Finite Group Databasebeta

← #11722975

Subgroups — #11722975

Order 864.

410 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 410 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C2×#6293C211yesmaximal, characteristic, complemented
2C2×C3×D3×Dic3C211yesmaximal, characteristic
3#88815C211yesmaximal, characteristic, complemented
4#88807C211yesmaximal, characteristic, complemented
5(C2×D3)⋊(C3⋊Dic3)C211yesmaximal, characteristic
6C3×#3724C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic3C211yesmaximal, characteristic
8#63964—31nomaximal
9#63896—31nomaximal
10#63851—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#6293—22no—
19C3×D3×Dic3—22no—
20C2×#634—31no—
21C22×C3×Dic3—31no—
22C2×C3×C4×D3—31no—
23C2×#627—31no—
24C2×#627—31no—
25C2×#627—61no—
26C2×D3×Dic3D311yescharacteristic
27#3757—31no—
28#3750—31no—
29C3×#217—31no—
30C3×(C2×D3)⋊C4—31no—
31C3×Dic3⋊C4—31no—
32C3×Dic3⋊C4—31no—
33#3724—31no—
34#3724D311yescharacteristic, complemented
35Dic3⋊Dic3—31no—
36Dic3⋊Dic3—31no—
37Dic3⋊Dic3—31no—
38Dic3⋊Dic3—31no—
39Dic3⋊Dic3—61no—
40(C2×D3)⋊Dic3—31no—
41(C2×D3)⋊Dic3—31no—
42(C2×D3)⋊Dic3—61no—
43(C2×D3)⋊Dic3D311yescharacteristic
44C22×C33C2311yescharacteristic
45C2×C32×D3—21no—
46C2×C32×D3—21no—
47C2×C32×D3Q812yescomplemented
48C3⋊(C3⋊Dic3)—21no—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3—21no—
51C3×C3⋊Dic3D412yes—
52C32×Dic3—21no—
53C32×Dic3—21no—
54#924—91no—
55#896—91no—
56#878—91no—
57C23×C32—31no—
58C22×C3×D3—21no—
59C22×C3×D3C2×D311yescharacteristic
60C22×C3×D3C2×D311yescharacteristic
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C3⋊Dic3—31no—
64C2×C3⋊Dic3—31no—
65C2×C3⋊Dic3—31no—
66C2×C3⋊Dic3—61no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—61no—
69C2×C3⋊Dic3—121no—
70C2×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊Dic3C2×D311yescharacteristic
73#634—62no—
74C2×C3×Dic3—21no—
75C2×C3×Dic3—21no—
76C2×C3×Dic3—31no—
77C2×C3×Dic3—31no—
78C2×C3×Dic3—31no—
79C2×C3×Dic3—31no—
80C2×C3×Dic3—31no—
81C2×C3×Dic3—31no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—62no—
86C2×C3×Dic3C2×D311yescharacteristic
87C2×C3×Dic3C2×D311yescharacteristic
88C2×C3×Dic3C2×D311yescharacteristic
89C2×C3×Dic3C2×D311yescharacteristic
90C3×C4×D3—62no—
91#627—62no—
92#627—62no—
93#627—64no—
94D3×Dic3—22no—
95C2×C33C2×D411yescharacteristic
96C2×C33C2×Q811yescharacteristic
97C2×C33#4111yescharacteristic
98C32×D3—22no—
99C22×C3×C4—91no—
100C22×Dic3—31no—
101C2×C4×D3—31no—
102C2×Dic6—91no—
103C2×Dic6—91no—
104C2×Dic6—91no—
105C2×Dic6—181no—
106C3×C4⋊C4—91no—
107C3×C22⋊C4—91no—
108#217—31no—
109#217—91no—
110#217—91no—
111#217—181no—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—91no—
114C4⋊Dic3—91no—
115C4⋊Dic3—91no—
116Dic3⋊C4—31no—
117Dic3⋊C4—31no—
118Dic3⋊C4—91no—
119Dic3⋊C4—91no—
120Dic3⋊C4—91no—
121Dic3⋊C4—91no—
122Dic3⋊C4—181no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—32no—
127C22×C32—41no—
128C22×C32—61no—
129C22×C32—61no—
130C22×C32C22×D311yescharacteristic
131C22×C32C22×D311yescharacteristic
132C22×C32C22×D311yescharacteristic
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3—21no—
137C2×C3×D3—22no—
138C2×C3×D3—41no—
139C2×C3×D3—41no—
140C2×C3×D3Dic612yescomplemented
141C2×C3×D3Dic612yescomplemented
142C32×C4—61no—
143C32×C4—61no—
144C3⋊Dic3—21no—
145C3⋊Dic3—21no—
146C3⋊Dic3—61no—
147C3⋊Dic3—62no—
148C3⋊Dic3—61no—
149C3⋊Dic3—61no—
150C3⋊Dic3—121no—
151C3⋊Dic3—121no—
152C3⋊Dic3—122no—
153C3⋊Dic3#6712yes—
154C3×Dic3—22no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3—21no—
158C3×Dic3—21no—
159C3×Dic3—32no—
160C3×Dic3—32no—
161C3×Dic3—41no—
162C3×Dic3—62no—
163C3×Dic3—62no—
164C3×Dic3—61no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—121no—
169C22⋊Q8—271noSylow
170C33C22⋊Q811yescharacteristic, complemented, Sylow
171C23×C3—31no—
172C23×C3—31no—
173C23×C3—61no—
