Finite Group Databasebeta

← #10593947

Subgroups — #10593947

Order 576.

475 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 130 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
1C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×#870C211yesmaximal, characteristic, complemented
3C3×C4×Dic6C211yesmaximal, characteristic, complemented
4#63900C211yesmaximal, characteristic, complemented
5#63897C212yesmaximal
6#63871C211yesmaximal, characteristic, complemented
7#63871C211yesmaximal, characteristic, complemented
8#63868C212yesmaximal, complemented
9#63849C211yesmaximal, characteristic, complemented
10#63845C212yesmaximal, complemented
11#63836C212yesmaximal, complemented
12#5694—31nomaximal
13#5581—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×#624C2212yes—
19C32×C42C2211yescharacteristic
20#3752C2211yescharacteristic
21#3752C2211yescharacteristic
22C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊Dic3C2211yescharacteristic
24C3×C4⋊Dic3C2211yescharacteristic
25C3×C4⋊Dic3C2211yescharacteristic
26C3×Dic3⋊C4C2212yes—
27C3×Dic3⋊C4C2212yes—
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2212yes—
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×#125—31no—
38C3×C4×Q8—31no—
39#946—31no—
40#946—31no—
41#898—32no—
42#897—31no—
43#897—31no—
44#894—32no—
45#894—32no—
46#892—31no—
47#892—31no—
48#892—32no—
49D3×C4⋊C4—31no—
50D3×C4⋊C4—31no—
51#890—32no—
52#889—32no—
53#876—32no—
54C4×D12—31no—
55C4×D12—31no—
56C4×D12—61no—
57#872—32no—
58#870D311yescharacteristic, complemented
59C4×Dic6D311yescharacteristic, complemented
60C22×C3⋊D3C2311yescharacteristic
61C22×C3⋊D3C2311yescharacteristic
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C4⋊(C3⋊D3)—22no—
68C4×C3⋊D3—22no—
69C4×C3⋊D3—22no—
70C2×C3×Dic3C2312yes—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2312yes—
74C3×Dic6—22no—
75#624—24no—
76#552—91noSylow
77C2×C3×Q8—31no—
78C2×D12—31no—
79C2×D12—31no—
80C2×D12—61no—
81C2×C4×D3—31no—
82C2×C4×D3—31no—
83C2×C4×D3—31no—
84C2×C4×D3—31no—
85C2×C4×D3—32no—
86C2×C4×D3—32no—
87C2×C4×D3—61no—
88C2×C4×D3—61no—
89C2×Dic6C2×D311yescharacteristic
90C3×C4⋊C4—32no—
91C3×C4⋊C4—32no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4—31no—
95C3×C4⋊C4—32no—
96C3×C42—21no—
97C3×C42—32no—
98C3×C42C2×D311yescharacteristic
99C3×C42C2×D311yescharacteristic
100(C2×D3)⋊C4—31no—
101(C2×D3)⋊C4—31no—
102(C2×D3)⋊C4—31no—
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4—61no—
111(C2×D3)⋊C4—61no—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—32no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3C2×D311yescharacteristic
118C4⋊Dic3C2×D311yescharacteristic
119C4⋊Dic3C2×D311yescharacteristic
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—32no—
123Dic3⋊C4—32no—
124Dic3⋊C4C2×D312yes—
125Dic3⋊C4C2×D312yes—
126Dic3⋊C4C2×D312yes—
127C4×Dic3—32no—
128C4×Dic3—32no—
129C4×Dic3C2×D312yes—
130C22×C32C2411yescharacteristic
131C2×C3⋊D3—21no—
132C2×C3⋊D3—21no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3#4112yes—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4#4112yes—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3×Dic3—22no—
143C3×Dic3—22no—
144C3×Dic3—22no—
145C3×Dic3—22no—
146#126—92no—
147#125—31no—
148#125—92no—
149#123—92no—
150C22⋊Q8—91no—
151C22⋊Q8—91no—
152C4×Q8—31no—
153C4×D4—91no—
154#117—92no—
155#117—91no—
156C2×C4⋊C4—91no—
157C22×D3—31no—
158C22×D3—31no—
159C22×D3—31no—
160C22×D3—31no—
161C22×D3—61no—
162C22×D3—61no—
