Finite Group Databasebeta

← #11722896

Subgroups — #11722896

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
1#89041C211yesmaximal, characteristic
2C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
3#88800C211yesmaximal, characteristic, complemented
4#88796C211yesmaximal, characteristic, complemented
5C3×#3719C211yesmaximal, characteristic, complemented
6C3×#3717C211yesmaximal, characteristic, complemented
7C3×#3712C211yesmaximal, characteristic, complemented
8#63951—31nomaximal
9#63801—31nomaximal
10#63800—31nomaximal
11#6308C2211yescharacteristic
12C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C3×C4×C3⋊D3C2211yescharacteristic
14C3×#634C2211yescharacteristic
15C32×Dic6C2211yescharacteristic
16#6295—21no—
17#6294—21no—
18#6293—21no—
19Dic3×C3⋊D3—21no—
20C32×C3⋊C8C2211yescharacteristic
21C32×C3⋊C8C2211yescharacteristic
22Q8×C3⋊D3—31no—
23#3799—31no—
24#3796—31no—
25#3795—31no—
26D3×Dic6—31no—
27D3×Dic6—31no—
28D3×Dic6—61no—
29#3746—31no—
30#3744—31no—
31C3×#216—31no—
32C3×#215—31no—
33C3×Dic12—31no—
34C3×#204—31no—
35C3×ΓL(1,𝔽25)—31no—
36C3×ΓL(1,𝔽25)—31no—
37#3719—31no—
38#3719—31no—
39#3719—61no—
40#3719D311yescharacteristic, complemented
41#3717D311yescharacteristic, complemented
42#3716—31no—
43#3716—31no—
44#3716—61no—
45#3712D311yescharacteristic, complemented
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3D411yescharacteristic
48C33×C4C2311yescharacteristic
49C3⋊(C3⋊Dic3)—21no—
50C3×C3⋊Dic3—21no—
51C3×C3⋊Dic3D411yescharacteristic
52C32×Dic3—21no—
53#918—91no—
54#915—91no—
55#909—91no—
56C32×Q8—31no—
57C4⋊(C3⋊D3)C2×D311yescharacteristic
58C4×C3⋊D3—61no—
59C4×C3⋊D3C2×D311yescharacteristic
60#634—31no—
61#634—31no—
62#634—31no—
63#634—61no—
64#634—61no—
65#634—61no—
66#634—61no—
67#634—121no—
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#627—61no—
82#627—61no—
83#627—62no—
84#626—61no—
85#626—61no—
86#625—61no—
87#624—61no—
88D3×Dic3—61no—
89D3×Dic3—61no—
90D3×Dic3—62no—
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—21no—
103C3×C3⋊D3—41no—
104#239—91no—
105D3×Q8—91no—
106D3×Q8—91no—
107D3×Q8—181no—
108#237—91no—
109#235—91no—
110C2×Dic6—91no—
111C3×Q16—91no—
112C3×SD16—91no—
113C3×M16—91no—
114#216—31no—
115#216—91no—
116#215—31no—
117#214—91no—
118Dic12—31no—
119Dic12—91no—
120Dic12—91no—
121Dic12—181no—
122#204—31no—
123#204—91no—
124#204—91no—
125#204—181no—
126ΓL(1,𝔽25)—31no—
127ΓL(1,𝔽25)—31no—
128C2×C3⋊D3—21no—
129C2×C3⋊D3D1211yescharacteristic
130C2×C3×D3—31no—
131C2×C3×D3—31no—
132C2×C3×D3—61no—
133C2×C3×D3—61no—
134C2×C3×D3—61no—
135C2×C3×D3—121no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—41no—
140C32×C4—61no—
141C32×C4C22×D311yescharacteristic
142C32×C4C22×D311yescharacteristic
143C32×C4C22×D311yescharacteristic
144C3⋊Dic3—21no—
145C3⋊Dic3—61no—
146C3⋊Dic3—62no—
147C3⋊Dic3—61no—
148C3⋊Dic3—61no—
149C3⋊Dic3—121no—
150C3⋊Dic3—121no—
151C3⋊Dic3—122no—
152C3⋊Dic3D1211yescharacteristic
153C3×Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—31no—
157C3×Dic3—31no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—121no—
162#137—271noSylow
163C33#13711yescharacteristic, complemented, Sylow
164C3×Q8—31no—
165C3×Q8—31no—
166C3×Q8—61no—
167C3×Q8—91no—
168C3×D4—91no—
169C2×C3×C4—91no—
170#67—181no—
171#67—181no—
