Finite Group Databasebeta

← #11722983

Subgroups — #11722983

Order 864.

430 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 76 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×Dic3×C3⋊D3C211yesmaximal, characteristic
3#88814C211yesmaximal, characteristic
4#88806C211yesmaximal, characteristic, complemented
5C3×#3724C211yesmaximal, characteristic, complemented
6C3×Dic3⋊Dic3C211yesmaximal, characteristic, complemented
7C3×#3722C211yesmaximal, characteristic
8#63899—31nomaximal
9#63896—31nomaximal
10#63865—31nomaximal
11C22×C3×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C2×C3×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊Dic3C2211yescharacteristic
15C2×C32×Dic3C2211yescharacteristic
16C2×C32×Dic3C2211yescharacteristic
17C2×C32×Dic3C2211yescharacteristic
18#6293—22no—
19Dic3×C3⋊D3—22no—
20C2×C4×C3⋊D3—31no—
21C2×#634—31no—
22C2×#627—31no—
23C2×#627—31no—
24C2×#627—61no—
25C2×D3×Dic3—31no—
26C2×D3×Dic3—31no—
27C2×D3×Dic3—61no—
28#3750—31no—
29C3×(C2×D3)⋊C4—31no—
30C3×(C2×D3)⋊C4—31no—
31C3×C4⋊Dic3—31no—
32C3×Dic3⋊C4—31no—
33C3×Dic3⋊C4—31no—
34C3×Dic3⋊C4—31no—
35#3724—31no—
36#3724—31no—
37#3724—61no—
38#3724D311yescharacteristic, complemented
39Dic3⋊Dic3D311yescharacteristic, complemented
40#3722—31no—
41#3722D311yescharacteristic
42(C2×D3)⋊Dic3—31no—
43(C2×D3)⋊Dic3—31no—
44C22×C33C2311yescharacteristic
45C2×C3×C3⋊D3—21no—
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3Q812yescomplemented
48C3⋊(C3⋊Dic3)D412yes—
49C3×C3⋊Dic3—21no—
50C3×C3⋊Dic3—21no—
51C32×Dic3—21no—
52C32×Dic3—21no—
53C32×Dic3—21no—
54#896—91no—
55#896—91no—
56#878—91no—
57C22×C3⋊D3C2×D311yescharacteristic
58C22×C3×D3—31no—
59C22×C3×D3—31no—
60C22×C3×D3—61no—
61C2×C32×C4—31no—
62C2×C32×C4—31no—
63C2×C32×C4—31no—
64C2×C3⋊Dic3—31no—
65C2×C3⋊Dic3—31no—
66C2×C3⋊Dic3—31no—
67C2×C3⋊Dic3—61no—
68C2×C3⋊Dic3—61no—
69C2×C3⋊Dic3—61no—
70C2×C3⋊Dic3—121no—
71C2×C3⋊Dic3C2×D311yescharacteristic
72C2×C3⋊Dic3C2×D311yescharacteristic
73C4×C3⋊D3—62no—
74#634—62no—
75C2×C3×Dic3—21no—
76C2×C3×Dic3—21no—
77C2×C3×Dic3—21no—
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×Dic3C2×D311yescharacteristic
85C2×C3×Dic3C2×D311yescharacteristic
86C2×C3×Dic3C2×D311yescharacteristic
87C2×C3×Dic3C2×D311yescharacteristic
88C2×C3×Dic3C2×D311yescharacteristic
89C2×C3×Dic3C2×D311yescharacteristic
90#627—62no—
91#627—62no—
92#627—64no—
93D3×Dic3—62no—
94D3×Dic3—62no—
95D3×Dic3—64no—
96C2×C33C2×D411yescharacteristic
97C2×C33C2×Q811yescharacteristic
98C2×C33#4111yescharacteristic
99C3×C3⋊D3—22no—
100C22×Dic3—91no—
101C2×C4×D3—91no—
102C2×C4×D3—91no—
103C2×C4×D3—181no—
104C2×Dic6—91no—
105C2×Dic6—91no—
106C2×Dic6—91no—
107C2×Dic6—181no—
108C3×C4⋊C4—91no—
109C3×C4⋊C4—91no—
110C3×C22⋊C4—91no—
111#217—91no—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—91no—
115(C2×D3)⋊C4—91no—
116C4⋊Dic3—31no—
117Dic3⋊C4—31no—
118Dic3⋊C4—31no—
119Dic3⋊C4—31no—
120Dic3⋊C4—91no—
121Dic3⋊C4—91no—
122Dic3⋊C4—91no—
123Dic3⋊C4—181no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32—41no—
128C22×C32C22×D311yescharacteristic
129C22×C32C22×D311yescharacteristic
130C22×C32C22×D311yescharacteristic
131C2×C3⋊D3—21no—
132C2×C3⋊D3—21no—
133C2×C3⋊D3Dic612yescomplemented
134C2×C3×D3—32no—
135C2×C3×D3—32no—
136C2×C3×D3—62no—
137C2×C3×D3—62no—
138C2×C3×D3—62no—
139C2×C3×D3—61no—
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C32×C4—61no—
144C32×C4—61no—
145C32×C4—61no—
146C3⋊Dic3—21no—
147C3⋊Dic3—21no—
148C3⋊Dic3—32no—
149C3⋊Dic3—32no—
150C3⋊Dic3—32no—
151C3⋊Dic3—62no—
152C3⋊Dic3—62no—
153C3⋊Dic3—62no—
154C3⋊Dic3—122no—
155C3×Dic3—22no—
156C3×Dic3—22no—
157C3×Dic3—21no—
158C3×Dic3—21no—
159C3×Dic3—21no—
160C3×Dic3—21no—
161C3×Dic3—21no—
162C3×Dic3—21no—
163C3×Dic3—41no—
164C3×Dic3—62no—
165C3×Dic3—61no—
166C3×Dic3—61no—
167C3×Dic3—61no—
168C3×Dic3—61no—
169C3×Dic3—121no—
170C22⋊Q8—271noSylow
171C33C22⋊Q811yescharacteristic, complemented, Sylow
172C23×C3—91no—
173C22×D3—31no—
