Finite Group Databasebeta

← C2×#88676 #11722543

Subgroups — C2×#88676 #11722543

Order 864.

880 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 140 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
1C23×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C22×((C3⋊D3)⋊C3)⋊C2C211yesmaximal, characteristic, complemented
3C2×#6235C211yesmaximal, characteristic, complemented
4C2×#638⋊C3C211yesmaximal, characteristic, complemented
5#88676C218yesmaximal, direct product factor
6C2×#6200C211yesmaximal, characteristic, complemented
7C2×#6198C211yesmaximal, characteristic, complemented
8C2×((C3⋊D3)⋊C3)⋊C4C211yesmaximal, characteristic, complemented
9C2×#3811—31nomaximal
10C2×D3×#67—31nomaximal
11C23×H(3,𝔽3)C2211yescharacteristic, complemented
12C22×#272C2211yescharacteristic, complemented
13C22×(C3⋊D3)⋊C3—24no—
14C22×(C3⋊D3)⋊C3C2211yescharacteristic
15C22×(C3⋊D3)⋊C3C2214yescomplemented
16C22×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
17C22×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
18C2×((C3⋊D3)⋊C3)⋊C2—24no—
19C2×((C3⋊D3)⋊C3)⋊C2C2214yescomplemented
20#6235C2214yescomplemented
21C2×#1084C2211yescharacteristic, complemented
22#638⋊C3C2214yescomplemented
23C2×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
24#6200C2214yescomplemented
25#6198C2214yescomplemented
26((C3⋊D3)⋊C3)⋊C4C2214yescomplemented
27C23×C3⋊D3D311yescharacteristic, complemented
28C23×C3×D3—31no—
29C22×D32—31no—
30C22×D32—31no—
31C2×#638D311yescharacteristic, complemented
32C2×C3×#67—31no—
33C2×C3×#67—31no—
34C2×C3×#67—31no—
35C2×C3×#67—61no—
36#3811—38no—
37D3×#67—38no—
38C2×#626—31no—
39C2×#626—31no—
40C2×#626—31no—
41C2×#625—31no—
42C2×#624—31no—
43C2×D3×Dic3—31no—
44C22×H(3,𝔽3)—22no—
45C22×H(3,𝔽3)C2311yescharacteristic
46C22×H(3,𝔽3)C2312yescomplemented
47C2×#272—22no—
48C2×#272C2312yescomplemented
49C2×(C3⋊D3)⋊C3—22no—
50C2×(C3⋊D3)⋊C3—22no—
51C2×(C3⋊D3)⋊C3—28no—
52C2×(C3⋊D3)⋊C3D414yescomplemented
53C2×(C3⋊D3)⋊C3C2312yes—
54C2×(C3⋊D3)⋊C3C2312yescomplemented
55C2×(C3⋊D3)⋊C3C2312yescomplemented
56((C3⋊D3)⋊C3)⋊C2—28no—
57#1084C2312yescomplemented
58(C3⋊Dic3)⋊C3C2312yescomplemented
59C22×#67—91no—
60C2×D3×D4—91no—
61C2×D3×D4—91no—
62C23×C32—21no—
63C23×C32C2×D311yescharacteristic, complemented
64C23×C32C2×D311yescharacteristic, complemented
65C22×C3⋊D3—24no—
66C22×C3⋊D3C2×D311yescharacteristic
67C22×C3⋊D3C2×D314yescomplemented
68C22×C3⋊D3C2×D311yescharacteristic, complemented
69C22×C3⋊D3C2×D311yescharacteristic, complemented
70C22×C3×D3—31no—
71C22×C3×D3—31no—
72C22×C3×D3—31no—
73C22×C3×D3—31no—
74C22×C3×D3—31no—
75C22×C3×D3—34no—
76C22×C3×D3—61no—
77C22×C3×D3—64no—
78C2×D32—34no—
79C2×D32—34no—
80C2×D32—64no—
81C2×D32—64no—
82#638C2×D314yescomplemented
83C2×C3⋊Dic3C2×D311yescharacteristic, complemented
84C3×#67—34no—
85C3×#67—34no—
86C3×#67—34no—
87C3×#67—64no—
88C2×C3×Dic3—31no—
89C2×C3×Dic3—31no—
90C2×C3×Dic3—31no—
91C2×C3×Dic3—61no—
92#626—34no—
93#626—34no—
94#626—34no—
95#625—34no—
96#624—34no—
97D3×Dic3—34no—
98C2×H(3,𝔽3)—22no—
99C2×H(3,𝔽3)C2×D412yescomplemented
100C2×H(3,𝔽3)C2411yescharacteristic
101#272—22no—
102(C3⋊D3)⋊C3—22no—
103(C3⋊D3)⋊C3—22no—
104(C3⋊D3)⋊C3C2×D414yescomplemented
105C24×C3—91no—
106C23×D3—31no—
107C23×D3—91no—
108C23×D3—91no—
