteristic
299C2C2×#1007⋊C311yescharacteristic
300C2C2×D3×#41⋊C311yescharacteristic
subgroupG/H
centerC23C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC2×C3×C4×Q8C2×C3
socleC23×C3C22×A4
prime radicalC22×C3×SL(2,𝔽3)C22
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #5569⋊C3C2direct (N × H)
5. C2×#5569C3semidirect
16. #5569C2×C3semidirect
29. C2×C3×SL(2,𝔽3)Q8semidirect
73. C3×SL(2,𝔽3)C2×Q8semidirect
102. C2×SL(2,𝔽3)Dic6semidirect
187. SL(2,𝔽3)C2×Dic6semidirect
277. C2×C3#557⋊C3semidirect
296. C3C2×#557⋊C3semidirect
297. C2#5569⋊C3direct (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×Dic3×SL(2,𝔽3)2
#5569⋊C32
C2×#55693
C2×#557⋊C33
C2×C3×Dic3⋊C44