Finite Group Databasebeta

← #10594608

Subgroups — #10594608

Order 576.

730 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C2×#3802C211yesmaximal, characteristic
2C2×C4×D32C211yesmaximal, characteristic, complemented
3C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
4C3×#894C212yesmaximal, complemented
5#63921C211yesmaximal, characteristic, complemented
6#63921C211yesmaximal, characteristic, complemented
7#63921C211yesmaximal, characteristic, complemented
8#63905C212yesmaximal, complemented
9#63891C212yesmaximal, complemented
10#63884C212yesmaximal, complemented
11#63883C211yesmaximal, characteristic, complemented
12D3×#123—32nomaximal
13C22×D32C2211yescharacteristic
14C22×D32C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C4×C3⋊D3C2211yescharacteristic
18C2×C3×D12C2212yes—
19C2×C3×C4×D3C2212yescomplemented
20C2×#625C2212yes—
21C2×#624C2211yescharacteristic
22C2×#624C2211yescharacteristic
23C2×D3×Dic3C2212yescomplemented
24#3802—24no—
25C4×D32—24no—
26C32×C4⋊C4C2211yescharacteristic, complemented
27C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
28#3750C2211yescharacteristic, complemented
29#3750C2211yescharacteristic, complemented
30C3×(C2×D3)⋊C4C2212yescomplemented
31C3×(C2×D3)⋊C4C2212yescomplemented
32C3×(C2×D3)⋊C4C2212yescomplemented
33C3×Dic3⋊C4C2212yescomplemented
34Dic3⋊Dic3C2212yescomplemented
35(C2×D3)⋊Dic3C2212yescomplemented
36(C2×D3)⋊Dic3C2212yescomplemented
37(C2×D3)⋊Dic3C2212yescomplemented
38C2×D3×D4—32no—
39C22×C4×D3—32no—
40C3×#123—32no—
41#935—32no—
42#929—32no—
43#894—32no—
44#894D312yescomplemented
45D3×C4⋊C4—32no—
46D3×C4⋊C4—32no—
47D3×C4⋊C4—61no—
48#886—32no—
49#883—32no—
50#883—32no—
51D3×C22⋊C4—32no—
52D3×C22⋊C4—32no—
53D3×C22⋊C4—32no—
54C22×C3⋊D3C2311yescharacteristic
55C22×C3×D3C2312yes—
56C22×C3×D3C2312yes—
57C2×D32—22no—
58C2×D32—22no—
59C2×D32—22no—
60C2×D32—22no—
61C2×D32—22no—
62C2×D32D414yes—
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C4×C3⋊D3—22no—
70C4×C3⋊D3—22no—
71C4×C3⋊D3—22no—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2312yes—
74C3×D12—24no—
75C3×C4×D3—24no—
76#625—24no—
77#624—22no—
78#624—22no—
79D3×Dic3—24no—
80C2×#123—91noSylow
81C23×D3—32no—
82C23×D3—32no—
83C2×C3×D4—32no—
84C22×C3×C4—32no—
85C2×#67—32no—
86C2×#67—32no—
87C22×Dic3—32no—
88D3×D4—68no—
89C2×D12C2×D312yes—
90C2×C4×D3—32no—
91C2×C4×D3—32no—
92C2×C4×D3—32no—
93C2×C4×D3—32no—
94C2×C4×D3—32no—
95C2×C4×D3—32no—
96C2×C4×D3—61no—
97C2×C4×D3—61no—
98C2×C4×D3—61no—
99C2×C4×D3—68no—
100C2×C4×D3C2×D312yescomplemented
101C3×C4⋊C4—21no—
102C3×C4⋊C4—32no—
103C3×C4⋊C4C2×D312yescomplemented
104C3×C22⋊C4—32no—
105C3×C22⋊C4—32no—
