Finite Group Databasebeta

← #11724622

Subgroups — #11724622

Order 864.

806 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C3×#3830C212yesmaximal, complemented
2C3×D4×C3⋊D3C211yesmaximal, characteristic, complemented
3#89049C212yesmaximal, complemented
4#89046C214yesmaximal, complemented
5C2×#6294C212yesmaximal, complemented
6C4×#1113C211yesmaximal, characteristic, complemented
7#89041C212yesmaximal, complemented
8#89040C211yesmaximal, characteristic, complemented
9#64312—31nomaximal
10D3×#237—32nomaximal
11C22×C3×C3⋊D3C2212yes—
12C2×#1113C2211yescharacteristic
13C33×D4C2211yescharacteristic, complemented
14C3×#638C2212yescomplemented
15C3×#638C2214yescomplemented
16C2×C3×C3⋊Dic3C2214yescomplemented
17C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
18C3×C4×C3⋊D3C2211yescharacteristic
19C3×C4×C3⋊D3C2212yescomplemented
20C3×#634C2212yescomplemented
21#6296C2212yescomplemented
22#6295C2214yescomplemented
23#6294C2212yes—
24#6294C2211yescharacteristic
25#6294C2214yescomplemented
26#6294C2212yescomplemented
27#3830D312yescomplemented
28D4×C3⋊D3D311yescharacteristic, complemented
29C3×#237—32no—
30C3×#237—32no—
31C3×#237—62no—
32C3×D3×D4—32no—
33C3×D3×D4—61no—
34#3811—32no—
35D3×#67—34no—
36C2×#624—32no—
37#3805—34no—
38#3804—34no—
39#3804—34no—
40C2×D3×Dic3—34no—
41C4×D32—31no—
42C4×D32—32no—
43#3799—32no—
44#3797—31no—
45#3796—32no—
46#3795—32no—
47D3×Dic6—32no—
48C22×C33C2312yes—
49C2×C3×C3⋊D3—24no—
50C2×C3×C3⋊D3C2312yes—
51C2×C3×C3⋊D3C2312yes—
52C2×C3×C3⋊D3C2311yescharacteristic
53#1113—22no—
54C33×C4C2311yescharacteristic
55C3×C3⋊Dic3C2312yes—
56C3×C3⋊Dic3C2314yes—
57C3×C3⋊Dic3C2311yescharacteristic
58D3×#41—92no—
59C2×#237—91no—
60C22×C3⋊D3C2×D312yes—
61C22×C3×D3—34no—
62C22×C3×D3—62no—
63C2×D32—32no—
64C2×D32—31no—
65C32×D4—21no—
66C32×D4—22no—
67C32×D4—41no—
68C32×D4C2×D312yescomplemented
69C32×D4C2×D311yescharacteristic, complemented
70#638C2×D312yescomplemented
71#638C2×D314yescomplemented
72C2×C3⋊Dic3C2×D314yescomplemented
73C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
74C4×C3⋊D3C2×D311yescharacteristic
75C4×C3⋊D3C2×D312yescomplemented
76#634C2×D312yescomplemented
77C3×#67—34no—
78C3×#67—34no—
79C3×#67—34no—
80C3×#67—62no—
81C3×#67—64no—
82C2×C3×Dic3—34no—
83C2×C3×Dic3—34no—
84C2×C3×Dic3—64no—
85C3×D12—32no—
86C3×D12—61no—
87C3×C4×D3—32no—
88C3×C4×D3—32no—
89C3×C4×D3—32no—
90C3×C4×D3—61no—
91C3×C4×D3—62no—
92C3×Dic6—32no—
93C3×Dic6—32no—
94C3×Dic6—62no—
95#627—34no—
96#627—32no—
97#626—34no—
98#626—34no—
99#625—34no—
100#624—32no—
101#624—31no—
102#624—34no—
103#624—32no—
104D3×Dic3—34no—
105D3×Dic3—32no—
106D3×Dic3—34no—
107D3×Dic3—34no—
108D3×Dic3—32no—
109D3×Dic3—32no—
110C2×C33—22no—
111C2×C33C2411yescharacteristic
112C3×C3⋊D3—22no—
