Finite Group Databasebeta

← #80145

Subgroups — #80145

Order 384.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C22×#237C211yesmaximal, characteristic, complemented
2C2×#916C212yesmaximal, complemented
3#5627C211yesmaximal, characteristic, complemented
4C3×#450C211yesmaximal, characteristic, complemented
5#5217C211yesmaximal, characteristic, complemented
6#5173C212yesmaximal, complemented
7#5050C211yesmaximal, characteristic, complemented
8#4781C212yesmaximal, complemented
9#4772C212yesmaximal, complemented
10#4738C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3002—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C23×Dic3—21no—
16C2×#239C2211yescharacteristic, complemented
17C2×#237—28no—
18C2×#237—21no—
19C2×#237—21no—
20C2×#237C2212yes—
21C2×#237C2212yes—
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C2×C3×SD16C2212yescomplemented
25C3×C22⋊Q8C2211yescharacteristic
26#946C2211yescharacteristic, complemented
27C2×#216C2212yescomplemented
28C2×#213C2212yescomplemented
29#930C2211yescharacteristic
30#916—28no—
31C2×#204C2212yescomplemented
32C2×D3×C8C2212yescomplemented
33#895C2211yescharacteristic, complemented
34#894—21no—
35#893—21no—
36#892C2211yescharacteristic, complemented
37#882—21no—
38#67⋊C4—21no—
39C3×#102C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic
41#832C2212yescomplemented
42#830C2211yescharacteristic
43(C2×D3)⋊C8C2212yescomplemented
44#816C2212yescomplemented
45#810C2212yescomplemented
46C22×#41—31no—
47C2×#135—32no—
48#525—31no—
49#452—32no—
50#451—31no—
51#450D311yescharacteristic, complemented
52#449—32no—
53#448—31no—
54#447—31no—
55#416—32no—
56C2×C22⋊C8—31no—
57C24×C3—21no—
58C23×D3D411yescharacteristic
59C2×C3×Q8C2311yescharacteristic
60C2×C3×D4—24no—
61C2×C3×D4—21no—
62C2×C3×D4C2312yes—
63C22×C3×C4C2311yescharacteristic
64C2×#67—28no—
65C2×#67—22no—
66C2×#67—22no—
67C2×#67—21no—
68C22×Dic3—28no—
69C22×Dic3—21no—
70C22×Dic3—22no—
71C22×Dic3—22no—
72C22×Dic3D411yescharacteristic
73#239—24no—
74#237—28no—
75#237—28no—
76#237—24no—
77#237—24no—
78#237—24no—
79#237D418yescomplemented
80C2×D12—21no—
81C2×D12—21no—
82C2×D12C2311yescharacteristic
83C2×C4×D3—24no—
84C2×C4×D3—21no—
85C2×C4×D3D411yescharacteristic
86C2×C4×D3D411yescharacteristic
87C2×C4×D3C2311yescharacteristic
88C2×C4×D3C2311yescharacteristic
89C2×Dic6—21no—
90C2×Dic6—24no—
91C2×Dic6C2312yes—
92C3×SD16—24no—
93C2×C3×C8C2312yes—
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97#216—24no—
98#213—24no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103C4⋊Dic3C2311yescharacteristic
104Dic3⋊C4—21no—
105C4×Dic3—21no—
106C2×C3⋊C8C2312yes—
107#204—24no—
108D3×C8—24no—
109C2×#41—31no—
110C2×#41—32no—
111C2×#41—32no—
112C2×#41—61no—
113C2×#41—61no—
114C2×#41—68no—
115C22×Q8—31no—
116C22×D4—61no—
117C22×D4C2×D311yescharacteristic
118C23×C4—31no—
119C23×C4—61no—
120#135—68no—
121C2×Q16—32no—
122C2×SD16—32no—
123C2×SD16C2×D312yescomplemented
124C2×D8—32no—
125C22×C8—32no—
126#123—61no—
127C22⋊Q8—31no—
128C22⋊Q8C2×D311yescharacteristic
129#121—31no—
130#121—31no—
131#120—61no—
132C4×D4—61no—
133C4×D4—61no—
134#117—31no—
135#103—32no—
136#103—32no—
137#102—32no—
138#102C2×D312yescomplemented
139C22⋊C8—31no—
140C22⋊C8—32no—
141C22⋊C8C2×D311yescharacteristic
142C23×C3—24no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—42no—
146C23×C3C2×D411yescharacteristic
147C22×D3—22no—
148C22×D3—22no—
149C22×D3—21no—
150C22×D3—41no—
151C22×D3C2×D411yescharacteristic
152C22×D3C2×D411yescharacteristic
153C3×Q8—22no—
154C3×D4—24no—
155C3×D4—22no—
156C3×D4C2×D414yes—
157C2×C3×C4—22no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4C2×D411yescharacteristic
161C2×C3×C4C2411yescharacteristic
162#67—28no—
163#67—28no—
164#67—42no—
165C2×Dic3—24no—
166C2×Dic3—28no—
167C2×Dic3—22no—
168C2×Dic3—22no—
169C2×Dic3—28no—
170C2×Dic3—21no—
171C2×Dic3C2×D411yescharacteristic
172C2×Dic3C2×D411yescharacteristic
173D12—22no—
