Finite Group Databasebeta

← #11722888

Subgroups — #11722888

Order 864.

328 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 58 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⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C3×#3796C211yesmaximal, characteristic
3#88809C211yesmaximal, characteristic, complemented
4#88799C211yesmaximal, characteristic, complemented
5#88794C211yesmaximal, characteristic, complemented
6C3×#3718C211yesmaximal, characteristic, complemented
7C3×#3715C211yesmaximal, characteristic, complemented
8#63951—31nomaximal
9#63943—31nomaximal
10#63929—31nomaximal
11#6308C2211yescharacteristic
12C3×C4×C3⋊D3C2211yescharacteristic
13C32×D12C2211yescharacteristic
14C32×Dic6C2211yescharacteristic
15C32×Dic6C2211yescharacteristic
16#6293—21no—
17Dic3×C3⋊D3—21no—
18C3×#626—21no—
19C3×D3×Dic3—21no—
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×C3⋊(C3⋊C8)C2211yescharacteristic
22Q8×C3⋊D3—31no—
23C3×#239—31no—
24C3×#237—31no—
25#3796D311yescharacteristic
26D3×Dic6—31no—
27D3×Dic6—31no—
28D3×Dic6—61no—
29#3756—31no—
30#3754—31no—
31C3×#216—31no—
32C3×#216—31no—
33C3×#215—31no—
34C3×#214—31no—
35#3719—31no—
36#3719—31no—
37#3719—61no—
38#3718D311yescharacteristic, complemented
39#3716—31no—
40#3716—31no—
41#3716—61no—
42#3715D311yescharacteristic, complemented
43#3712—31no—
44#3711—31no—
45#3711—31no—
46C2×C3×C3⋊D3D411yescharacteristic
47C2×C32×D3—21no—
48C33×C4C2311yescharacteristic
49C3⋊(C3⋊Dic3)—21no—
50C3×C3⋊Dic3D411yescharacteristic
51C32×Dic3—21no—
52C32×Dic3—21no—
53#951—91no—
54#918—91no—
55#915—91no—
56C32×Q8—31no—
57C32×Q8—31no—
58C32×D4—31no—
59C4×C3⋊D3—61no—
60C4×C3⋊D3C2×D311yescharacteristic
61#634—31no—
62#634—31no—
63#634—31no—
64#634—61no—
65#634—61no—
66#634—61no—
67#634—61no—
68#634—121no—
69C3×#67—61no—
70C2×C3×Dic3—61no—
71C3×D12—21no—
72C3×D12—61no—
73C3×D12C2×D311yescharacteristic
74C3×D12C2×D311yescharacteristic
75C3×C4×D3—31no—
76C3×C4×D3—31no—
77C3×C4×D3—61no—
78C3×C4×D3—61no—
79C3×Dic6—21no—
80C3×Dic6—21no—
81C3×Dic6C2×D311yescharacteristic
82C3×Dic6C2×D311yescharacteristic
83C3×Dic6C2×D311yescharacteristic
84C3×Dic6C2×D311yescharacteristic
85#627—61no—
86#627—61no—
87#627—62no—
88#626—21no—
89D3×Dic3—21no—
90D3×Dic3—61no—
91D3×Dic3—61no—
92D3×Dic3—62no—
93C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3⋊(C3⋊C8)C2×D311yescharacteristic
95C3×C3⋊C8—31no—
96C3×C3⋊C8—31no—
97C3×C3⋊C8—31no—
98C3×C3⋊C8—31no—
99C3×C3⋊C8—61no—
100C3×C3⋊C8—61no—
101C2×C33C2×D411yescharacteristic
102C3×C3⋊D3—21no—
103C32×D3—41no—
104C3×#41—91no—
105#239—31no—
106D3×Q8—91no—
107D3×Q8—91no—
108D3×Q8—181no—
109#237—31no—
110C2×Dic6—91no—
111C3×Q16—91no—
112C3×SD16—91no—
113#216—31no—
114#216—31no—
115#216—91no—
116#216—91no—
117#216—181no—
118#215—31no—
119#214—31no—
120#214—91no—
121#214—91no—
122#214—181no—
123#208—91no—
124Dic12—91no—
125#204—91no—
126ΓL(1,𝔽25)—91no—
127ΓL(1,𝔽25)—91no—
128C22×C32—61no—
129C2×C3⋊D3#6711yescharacteristic
130C2×C3×D3—21no—
131C2×C3×D3—21no—
132C2×C3×D3—31no—
133C2×C3×D3—31no—
134C2×C3×D3—41no—
135C2×C3×D3—61no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—41no—
140C32×C4—61no—
141C32×C4—61no—
142C32×C4C22×D311yescharacteristic
143C32×C4C22×D311yescharacteristic
144C32×C4C22×D311yescharacteristic
145C3⋊Dic3—61no—
146C3⋊Dic3—62no—
147C3⋊Dic3—61no—
148C3⋊Dic3—61no—
149C3⋊Dic3—121no—
150C3⋊Dic3—121no—
151C3⋊Dic3—122no—
152C3⋊Dic3#6711yescharacteristic
153C3×Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3—21no—