174C22×D3D3211yescharacteristic
175C2×C3×C4—31no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—61no—
180C2×C3×C4—61no—
181C2×C3×C4—91no—
182C2×C3×C4—91no—
183C2×C3×C4—91no—
184C2×C3×C4—182no—
185C2×Dic3—31no—
186C2×Dic3—31no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—62no—
195C2×Dic3—91no—
196C2×Dic3—91no—
197C2×Dic3—91no—
198C2×Dic3—181no—
199C2×Dic3—181no—
200C2×Dic3—181no—
201C2×Dic3—361no—
202C2×Dic3D3211yescharacteristic
203C2×Dic3D3211yescharacteristic
204C4×D3—62no—
205Dic6—182no—
206Dic6—182no—
207Dic6—182no—
208Dic6—184no—
209C2×C32—21no—
210C2×C32—21no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32—41no—
221C2×C32—62no—
222C2×C32C2×Dic611yescharacteristic
223C2×C32C2×Dic611yescharacteristic
224C2×C32#23511yescharacteristic
225C2×C32#23511yescharacteristic
226C2×C32D3×D411yescharacteristic
227C2×C32D3×D411yescharacteristic
228C2×C32#23711yescharacteristic
229C2×C32D3×Q811yescharacteristic
230C2×C32C2×#6711yescharacteristic
231C3×D3—22no—
232C3×D3—22no—
233C3×D3—42no—
234C2×Q8—271no—
235C22×C4—91no—
236C4⋊C4—91no—
237C4⋊C4—271no—
238C4⋊C4—271no—
239C22⋊C4—91no—
240C22⋊C4—271no—
241C22×C3—21no—
242C22×C3—21no—
243C22×C3—21no—
244C22×C3—32no—
245C22×C3—32no—
246C22×C3—41no—
247C22×C3—62no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—121no—
253C22×C3—121no—
254C22×C3C2×D3211yescharacteristic
255C22×C3C2×D3211yescharacteristic
256C22×C3C2×D3211yescharacteristic
257C2×D3—21no—
258C2×D3—21no—
259C2×D3#62712yescomplemented
260C3×C4—62no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—61no—
265C3×C4—92no—
266C3×C4—121no—
267C3×C4—181no—
268C3×C4—181no—
269Dic3—21no—
270Dic3—21no—
271Dic3—32no—
272Dic3—32no—
273Dic3—62no—
274Dic3—62no—
275Dic3—61no—
276Dic3—61no—
277Dic3—61no—
278Dic3—61no—
279Dic3—121no—
280Dic3—182no—
281Dic3—181no—
282Dic3—181no—
283Dic3—181no—
284Dic3—361no—
285Dic3—361no—
286Dic3—362no—
287C32—21no—
288C32—21no—
289C32—21no—
290C32—41no—
291C32#87811yescharacteristic, complemented
292C32#89611yescharacteristic, complemented
293C32#92411yescharacteristic, complemented
294C23—31no—
295Q8—542no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—91no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—182no—
302C2×C4—271no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—41no—
313C2×C3—41no—
314C2×C3—41no—
315C2×C3—62no—
316C2×C3—62no—
317C2×C3—122no—
318C2×C3D3×Dic611yescharacteristic
319C2×C3D3×Dic611yescharacteristic
320C2×C3#379511yescharacteristic
321C2×C3#379811yescharacteristic
322C2×C3#380411yescharacteristic
323C2×C3C2×#62711yescharacteristic
324C2×C3D3×#6711yescharacteristic
325C2×C3D3×#6711yescharacteristic
326C2×C3#381111yescharacteristic
327D3—22no—
328C22—32no—
329C22—61no—
330C22—61no—
331C22D3311yescharacteristic
332C4—61no—
333C4—61no—
334C4—92no—
335C4—181no—
336C4—181no—
337C4—541no—
338C3—21no—
339C3—21no—
340C3—21no—
341C3—41no—
342C3#6385111yescharacteristic, complemented
343C3#6389611yescharacteristic, complemented
344C3#6396411yescharacteristic, complemented
345C2—62no—
346C2#8895211yescharacteristic
347C2D3×#62711yescharacteristic
348C2#8896011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C32×D3C22
hypocenterC33C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×#6293C2semidirect
3. #88815C2semidirect
4. #88807C2semidirect
6. C3×#3724C2semidirect
34. #3724D3semidirect
47. C2×C32×D3Q8semidirect
140. C2×C3×D3Dic6semidirect
141. C2×C3×D3Dic6semidirect
170. C33C22⋊Q8semidirect
259. C2×D3#627semidirect
291. C32#878semidirect
292. C32#896semidirect
293. C32#924semidirect
342. C3#63851semidirect
343. C3#63896semidirect
344. C3#63964semidirect

Maximal subgroups

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

subgroupindex
C2×#62932
C2×C3×D3×Dic32
#888152
#888072
(C2×D3)⋊(C3⋊Dic3)2
C3×#37242
C3×(C2×D3)⋊Dic32
#639643
#638963
#638513