Finite Group Databasebeta

← #11723022

Subgroups — #11723022

Order 864.

346 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 70 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#88815C211yesmaximal, characteristic, complemented
2Dic3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
3(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
4Dic3×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic
8#63956—31nomaximal
9#63862—31nomaximal
10#63861—31nomaximal
11C22×C32×D3C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#3757—31no—
19C4⋊(C3⋊Dic3)—31no—
20C4×C3⋊Dic3—31no—
21C3×#217—31no—
22C3×#217—31no—
23C3×(C2×D3)⋊C4—31no—
24C3×(C2×D3)⋊C4—31no—
25C3×Dic3⋊C4—31no—
26C3×Dic3⋊C4—31no—
27#3724—31no—
28#3724D311yescharacteristic, complemented
29Dic3⋊Dic3—31no—
30Dic3⋊Dic3—31no—
31Dic3⋊Dic3—31no—
32Dic3⋊Dic3—31no—
33Dic3⋊Dic3—61no—
34(C2×D3)⋊Dic3—31no—
35(C2×D3)⋊Dic3—31no—
36(C2×D3)⋊Dic3—61no—
37(C2×D3)⋊Dic3D311yescharacteristic
38(C2×D3)⋊Dic3D311yescharacteristic
39Dic32—31no—
40Dic32—31no—
41Dic32—61no—
42C22×C33C2311yescharacteristic
43C2×C32×D3—21no—
44C2×C32×D3—21no—
45C2×C32×D3—21no—
46C3⋊(C3⋊Dic3)—21no—
47C3×C3⋊Dic3—21no—
48C3×C3⋊Dic3—21no—
49C3×C3⋊Dic3—21no—
50C32×Dic3—21no—
51C32×Dic3—21no—
52#898—91no—
53#879—91no—
54#879—91no—
55C23×C32—31no—
56C22×C3×D3—21no—
57C22×C3×D3C2×D311yescharacteristic
58C22×C3×D3C2×D311yescharacteristic
59C2×C32×C4—31no—
60C2×C32×C4—31no—
61C2×C3⋊Dic3—31no—
62C2×C3⋊Dic3—31no—
63C2×C3⋊Dic3—31no—
64C2×C3⋊Dic3—61no—
65C2×C3⋊Dic3—61no—
66C2×C3⋊Dic3—61no—
67C2×C3⋊Dic3—121no—
68C2×C3⋊Dic3C2×D311yescharacteristic
69C2×C3⋊Dic3C2×D311yescharacteristic
70C2×C3⋊Dic3C2×D311yescharacteristic
71C2×C3×Dic3—21no—
72C2×C3×Dic3—21no—
73C2×C3×Dic3—31no—
74C2×C3×Dic3—31no—
75C2×C3×Dic3—31no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3C2×D311yescharacteristic
83C2×C3×Dic3C2×D311yescharacteristic
84C2×C3×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C2×D311yescharacteristic
86C2×C33#4111yescharacteristic
87C2×C33#4111yescharacteristic
88C2×C33#4111yescharacteristic
89C32×D3—41no—
90C3×C4⋊C4—91no—
91C3×C22⋊C4—91no—
92C3×C22⋊C4—91no—
93#217—31no—
94#217—31no—
95#217—91no—
96#217—91no—
97#217—181no—
98(C2×D3)⋊C4—31no—
99(C2×D3)⋊C4—31no—
100(C2×D3)⋊C4—91no—
101C4⋊Dic3—91no—
102C4⋊Dic3—91no—
103C4⋊Dic3—91no—
104C4⋊Dic3—181no—
105Dic3⋊C4—31no—
106Dic3⋊C4—31no—
107Dic3⋊C4—91no—
108Dic3⋊C4—91no—
109Dic3⋊C4—91no—
110C4×Dic3—91no—
111C4×Dic3—91no—
112C4×Dic3—91no—
113C4×Dic3—181no—
114C22×C32—21no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—41no—
118C22×C32—61no—
119C22×C32—61no—
120C22×C32—61no—
121C22×C32C22×D311yescharacteristic
122C22×C32C22×D311yescharacteristic
123C22×C32C22×D311yescharacteristic
124C2×C3×D3—22no—
125C2×C3×D3—21no—
126C2×C3×D3—21no—
127C2×C3×D3—21no—
128C2×C3×D3—21no—
129C2×C3×D3—21no—
130C2×C3×D3—21no—
131C2×C3×D3—41no—
132C2×C3×D3—41no—
133C32×C4—61no—
134C32×C4—61no—
135C3⋊Dic3—21no—
136C3⋊Dic3—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—61no—
139C3⋊Dic3—62no—
140C3⋊Dic3—61no—
141C3⋊Dic3—61no—
142C3⋊Dic3—121no—
143C3⋊Dic3—121no—
144C3⋊Dic3—122no—
145C3×Dic3—22no—
146C3×Dic3—22no—
147C3×Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3—21no—
150C3×Dic3—21no—
151C3×Dic3—62no—
152C3×Dic3—61no—
153C3×Dic3—61no—
154C3×Dic3—61no—
155C3×Dic3—61no—
156C3×Dic3—61no—
157C3×Dic3—61no—
158C3×Dic3—121no—
159C3×Dic3—121no—
160#126—271noSylow
161C33#12611yescharacteristic, complemented, Sylow
162C23×C3—31no—
163C23×C3—31no—
164C23×C3—61no—
165C22×D3D3211yescharacteristic