158C3×Dic3—31no—
159C3×Dic3—31no—
160C3×Dic3—41no—
161C3×Dic3—61no—
162#137—271noSylow
163C33#13711yescharacteristic, complemented, Sylow
164C3×Q8—31no—
165C3×Q8—31no—
166C3×Q8—31no—
167C3×Q8—31no—
168C3×Q8—61no—
169C3×Q8—61no—
170C3×D4—31no—
171C3×D4—31no—
172C3×D4—61no—
173C3×D4—181no—
174C2×C3×C4—91no—
175C2×C3×C4—181no—
176#67—61no—
177C2×Dic3—61no—
178C2×Dic3—181no—
179D12—61no—
180D12D3211yescharacteristic
181C4×D3—31no—
182C4×D3—31no—
183C4×D3—61no—
184C4×D3—61no—
185C4×D3—181no—
186C4×D3—181no—
187C4×D3—182no—
188Dic6—91no—
189Dic6—91no—
190Dic6—91no—
191Dic6—181no—
192Dic6—181no—
193Dic6—181no—
194Dic6—181no—
195Dic6—181no—
196Dic6—181no—
197Dic6—182no—
198Dic6—361no—
199Dic6D3211yescharacteristic
200Dic6D3211yescharacteristic
201C3×C8—91no—
202C3×C8—91no—
203C3⋊C8—31no—
204C3⋊C8—31no—
205C3⋊C8—31no—
206C3⋊C8—31no—
207C3⋊C8—61no—
208C3⋊C8—61no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32—121no—
214C2×C32D3×D411yescharacteristic
215C2×C32D3×D411yescharacteristic
216C2×C32C2×#6711yescharacteristic
217C3⋊D3—21no—
218C3×D3—41no—
219C3×D3—41no—
220C3×D3—62no—
221C3×D3—61no—
222C3×D3—61no—
223C3×D3—81no—
224#41—91no—
225C2×Q8—271no—
226Q16—91no—
227Q16—271no—
228SD16—91no—
229SD16—271no—
230M16—271no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—91no—
234C22×C3—121no—
235C2×D3—21no—
236C2×D3—31no—
237C2×D3—31no—
238C2×D3—61no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—41no—
243C3×C4—62no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—91no—
249C3×C4—121no—
250C3×C4C2×D3211yescharacteristic
251C3×C4C2×D3211yescharacteristic
252C3×C4C2×D3211yescharacteristic
253Dic3—21no—
254Dic3—21no—
255Dic3—31no—
256Dic3—31no—
257Dic3—61no—
258Dic3—182no—
259Dic3—181no—
260Dic3—181no—
261Dic3—181no—
262Dic3—361no—
263Dic3—361no—
264Dic3—362no—
265C32—21no—
266C32—21no—
267C32—21no—
268C32—41no—
269C32#91511yescharacteristic, complemented
270C32#91811yescharacteristic, complemented
271C32#95111yescharacteristic, complemented
272Q8—31no—
273Q8—31no—
274Q8—271no—
275Q8—541no—
276D4—31no—
277D4—181no—
278C2×C4—91no—
279C2×C4—181no—
280C2×C4—541no—
281C8—91no—
282C8—91no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—121no—
288C2×C3—121no—
289C2×C3—181no—
290C2×C3—241no—
291C2×C3D3×#6711yescharacteristic
292C2×C3D3×#6711yescharacteristic
293C2×C3#381111yescharacteristic
294D3—41no—
295D3—62no—
296D3—61no—
297D3—61no—
298C22—61no—
299C22—91no—
300C4—61no—
301C4—61no—
302C4—91no—
303C4—541no—
304C4D3311yescharacteristic
305C3—21no—
306C3—21no—
307C3—21no—
308C3—41no—
309C3#6392911yescharacteristic, complemented
310C3#6394311yescharacteristic, complemented
311C3#6395111yescharacteristic, complemented
312C2—121no—
313C2—181no—
314C2#8895211yescharacteristic
subgroupG/H
centerC2#88952
commutator, Fitting, cosocleC33×C4C23
Frattini, hypercenterC4D33
socleC2×C33C2×D4
prime radicalC3×#3796C2
hypocenterC33#137
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. C3⋊D3×Dic6C2semidirect
3. #88809C2semidirect
4. #88799C2semidirect
5. #88794C2semidirect
6. C3×#3718C2semidirect
7. C3×#3715C2semidirect
38. #3718D3semidirect
42. #3715D3semidirect
163. C33#137semidirect
269. C32#915semidirect
270. C32#918semidirect
271. C32#951semidirect
309. C3#63929semidirect
310. C3#63943semidirect
311. C3#63951semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×Dic62
C3×#37962
#888092
#887992
#887942
C3×#37182
C3×#37152
#639513
#639433
#639293