Finite Group Databasebeta

← #11723713

Subgroups — #11723713

Order 864.

338 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 59 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×#3832C211yesmaximal, characteristic, complemented
2#89041C211yesmaximal, characteristic, complemented
3C3×#3756C211yesmaximal, characteristic, complemented
4C3×#3755C211yesmaximal, characteristic, complemented
5#88811C211yesmaximal, characteristic, complemented
6#88810C211yesmaximal, characteristic, complemented
7#88808C211yesmaximal, characteristic, complemented
8#63950—31nomaximal
9#63949—31nomaximal
10#63947—31nomaximal
11C33×Q8C2211yescharacteristic
12C3×C4⋊(C3⋊D3)—21no—
13C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4⋊(C3⋊D3)C2211yescharacteristic
15C3×C4×C3⋊D3—21no—
16C3×C4×C3⋊D3C2211yescharacteristic
17C3×#634C2211yescharacteristic
18#6295—21no—
19#6294—21no—
20C3×C3⋊(C3⋊C8)C2211yescharacteristic
21C3×C3⋊(C3⋊C8)C2211yescharacteristic
22#3832D311yescharacteristic, complemented
23C3×#239—31no—
24C3×#239—31no—
25C3×#239—61no—
26#3799—31no—
27#3796—31no—
28#3795—31no—
29#3756D311yescharacteristic, complemented
30#3755D311yescharacteristic, complemented
31C3×#216—31no—
32C3×#216—31no—
33C3×#216—61no—
34C3×#215—31no—
35C3×#215—31no—
36C3×#215—61no—
37#3717—31no—
38#3716—31no—
39#3715—31no—
40#3714—31no—
41#3714—31no—
42#3713—31no—
43#3710—31no—
44D3×C3⋊C8—31no—
45D3×C3⋊C8—31no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3D411yescharacteristic
49C33×C4—21no—
50C33×C4C2311yescharacteristic
51C3×C3⋊Dic3—21no—
52C3×C3⋊Dic3D411yescharacteristic
53#950—91no—
54#919—91no—
55#916—91no—
56C32×Q8—21no—
57C32×Q8—21no—
58C32×Q8—21no—
59C32×Q8—41no—
60C32×Q8C2×D311yescharacteristic
61C32×Q8C2×D311yescharacteristic
62C32×Q8C2×D311yescharacteristic
63C4⋊(C3⋊D3)—21no—
64C4⋊(C3⋊D3)C2×D311yescharacteristic
65C4⋊(C3⋊D3)C2×D311yescharacteristic
66C4×C3⋊D3—21no—
67C4×C3⋊D3C2×D311yescharacteristic
68#634C2×D311yescharacteristic
69C3×D12—31no—
70C3×D12—31no—
71C3×D12—31no—
72C3×D12—31no—
73C3×D12—61no—
74C3×D12—61no—
75C3×D12—61no—
76C3×D12—61no—
77C3×D12—62no—
78C3×C4×D3—31no—
79C3×C4×D3—31no—
80C3×C4×D3—61no—
81C3×C4×D3—61no—
82C3×C4×D3—61no—
83C3×C4×D3—62no—
84C3×Dic6—31no—
85C3×Dic6—31no—
86C3×Dic6—61no—
87#626—61no—
88#626—61no—
89#625—61no—
90#624—61no—
91D3×Dic3—61no—
92D3×Dic3—61no—
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—
103C3×C3⋊D3—41no—
104C3×C3⋊D3—41no—
105C3×#41—91no—
106#239—31no—
107#239—31no—
108#239—61no—
109#239—91no—
110#237—91no—
111#235—91no—
112C3×Q16—91no—
113C3×SD16—91no—
114#216—31no—
115#216—31no—
116#216—61no—
117#215—31no—
118#215—31no—
119#215—61no—
120#215—91no—
121#214—91no—
122#213—91no—
123#213—91no—
124C2×C3⋊C8—91no—
125D24—91no—
126#204—91no—
127D3×C8—91no—
128D3×C8—91no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C2×C3⋊D3#6711yescharacteristic
132C2×C3×D3—31no—
133C2×C3×D3—31no—
134C2×C3×D3—61no—
135C2×C3×D3—62no—
136C2×C3×D3—61no—
137C2×C3×D3—61no—
138C2×C3×D3—61no—
139C2×C3×D3—61no—
140C2×C3×D3—121no—
141C32×C4—21no—
142C32×C4—22no—
143C32×C4—21no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—21no—
148C32×C4—41no—
149C32×C4—41no—
150C32×C4—41no—
151C32×C4—42no—
152C32×C4C22×D311yescharacteristic
153C32×C4C22×D311yescharacteristic
154C32×C4C22×D311yescharacteristic
155C3⋊Dic3—21no—
156C3⋊Dic3#6711yescharacteristic
157C3×Dic3—31no—
158C3×Dic3—31no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—121no—
163#135—271noSylow
164C33#13511yescharacteristic, complemented, Sylow