172C2×Dic3—181no—
173C2×Dic3—181no—
174D12—31no—
175D12—31no—
176D12—61no—
177D12—181no—
178C4×D3—31no—
179C4×D3—31no—
180C4×D3—61no—
181C4×D3—181no—
182C4×D3—181no—
183C4×D3—181no—
184C4×D3—181no—
185C4×D3—182no—
186Dic6—31no—
187Dic6—31no—
188Dic6—61no—
189Dic6—91no—
190Dic6—91no—
191Dic6—91no—
192Dic6—181no—
193Dic6—181no—
194Dic6—181no—
195Dic6—181no—
196Dic6—181no—
197Dic6—181no—
198Dic6—182no—
199Dic6—361no—
200Dic6D3211yescharacteristic
201C3×C8—31no—
202C3×C8—31no—
203C3×C8—31no—
204C3×C8—31no—
205C3×C8—61no—
206C3×C8—61no—
207C3⋊C8D3211yescharacteristic
208C3⋊C8D3211yescharacteristic
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—41no—
213C2×C32C2×D1211yescharacteristic
214C2×C32D3×D411yescharacteristic
215C2×C32D3×D411yescharacteristic
216C3⋊D3—21no—
217C3⋊D3—41no—
218C3×D3—62no—
219C3×D3—61no—
220C3×D3—61no—
221C3×D3—121no—
222C3×D3—121no—
223C3×D3—241no—
224#41—271no—
225C2×Q8—271no—
226Q16—91no—
227Q16—271no—
228SD16—91no—
229SD16—271no—
230M16—91no—
231C22×C3—91no—
232C22×C3—181no—
233C2×D3—31no—
234C2×D3—31no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—121no—
239C3×C4—21no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—41no—
243C3×C4—62no—
244C3×C4—61no—
245C3×C4—61no—
246C3×C4—91no—
247C3×C4—181no—
248C3×C4C2×D3211yescharacteristic
249C3×C4C2×D3211yescharacteristic
250C3×C4C2×D3211yescharacteristic
251Dic3—21no—
252Dic3—31no—
253Dic3—31no—
254Dic3—61no—
255Dic3—61no—
256Dic3—61no—
257Dic3—121no—
258Dic3—182no—
259Dic3—181no—
260Dic3—181no—
261Dic3—181no—
262Dic3—361no—
263Dic3—361no—
264Dic3—362no—
265C32—21no—
266C32—21no—
267C32—21no—
268C32—41no—
269C32#90911yescharacteristic, complemented
270C32#91511yescharacteristic, complemented
271C32#91811yescharacteristic, complemented
272Q8—31no—
273Q8—91no—
274Q8—271no—
275Q8—541no—
276D4—91no—
277D4—541no—
278C2×C4—91no—
279C2×C4—541no—
280C2×C4—541no—
281C8—31no—
282C8—31no—
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—41no—
287C2×C3—181no—
288C2×C3—361no—
289C2×C3D3×D1211yescharacteristic
290C2×C3D3×D1211yescharacteristic
291C2×C3#381111yescharacteristic
292D3—62no—
293D3—61no—
294D3—61no—
295D3—121no—
296D3—121no—
297D3—241no—
298C22—91no—
299C22—181no—
300C4—61no—
301C4—91no—
302C4—181no—
303C4—541no—
304C4D3311yescharacteristic
305C3—21no—
306C3—21no—
307C3—21no—
308C3—41no—
309C3#6380011yescharacteristic, complemented
310C3#6380111yescharacteristic, complemented
311C3#6395111yescharacteristic, complemented
312C2—181no—
313C2—361no—
314C2#8895511yescharacteristic
subgroupG/H
centerC2#88955
commutator, Fitting, cosocleC33×C4C23
Frattini, hypercenterC4D33
socleC2×C33C2×D4
prime radical#89041C2
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
2. C3⋊D3×Dic6C2semidirect
3. #88800C2semidirect
4. #88796C2semidirect
5. C3×#3719C2semidirect
6. C3×#3717C2semidirect
7. C3×#3712C2semidirect
40. #3719D3semidirect
41. #3717D3semidirect
45. #3712D3semidirect
163. C33#137semidirect
269. C32#909semidirect
270. C32#915semidirect
271. C32#918semidirect
309. C3#63800semidirect
310. C3#63801semidirect
311. C3#63951semidirect

Maximal subgroups

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

subgroupindex
#890412
C3⋊D3×Dic62
#888002
#887962
C3×#37192
C3×#37172
C3×#37122
#639513
#638013
#638003