113C3×C3⋊D3—22no—
114C3×C3⋊D3#4112yescomplemented
115C3×#41—92no—
116C2×C3×D4—91no—
117C2×#67—92no—
118C22×Dic3—92no—
119#239—92no—
120D3×Q8—92no—
121#237—32no—
122#237—32no—
123#237—62no—
124#237—92no—
125#237—94no—
126#237—94no—
127D3×D4—32no—
128D3×D4—61no—
129D3×D4—94no—
130#235—92no—
131#235—94no—
132C2×C4×D3—91no—
133C2×C4×D3—92no—
134C2×C4×D3—94no—
135C2×Dic6—91no—
136C22×C32—24no—
137C22×C32—22no—
138C22×C32—42no—
139C22×C32C22×D312yes—
140C22×C32C22×D314yes—
141C2×C3⋊D3—24no—
142C2×C3⋊D3C22×D312yes—
143C2×C3⋊D3C22×D312yes—
144C2×C3⋊D3C22×D311yescharacteristic
145C2×C3×D3—32no—
146C2×C3×D3—34no—
147C2×C3×D3—32no—
148C2×C3×D3—32no—
149C2×C3×D3—61no—
150C2×C3×D3—62no—
151C2×C3×D3—62no—
152C2×C3×D3—68no—
153C2×C3×D3—124no—
154D32—64no—
155D32—62no—
156C32×C4—22no—
157C32×C4—21no—
158C32×C4—41no—
159C32×C4C22×D311yescharacteristic
160C32×C4C22×D312yes—
161C3⋊Dic3C22×D312yes—
162C3⋊Dic3C22×D314yes—
163C3⋊Dic3C22×D311yescharacteristic
164C3×Dic3—32no—
165C3×Dic3—34no—
166C3×Dic3—32no—
167C3×Dic3—32no—
168C3×Dic3—34no—
169C3×Dic3—64no—
170C3×Dic3—62no—
171C3×Dic3—61no—
172C2×#41—271noSylow
173C33C2×#4111yescharacteristic, complemented, Sylow
174C23×C3—92no—
175C22×D3—34no—
176C22×D3—62no—
177C22×D3—92no—
178C22×D3—91no—
179C3×Q8—92no—
180C3×D4—21no—
181C3×D4—22no—
182C3×D4—41no—
183C3×D4—94no—
184C3×D4—92no—
185C3×D4—91no—
186C3×D4D3212yescomplemented
187C3×D4D3211yescharacteristic, complemented
188C2×C3×C4—94no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191#67—34no—
192#67—34no—
193#67—34no—
194#67—62no—
195#67—64no—
196#67—94no—
197#67—94no—
198C2×Dic3—34no—
199C2×Dic3—34no—
200C2×Dic3—64no—
201C2×Dic3—92no—
202C2×Dic3—92no—
203C2×Dic3—91no—
204C2×Dic3—94no—
205D12—32no—
206D12—61no—
207D12—94no—
208C4×D3—32no—
209C4×D3—32no—
210C4×D3—32no—
211C4×D3—61no—
212C4×D3—62no—
213C4×D3—92no—
214C4×D3—94no—
215C4×D3—92no—
216C4×D3—94no—
217C4×D3—92no—
218C4×D3—94no—
219Dic6—32no—
220Dic6—32no—
221Dic6—62no—
222Dic6—92no—
223Dic6—94no—
224C2×C32—22no—
225C2×C32—24no—
226C2×C32—22no—
227C2×C32—21no—
228C2×C32—41no—
229C2×C32—42no—
230C2×C32—44no—
231C2×C32—82no—
232C2×C32C23×D311yescharacteristic
233C2×C32C23×D312yes—
234C3⋊D3—22no—
235C3⋊D3—22no—
236C3⋊D3#23712yescomplemented
237C3×D3—34no—
238C3×D3—62no—
239C3×D3—62no—
240C3×D3—62no—
241C3×D3—64no—
242C3×D3—122no—
243C3×D3—122no—
244#41—92no—
245#41—274no—
246#41—272no—
247C2×Q8—271no—
248C2×D4—91no—
249C2×D4—272no—
250C22×C4—271no—
251C22×C4—272no—
252C22×C3—24no—
253C22×C3—22no—
254C22×C3—42no—
255C22×C3—91no—
256C22×C3—92no—
257C22×C3—92no—
258C22×C3—184no—
259C22×C3C2×D3212yes—
260C22×C3C2×D3214yes—
261C2×D3—32no—
262C2×D3—32no—
263C2×D3—32no—
264C2×D3—34no—
265C2×D3—61no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—68no—
