Finite Group Databasebeta

← #10594550

Subgroups — #10594550

Order 576.

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

normal subgroups only

This group has 463 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#64096C211yesmaximal, characteristic, complemented
2C3×#892C211yesmaximal, characteristic, complemented
3C3×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4#63905C212yesmaximal
5C4×#625C211yesmaximal, characteristic
6#63872C211yesmaximal, characteristic, complemented
7#63867C212yesmaximal, complemented
8#63862C212yesmaximal, complemented
9#63857C211yesmaximal, characteristic, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63855C212yesmaximal, complemented
12#5529—31nomaximal
13#5513—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×C4×D3C2212yes—
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×C4×D3C2211yescharacteristic
18C2×#625C2211yescharacteristic
19C32×C4⋊C4C2211yescharacteristic
20#3750C2212yes—
21C4×C3⋊Dic3C2212yes—
22C4×C3⋊Dic3C2211yescharacteristic
23C3×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic3C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×Dic3⋊C4C2212yes—
27C3×C4×Dic3C2212yes—
28#3724C2211yescharacteristic
29Dic3⋊Dic3C2212yes—
30Dic3⋊Dic3C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2211yescharacteristic
37C3×#117—31no—
38C2×C3×C4⋊C4—31no—
39#929—32no—
40C4×#67—31no—
41C4×#67—31no—
42#926—32no—
43#924—31no—
44#924—31no—
45#898—32no—
46#897—31no—
47#897—31no—
48#892D311yescharacteristic, complemented
49D3×C4⋊C4D311yescharacteristic, complemented
50#890—31no—
51#890—31no—
52#889—32no—
53#887—31no—
54#887—31no—
55#887—61no—
56#883—32no—
57#879—32no—
58#872—32no—
59#870—32no—
60C22×C3×D3C2311yescharacteristic
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67#634—22no—
68C2×C3×Dic3C2312yes—
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C3×C4×D3—24no—
73C3×C4×D3—22no—
74C3×C4×D3—22no—
75#625—22no—
76#552—91noSylow
77C22×C3×C4—31no—
78C22×C3×C4—31no—
79C22×C3×C4—32no—
80C2×#67—31no—
81C2×#67—31no—
82C2×C4×D3C2×D312yes—
83C2×C4×D3C2×D311yescharacteristic
84C2×C4×D3C2×D311yescharacteristic
85C2×Dic6—31no—
86C2×Dic6—31no—
87C2×Dic6—61no—
88C3×C4⋊C4—21no—
89C3×C4⋊C4—32no—
90C3×C4⋊C4—31no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4C2×D311yescharacteristic
93C3×C4⋊C4C2×D311yescharacteristic
94C3×C22⋊C4—32no—
95C3×C42—32no—
96#217—31no—
97#217—32no—
98#217—32no—
99#217—31no—
100(C2×D3)⋊C4—31no—
101(C2×D3)⋊C4—32no—
102(C2×D3)⋊C4—32no—
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4C2×D312yes—
105C4⋊Dic3—32no—
106C4⋊Dic3—32no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3C2×D311yescharacteristic
110C4⋊Dic3C2×D311yescharacteristic
111Dic3⋊C4—32no—
112Dic3⋊C4—32no—
113Dic3⋊C4—31no—
114Dic3⋊C4—32no—
115Dic3⋊C4—32no—
116Dic3⋊C4—31no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—62no—
120Dic3⋊C4C2×D312yes—
121C4×Dic3—32no—
122C4×Dic3—31no—
123C4×Dic3—31no—
124C4×Dic3—32no—
125C4×Dic3—62no—
126C4×Dic3—61no—
127C4×Dic3C2×D312yes—
128C22×C32C2411yescharacteristic
129C2×C3×D3—21no—
130C2×C3×D3—21no—
131C2×C3×D3—21no—
132C2×C3×D3—21no—
133C2×C3×D3—21no—
134C2×C3×D3#4112yes—
135C32×C4—22no—
136C32×C4#4112yes—
137C3⋊Dic3—22no—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3×Dic3—21no—
141C3×Dic3—21no—
142C3×Dic3—22no—
143C3×Dic3—22no—
144#126—92no—
145#125—92no—
146#125—91no—
147#123—92no—
148C22⋊Q8—91no—
149C22⋊Q8—91no—
150C4×Q8—91no—
151C4×D4—91no—
152#117—31no—
153#117—92no—
154C2×C4⋊C4—31no—
155C23×C3—31no—
