Finite Group Databasebeta

← #10595387

Subgroups — #10595387

Order 576.

462 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 132 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#64156C211yesmaximal, characteristic
2C3×Q8×Dic3C211yesmaximal, characteristic, complemented
3C3×#944C211yesmaximal, characteristic, complemented
4C4×#627C211yesmaximal, characteristic, complemented
5#63895C212yesmaximal, complemented
6#63894C211yesmaximal, characteristic
7#63868C212yesmaximal, complemented
8#63864C211yesmaximal, characteristic
9#63848C212yesmaximal, complemented
10#63843C211yesmaximal, characteristic, complemented
11#63836C212yesmaximal, complemented
12#5732—31nomaximal
13#5568—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×Dic6C2211yescharacteristic
17C2×#627C2211yescharacteristic
18C2×#624C2212yes—
19#3752C2212yes—
20#3752C2212yes—
21C4×C3⋊Dic3C2211yescharacteristic
22C3×C4⋊Dic3C2211yescharacteristic
23C3×C4⋊Dic3C2212yes—
24C3×Dic3⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×C4×Dic3C2211yescharacteristic
27C3×C4×Dic3C2212yes—
28C3×C4×Dic3C2211yescharacteristic
29#3724C2211yescharacteristic
30Dic3⋊Dic3C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2211yescharacteristic
33#3722C2212yes—
34#3722C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×C4⋊Q8—31no—
38C3×C4×Q8—31no—
39#947—31no—
40#947—31no—
41#947—31no—
42#947—61no—
43Q8×Dic3—31no—
44Q8×Dic3D311yescharacteristic, complemented
45#944D311yescharacteristic, complemented
46#898—32no—
47#898—32no—
48#898—32no—
49#897—32no—
50#896—32no—
51#893—31no—
52#892—32no—
53#887—31no—
54#876—32no—
55#875—31no—
56C4×D12—31no—
57#872—32no—
58C4×Dic6—31no—
59C4×Dic6—31no—
60C22×C3⋊D3C2312yes—
61C32×Q8—22no—
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C4⋊(C3⋊D3)—22no—
67C2×C3×Dic3C2312yes—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×Dic6—22no—
74#627—22no—
75#624—24no—
76#556—91noSylow
77C2×C3×Q8—21no—
78C2×C3×Q8—31no—
79C2×C3×Q8C2×D311yescharacteristic
80C2×C3×Q8C2×D311yescharacteristic
81C2×D12—31no—
82C2×D12—31no—
83C2×D12—61no—
84C2×C4×D3—32no—
85C2×C4×D3—32no—
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6C2×D311yescharacteristic
89C3×C4⋊C4—31no—
90C3×C4⋊C4—32no—
91C3×C4⋊C4—32no—
92C3×C4⋊C4—32no—
93C3×C42—31no—
94C3×C42—31no—
95C3×C42—32no—
96(C2×D3)⋊C4—32no—
97(C2×D3)⋊C4—32no—
98(C2×D3)⋊C4—32no—
99(C2×D3)⋊C4—32no—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4—32no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—62no—
107(C2×D3)⋊C4—62no—
108C4⋊Dic3—31no—
109C4⋊Dic3—31no—
110C4⋊Dic3—32no—
111C4⋊Dic3C2×D311yescharacteristic
112C4⋊Dic3C2×D312yes—
113Dic3⋊C4—31no—
114Dic3⋊C4—31no—
115Dic3⋊C4—31no—
116Dic3⋊C4—31no—
117Dic3⋊C4—32no—
118Dic3⋊C4C2×D312yes—
119Dic3⋊C4C2×D312yes—
120C4×Dic3—31no—
121C4×Dic3—31no—
122C4×Dic3—32no—
123C4×Dic3—32no—
124C4×Dic3—61no—
125C4×Dic3C2×D311yescharacteristic
126C4×Dic3C2×D311yescharacteristic
127C4×Dic3C2×D312yes—
128C22×C32C2411yescharacteristic
129C2×C3⋊D3—22no—
130C2×C3⋊D3—22no—
131C2×C3⋊D3—22no—
132C32×C4—22no—
133C32×C4#4112yes—
134C3⋊Dic3—21no—
135C3⋊Dic3—21no—
136C3×Dic3—22no—
137C3×Dic3—21no—
138C3×Dic3—22no—
139C3×Dic3—21no—
140C3×Dic3—21no—
141C3×Dic3#4112yes—
142C4⋊Q8—31no—
143#126—92no—
144#126—92no—
145#124—91no—
146#124—91no—
147C22⋊Q8—92no—
148C4×Q8—31no—
149C4×Q8—91no—
150C4×Q8—91no—
151C4×D4—91no—
152#117—92no—