106C3×C22⋊C4—32no—
107#217—32no—
108#217—32no—
109#217—32no—
110(C2×D3)⋊C4—32no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4C2×D312yescomplemented
114(C2×D3)⋊C4C2×D312yescomplemented
115(C2×D3)⋊C4C2×D312yescomplemented
116C4⋊Dic3—32no—
117C4⋊Dic3—32no—
118C4⋊Dic3—61no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—61no—
123Dic3⋊C4—61no—
124Dic3⋊C4C2×D312yescomplemented
125C22×C32C2411yescharacteristic
126C2×C3⋊D3C2×D412yes—
127C2×C3⋊D3#4112yes—
128C2×C3⋊D3#4112yes—
129C2×C3×D3—22no—
130C2×C3×D3—22no—
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—22no—
134C2×C3×D3C2×D414yes—
135D32—28no—
136D32—44no—
137C32×C4—21no—
138C32×C4—21no—
139C32×C4—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—22no—
145C22×D4—91no—
146C23×C4—91no—
147#123—32no—
148#123—92no—
149#123—92no—
150#123—92no—
151C2×C4⋊C4—91no—
152C2×C4⋊C4—91no—
153C2×C22⋊C4—91no—
154C2×C22⋊C4—91no—
155C2×C22⋊C4—91no—
156C23×C3—32no—
157C23×C3—32no—
158C22×D3—32no—
159C22×D3—38no—
160C22×D3—61no—
161C22×D3—64no—
162C22×D3—64no—
163C22×D3—64no—
164C22×D3—64no—
165C22×D3—64no—
166C22×D3C22×D312yes—
167C22×D3C22×D312yes—
168C3×D4—64no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—32no—
173C2×C3×C4—32no—
174C2×C3×C4—64no—
175C2×C3×C4C22×D312yes—
176C2×C3×C4C22×D312yes—
177C2×C3×C4C22×D312yes—
178#67—64no—
179#67—64no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—32no—
183C2×Dic3—61no—
184C2×Dic3—61no—
185C2×Dic3—61no—
186C2×Dic3—64no—
187C2×Dic3C22×D312yes—
188C2×Dic3C22×D312yes—
189D12—24no—
190C4×D3—24no—
191C4×D3—64no—
192C4×D3—64no—
193C4×D3—64no—
194C4×D3—64no—
195C4×D3—64no—
196C4×D3—64no—
197C4×D3—64no—
198C4×D3—122no—
199C4×D3—122no—
200C2×C32C22×D411yescharacteristic
201C2×C32C2×#4111yescharacteristic
202C2×C32C2×#4111yescharacteristic
203C3⋊D3#12314yescomplemented
204C3×D3—24no—
205C3×D3—42no—
206C24—91no—
207C24—91no—
208C2×D4—32no—
209C2×D4—92no—
210C2×D4—184no—
211C22×C4—32no—
212C22×C4—91no—
213C22×C4—91no—
214C22×C4—92no—
215C22×C4—91no—
216C22×C4—91no—
217C22×C4—91no—
218C22×C4—184no—
219C4⋊C4—32no—
220C4⋊C4—92no—
221C4⋊C4—91no—
222C4⋊C4—91no—
223C4⋊C4—91no—
224C4⋊C4D3211yescharacteristic, complemented
225C22⋊C4—32no—
226C22⋊C4—32no—
227C22⋊C4—32no—
228C22⋊C4—92no—
229C22⋊C4—92no—
230C22⋊C4—92no—
231C22×C3—21no—
232C22×C3—34no—
233C22×C3—62no—
234C22×C3—62no—
235C22×C3—62no—
236C22×C3—62no—
237C22×C3—62no—
238C22×C3C23×D312yes—
239C2×D3—22no—
240C2×D3—22no—
241C2×D3—22no—
242C2×D3—22no—
243C2×D3—22no—
244C2×D3—34no—
245C2×D3—34no—
246C2×D3—34no—
247C2×D3—62no—
248C2×D3—62no—
249C2×D3—62no—
250C2×D3—616no—
251C2×D3—128no—
252C2×D3D3×D414yes—
253C3×C4—22no—