156C23×C3—31no—
157C22×D3C22×D311yescharacteristic
158C22×D3C22×D311yescharacteristic
159C2×C3×C4—21no—
160C2×C3×C4—22no—
161C2×C3×C4—31no—
162C2×C3×C4—31no—
163C2×C3×C4—32no—
164C2×C3×C4—32no—
165C2×C3×C4—64no—
166C2×C3×C4—62no—
167C2×C3×C4—62no—
168C2×C3×C4C22×D312yes—
169C2×C3×C4C22×D312yes—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172#67—62no—
173#67—62no—
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—61no—
181C2×Dic3—62no—
182C2×Dic3—61no—
183C2×Dic3C22×D311yescharacteristic
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D311yescharacteristic
187C4×D3—24no—
188C4×D3—22no—
189C4×D3—22no—
190Dic6—64no—
191Dic6—62no—
192Dic6—62no—
193C2×C32C2×#4111yescharacteristic
194C2×C32C2×#4111yescharacteristic
195C2×C322−1+411yescharacteristic
196C3×D3—22no—
197C3×D3—41no—
198C2×Q8—91no—
199C2×D4—91no—
200C22×C4—32no—
201C22×C4—31no—
202C22×C4—31no—
203C4⋊C4—31no—
204C4⋊C4—32no—
205C4⋊C4—31no—
206C4⋊C4—92no—
207C4⋊C4—92no—
208C4⋊C4—91no—
209C4⋊C4—92no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4D3211yescharacteristic
213C22⋊C4—32no—
214C22⋊C4—91no—
215C22⋊C4—92no—
216C22⋊C4—91no—
217C22⋊C4—92no—
218C42—32no—
219C42—92no—
220C42—91no—
221C22×C3—21no—
222C22×C3—32no—
223C22×C3—61no—
224C22×C3—61no—
225C22×C3—61no—
226C22×C3—61no—
227C22×C3—61no—
228C22×C3C23×D311yescharacteristic
229C22×C3C23×D311yescharacteristic
230C2×D3—21no—
231C2×D3—21no—
232C2×D3—21no—
233C2×D3—21no—
234C2×D3—21no—
235C2×D3#23712yes—
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—42no—
240C3×C4—61no—
241C3×C4—61no—
242C3×C4—62no—
243C3×C4—62no—
244C3×C4#23512yes—
245C3×C4#23512yes—
246Dic3—21no—
247Dic3—21no—
248Dic3—22no—
249Dic3—22no—
250Dic3—64no—
251Dic3—62no—
252Dic3—62no—
253Dic3—62no—
254Dic3—61no—
255Dic3—61no—
256Dic3—61no—
257Dic3—61no—
258Dic3—121no—
259C32#55211yescharacteristic, complemented, Sylow
260C23—31no—
261C23—31no—
262Q8—182no—
263D4—182no—
264C2×C4—31no—
265C2×C4—32no—
266C2×C4—31no—
267C2×C4—32no—
268C2×C4—64no—
269C2×C4—62no—
270C2×C4—62no—
271C2×C4—91no—
272C2×C4—92no—
273C2×C4—91no—
274C2×C4C2×D3212yes—
275C2×C4C2×D3211yescharacteristic
276C2×C3—21no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—62no—
280C2×C3—121no—
281C2×C3C2×#23511yescharacteristic
282C2×C3C2×#23511yescharacteristic
283C2×C3C2×#23711yescharacteristic
284C2×C3#100711yescharacteristic
285C2×C3D3×#4111yescharacteristic
286C2×C3#101011yescharacteristic
287D3—22no—
288D3—41no—
289C22—32no—
290C22—61no—
291C22—61no—
292C22—61no—
293C22—61no—
294C22—61no—
295C22C22×D3211yescharacteristic
296C4—22no—
297C4—61no—
298C4—61no—
299C4—62no—
300C4—62no—
301C4—182no—
302C4—181no—
303C4—181no—
304C4#379812yes—
305C3—21no—
306C3#551311yescharacteristic, complemented
307C3#552911yescharacteristic, complemented
308C2—62no—
309C2—121no—
310C2C2×#379811yescharacteristic
311C2D3×#23711yescharacteristic
312C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#625C22
hypercenterC4⋊C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64096C2semidirect
2. C3×#892C2semidirect
3. C3×D3×C4⋊C4C2semidirect
6. #63872C2semidirect
7. #63867C2semidirect
8. #63862C2semidirect
9. #63857C2semidirect
10. #63857C2semidirect
11. #63855C2semidirect
48. #892D3semidirect
49. D3×C4⋊C4D3semidirect
259. C32#552semidirect
306. C3#5513semidirect
307. C3#5529semidirect

Maximal subgroups

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

subgroupindex
#640962
C3×#8922
C3×D3×C4⋊C42
#639052
C4×#6252
#638722
#638672
#638622
#638572
#638552
#55293
#55133