109C23×D3—91no—
110C23×D3—91no—
111C2×C3×D4—91no—
112C2×C3×D4—91no—
113C2×#67—31no—
114C2×#67—31no—
115C2×#67—31no—
116C2×#67—61no—
117C2×#67—91no—
118C2×#67—91no—
119C2×#67—91no—
120C2×#67—91no—
121C2×#67—98no—
122C2×#67—91no—
123C22×Dic3—91no—
124D3×D4—98no—
125D3×D4—98no—
126C2×D12—91no—
127C2×D12—91no—
128C2×C4×D3—91no—
129C2×C4×D3—91no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—21no—
133C22×C32—22no—
134C22×C32—42no—
135C22×C32C22×D311yescharacteristic
136C22×C32C22×D311yescharacteristic
137C22×C32C22×D312yescomplemented
138C22×C32C22×D312yescomplemented
139C2×C3⋊D3—22no—
140C2×C3⋊D3—22no—
141C2×C3⋊D3—28no—
142C2×C3⋊D3#6714yescomplemented
143C2×C3⋊D3C22×D312yes—
144C2×C3⋊D3C22×D312yescomplemented
145C2×C3⋊D3C22×D312yescomplemented
146C2×C3×D3—34no—
147C2×C3×D3—32no—
148C2×C3×D3—32no—
149C2×C3×D3—32no—
150C2×C3×D3—32no—
151C2×C3×D3—32no—
152C2×C3×D3—62no—
153C2×C3×D3—68no—
154C2×C3×D3—62no—
155C2×C3×D3—62no—
156C2×C3×D3—62no—
157C2×C3×D3—62no—
158C2×C3×D3—122no—
159D32—68no—
160D32—68no—
161C3⋊Dic3C22×D312yescomplemented
162C3×Dic3—32no—
163C3×Dic3—32no—
164C3×Dic3—32no—
165C3×Dic3—62no—
166C22×D4—271noSylow
167H(3,𝔽3)C22×D411yescharacteristic, complemented, Sylow
168C23×C3—31no—
169C23×C3—31no—
170C23×C3—61no—
171C23×C3—91no—
172C23×C3—91no—
173C23×C3—94no—
174C23×C3—91no—
175C23×C3—91no—
176C23×C3—184no—
177C23×C3D3211yescharacteristic
178C22×D3—31no—
179C22×D3—31no—
180C22×D3—31no—
181C22×D3—31no—
182C22×D3—31no—
183C22×D3—34no—
184C22×D3—61no—
185C22×D3—64no—
186C22×D3—91no—
187C22×D3—91no—
188C22×D3—94no—
189C22×D3—91no—
190C22×D3—94no—
191C22×D3—94no—
192C22×D3—94no—
193C22×D3—184no—
194C22×D3—184no—
195C22×D3—184no—
196C22×D3—184no—
197C3×D4—94no—
198C3×D4—94no—
199C2×C3×C4—91no—
200C2×C3×C4—91no—
201#67—34no—
202#67—34no—
203#67—34no—
204#67—64no—
205#67—94no—
206#67—94no—
207#67—94no—
208#67—94no—
209#67—94no—
210C2×Dic3—31no—
211C2×Dic3—31no—
212C2×Dic3—31no—
213C2×Dic3—61no—
214C2×Dic3—91no—
215C2×Dic3—94no—
216D12—94no—
217D12—94no—
218C4×D3—94no—
219C4×D3—94no—
220C2×C32—22no—
221C2×C32—22no—
222C2×C32—21no—
223C2×C32—22no—
224C2×C32—42no—
225C2×C32D3×D412yescomplemented
226C2×C32C2×#6712yescomplemented
227C2×C32C23×D311yescharacteristic
228C2×C32C23×D311yescharacteristic
229C3⋊D3—22no—
230C3⋊D3—22no—
231C3⋊D3C2×#6714yescomplemented
232C3×D3—34no—
233C3×D3—62no—
234C3×D3—62no—
235C3×D3—62no—
236C3×D3—62no—
237C3×D3—122no—
238C24—91no—
239C24—271no—
240C2×D4—91no—
241C2×D4—91no—
242C2×D4—271no—
243C2×D4—271no—
244C2×D4—278no—
245C22×C4—271no—
246C22×C3—22no—
247C22×C3—31no—
248C22×C3—31no—
249C22×C3—32no—
250C22×C3—32no—
251C22×C3—62no—
252C22×C3—62no—
253C22×C3—61no—
254C22×C3—62no—
255C22×C3—92no—
256C22×C3—94no—
257C22×C3—92no—
258C22×C3—92no—
259C22×C3—92no—
260C22×C3—122no—
261C22×C3—182no—
262C22×C3—182no—
263C22×C3—188no—
264C22×C3—182no—
265C22×C3C2×D3211yescharacteristic
266C22×C3C2×D3212yes—
267C2×D3—32no—
268C2×D3—34no—
269C2×D3—32no—
270C2×D3—32no—
271C2×D3—32no—
272C2×D3—32no—
273C2×D3—62no—
274C2×D3—68no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—92no—
280C2×D3—94no—
281C2×D3—92no—
282C2×D3—92no—
283C2×D3—122no—
284C2×D3—188no—
285C2×D3—182no—
286C2×D3—182no—
287C2×D3—188no—
288C2×D3—188no—
