Finite Group Databasebeta

← #10594970

Subgroups — #10594970

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×#638C211yesmaximal, characteristic
2C3×#926C211yesmaximal, characteristic, complemented
3C3×#924C211yesmaximal, characteristic, complemented
4#63975C212yesmaximal
5#63956C212yesmaximal, complemented
6#63900C211yesmaximal, characteristic, complemented
7#63872C211yesmaximal, characteristic, complemented
8#63871C211yesmaximal, characteristic, complemented
9#63867C212yesmaximal, complemented
10#63843C211yesmaximal, characteristic, complemented
11#63836C212yesmaximal, complemented
12#5639—31nomaximal
13#5529—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×Dic6C2211yescharacteristic
18C2×#624C2212yes—
19#3757C2211yescharacteristic
20#3752C2211yescharacteristic
21#3750C2211yescharacteristic
22C4×C3⋊Dic3C2211yescharacteristic
23C3×#217C2212yes—
24C3×#217C2212yes—
25C3×C4⋊Dic3C2211yescharacteristic
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2212yes—
29C3×C4×Dic3C2212yes—
30#3724C2212yes—
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36Dic32C2212yes—
37C3×C22⋊Q8—31no—
38C3×#117—31no—
39#946—31no—
40Q8×Dic3—31no—
41C4×#67—31no—
42C4×#67—31no—
43C4×#67—61no—
44#926D311yescharacteristic, complemented
45#924D311yescharacteristic, complemented
46#897—31no—
47#892—32no—
48D3×C4⋊C4—31no—
49D3×C4⋊C4—31no—
50#890—31no—
51#890—31no—
52#889—32no—
53#887—31no—
54#883—32no—
55#883—32no—
56#879—32no—
57#879—32no—
58#872—32no—
59#870—32no—
60C23×C32C2311yescharacteristic
61C22×C3⋊D3C2311yescharacteristic
62C2×C32×C4—22no—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65#638—22no—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
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—
78C22×C3×C4—21no—
79C22×C3×C4C2×D311yescharacteristic
80C22×C3×C4C2×D311yescharacteristic
81C2×#67—31no—
82C2×#67—31no—
83C2×#67—61no—
84C2×C4×D3—31no—
85C2×C4×D3—31no—
86C2×C4×D3—32no—
87C2×C4×D3—32no—
88C2×C4×D3—61no—
89C2×Dic6C2×D311yescharacteristic
90C3×C4⋊C4—31no—
91C3×C4⋊C4—32no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C22⋊C4—32no—
95C3×C22⋊C4—32no—
96C3×C42—32no—
97#217—31no—
98#217—31no—
99#217—61no—
100#217C2×D312yes—
101#217C2×D312yes—
102(C2×D3)⋊C4—31no—
103(C2×D3)⋊C4—31no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—61no—
107C4⋊Dic3—32no—
108C4⋊Dic3—32no—
109C4⋊Dic3—32no—
110C4⋊Dic3—32no—
111C4⋊Dic3C2×D311yescharacteristic
112C4⋊Dic3C2×D311yescharacteristic
113C4⋊Dic3C2×D311yescharacteristic
114Dic3⋊C4—31no—
115Dic3⋊C4—31no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—61no—
123Dic3⋊C4C2×D312yes—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—32no—
127C4×Dic3—32no—
128C4×Dic3—61no—
129C4×Dic3C2×D312yes—
130C22×C32—21no—
131C22×C32—21no—
132C22×C32—21no—
133C22×C32C2411yescharacteristic
134C2×C3⋊D3—21no—
135C2×C3⋊D3—21no—
136C2×C3⋊D3#4112yes—
137C32×C4—21no—
138C32×C4#4112yes—
139C3⋊Dic3—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3×Dic3—22no—
143C3×Dic3—22no—
144C3×Dic3—22no—
145C3×Dic3—22no—
146#126—92no—
147#125—92no—
148#125—91no—
149#123—92no—
150C22⋊Q8—31no—
151C22⋊Q8—91no—
152C4×Q8—91no—
153C4×D4—91no—
154#117—31no—
155#117—92no—
156C2×C4⋊C4—91no—
157C23×C3—21no—
158C23×C3C22×D311yescharacteristic
159C23×C3C22×D311yescharacteristic
160C22×D3—31no—
161C22×D3—31no—
162C22×D3—61no—
163C3×Q8—62no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—32no—
169C2×C3×C4—32no—
170C2×C3×C4—32no—
171C2×C3×C4—32no—