269C2×D3—124no—
270C2×D3—184no—
271C2×D3—182no—
272C3×C4—22no—
273C3×C4—21no—
274C3×C4—41no—
275C3×C4—91no—
276C3×C4—94no—
277C3×C4—92no—
278C3×C4C2×D3211yescharacteristic
279C3×C4C2×D3212yes—
280Dic3—34no—
281Dic3—34no—
282Dic3—32no—
283Dic3—32no—
284Dic3—32no—
285Dic3—64no—
286Dic3—62no—
287Dic3—61no—
288C32—22no—
289C32—21no—
290C32—41no—
291C32C2×#23711yescharacteristic, complemented
292C32D3×#4112yescomplemented
293C23—92no—
294C23—271no—
295Q8—92no—
296Q8—272no—
297D4—92no—
298D4—94no—
299D4—91no—
300D4—274no—
301D4#111311yescharacteristic, complemented
302C2×C4—94no—
303C2×C4—91no—
304C2×C4—92no—
305C2×C4—271no—
306C2×C4—272no—
307C2×C4—272no—
308C2×C4—274no—
309C2×C3—24no—
310C2×C3—22no—
311C2×C3—22no—
312C2×C3—21no—
313C2×C3—42no—
314C2×C3—44no—
315C2×C3—41no—
316C2×C3—82no—
317C2×C3—92no—
318C2×C3—182no—
319C2×C3—182no—
320C2×C3C22×D3212yes—
321C2×C3C22×D3211yescharacteristic
322D3—34no—
323D3—62no—
324D3—64no—
325D3—62no—
326D3—62no—
327D3—122no—
328D3—122no—
329C22—91no—
330C22—92no—
331C22—92no—
332C22—184no—
333C22—542no—
334C22C2×#111312yes—
335C4—92no—
336C4—91no—
337C4—94no—
338C4C2×#111311yescharacteristic
339C3—21no—
340C3—22no—
341C3—41no—
342C3D3×#23712yescomplemented
343C3#6431211yescharacteristic, complemented
344C2—22no—
345C2—92no—
346C2—182no—
347C2—182no—
348C2C22×#111311yescharacteristic
subgroupG/H
center, FrattiniC2C22×#1113
commutator, socle, cosocleC2×C33C24
FittingC33×D4C22
hypercenterD4#1113
hypocenterC33C2×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
1. C3×#3830C2semidirect
2. C3×D4×C3⋊D3C2semidirect
3. #89049C2semidirect
4. #89046C2semidirect
5. C2×#6294C2semidirect
6. C4×#1113C2semidirect
7. #89041C2semidirect
8. #89040C2semidirect
13. C33×D4C22semidirect
14. C3×#638C22semidirect
15. C3×#638C22semidirect
16. C2×C3×C3⋊Dic3C22semidirect
17. C3×C4⋊(C3⋊D3)C22semidirect
19. C3×C4×C3⋊D3C22semidirect
20. C3×#634C22semidirect
21. #6296C22semidirect
22. #6295C22semidirect
25. #6294C22semidirect
26. #6294C22semidirect
27. #3830D3semidirect
28. D4×C3⋊D3D3semidirect
68. C32×D4C2×D3semidirect
69. C32×D4C2×D3semidirect
70. #638C2×D3semidirect
71. #638C2×D3semidirect
72. C2×C3⋊Dic3C2×D3semidirect
73. C4⋊(C3⋊D3)C2×D3semidirect
75. C4×C3⋊D3C2×D3semidirect
76. #634C2×D3semidirect
114. C3×C3⋊D3#41semidirect
173. C33C2×#41semidirect
186. C3×D4D32semidirect
187. C3×D4D32semidirect
236. C3⋊D3#237semidirect
291. C32C2×#237semidirect
292. C32D3×#41semidirect
301. D4#1113semidirect
342. C3D3×#237semidirect
343. C3#64312semidirect

Maximal subgroups

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

subgroupindex
C3×#38302
C3×D4×C3⋊D32
#890492
#890462
C2×#62942
C4×#11132
#890412
#890402
#643123
D3×#2373