Finite Group Databasebeta

← C2×#88671 #11722538

Subgroups — C2×#88671 #11722538

Order 864.

628 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 124 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×#6235C211yesmaximal, characteristic, complemented
2C2×#638⋊C3C211yesmaximal, characteristic, complemented
3C22×(C3⋊Dic3)⋊C3C211yesmaximal, characteristic, complemented
4C2×#6200C211yesmaximal, characteristic, complemented
5C2×#6199C211yesmaximal, characteristic, complemented
6#88671C218yesmaximal, direct product factor
7C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
8C2×#6196C211yesmaximal, characteristic, complemented
9C2×#3805—31nomaximal
10C2×#3804—31nomaximal
11C23×H(3,𝔽3)C2211yescharacteristic
12C22×#272C2211yescharacteristic
13C22×(C3⋊D3)⋊C3C2211yescharacteristic
14#6235C2214yescomplemented
15C2×#1084C2211yescharacteristic
16#638⋊C3C2214yescomplemented
17C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
18C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
19C2×(C3⋊Dic3)⋊C3C2211yescharacteristic
20C2×(C3⋊Dic3)⋊C3C2214yescomplemented
21#6200C2214yescomplemented
22#6199C2214yescomplemented
23((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
24#6196C2214yescomplemented
25C2×#638D311yescharacteristic, complemented
26C22×C3⋊Dic3D311yescharacteristic, complemented
27C2×C3×#67—31no—
28C2×C3×#67—31no—
29C2×C3×#67—31no—
30C2×C3×#67—61no—
31C22×C3×Dic3—31no—
32C2×#627—31no—
33C2×#627—31no—
34C2×#626—31no—
35C2×#625—31no—
36C2×#624—31no—
37#3805—38no—
38#3804—38no—
39C2×D3×Dic3—31no—
40C2×D3×Dic3—31no—
41C2×D3×Dic3—31no—
42C22×H(3,𝔽3)—22no—
43C22×H(3,𝔽3)C2312yes—
44C22×H(3,𝔽3)C2311yescharacteristic
45C2×#272—22no—
46C2×#272C2312yes—
47C2×(C3⋊D3)⋊C3—22no—
48C2×(C3⋊D3)⋊C3C2312yes—
49#1084C2312yes—
50(C3⋊Dic3)⋊C3C2312yes—
51(C3⋊Dic3)⋊C3C2312yes—
52(C3⋊Dic3)⋊C3C2312yes—
53C2×#237—91no—
54C2×#237—91no—
55C2×#235—91no—
56C23×C32—21no—
57C23×C32C2×D311yescharacteristic
58C23×C32C2×D311yescharacteristic
59C22×C3⋊D3C2×D311yescharacteristic
60C22×C3×D3—31no—
61C22×C3×D3—31no—
62C22×C3×D3—31no—
63C22×C3×D3—61no—
64#638C2×D314yescomplemented
65C2×C3⋊Dic3C2×D311yescharacteristic
66C2×C3⋊Dic3C2×D311yescharacteristic
67C2×C3⋊Dic3C2×D311yescharacteristic
68C2×C3⋊Dic3C2×D314yescomplemented
69C3×#67—34no—
70C3×#67—34no—
71C3×#67—34no—
72C3×#67—64no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—34no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—61no—
80#627—34no—
81#627—34no—
82#626—34no—
83#625—34no—
84#624—34no—
85D3×Dic3—34no—
86D3×Dic3—34no—
87D3×Dic3—34no—
88C2×H(3,𝔽3)—22no—
89C2×H(3,𝔽3)#4112yescomplemented
90C2×H(3,𝔽3)C2411yescharacteristic
91#272—22no—
92(C3⋊D3)⋊C3—22no—
93C2×C3×D4—91no—
94C2×C3×D4—91no—
95C22×C3×C4—91no—
96C2×#67—31no—
97C2×#67—31no—
98C2×#67—31no—
99C2×#67—61no—
100C2×#67—91no—
101C2×#67—91no—
102C2×#67—91no—
103C22×Dic3—31no—
104C22×Dic3—91no—
105C22×Dic3—91no—
106C22×Dic3—91no—
107#237—98no—
108#237—98no—
109#235—98no—
110C2×D12—91no—
111C2×C4×D3—91no—
112C2×C4×D3—91no—
113C2×C4×D3—91no—
114C2×C4×D3—91no—
115C2×Dic6—91no—
116C2×Dic6—91no—
117C2×Dic6—91no—
118C22×C32—22no—
119C22×C32—22no—
120C22×C32—21no—
121C22×C32—22no—
122C22×C32—42no—
123C22×C32C22×D312yes—
124C22×C32C22×D312yes—
125C22×C32C22×D311yescharacteristic
126C22×C32C22×D311yescharacteristic
127C2×C3⋊D3—22no—
128C2×C3⋊D3C22×D312yes—
129C2×C3×D3—32no—
130C2×C3×D3—32no—
131C2×C3×D3—32no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—122no—
137C3⋊Dic3C22×D312yes—
138C3⋊Dic3C22×D312yes—
139C3⋊Dic3C22×D312yes—
140C3×Dic3—32no—
141C3×Dic3—32no—
142C3×Dic3—32no—
143C3×Dic3—32no—
144C3×Dic3—32no—
145C3×Dic3—62no—
146C2×#41—271noSylow
147H(3,𝔽3)C2×#4111yescharacteristic, complemented, Sylow
148C23×C3—31no—
149C23×C3—31no—
150C23×C3—61no—
151C23×C3—91no—
152C23×C3—91no—
153C23×C3D3211yescharacteristic
154C22×D3—31no—
155C22×D3—31no—
