Finite Group Databasebeta

← #10591293

Subgroups — #10591293

Order 576.

442 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 102 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
1C22×#627C211yesmaximal, characteristic, complemented
2C2×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
3C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
4C2×(C2×D3)⋊Dic3C211yesmaximal, characteristic
5#63658C211yesmaximal, characteristic, complemented
6C3×C3⋊#95C211yesmaximal, characteristic, complemented
7#63579C211yesmaximal, characteristic, complemented
8#4947—31nomaximal
9#4673—31nomaximal
10C23×C3×D3C2211yescharacteristic
11C22×C32×C4C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3⋊Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C2×#627—22no—
18C2×#627—22no—
19C2×#627—22no—
20C3×(C2×D3)⋊C4—22no—
21(C2×D3)⋊Dic3—22no—
22(C2×D3)⋊Dic3—22no—
23C22×Dic6—31no—
24C22×Dic6—31no—
25C2×C3×C22⋊C4—31no—
26C2×#217—31no—
27C2×#217—31no—
28C2×(C2×D3)⋊C4—31no—
29C2×(C2×D3)⋊C4—31no—
30C2×(C2×D3)⋊C4D311yescharacteristic
31C3×#95—31no—
32C3⋊#95—31no—
33C3⋊#95—31no—
34C3⋊#95—31no—
35C3⋊#95—31no—
36C3⋊#95—61no—
37C3⋊#95D311yescharacteristic, complemented
38C23×C32C2311yescharacteristic
39C22×C3×D3—21no—
40C22×C3×D3—21no—
41C22×C3×D3—21no—
42C22×C3×D3—21no—
43C22×C3×D3—21no—
44C22×C3×D3—21no—
45C22×C3×D3Q812yescomplemented
46C2×C32×C4—21no—
47C2×C32×C4—21no—
48C2×C32×C4—21no—
49C2×C3⋊Dic3—21no—
50C2×C3⋊Dic3—21no—
51C2×C3⋊Dic3—21no—
52C2×C3⋊Dic3—21no—
53C2×C3⋊Dic3—21no—
54C2×C3⋊Dic3D412yes—
55C2×C3×Dic3—21no—
56C2×C3×Dic3—21no—
57C2×C3×Dic3—21no—
58C2×C3×Dic3—21no—
59C2×C3×Dic3—21no—
60C2×C3×Dic3—21no—
61C2×C3×Dic3—21no—
62C2×C3×Dic3D412yes—
63C2×C3×Dic3D412yes—
64#627—28no—
65#393—91noSylow
66C24×C3—31no—
67C23×D3C2×D311yescharacteristic
68C22×C3×C4—21no—
69C22×C3×C4—31no—
70C22×C3×C4—31no—
71C22×C3×C4—31no—
72C22×C3×C4C2×D311yescharacteristic
73C22×C3×C4C2×D311yescharacteristic
74C22×Dic3—31no—
75C22×Dic3—31no—
76C22×Dic3—31no—
77C22×Dic3—31no—
78C22×Dic3—61no—
79C22×Dic3—61no—
80C22×Dic3C2×D311yescharacteristic
81C22×Dic3C2×D311yescharacteristic
82C22×Dic3C2×D311yescharacteristic
83C2×Dic6—62no—
84C2×Dic6—62no—
85C2×Dic6—62no—
86C2×Dic6—62no—
87C2×Dic6—62no—
88C2×Dic6—62no—
89C3×C22⋊C4—62no—
90#217—62no—
91#217—62no—
92(C2×D3)⋊C4—22no—
93(C2×D3)⋊C4—62no—
94(C2×D3)⋊C4—62no—
95C22×C32C2×D411yescharacteristic
96C22×C32C2×D411yescharacteristic
97C22×C32C2×D411yescharacteristic
98C22×C32C2×Q811yescharacteristic
99C22×C32#4111yescharacteristic
100C22×C32#4111yescharacteristic
101C22×C32#4111yescharacteristic
102C2×C3×D3—22no—
103C2×C3×D3—22no—
104C2×C3×D3—22no—
105C2×C3×D3—41no—
106C2×C3×D3—41no—
107C2×C3×D3—41no—
108C2×C3×D3—41no—
109C32×C4—41no—
110C3⋊Dic3—22no—
111C3⋊Dic3—41no—
112C3×Dic3—22no—
113C3×Dic3—22no—
114C3×Dic3—41no—
115C22×Q8—91no—
116C2×C22⋊C4—31no—
117C2×C22⋊C4—91no—
118C2×C22⋊C4—91no—
119#95—31no—
120#95—91no—
121#95—91no—
122C23×C3—21no—
123C23×C3—32no—
124C23×C3—61no—
125C23×C3—61no—
126C23×C3—61no—
127C23×C3—61no—
128C23×C3—61no—
129C23×C3—61no—
130C23×C3C22×D311yescharacteristic
131C23×C3C22×D311yescharacteristic
132C22×D3—21no—
133C22×D3—21no—
134C22×D3—21no—
135C22×D3—21no—
136C22×D3—21no—
137C22×D3—21no—
138C22×D3Dic612yescomplemented
139C2×C3×C4—22no—
140C2×C3×C4—21no—
141C2×C3×C4—21no—
142C2×C3×C4—21no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4—21no—
146C2×C3×C4—32no—
147C2×C3×C4—32no—
148C2×C3×C4—41no—
149C2×C3×C4—41no—
150C2×C3×C4—61no—
151C2×C3×C4—61no—
152C2×C3×C4—61no—
153C2×C3×C4—61no—
154C2×C3×C4—61no—
155C2×C3×C4—61no—
156C2×C3×C4—61no—
157C2×Dic3—21no—
158C2×Dic3—21no—
159C2×Dic3—21no—
160C2×Dic3—21no—
161C2×Dic3—21no—
162C2×Dic3—21no—
163C2×Dic3—21no—
164C2×Dic3—32no—
165C2×Dic3—32no—
166C2×Dic3—62no—
167C2×Dic3—61no—
168C2×Dic3—61no—
169C2×Dic3—61no—
170C2×Dic3—61no—
171C2×Dic3—62no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—61no—
175C2×Dic3—61no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—121no—