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—61no—
171C2×C3×C4—61no—
172C2×C3×C4—91no—
173C2×C3×C4—91no—
174C2×C3×C4—91no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—31no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3—61no—
184C2×Dic3—91no—
185C2×Dic3—91no—
186C2×Dic3—91no—
187C2×Dic3—181no—
188C2×Dic3—181no—
189C2×Dic3—181no—
190C2×Dic3—361no—
191C2×Dic3D3211yescharacteristic
192C2×Dic3D3211yescharacteristic
193C2×C32—21no—
194C2×C32—21no—
195C2×C32—21no—
196C2×C32—21no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—41no—
203C2×C32—41no—
204C2×C32—41no—
205C2×C32—121no—
206C2×C32#23511yescharacteristic
207C2×C32#23511yescharacteristic
208C2×C32#23511yescharacteristic
209C2×C32#23711yescharacteristic
210C2×C32#23711yescharacteristic
211C2×C32#23711yescharacteristic
212C2×C32#23711yescharacteristic
213C2×C32#23711yescharacteristic
214C2×C32#23911yescharacteristic
215C3×D3—42no—
216C3×D3—41no—
217C3×D3—41no—
218C4⋊C4—91no—
219C4⋊C4—271no—
220C4⋊C4—271no—
221C22⋊C4—91no—
222C22⋊C4—91no—
223C22⋊C4—271no—
224C42—271no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—41no—
229C22×C3—62no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—61no—
233C22×C3—61no—
234C22×C3—61no—
235C22×C3—61no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3C2×D3211yescharacteristic
239C22×C3C2×D3211yescharacteristic
240C22×C3C2×D3211yescharacteristic
241C2×D3—21no—
242C2×D3—21no—
243C2×D3—21no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—181no—
251C3×C4—181no—
252C3×C4—181no—
253Dic3—21no—
254Dic3—21no—
255Dic3—62no—
256Dic3—61no—
257Dic3—61no—
258Dic3—61no—
259Dic3—61no—
260Dic3—61no—
261Dic3—61no—
262Dic3—121no—
263Dic3—121no—
264Dic3—182no—
265Dic3—181no—
266Dic3—181no—
267Dic3—181no—
268Dic3—361no—
269Dic3—361no—
270Dic3—362no—
271C32—21no—
272C32—21no—
273C32—21no—
274C32—41no—
275C32#87911yescharacteristic, complemented
276C32#87911yescharacteristic, complemented
277C32#89811yescharacteristic, complemented
278C23—31no—
279C2×C4—31no—
280C2×C4—31no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—271no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—122no—
298C2×C3—121no—
299C2×C3—121no—
300C2×C3#379511yescharacteristic
301C2×C3#379611yescharacteristic
302C2×C3#379811yescharacteristic
303C2×C3#379911yescharacteristic
304C2×C3#380411yescharacteristic
305C2×C3#380411yescharacteristic
306C2×C3#380511yescharacteristic
307C2×C3#380511yescharacteristic
308C2×C3#380511yescharacteristic
309D3—41no—
310C22—61no—
311C22—61no—
312C22—61no—
313C22D3311yescharacteristic
314C4—61no—
315C4—61no—
316C4—181no—
317C4—181no—
318C4—181no—
319C4—541no—
320C3—21no—
321C3—21no—
322C3—21no—
323C3—41no—
324C3#6386111yescharacteristic, complemented
325C3#6386211yescharacteristic, complemented
326C3#6395611yescharacteristic, complemented
327C2—121no—
328C2#8896011yescharacteristic
329C2#8896211yescharacteristic
330C2#8896311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C32×D3C22
hypocenterC33#126
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. #88815C2semidirect
2. Dic3⋊(C3⋊Dic3)C2semidirect
4. Dic3×C3⋊Dic3C2semidirect
5. C3×#3724C2semidirect
28. #3724D3semidirect
161. C33#126semidirect
275. C32#879semidirect
276. C32#879semidirect
277. C32#898semidirect
324. C3#63861semidirect
325. C3#63862semidirect
326. C3#63956semidirect

Maximal subgroups

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

subgroupindex
#888152
Dic3⋊(C3⋊Dic3)2
(C2×D3)⋊(C3⋊Dic3)2
Dic3×C3⋊Dic32
C3×#37242
C3×(C2×D3)⋊Dic32
#639563
#638623
#638613