Finite Group Databasebeta

← #80182

Subgroups — #80182

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×#239C211yesmaximal, characteristic, complemented
2C2×#919C212yesmaximal, complemented
3#5627C211yesmaximal, characteristic, complemented
4C3×#451C211yesmaximal, characteristic, complemented
5#5227C211yesmaximal, characteristic, complemented
6#5187C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic, complemented
8#4810C212yesmaximal, complemented
9#4804C212yesmaximal, complemented
10#4732C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3002—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#239—28no—
15C2×#239—21no—
16C2×#239—21no—
17C2×#239C2212yes—
18C2×#239C2212yes—
19C2×#239C2211yescharacteristic, complemented
20C22×D12—21no—
21C22×D12C2211yescharacteristic
22C22×C4×D3—21no—
23C22×C4×D3C2211yescharacteristic
24C2×C3×Q16C2212yescomplemented
25C3×C22⋊Q8C2211yescharacteristic
26#946C2211yescharacteristic, complemented
27C2×#215C2212yescomplemented
28C2×#215C2212yescomplemented
29#930C2211yescharacteristic
30#919—28no—
31C2×D24C2212yescomplemented
32C2×D3×C8C2212yescomplemented
33#895C2211yescharacteristic, complemented
34#894—21no—
35#893—21no—
36#892C2211yescharacteristic, complemented
37#882—21no—
38#67⋊C4—21no—
39C3×#103C2212yescomplemented
40C3×C22⋊C8C2211yescharacteristic
41#835C2212yescomplemented
42#830C2211yescharacteristic
43#821C2212yescomplemented
44(C2×D3)⋊C8C2212yescomplemented
45#809C2212yescomplemented
46C22×#41—31no—
47C2×#135—32no—
48#525—31no—
49#452—32no—
50#451D311yescharacteristic, complemented
51#450—31no—
52#449—32no—
53#448—31no—
54#447—31no—
55#416—32no—
56C2×C22⋊C8—31no—
57C23×D3—21no—
58C23×D3D411yescharacteristic
59C2×C3×Q8—24no—
60C2×C3×Q8—21no—
61C2×C3×Q8C2311yescharacteristic
62C2×C3×Q8C2312yes—
63C22×C3×C4—21no—
64C22×C3×C4C2311yescharacteristic
65C2×#67—21no—
66C22×Dic3D411yescharacteristic
67#239—28no—
68#239—28no—
69#239—24no—
70#239—24no—
71#239—24no—
72#239—24no—
73#239D418yescomplemented
74C2×D12—28no—
75C2×D12—22no—
76C2×D12—22no—
77C2×D12—21no—
78C2×D12—24no—
79C2×D12—21no—
80C2×D12—21no—
81C2×D12C2311yescharacteristic
82C2×D12C2312yes—
83C2×C4×D3—28no—
84C2×C4×D3—22no—
85C2×C4×D3—22no—
86C2×C4×D3—24no—
87C2×C4×D3—21no—
88C2×C4×D3D411yescharacteristic
89C2×C4×D3D411yescharacteristic
90C2×C4×D3C2311yescharacteristic
91C2×C4×D3C2311yescharacteristic
92C3×Q16—24no—
93C2×C3×C8C2312yes—
94C3×C4⋊C4—21no—
95C3×C4⋊C4C2311yescharacteristic
96C3×C22⋊C4—21no—
97#215—24no—
98#215—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—
107D24—24no—
108D3×C8—24no—
109C2×#41—32no—
110C2×#41—31no—
111C2×#41—32no—
112C2×#41—61no—
113C2×#41—61no—
114C2×#41—68no—
115C22×Q8C2×D311yescharacteristic
116C22×D4—31no—
117C22×D4—61no—
118C23×C4—31no—
119C23×C4—61no—
120#135—68no—
121C2×Q16C2×D312yescomplemented
122C2×SD16—32no—
123C2×SD16—32no—
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#103C2×D312yescomplemented
137#102—32no—
138#102—32no—
139C22⋊C8—31no—
140C22⋊C8—32no—
141C22⋊C8C2×D311yescharacteristic
142C23×C3C2×D411yescharacteristic
143C22×D3—24no—
144C22×D3—22no—
145C22×D3—22no—
146C22×D3—22no—
147C22×D3—21no—
148C22×D3—41no—
149C22×D3—42no—
150C22×D3—42no—
151C22×D3C2×D411yescharacteristic
152C22×D3C2×D411yescharacteristic
153C3×Q8—24no—
154C3×Q8—22no—
155C3×Q8—22no—
156C3×Q8C2×D414yes—
157C2×C3×C4—24no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—21no—
161C2×C3×C4—21no—
162C2×C3×C4C2×D411yescharacteristic
163C2×C3×C4C2411yescharacteristic
164#67—42no—
165C2×Dic3—22no—
166C2×Dic3—21no—
167C2×Dic3C2×D411yescharacteristic
168C2×Dic3C2×D411yescharacteristic
169D12—28no—
170D12—24no—