172C2×C3×C4—42no—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176C2×C3×C4C22×D311yescharacteristic
177#67—62no—
178#67—62no—
179#67—64no—
180C2×Dic3—31no—
181C2×Dic3—31no—
182C2×Dic3—31no—
183C2×Dic3—31no—
184C2×Dic3—31no—
185C2×Dic3—31no—
186C2×Dic3—61no—
187C2×Dic3—61no—
188C2×Dic3—61no—
189C2×Dic3C22×D312yes—
190C2×Dic3C22×D312yes—
191C2×Dic3C22×D312yes—
192C2×Dic3C22×D312yes—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—62no—
196C4×D3—62no—
197C4×D3—122no—
198Dic6—22no—
199C2×C32—41no—
200C2×C32C2×#4111yescharacteristic
201C2×C32C2×#4111yescharacteristic
202C2×C322−1+411yescharacteristic
203C3⋊D3—22no—
204C2×Q8—31no—
205C2×D4—91no—
206C22×C4—92no—
207C22×C4—91no—
208C22×C4D3211yescharacteristic
209C4⋊C4—31no—
210C4⋊C4—31no—
211C4⋊C4—31no—
212C4⋊C4—32no—
213C4⋊C4—92no—
214C4⋊C4—92no—
215C4⋊C4—92no—
216C4⋊C4—92no—
217C4⋊C4—92no—
218C4⋊C4—91no—
219C22⋊C4—32no—
220C22⋊C4—32no—
221C22⋊C4—92no—
222C22⋊C4—91no—
223C22⋊C4—91no—
224C42—32no—
225C42—92no—
226C42—91no—
227C22×C3—22no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—41no—
236C22×C3—41no—
237C22×C3C23×D311yescharacteristic
238C22×C3C23×D311yescharacteristic
239C2×D3—32no—
240C2×D3—32no—
241C2×D3—62no—
242C2×D3—61no—
243C2×D3—61no—
244C2×D3—61no—
245C2×D3—61no—
246C2×D3—121no—
247C2×D3—121no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—22no—
251C3×C4—41no—
252C3×C4—62no—
253C3×C4—62no—
254C3×C4—62no—
255C3×C4—62no—
256C3×C4#23512yes—
257C3×C4#23712yes—
258Dic3—22no—
259Dic3—22no—
260Dic3—22no—
261Dic3—22no—
262Dic3—62no—
263Dic3—61no—
264Dic3—61no—
265Dic3—61no—
266Dic3—61no—
267Dic3—61no—
268Dic3—61no—
269Dic3—121no—
270Dic3—121no—
271C32#55211yescharacteristic, complemented, Sylow
272C23—91no—
273C23C2×D3211yescharacteristic
274Q8—62no—
275D4—182no—
276C2×C4—22no—
277C2×C4—32no—
278C2×C4—32no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—91no—
282C2×C4—91no—
283C2×C4—91no—
284C2×C4—184no—
285C2×C4—182no—
286C2×C4C2×D3211yescharacteristic
287C2×C4C2×D3211yescharacteristic
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—42no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3C2×#23511yescharacteristic
295C2×C3C2×#23711yescharacteristic
296C2×C3#100711yescharacteristic
297C2×C3D3×#4111yescharacteristic
298C2×C3D3×#4111yescharacteristic
299C2×C3#101011yescharacteristic
300D3—62no—
301D3—62no—
302D3—122no—
303C22—21no—
304C22—21no—
305C22—21no—
306C22—92no—
307C22—181no—
308C22—181no—
309C22C22×D3211yescharacteristic
310C4—21no—
311C4—62no—
312C4—62no—
313C4—62no—
314C4—62no—
315C4—181no—
316C4—181no—
317C4—181no—
318C4#380412yes—
319C3—21no—
320C3#552911yescharacteristic, complemented
321C3#563911yescharacteristic, complemented
322C2—41no—
323C2—182no—
324C2#6429711yescharacteristic
325C2#6430611yescharacteristic
326C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
prime radicalC2×#638C22
hypercenterC22×C4D32
hypocenterC32#552
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C3×#926C2semidirect
3. C3×#924C2semidirect
5. #63956C2semidirect
6. #63900C2semidirect
7. #63872C2semidirect
8. #63871C2semidirect
9. #63867C2semidirect
10. #63843C2semidirect
11. #63836C2semidirect
44. #926D3semidirect
45. #924D3semidirect
271. C32#552semidirect
320. C3#5529semidirect
321. C3#5639semidirect

Maximal subgroups

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

subgroupindex
C4×#6382
C3×#9262
C3×#9242
#639752
#639562
#639002
#638722
#638712
#638672
#638432
#638362
#56393
#55293