153C22×D3—32no—
154C22×D3—32no—
155C22×D3—62no—
156C3×Q8—22no—
157C3×Q8—22no—
158C3×Q8—42no—
159C3×Q8—62no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4C22×D312yes—
169C2×C3×C4C22×D312yes—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172C2×Dic3—31no—
173C2×Dic3—31no—
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3C22×D312yes—
179C2×Dic3C22×D312yes—
180C2×Dic3C22×D311yescharacteristic
181C2×Dic3C22×D311yescharacteristic
182C2×Dic3C22×D311yescharacteristic
183C2×Dic3C22×D311yescharacteristic
184D12—62no—
185D12—62no—
186D12—122no—
187C4×D3—64no—
188C4×D3—64no—
189Dic6—22no—
190Dic6—62no—
191Dic6—62no—
192C2×C32C2×#4111yescharacteristic
193C2×C32C2×#4111yescharacteristic
194C2×C322−1+411yescharacteristic
195C3⋊D3—42no—
196C2×Q8—31no—
197C2×Q8—91no—
198C2×Q8D3211yescharacteristic
199C2×D4—91no—
200C22×C4—92no—
201C4⋊C4—32no—
202C4⋊C4—31no—
203C4⋊C4—32no—
204C4⋊C4—32no—
205C4⋊C4—91no—
206C4⋊C4—92no—
207C4⋊C4—91no—
208C4⋊C4—91no—
209C22⋊C4—92no—
210C22⋊C4—92no—
211C22⋊C4—92no—
212C22⋊C4—92no—
213C22⋊C4—92no—
214C42—32no—
215C42—31no—
216C42—31no—
217C42—92no—
218C42—91no—
219C22×C3—21no—
220C22×C3C23×D311yescharacteristic
221C22×C3C23×D311yescharacteristic
222C2×D3—62no—
223C2×D3—62no—
224C2×D3—62no—
225C2×D3—62no—
226C2×D3—62no—
227C2×D3—62no—
228C2×D3—122no—
229C2×D3—122no—
230C2×D3—122no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—22no—
234C3×C4—32no—
235C3×C4—42no—
236C3×C4—62no—
237C3×C4—61no—
238C3×C4—62no—
239C3×C4—61no—
240C3×C4—61no—
241C3×C4#23512yes—
242C3×C4#23712yes—
243Dic3—22no—
244Dic3—21no—
245Dic3—21no—
246Dic3—21no—
247Dic3—22no—
248Dic3—61no—
249Dic3—61no—
250Dic3—61no—
251Dic3—61no—
252Dic3—121no—
253Dic3—121no—
254Dic3#23912yes—
255C32#55611yescharacteristic, complemented, Sylow
256C23—92no—
257Q8—22no—
258Q8—62no—
259Q8—182no—
260D4—182no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—32no—
264C2×C4—31no—
265C2×C4—31no—
266C2×C4—32no—
267C2×C4—91no—
268C2×C4—91no—
269C2×C4—184no—
270C2×C4C2×D3212yes—
271C2×C4C2×D3211yescharacteristic
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—21no—
275C2×C3C2×#23511yescharacteristic
276C2×C3C2×#23711yescharacteristic
277C2×C3C2×#23911yescharacteristic
278C2×C3#100711yescharacteristic
279C2×C3#100711yescharacteristic
280C2×C3D3×#4111yescharacteristic
281D3—122no—
282D3—122no—
283D3—242no—
284C22—182no—
285C22—182no—
286C22—182no—
287C22C22×D3211yescharacteristic
288C4—22no—
289C4—32no—
290C4—61no—
291C4—61no—
292C4—62no—
293C4—61no—
294C4—62no—
295C4—181no—
296C4—181no—
297C4#380412yes—
298C3—21no—
299C3#556811yescharacteristic, complemented
300C3#573211yescharacteristic, complemented
301C2—362no—
302C2#6431611yescharacteristic
303C2D3×#23911yescharacteristic
304C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC2×Q8D32
hypocenterC32#556
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C3×Q8×Dic3C2semidirect
3. C3×#944C2semidirect
4. C4×#627C2semidirect
5. #63895C2semidirect
7. #63868C2semidirect
9. #63848C2semidirect
10. #63843C2semidirect
11. #63836C2semidirect
44. Q8×Dic3D3semidirect
45. #944D3semidirect
255. C32#556semidirect
299. C3#5568semidirect
300. C3#5732semidirect

Maximal subgroups

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

subgroupindex
#641562
C3×Q8×Dic32
C3×#9442
C4×#6272
#638952
#638942
#638682
#638642
#638482
#638432
#638362
#57323
#55683