174C22×D3—31no—
175C22×D3—61no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—61no—
183C2×C3×C4—61no—
184C2×C3×C4—61no—
185C2×C3×C4—91no—
186C2×C3×C4—91no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—61no—
192C2×Dic3—61no—
193C2×Dic3—91no—
194C2×Dic3—91no—
195C2×Dic3—91no—
196C2×Dic3—181no—
197C2×Dic3—181no—
198C2×Dic3—181no—
199C2×Dic3—182no—
200C2×Dic3—361no—
201C2×Dic3D3211yescharacteristic
202C2×Dic3D3211yescharacteristic
203C2×Dic3D3211yescharacteristic
204C4×D3—182no—
205C4×D3—182no—
206C4×D3—184no—
207Dic6—182no—
208Dic6—182no—
209Dic6—182no—
210Dic6—184no—
211C2×C32—21no—
212C2×C32—21no—
213C2×C32—21no—
214C2×C32—21no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—41no—
221C2×C32—41no—
222C2×C32—41no—
223C2×C32C2×Dic611yescharacteristic
224C2×C32#23511yescharacteristic
225C2×C32#23511yescharacteristic
226C2×C32D3×D411yescharacteristic
227C2×C32D3×D411yescharacteristic
228C2×C32D3×D411yescharacteristic
229C2×C32#23711yescharacteristic
230C2×C32D3×Q811yescharacteristic
231C2×C32D3×Q811yescharacteristic
232C3⋊D3—22no—
233C3×D3—64no—
234C3×D3—62no—
235C3×D3—62no—
236C2×Q8—271no—
237C22×C4—271no—
238C4⋊C4—91no—
239C4⋊C4—91no—
240C4⋊C4—271no—
241C22⋊C4—91no—
242C22⋊C4—271no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—41no—
247C22×C3—92no—
248C22×C3—181no—
249C22×C3—181no—
250C22×C3C2×D3211yescharacteristic
251C22×C3C2×D3211yescharacteristic
252C22×C3C2×D3211yescharacteristic
253C2×D3—32no—
254C2×D3—32no—
255C2×D3—62no—
256C2×D3—62no—
257C2×D3—62no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—61no—
261C2×D3—61no—
262C3×C4—62no—
263C3×C4—62no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—121no—
271C3×C4—181no—
272C3×C4—181no—
273Dic3—21no—
274Dic3—21no—
275Dic3—21no—
276Dic3—62no—
277Dic3—61no—
278Dic3—61no—
279Dic3—61no—
280Dic3—61no—
281Dic3—92no—
282Dic3—92no—
283Dic3—92no—
284Dic3—121no—
285Dic3—182no—
286Dic3—182no—
287Dic3—182no—
288Dic3—362no—
289C32—21no—
290C32—21no—
291C32—21no—
292C32—41no—
293C32#87811yescharacteristic, complemented
294C32#89611yescharacteristic, complemented
295C32#89611yescharacteristic, complemented
296C23—91no—
297Q8—542no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—31no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—271no—
304C2×C4—542no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—21no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—41no—
315C2×C3—41no—
316C2×C3—41no—
317C2×C3—182no—
318C2×C3D3×Dic611yescharacteristic
319C2×C3D3×Dic611yescharacteristic
320C2×C3#379511yescharacteristic
321C2×C3#379711yescharacteristic
322C2×C3#379811yescharacteristic
323C2×C3#380211yescharacteristic
324C2×C3#380411yescharacteristic
325C2×C3#381111yescharacteristic
326C2×C3#381111yescharacteristic
327D3—64no—
328D3—62no—
329D3—62no—
330C22—92no—
331C22—181no—
332C22—181no—
333C22D3311yescharacteristic
334C4—61no—
335C4—61no—
336C4—61no—
337C4—181no—
338C4—181no—
339C4—272no—
340C3—21no—
341C3—21no—
342C3—21no—
343C3—41no—
344C3#6386511yescharacteristic, complemented
345C3#6389611yescharacteristic, complemented
346C3#6389911yescharacteristic, complemented
347C2—182no—
348C2#8895411yescharacteristic
349C2#8895711yescharacteristic
350C2#8896011yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D33
commutator, Fitting, socle, cosocleC22×C33C23
prime radicalC22×C3×C3⋊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
4. #88806C2semidirect
5. C3×#3724C2semidirect
6. C3×Dic3⋊Dic3C2semidirect
38. #3724D3semidirect
39. Dic3⋊Dic3D3semidirect
47. C2×C3×C3⋊D3Q8semidirect
133. C2×C3⋊D3Dic6semidirect
171. C33C22⋊Q8semidirect
293. C32#878semidirect
294. C32#896semidirect
295. C32#896semidirect
344. C3#63865semidirect
345. C3#63896semidirect
346. C3#63899semidirect

Maximal subgroups

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

subgroupindex
C2×#62932
C2×Dic3×C3⋊D32
#888142
#888062
C3×#37242
C3×Dic3⋊Dic32
C3×#37222
#638993
#638963
#638653