Finite Group Databasebeta

← #11722865

Subgroups — #11722865

Order 864.

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

normal subgroups only

This group has 346 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
1#89041C211yesmaximal, characteristic, complemented
2#89018C211yesmaximal, characteristic, complemented
3#88797C211yesmaximal, characteristic, complemented
4#88791C211yesmaximal, characteristic, complemented
5C3×#3719C211yesmaximal, characteristic, complemented
6C3×#3717C211yesmaximal, characteristic, complemented
7C3×#3710C211yesmaximal, characteristic, complemented
8#63950—31nomaximal
9#63808—31nomaximal
10#63806—31nomaximal
11C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C4×C3⋊D3C2211yescharacteristic
14C3×#634C2211yescharacteristic
15C32×Dic6C2211yescharacteristic
16#6295—21no—
17#6294—21no—
18#6292—21no—
19#6289—21no—
20C32×C3⋊C8C2211yescharacteristic
21C32×C3⋊C8C2211yescharacteristic
22#3832—31no—
23#3799—31no—
24#3799—31no—
25#3799—31no—
26#3799—61no—
27#3796—31no—
28#3795—31no—
29#3745—31no—
30#3744—31no—
31C3×#216—31no—
32C3×#215—31no—
33C3×Dic12—31no—
34C3×#204—31no—
35C3×D3×C8—31no—
36C3×D3×C8—31no—
37#3719D311yescharacteristic, complemented
38#3717—31no—
39#3717—31no—
40#3717—61no—
41#3717D311yescharacteristic, complemented
42#3714—31no—
43#3714—31no—
44#3714—61no—
45#3710D311yescharacteristic, complemented
46C2×C3⋊(C3⋊D3)—21no—
47C2×C3×C3⋊D3—21no—
48C2×C3×C3⋊D3D411yescharacteristic
49C33×C4C2311yescharacteristic
50C3×C3⋊Dic3—21no—
51C3×C3⋊Dic3D411yescharacteristic
52C32×Dic3—21no—
53#919—91no—
54#916—91no—
55#904—91no—
56C32×Q8—31no—
57C4⋊(C3⋊D3)—31no—
58C4⋊(C3⋊D3)—31no—
59C4⋊(C3⋊D3)—31no—
60C4⋊(C3⋊D3)—61no—
61C4⋊(C3⋊D3)—61no—
62C4⋊(C3⋊D3)—61no—
63C4⋊(C3⋊D3)—61no—
64C4⋊(C3⋊D3)—121no—
65C4⋊(C3⋊D3)C2×D311yescharacteristic
66C4×C3⋊D3—61no—
67C4×C3⋊D3C2×D311yescharacteristic
68#634C2×D311yescharacteristic
69C3×D12—31no—
70C3×D12—31no—
71C3×D12—61no—
72C3×C4×D3—31no—
73C3×C4×D3—31no—
74C3×C4×D3—61no—
75C3×Dic6—21no—
76C3×Dic6—31no—
77C3×Dic6—31no—
78C3×Dic6—61no—
79C3×Dic6C2×D311yescharacteristic
80C3×Dic6C2×D311yescharacteristic
81#626—61no—
82#626—61no—
83#626—62no—
84#626—61no—
85#626—61no—
86#625—61no—
87#624—61no—
88#624—61no—
89#624—62no—
90#624—61no—
91D3×Dic3—61no—
92D3×Dic3—61no—
93C32×C8—31no—
94C32×C8—31no—
95C3×C3⋊C8—21no—
96C3×C3⋊C8—21no—
97C3×C3⋊C8C2×D311yescharacteristic
98C3×C3⋊C8C2×D311yescharacteristic
99C3×C3⋊C8C2×D311yescharacteristic
100C3×C3⋊C8C2×D311yescharacteristic
101C2×C33C2×D411yescharacteristic
102C3⋊(C3⋊D3)—41no—
103C3×C3⋊D3—21no—
104C3×C3⋊D3—41no—
105#239—91no—
106#239—91no—
107#239—91no—
108#239—181no—
109#237—91no—
110#235—91no—
111#235—91no—
112C3×Q16—91no—
113C3×SD16—91no—
114C2×C3×C8—91no—
115#216—31no—
116#215—31no—
117#215—91no—
118#213—91no—
119Dic12—31no—
120D24—91no—
121D24—91no—
122D24—181no—
123#204—31no—
124#204—91no—
125#204—91no—
126#204—181no—
127D3×C8—31no—
128D3×C8—31no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—61no—
131C2×C3⋊D3—62no—
132C2×C3⋊D3—61no—
133C2×C3⋊D3—61no—
134C2×C3⋊D3—121no—
135C2×C3⋊D3—121no—
136C2×C3⋊D3—122no—
137C2×C3⋊D3D1211yescharacteristic
138C2×C3×D3—31no—
139C2×C3×D3—31no—
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—121no—
144C32×C4—21no—
145C32×C4—21no—
146C32×C4—21no—
147C32×C4—41no—
148C32×C4—61no—
149C32×C4C22×D311yescharacteristic
150C32×C4C22×D311yescharacteristic
151C32×C4C22×D311yescharacteristic
152C3⋊Dic3—21no—
153C3⋊Dic3D1211yescharacteristic
154C3×Dic3—22no—
155C3×Dic3—21no—
156C3×Dic3—21no—
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—31no—
166C3×Q8—31no—
167C3×Q8—61no—
168C3×Q8—91no—