165C3×Q8—21no—
166C3×Q8—21no—
167C3×Q8—21no—
168C3×Q8—41no—
169C3×Q8—91no—
170C3×Q8D3211yescharacteristic
171C3×Q8D3211yescharacteristic
172C3×Q8D3211yescharacteristic
173C3×D4—91no—
174C3×D4—91no—
175C3×D4—181no—
176C2×C3×C4—91no—
177C2×C3×C4—181no—
178#67—181no—
179#67—181no—
180C2×Dic3—181no—
181D12—31no—
182D12—31no—
183D12—31no—
184D12—31no—
185D12—61no—
186D12—61no—
187D12—61no—
188D12—61no—
189D12—62no—
190D12—181no—
191C4×D3—31no—
192C4×D3—31no—
193C4×D3—61no—
194C4×D3—61no—
195C4×D3—61no—
196C4×D3—62no—
197C4×D3—181no—
198C4×D3—181no—
199Dic6—31no—
200Dic6—31no—
201Dic6—61no—
202C3×C8—91no—
203C3×C8—91no—
204C3⋊C8—31no—
205C3⋊C8—31no—
206C3⋊C8—31no—
207C3⋊C8—31no—
208C3⋊C8—61no—
209C3⋊C8—61no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—41no—
214C2×C32D3×D411yescharacteristic
215C2×C32D3×D411yescharacteristic
216C2×C32C2×#6711yescharacteristic
217C3⋊D3—21no—
218C3⋊D3—41no—
219C3⋊D3—41no—
220C3×D3—61no—
221C3×D3—61no—
222C3×D3—121no—
223C3×D3—122no—
224C3×D3—121no—
225C3×D3—121no—
226C3×D3—121no—
227C3×D3—121no—
228C3×D3—241no—
229#41—91no—
230#41—271no—
231Q16—91no—
232SD16—91no—
233SD16—271no—
234D8—271no—
235C2×C8—271no—
236C22×C3—91no—
237C22×C3—181no—
238C22×C3—181no—
239C2×D3—31no—
240C2×D3—31no—
241C2×D3—61no—
242C2×D3—62no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—61no—
247C2×D3—121no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—21no—
254C3×C4—21no—
255C3×C4—41no—
256C3×C4—41no—
257C3×C4—42no—
258C3×C4—41no—
259C3×C4—91no—
260C3×C4—181no—
261C3×C4C2×D3211yescharacteristic
262C3×C4C2×D3211yescharacteristic
263C3×C4C2×D3211yescharacteristic
264Dic3—31no—
265Dic3—31no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270C32—21no—
271C32—21no—
272C32—21no—
273C32—41no—
274C32#91611yescharacteristic, complemented
275C32#91911yescharacteristic, complemented
276C32#95011yescharacteristic, complemented
277Q8—91no—
278Q8#111311yescharacteristic
279D4—91no—
280D4—91no—
281D4—181no—
282D4—541no—
283C2×C4—91no—
284C2×C4—181no—
285C2×C4—541no—
286C8—91no—
287C8—91no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—181no—
293C2×C3—361no—
294C2×C3—361no—
295C2×C3D3×#6711yescharacteristic
296C2×C3D3×#6711yescharacteristic
297C2×C3#381111yescharacteristic
298D3—61no—
299D3—61no—
300D3—121no—
301D3—122no—
302D3—121no—
303D3—121no—
304D3—121no—
305D3—121no—
306D3—241no—
307C22—91no—
308C22—181no—
309C22—181no—
310C4—21no—
311C4—91no—
312C4—181no—
313C4C2×#111311yescharacteristic
314C3—21no—
315C3—21no—
316C3—21no—
317C3—41no—
318C3#6394711yescharacteristic, complemented
319C3#6394911yescharacteristic, complemented
320C3#6395011yescharacteristic, complemented
321C2—181no—
322C2—361no—
323C2—361no—
324C2#8904911yescharacteristic
subgroupG/H
centerC2#89049
commutator, cosocleC33×C4C23
FrattiniC4C2×#1113
FittingC33×Q8C22
socleC2×C33C2×D4
hypercenterQ8#1113
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
1. C3×#3832C2semidirect
2. #89041C2semidirect
3. C3×#3756C2semidirect
4. C3×#3755C2semidirect
5. #88811C2semidirect
6. #88810C2semidirect
7. #88808C2semidirect
22. #3832D3semidirect
29. #3756D3semidirect
30. #3755D3semidirect
164. C33#135semidirect
274. C32#916semidirect
275. C32#919semidirect
276. C32#950semidirect
318. C3#63947semidirect
319. C3#63949semidirect
320. C3#63950semidirect

Maximal subgroups

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

subgroupindex
C3×#38322
#890412
C3×#37562
C3×#37552
#888112
#888102
#888082
#639503
#639493
#639473