289C2×D3—188no—
290C3×C4—92no—
291C3×C4—92no—
292Dic3—32no—
293Dic3—32no—
294Dic3—32no—
295Dic3—62no—
296Dic3—92no—
297C32—21no—
298C32C2×D3×D411yescharacteristic, complemented
299C32C22×#6711yescharacteristic, complemented
300C23—91no—
301C23—91no—
302C23—91no—
303C23—91no—
304C23—94no—
305C23—184no—
306C23—274no—
307C23—544no—
308C23((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
309D4—94no—
310D4—94no—
311D4—274no—
312D4—274no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—274no—
316C2×C3—22no—
317C2×C3—31no—
318C2×C3—31no—
319C2×C3—32no—
320C2×C3—32no—
321C2×C3—62no—
322C2×C3—62no—
323C2×C3—61no—
324C2×C3—62no—
325C2×C3—94no—
326C2×C3—122no—
327C2×C3—182no—
328C2×C3—182no—
329C2×C3—182no—
330C2×C3D3×#6712yes—
331C2×C3C22×D3211yescharacteristic
332D3—34no—
333D3—62no—
334D3—62no—
335D3—62no—
336D3—62no—
337D3—94no—
338D3—122no—
339D3—182no—
340D3—182no—
341C22—22no—
342C22—92no—
343C22—94no—
344C22—92no—
345C22—92no—
346C22—92no—
347C22—188no—
348C22—182no—
349C22—182no—
350C22—182no—
351C22—548no—
352C22C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
353C22C2×((C3⋊D3)⋊C3)⋊C212yescomplemented
354C4—92no—
355C4—92no—
356C3—31no—
357C3—31no—
358C3—61no—
359C3C2×D3×#6711yescharacteristic
360C2—22no—
361C2—94no—
362C2—182no—
363C2—182no—
364C2—182no—
365C2#8867612yesdirect product factor
366C2C22×((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)C22×D4
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. C23×(C3⋊D3)⋊C3C2semidirect
2. C22×((C3⋊D3)⋊C3)⋊C2C2semidirect
3. C2×#6235C2semidirect
4. C2×#638⋊C3C2semidirect
5. #88676C2direct (N × H)
6. C2×#6200C2semidirect
7. C2×#6198C2semidirect
8. C2×((C3⋊D3)⋊C3)⋊C4C2semidirect
11. C23×H(3,𝔽3)C22semidirect
12. C22×#272C22semidirect
15. C22×(C3⋊D3)⋊C3C22semidirect
16. C22×(C3⋊D3)⋊C3C22semidirect
17. C22×(C3⋊D3)⋊C3C22semidirect
19. C2×((C3⋊D3)⋊C3)⋊C2C22semidirect
20. #6235C22semidirect
21. C2×#1084C22semidirect
22. #638⋊C3C22semidirect
23. C2×(C3⋊Dic3)⋊C3C22semidirect
24. #6200C22semidirect
25. #6198C22semidirect
26. ((C3⋊D3)⋊C3)⋊C4C22semidirect
27. C23×C3⋊D3D3semidirect
31. C2×#638D3semidirect
46. C22×H(3,𝔽3)C23semidirect
48. C2×#272C23semidirect
52. C2×(C3⋊D3)⋊C3D4semidirect
54. C2×(C3⋊D3)⋊C3C23semidirect
55. C2×(C3⋊D3)⋊C3C23semidirect
57. #1084C23semidirect
58. (C3⋊Dic3)⋊C3C23semidirect
63. C23×C32C2×D3semidirect
64. C23×C32C2×D3semidirect
67. C22×C3⋊D3C2×D3semidirect
68. C22×C3⋊D3C2×D3semidirect
69. C22×C3⋊D3C2×D3semidirect
82. #638C2×D3semidirect
83. C2×C3⋊Dic3C2×D3semidirect
99. C2×H(3,𝔽3)C2×D4semidirect
104. (C3⋊D3)⋊C3C2×D4semidirect
137. C22×C32C22×D3semidirect
138. C22×C32C22×D3semidirect
142. C2×C3⋊D3#67semidirect
144. C2×C3⋊D3C22×D3semidirect
145. C2×C3⋊D3C22×D3semidirect
161. C3⋊Dic3C22×D3semidirect
167. H(3,𝔽3)C22×D4semidirect
225. C2×C32D3×D4semidirect
226. C2×C32C2×#67semidirect
231. C3⋊D3C2×#67semidirect
298. C32C2×D3×D4semidirect
299. C32C22×#67semidirect
308. C23((C3⋊D3)⋊C3)⋊C2semidirect
353. C22C2×((C3⋊D3)⋊C3)⋊C2semidirect
365. C2#88676direct (N × H)

Maximal subgroups

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

subgroupindex
C23×(C3⋊D3)⋊C32
C22×((C3⋊D3)⋊C3)⋊C22
C2×#62352
C2×#638⋊C32
#886762
C2×#62002
C2×#61982
C2×((C3⋊D3)⋊C3)⋊C42
C2×#38113
C2×D3×#673