179C2×Dic3—121no—
180C2×Dic3—121no—
181C2×Dic3—121no—
182C2×Dic3D1212yes—
183C2×Dic3#6712yes—
184Dic6—68no—
185Dic6—68no—
186C2×C32#12011yescharacteristic
187C2×C32C22⋊Q811yescharacteristic
188C2×C32C22⋊Q811yescharacteristic
189C2×C32C22⋊Q811yescharacteristic
190C2×C32#12411yescharacteristic
191C2×C32#12411yescharacteristic
192C2×C32#12411yescharacteristic
193C3×D3—42no—
194C24—31no—
195C2×Q8—182no—
196C2×Q8—182no—
197C2×Q8—182no—
198C22×C4—31no—
199C22×C4—31no—
200C22×C4—31no—
201C22×C4—91no—
202C22×C4—91no—
203C22×C4D3211yescharacteristic
204C22⋊C4—62no—
205C22⋊C4—182no—
206C22⋊C4—182no—
207C22×C3—21no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—62no—
215C22×C3—62no—
216C22×C3—62no—
217C22×C3—121no—
218C22×C3—121no—
219C22×C3—121no—
220C22×C3—121no—
221C22×C3C2×Dic611yescharacteristic
222C22×C3C2×D1211yescharacteristic
223C22×C3#23511yescharacteristic
224C22×C3#23511yescharacteristic
225C22×C3#23511yescharacteristic
226C22×C3D3×D411yescharacteristic
227C22×C3D3×D411yescharacteristic
228C22×C3D3×D411yescharacteristic
229C22×C3D3×D411yescharacteristic
230C22×C3#23711yescharacteristic
231C22×C3#23711yescharacteristic
232C22×C3#23711yescharacteristic
233C22×C3D3×Q811yescharacteristic
234C22×C3C2×#6711yescharacteristic
235C2×D3—22no—
236C2×D3—22no—
237C2×D3—22no—
238C2×D3—41no—
239C2×D3—41no—
240C2×D3—41no—
241C2×D3—41no—
242C3×C4—42no—
243C3×C4—41no—
244C3×C4—41no—
245C3×C4—62no—
246C3×C4—62no—
247C3×C4—121no—
248Dic3—22no—
249Dic3—22no—
250Dic3—41no—
251Dic3—62no—
252Dic3—62no—
253Dic3—122no—
254Dic3—122no—
255Dic3—121no—
256Dic3—121no—
257C32#39311yescharacteristic, complemented, Sylow
258C23—32no—
259C23—61no—
260C23—61no—
261C23—61no—
262C23—61no—
263C23—61no—
264C23—61no—
265C23C2×D3211yescharacteristic
266Q8—188no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—32no—
271C2×C4—32no—
272C2×C4—61no—
273C2×C4—61no—
274C2×C4—61no—
275C2×C4—61no—
276C2×C4—61no—
277C2×C4—61no—
278C2×C4—61no—
279C2×C4—92no—
280C2×C4—181no—
281C2×C4—181no—
282C2×C4—181no—
283C2×C4—181no—
284C2×C4—181no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—122no—
293C2×C3#87511yescharacteristic
294C2×C3#87811yescharacteristic
295C2×C3#87811yescharacteristic
296C2×C3#88211yescharacteristic
297C2×C3#88511yescharacteristic
298C2×C3#88511yescharacteristic
299C2×C3#88511yescharacteristic
300C2×C3#88511yescharacteristic
301C2×C3#89611yescharacteristic
302C2×C3#89611yescharacteristic
303C2×C3#89711yescharacteristic
304C2×C3#92411yescharacteristic
305C2×C3#93611yescharacteristic
306C2×C3#93711yescharacteristic
307D3—42no—
308C22—62no—
309C22—62no—
310C22—62no—
311C22—121no—
312C22—121no—
313C22—121no—
314C22—121no—
315C22D3×Dic611yescharacteristic
316C22#379811yescharacteristic
317C22D3×D1211yescharacteristic
318C22#380211yescharacteristic
319C22#380411yescharacteristic
320C22#380511yescharacteristic
321C22D3×#6711yescharacteristic
322C4—41no—
323C4—62no—
324C4—62no—
325C4—121no—
326C4—182no—
327C4—361no—
328C3—21no—
329C3#467311yescharacteristic, complemented
330C3#494711yescharacteristic, complemented
331C2—122no—
332C2#6385111yescharacteristic
333C2#6385211yescharacteristic
334C2#6389311yescharacteristic
335C2#6389911yescharacteristic
336C2#6391511yescharacteristic
337C2#6392211yescharacteristic
338C2#6395911yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, cosocleC23×C32C23
FittingC22×C32×C4C22
prime radicalC23×C3×D3C22
hypercenterC22×C4D32
hypocenterC32#393
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#627C2semidirect
5. #63658C2semidirect
6. C3×C3⋊#95C2semidirect
7. #63579C2semidirect
37. C3⋊#95D3semidirect
45. C22×C3×D3Q8semidirect
138. C22×D3Dic6semidirect
257. C32#393semidirect
329. C3#4673semidirect
330. C3#4947semidirect

Maximal subgroups

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

subgroupindex
C22×#6272
C2×C3×(C2×D3)⋊C42
C2×(C2×D3)⋊Dic32
#636582
C3×C3⋊#952
#635792
#49473
#46733