171D12—28no—
172D12—22no—
173D12—22no—
174D12—42no—
175D12—42no—
176D12C2×D414yes—
177C4×D3—28no—
178C4×D3—22no—
179C4×D3—22no—
180C4×D3—28no—
181C4×D3—22no—
182C4×D3—42no—
183C4×D3C2×D414yes—
184C3×C8—22no—
185C3⋊C8—22no—
186C24—31no—
187C24—61no—
188#41—38no—
189#41—64no—
190#41—64no—
191#41—64no—
192#41—68no—
193#41—68no—
194#41—64no—
195C2×Q8—24no—
196C2×Q8—21no—
197C2×Q8C22×D311yescharacteristic
198C2×Q8C22×D312yes—
199C2×D4—32no—
200C2×D4—31no—
201C2×D4—64no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—68no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4—61no—
208C2×D4—61no—
209C22×C4—21no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—31no—
215C22×C4—64no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4—61no—
219C22×C4—68no—
220C22×C4C22×D311yescharacteristic
221Q16—24no—
222SD16—64no—
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#13512yescomplemented
241C22×C3C22×D411yescharacteristic
242C2×D3—21no—
243C2×D3—21no—
244C2×D3—21no—
245C2×D3—24no—
246C2×D3—21no—
247C2×D3—21no—
248C2×D3—44no—
249C2×D3—48no—
250C2×D3—42no—
251C2×D3—42no—
252C2×D3—41no—
253C2×D3—41no—
254C2×D3—41no—
255C2×D3#12012yes—
256C3×C4—24no—
257C3×C4—21no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4C22×D412yes—
261Dic3—21no—
262Dic3—41no—
263Dic3#12012yes—
264C23—31no—
265C23—31no—
266C23—62no—
267C23—64no—
268C23—62no—
269C23—62no—
270C23—61no—
271C23—121no—
272C23—122no—
273C23—122no—
274C23D3×D411yescharacteristic
275Q8—24no—
276Q8—22no—
277Q8—22no—
278Q8D3×D414yes—
279D4—34no—
280D4—68no—
281D4—68no—
282D4—64no—
283D4—62no—
284D4—62no—
285D4—122no—
286D4—122no—
287D4—122no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—31no—
294C2×C4—34no—
295C2×C4—31no—
296C2×C4—62no—
297C2×C4—68no—
298C2×C4—62no—
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×C3C2×#12011yescharacteristic
310C2×C3C2×#13511yescharacteristic
311C2×C3#57511yescharacteristic
312D3—22no—
313D3—41no—
314D3—44no—
315D3—81no—
316C22—21no—
317C22—21no—
318C22—32no—
319C22—61no—
320C22—61no—
321C22—61no—
322C22—61no—
323C22—64no—
324C22—61no—
325C22—124no—
326C22—128no—
327C22—122no—
328C22—122no—
329C22—121no—
330C22—121no—
331C22—121no—
332C22#91912yescomplemented
333C22C2×D3×D411yescharacteristic
334C4—24no—
335C4—21no—
336C4—32no—
337C4—41no—
338C4—41no—
339C4—61no—
340C4—121no—
341C4C2×D3×D412yes—
342C3#300211yescharacteristic, complemented, Sylow
343C2—22no—
344C2—62no—
345C2—121no—
346C2—124no—
347C2—241no—
348C2D3×#12011yescharacteristic
349C2C2×#91911yescharacteristic
350C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#451C2
socleC22×C3C22×D4
hypercenter#451D3
hypocenterC3#3002
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#239C2semidirect
2. C2×#919C2semidirect
3. #5627C2semidirect
4. C3×#451C2semidirect
5. #5227C2semidirect
6. #5187C2semidirect
7. #5046C2semidirect
8. #4810C2semidirect
9. #4804C2semidirect
10. #4732C2semidirect
11. D3×C22⋊C8C2semidirect
19. C2×#239C22semidirect
24. C2×C3×Q16C22semidirect
26. #946C22semidirect
27. C2×#215C22semidirect
28. C2×#215C22semidirect
31. C2×D24C22semidirect
32. C2×D3×C8C22semidirect
33. #895C22semidirect
36. #892C22semidirect
39. C3×#103C22semidirect
41. #835C22semidirect
43. #821C22semidirect
44. (C2×D3)⋊C8C22semidirect
45. #809C22semidirect
50. #451D3semidirect
73. #239D4semidirect
121. C2×Q16C2×D3semidirect
136. #103C2×D3semidirect
240. C22×C3#135semidirect
332. C22#919semidirect
342. C3#3002semidirect

Maximal subgroups

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

subgroupindex
C22×#2392
C2×#9192
#56272
C3×#4512
#52272
#51872
#50462
#48102
#48042
#47322
D3×C22⋊C82
#30023