163C3×Q8—62no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—32no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4C22×D311yescharacteristic
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177C2×Dic3—31no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—31no—
181C2×Dic3—61no—
182C2×Dic3—61no—
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D312yes—
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D312yes—
187D12—62no—
188D12—62no—
189D12—64no—
190C4×D3—64no—
191C4×D3—64no—
192C4×D3—62no—
193C4×D3—62no—
194C4×D3—62no—
195C4×D3—62no—
196C4×D3—122no—
197C4×D3—122no—
198Dic6—22no—
199C2×C32C2×#4111yescharacteristic
200C2×C32C2×#4111yescharacteristic
201C2×C322−1+411yescharacteristic
202C3⋊D3—22no—
203C3⋊D3—41no—
204C2×Q8—31no—
205C2×D4—91no—
206C22×C4—92no—
207C22×C4—91no—
208C22×C4—91no—
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—32no—
214C4⋊C4—32no—
215C4⋊C4—92no—
216C4⋊C4—92no—
217C4⋊C4—92no—
218C4⋊C4—91no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C22⋊C4—91no—
223C22⋊C4—91no—
224C42—32no—
225C42—92no—
226C42D3211yescharacteristic
227C22×C3—21no—
228C22×C3C23×D311yescharacteristic
229C22×C3C23×D311yescharacteristic
230C2×D3—32no—
231C2×D3—32no—
232C2×D3—62no—
233C2×D3—61no—
234C2×D3—61no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—62no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—61no—
242C2×D3—61no—
243C2×D3—61no—
244C2×D3—121no—
245C2×D3—121no—
246C2×D3—121no—
247C2×D3—121no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—21no—
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—41no—
255C3×C4—62no—
256C3×C4—62no—
257C3×C4—62no—
258C3×C4—62no—
259C3×C4#23512yes—
260C3×C4#23912yes—
261Dic3—22no—
262Dic3—22no—
263Dic3—22no—
264Dic3—22no—
265Dic3—61no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270Dic3—121no—
271C32#55211yescharacteristic, complemented, Sylow
272C23—91no—
273C23—91no—
274Q8—62no—
275D4—182no—
276C2×C4—32no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—91no—
281C2×C4—91no—
282C2×C4—184no—
283C2×C4—182no—
284C2×C4—182no—
285C2×C4C2×D3211yescharacteristic
286C2×C4C2×D3211yescharacteristic
287C2×C4C2×D3211yescharacteristic
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3C2×#23511yescharacteristic
292C2×C3C2×#23911yescharacteristic
293C2×C3#100711yescharacteristic
294C2×C3D3×#4111yescharacteristic
295C2×C3D3×#4111yescharacteristic
296C2×C3#101011yescharacteristic
297D3—62no—
298D3—62no—
299D3—122no—
300D3—122no—
301D3—121no—
302D3—121no—
303C22—92no—
304C22—181no—
305C22—181no—
306C22—181no—
307C22—181no—
308C22—181no—
309C22C22×D3211yescharacteristic
310C4—21no—
311C4—21no—
312C4—62no—
313C4—62no—
314C4—62no—
315C4—62no—
316C4—181no—
317C4—181no—
318C4#379912yes—
319C3—21no—
320C3#558111yescharacteristic, complemented
321C3#569411yescharacteristic, complemented
322C2—182no—
323C2—361no—
324C2#6429711yescharacteristic
325C2C2×#379911yescharacteristic
326C2#6430611yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radicalC2×C4⋊(C3⋊D3)C22
hypercenterC42D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×#870C2semidirect
3. C3×C4×Dic6C2semidirect
4. #63900C2semidirect
6. #63871C2semidirect
7. #63871C2semidirect
8. #63868C2semidirect
9. #63849C2semidirect
10. #63845C2semidirect
11. #63836C2semidirect
58. #870D3semidirect
59. C4×Dic6D3semidirect
271. C32#552semidirect
320. C3#5581semidirect
321. C3#5694semidirect

Maximal subgroups

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

subgroupindex
C4×C4⋊(C3⋊D3)2
C3×#8702
C3×C4×Dic62
#639002
#638972
#638712
#638682
#638492
#638452
#638362
#56943
#55813