174D12—42no—
175D12—42no—
176C4×D3—22no—
177C4×D3—22no—
178C4×D3—22no—
179C4×D3—42no—
180C4×D3C2×D414yes—
181Dic6—24no—
182Dic6—22no—
183Dic6C2×D414yes—
184C3×C8—22no—
185C3⋊C8—22no—
186C24—21no—
187C24—31no—
188#41—38no—
189#41—64no—
190#41—64no—
191#41—64no—
192#41—68no—
193#41—68no—
194#41—64no—
195C2×Q8—32no—
196C2×Q8—64no—
197C2×Q8—61no—
198C2×Q8C22×D311yescharacteristic
199C2×D4—24no—
200C2×D4—21no—
201C2×D4—31no—
202C2×D4—61no—
203C2×D4—61no—
204C2×D4—62no—
205C2×D4—62no—
206C2×D4—68no—
207C2×D4—61no—
208C2×D4C22×D312yes—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—64no—
215C22×C4—61no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4—68no—
219C22×C4—61no—
220C22×C4C22×D311yescharacteristic
221Q16—64no—
222SD16—24no—
223SD16—64no—
224D8—64no—
225C2×C8—32no—
226C2×C8—64no—
227C2×C8C22×D312yes—
228C4⋊C4—21no—
229C4⋊C4—31no—
230C4⋊C4—61no—
231C4⋊C4C22×D311yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—61no—
234C22⋊C4—61no—
235C22⋊C4—61no—
236C22⋊C4—61no—
237C42—61no—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3—24no—
241C22×C3—48no—
242C22×C3—42no—
243C22×C3—42no—
244C22×C3#13512yescomplemented
245C22×C3C22×D411yescharacteristic
246C2×D3—21no—
247C2×D3—21no—
248C2×D3—21no—
249C2×D3—21no—
250C2×D3—21no—
251C2×D3—44no—
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3#12012yes—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4—41no—
259C3×C4C22×D412yes—
260Dic3—24no—
261Dic3—21no—
262Dic3—41no—
263Dic3#12012yes—
264C23—24no—
265C23—22no—
266C23—31no—
267C23—31no—
268C23—42no—
269C23—42no—
270C23—62no—
271C23—62no—
272C23—61no—
273C23—121no—
274C23D3×D411yescharacteristic
275Q8—22no—
276Q8—34no—
277Q8—64no—
278Q8—62no—
279D4—24no—
280D4—22no—
281D4—68no—
282D4—68no—
283D4—62no—
284D4—122no—
285D4—122no—
286D4—122no—
287D4D3×D414yes—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—31no—
292C2×C4—34no—
293C2×C4—31no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—62no—
297C2×C4—68no—
298C2×C4—64no—
299C2×C4—68no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—61no—
303C2×C4—122no—
304C2×C4D3×D411yescharacteristic
305C2×C4C23×D311yescharacteristic
306C8—22no—
307C8—62no—
308C2×C3—22no—
309C2×C3—44no—
310C2×C3C2×#12011yescharacteristic
311C2×C3C2×#13511yescharacteristic
312C2×C3#57511yescharacteristic
313D3—22no—
314D3—41no—
315D3—81no—
316C22—21no—
317C22—21no—
318C22—24no—
319C22—32no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—61no—
324C22—61no—
325C22—61no—
326C22—61no—
327C22—61no—
328C22—124no—
329C22—121no—
330C22—121no—
331C22—121no—
332C22#91612yescomplemented
333C22C2×D3×D411yescharacteristic
334C4—21no—
335C4—32no—
336C4—41no—
337C4—41no—
338C4—61no—
339C4—64no—
340C4—121no—
341C4C2×D3×D412yes—
342C3#300211yescharacteristic, complemented, Sylow
343C2—22no—
344C2—44no—
345C2—62no—
346C2—121no—
347C2—241no—
348C2D3×#12011yescharacteristic
349C2C2×#91611yescharacteristic
350C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
hypercenter#450D3
hypocenterC3#3002
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#916C2semidirect
3. #5627C2semidirect
4. C3×#450C2semidirect
5. #5217C2semidirect
6. #5173C2semidirect
7. #5050C2semidirect
8. #4781C2semidirect
9. #4772C2semidirect
10. #4738C2semidirect
11. D3×C22⋊C8C2semidirect
16. C2×#239C22semidirect
24. C2×C3×SD16C22semidirect
26. #946C22semidirect
27. C2×#216C22semidirect
28. C2×#213C22semidirect
31. C2×#204C22semidirect
32. C2×D3×C8C22semidirect
33. #895C22semidirect
36. #892C22semidirect
39. C3×#102C22semidirect
41. #832C22semidirect
43. (C2×D3)⋊C8C22semidirect
44. #816C22semidirect
45. #810C22semidirect
51. #450D3semidirect
79. #237D4semidirect
123. C2×SD16C2×D3semidirect
138. #102C2×D3semidirect
244. C22×C3#135semidirect
332. C22#916semidirect
342. C3#3002semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×#9162
#56272
C3×#4502
#52172
#51732
#50502
#47812
#47722
#47382
D3×C22⋊C82
#30023