156C22×D3—31no—
157C22×D3—61no—
158C22×D3—91no—
159C3×D4—94no—
160C3×D4—94no—
161C2×C3×C4—91no—
162C2×C3×C4—91no—
163C2×C3×C4—91no—
164C2×C3×C4—94no—
165C2×C3×C4—91no—
166#67—34no—
167#67—34no—
168#67—34no—
169#67—64no—
170#67—94no—
171#67—94no—
172#67—94no—
173C2×Dic3—31no—
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—34no—
179C2×Dic3—61no—
180C2×Dic3—91no—
181C2×Dic3—91no—
182C2×Dic3—91no—
183C2×Dic3—94no—
184C2×Dic3—94no—
185C2×Dic3—94no—
186D12—94no—
187C4×D3—94no—
188C4×D3—94no—
189C4×D3—94no—
190C4×D3—94no—
191Dic6—94no—
192Dic6—94no—
193Dic6—94no—
194C2×C32—22no—
195C2×C32—22no—
196C2×C32—21no—
197C2×C32—22no—
198C2×C32—42no—
199C2×C32#23512yescomplemented
200C2×C32#23712yescomplemented
201C2×C32C23×D311yescharacteristic
202C2×C32C23×D311yescharacteristic
203C3⋊D3—22no—
204C3×D3—62no—
205C3×D3—62no—
206C3×D3—62no—
207C3×D3—122no—
208#41—278no—
209C2×Q8—271no—
210C2×D4—91no—
211C2×D4—91no—
212C2×D4—271no—
213C22×C4—91no—
214C22×C4—271no—
215C22×C4—271no—
216C22×C3—22no—
217C22×C3—32no—
218C22×C3—32no—
219C22×C3—31no—
220C22×C3—31no—
221C22×C3—62no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3—61no—
225C22×C3—92no—
226C22×C3—92no—
227C22×C3—122no—
228C22×C3—182no—
229C22×C3—182no—
230C22×C3C2×D3212yes—
231C22×C3C2×D3211yescharacteristic
232C2×D3—32no—
233C2×D3—32no—
234C2×D3—32no—
235C2×D3—62no—
236C2×D3—62no—
237C2×D3—62no—
238C2×D3—62no—
239C2×D3—92no—
240C2×D3—122no—
241C2×D3—182no—
242C3×C4—92no—
243C3×C4—92no—
244C3×C4—92no—
245C3×C4—92no—
246Dic3—32no—
247Dic3—32no—
248Dic3—32no—
249Dic3—32no—
250Dic3—32no—
251Dic3—62no—
252Dic3—92no—
253Dic3—92no—
254Dic3—92no—
255C32—21no—
256C32C2×#23511yescharacteristic, complemented
257C32C2×#23711yescharacteristic, complemented
258C23—91no—
259C23—91no—
260C23((C3⋊D3)⋊C3)⋊C211yescharacteristic
261Q8—274no—
262D4—94no—
263D4—94no—
264D4—274no—
265C2×C4—94no—
266C2×C4—91no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—91no—
270C2×C4—274no—
271C2×C4—274no—
272C2×C3—22no—
273C2×C3—31no—
274C2×C3—31no—
275C2×C3—32no—
276C2×C3—32no—
277C2×C3—62no—
278C2×C3—62no—
279C2×C3—61no—
280C2×C3—62no—
281C2×C3—122no—
282C2×C3—182no—
283C2×C3—182no—
284C2×C3#380412yes—
285C2×C3C22×D3211yescharacteristic
286D3—62no—
287D3—62no—
288D3—62no—
289D3—122no—
290D3—182no—
291C22—22no—
292C22—92no—
293C22—92no—
294C22—182no—
295C22—182no—
296C22C2×((C3⋊D3)⋊C3)⋊C212yes—
297C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
298C4—92no—
299C4—92no—
300C4—92no—
301C4—92no—
302C3—31no—
303C3—31no—
304C3—61no—
305C3C2×#380411yescharacteristic
306C2—22no—
307C2—182no—
308C2—182no—
309C2#8867112yesdirect product factor
310C2C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
subgroupG/H
centerC22C2×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC2×H(3,𝔽3)C24
FrattiniC2×C3C22×D32
FittingC23×H(3,𝔽3)C22
socleC22×C3C2×D32
hypercenterC23((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#41
LCSG ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
1. C2×#6235C2semidirect
2. C2×#638⋊C3C2semidirect
3. C22×(C3⋊Dic3)⋊C3C2semidirect
4. C2×#6200C2semidirect
5. C2×#6199C2semidirect
6. #88671C2direct (N × H)
7. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
8. C2×#6196C2semidirect
14. #6235C22semidirect
16. #638⋊C3C22semidirect
20. C2×(C3⋊Dic3)⋊C3C22semidirect
21. #6200C22semidirect
22. #6199C22semidirect
23. ((C3⋊D3)⋊C3)⋊C4C22semidirect
24. #6196C22semidirect
25. C2×#638D3semidirect
26. C22×C3⋊Dic3D3semidirect
64. #638C2×D3semidirect
68. C2×C3⋊Dic3C2×D3semidirect
89. C2×H(3,𝔽3)#41semidirect
147. H(3,𝔽3)C2×#41semidirect
199. C2×C32#235semidirect
200. C2×C32#237semidirect
256. C32C2×#235semidirect
257. C32C2×#237semidirect
309. C2#88671direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#62352
C2×#638⋊C32
C22×(C3⋊Dic3)⋊C32
C2×#62002
C2×#61992
#886712
C2×((C3⋊D3)⋊C3)⋊C42
C2×#61962
C2×#38053
C2×#38043