254C3×C4—22no—
255C3×C4—22no—
256C3×C4—22no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4—62no—
260C3×C4—62no—
261Dic3—22no—
262Dic3—22no—
263Dic3—62no—
264Dic3—62no—
265Dic3—62no—
266Dic3—62no—
267Dic3—121no—
268Dic3—121no—
269C32C2×#12311yescharacteristic, complemented, Sylow
270C23—32no—
271C23—32no—
272C23—94no—
273C23—91no—
274C23—182no—
275C23—182no—
276C23—182no—
277C23—182no—
278C23—182no—
279D4—64no—
280D4—184no—
281C2×C4—32no—
282C2×C4—32no—
283C2×C4—64no—
284C2×C4—91no—
285C2×C4—91no—
286C2×C4—91no—
287C2×C4—182no—
288C2×C4—182no—
289C2×C4—182no—
290C2×C4—182no—
291C2×C4—182no—
292C2×C4—184no—
293C2×C4C2×D3211yescharacteristic
294C2×C4C2×D3211yescharacteristic
295C2×C4C2×D3211yescharacteristic
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—64no—
300C2×C3—122no—
301C2×C3C2×D3×D412yes—
302C2×C3D3×#4112yes—
303C2×C3D3×#4112yes—
304D3—24no—
305D3—38no—
306D3—42no—
307D3—64no—
308C22—34no—
309C22—62no—
310C22—62no—
311C22—62no—
312C22—62no—
313C22—62no—
314C22—92no—
315C22—92no—
316C22—92no—
317C22—188no—
318C22—364no—
319C22C22×D3211yescharacteristic
320C4—21no—
321C4—21no—
322C4—21no—
323C4—62no—
324C4—62no—
325C4—181no—
326C4—181no—
327C4—181no—
328C3—21no—
329C3D3×#12312yescomplemented
330C2—64no—
331C2—94no—
332C2—122no—
333C2#6430611yescharacteristic
334C2D32×D411yescharacteristic
335C2#6431911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC2×#3802C2
hypercenterC4⋊C4D32
hypocenterC32C2×#123
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×D32C2semidirect
3. C4⋊C4×C3⋊D3C2semidirect
4. C3×#894C2semidirect
5. #63921C2semidirect
6. #63921C2semidirect
7. #63921C2semidirect
8. #63905C2semidirect
9. #63891C2semidirect
10. #63884C2semidirect
11. #63883C2semidirect
19. C2×C3×C4×D3C22semidirect
23. C2×D3×Dic3C22semidirect
26. C32×C4⋊C4C22semidirect
27. C4⋊(C3⋊Dic3)C22semidirect
28. #3750C22semidirect
29. #3750C22semidirect
30. C3×(C2×D3)⋊C4C22semidirect
31. C3×(C2×D3)⋊C4C22semidirect
32. C3×(C2×D3)⋊C4C22semidirect
33. C3×Dic3⋊C4C22semidirect
34. Dic3⋊Dic3C22semidirect
35. (C2×D3)⋊Dic3C22semidirect
36. (C2×D3)⋊Dic3C22semidirect
37. (C2×D3)⋊Dic3C22semidirect
44. #894D3semidirect
100. C2×C4×D3C2×D3semidirect
103. C3×C4⋊C4C2×D3semidirect
113. (C2×D3)⋊C4C2×D3semidirect
114. (C2×D3)⋊C4C2×D3semidirect
115. (C2×D3)⋊C4C2×D3semidirect
124. Dic3⋊C4C2×D3semidirect
203. C3⋊D3#123semidirect
224. C4⋊C4D32semidirect
269. C32C2×#123semidirect
329. C3D3×#123semidirect

Maximal subgroups

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

subgroupindex
C2×#38022
C2×C4×D322
C4⋊C4×C3⋊D32
C3×#8942
#639212
#639052
#638912
#638842
#638832
D3×#1233