169C3×D4—91no—
170C2×C3×C4—91no—
171#67—181no—
172#67—181no—
173#67—181no—
174C2×Dic3—181no—
175D12—31no—
176D12—31no—
177D12—61no—
178D12—91no—
179D12—91no—
180D12—91no—
181D12—181no—
182D12—181no—
183D12—181no—
184D12—181no—
185D12—181no—
186D12—182no—
187D12—181no—
188D12—361no—
189C4×D3—31no—
190C4×D3—31no—
191C4×D3—61no—
192C4×D3—181no—
193C4×D3—181no—
194C4×D3—181no—
195C4×D3—181no—
196C4×D3—182no—
197C4×D3—181no—
198Dic6—31no—
199Dic6—31no—
200Dic6—61no—
201Dic6D3211yescharacteristic
202C3×C8—31no—
203C3×C8—31no—
204C3×C8—31no—
205C3×C8—31no—
206C3×C8—61no—
207C3×C8—61no—
208C3⋊C8D3211yescharacteristic
209C3⋊C8D3211yescharacteristic
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—41no—
214C2×C32C2×D1211yescharacteristic
215C2×C32D3×D411yescharacteristic
216C2×C32D3×D411yescharacteristic
217C3⋊D3—21no—
218C3⋊D3—41no—
219C3⋊D3—121no—
220C3⋊D3—122no—
221C3⋊D3—121no—
222C3⋊D3—121no—
223C3⋊D3—241no—
224C3⋊D3—241no—
225C3⋊D3—242no—
226C3×D3—61no—
227C3×D3—61no—
228C3×D3—121no—
229C3×D3—121no—
230C3×D3—121no—
231C3×D3—241no—
232#41—271no—
233#41—271no—
234Q16—91no—
235SD16—91no—
236SD16—271no—
237D8—271no—
238C2×C8—91no—
239C22×C3—91no—
240C22×C3—181no—
241C2×D3—31no—
242C2×D3—31no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—121no—
247C2×D3—182no—
248C2×D3—181no—
249C2×D3—181no—
250C2×D3—181no—
251C2×D3—361no—
252C2×D3—361no—
253C2×D3—362no—
254C3×C4—21no—
255C3×C4—21no—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—62no—
259C3×C4—61no—
260C3×C4—61no—
261C3×C4—91no—
262C3×C4—181no—
263C3×C4C2×D3211yescharacteristic
264C3×C4C2×D3211yescharacteristic
265C3×C4C2×D3211yescharacteristic
266Dic3—21no—
267Dic3—31no—
268Dic3—31no—
269Dic3—61no—
270Dic3—61no—
271Dic3—61no—
272Dic3—121no—
273C32—21no—
274C32—21no—
275C32—21no—
276C32—41no—
277C32#90411yescharacteristic, complemented
278C32#91611yescharacteristic, complemented
279C32#91911yescharacteristic, complemented
280Q8—31no—
281Q8—91no—
282D4—91no—
283D4—271no—
284D4—541no—
285D4—541no—
286C2×C4—91no—
287C2×C4—541no—
288C2×C4—541no—
289C8—31no—
290C8—31no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—41no—
295C2×C3—181no—
296C2×C3—361no—
297C2×C3D3×D1211yescharacteristic
298C2×C3D3×D1211yescharacteristic
299C2×C3#381111yescharacteristic
300D3—61no—
301D3—61no—
302D3—121no—
303D3—121no—
304D3—121no—
305D3—241no—
306D3—362no—
307D3—361no—
308D3—361no—
309D3—361no—
310D3—721no—
311D3—721no—
312D3—722no—
313C22—91no—
314C22—181no—
315C22—541no—
316C4—61no—
317C4—91no—
318C4—181no—
319C4D3311yescharacteristic
320C3—21no—
321C3—21no—
322C3—21no—
323C3—41no—
324C3#6380611yescharacteristic, complemented
325C3#6380811yescharacteristic, complemented
326C3#6395011yescharacteristic, complemented
327C2—181no—
328C2—361no—
329C2—1081no—
330C2#8895511yescharacteristic
subgroupG/H
centerC2#88955
commutator, Fitting, cosocleC33×C4C23
Frattini, hypercenterC4D33
socleC2×C33C2×D4
hypocenterC33#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

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

NH ≅ G/Ntype
1. #89041C2semidirect
2. #89018C2semidirect
3. #88797C2semidirect
4. #88791C2semidirect
5. C3×#3719C2semidirect
6. C3×#3717C2semidirect
7. C3×#3710C2semidirect
37. #3719D3semidirect
41. #3717D3semidirect
45. #3710D3semidirect
164. C33#135semidirect
277. C32#904semidirect
278. C32#916semidirect
279. C32#919semidirect
324. C3#63806semidirect
325. C3#63808semidirect
326. C3#63950semidirect

Maximal subgroups

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

subgroupindex
#890412
#890182
#887972
#887912
C3×#37192
C3×#37172
C3×#37102
#639503
#638083
#638063