Finite Group Databasebeta

← #80001

Subgroups — #80001

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×#912C212yesmaximal, complemented
3#5606C211yesmaximal, characteristic, complemented
4C3×#447C211yesmaximal, characteristic, complemented
5#5218C211yesmaximal, characteristic, complemented
6#5158C212yesmaximal, complemented
7#5040C211yesmaximal, characteristic, complemented
8#4783C212yesmaximal, complemented
9#4772C212yesmaximal, complemented
10#4739C211yesmaximal, characteristic, complemented
11D3×C22⋊C8C211yesmaximal, characteristic, complemented
12#3002—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C23×Dic3—21no—
16C2×#237—28no—
17C2×#237—21no—
18C2×#237—21no—
19C2×#237C2212yes—
20C2×#237C2212yes—
21C2×#237C2211yescharacteristic, complemented
22C22×C4×D3C2211yescharacteristic
23C22×Dic6C2211yescharacteristic
24C2×C3×D8C2212yescomplemented
25C3×#121C2211yescharacteristic
26#938C2211yescharacteristic, complemented
27#937—21no—
28D4×Dic3—21no—
29C2×#214C2212yescomplemented
30C2×#214C2212yescomplemented
31#924C2211yescharacteristic
32#912—28no—
33C2×Dic12C2212yescomplemented
34C2×D3×C8C2212yescomplemented
35#897C2211yescharacteristic, complemented
36#892C2211yescharacteristic, complemented
37#883—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#450—31no—
52#449—32no—
53#448—31no—
54#447D311yescharacteristic, complemented
55#416—32no—
56C2×C22⋊C8—31no—
57C24×C3—21no—
58C23×D3D411yescharacteristic
59C2×C3×D4—24no—
60C2×C3×D4—21no—
61C2×C3×D4—21no—
62C2×C3×D4C2311yescharacteristic
63C2×C3×D4C2312yes—
64C22×C3×C4C2311yescharacteristic
65C2×#67—28no—
66C2×#67—22no—
67C2×#67—22no—
68C2×#67—21no—
69C2×#67—21no—
70C22×Dic3—28no—
71C22×Dic3—21no—
72C22×Dic3—22no—
73C22×Dic3—22no—
74C22×Dic3—21no—
75C22×Dic3D411yescharacteristic
76#237—28no—
77#237—28no—
78#237—24no—
79#237—24no—
80#237—24no—
81#237—24no—
82#237D418yescomplemented
83C2×C4×D3—24no—
84C2×C4×D3D411yescharacteristic
85C2×C4×D3D411yescharacteristic
86C2×C4×D3C2311yescharacteristic
87C2×C4×D3C2311yescharacteristic
88C2×Dic6—21no—
89C2×Dic6—24no—
90C2×Dic6C2311yescharacteristic
91C2×Dic6C2312yes—
92C3×D8—24no—
93C2×C3×C8C2312yes—
94C3×C4⋊C4C2311yescharacteristic
95C3×C22⋊C4—21no—
96#217—21no—
97#217—21no—
98#214—24no—
99#214—24no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102C4⋊Dic3—21no—
103C4⋊Dic3C2311yescharacteristic
104Dic3⋊C4—21no—
105C4×Dic3—21no—
106C2×C3⋊C8C2312yes—
107Dic12—24no—
108D3×C8—24no—
109C2×#41—32no—
110C2×#41—31no—
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×SD16—32no—
124C2×D8C2×D312yescomplemented
125C22×C8—32no—
126#123—61no—
127C22⋊Q8—31no—
128C22⋊Q8—31no—
129#121—31no—
130#121C2×D311yescharacteristic
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—21no—
145C23×C3—42no—
146C23×C3—42no—
147C23×C3C2×D411yescharacteristic
148C22×D3—22no—
149C22×D3—22no—
150C22×D3—41no—
151C22×D3C2×D411yescharacteristic
152C22×D3C2×D411yescharacteristic
153C3×D4—24no—
154C3×D4—22no—
155C3×D4—22no—
156C3×D4—42no—
157C3×D4C2×D414yes—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4C2×D411yescharacteristic
161C2×C3×C4C2411yescharacteristic
162#67—28no—
163#67—28no—
164#67—42no—
165#67—42no—
166C2×Dic3—24no—
167C2×Dic3—28no—
168C2×Dic3—22no—
169C2×Dic3—22no—
170C2×Dic3—28no—
171C2×Dic3—21no—
172C2×Dic3—21no—
173C2×Dic3—42no—
174C2×Dic3C2×D411yescharacteristic
175C2×Dic3C2×D411yescharacteristic
176C4×D3—22no—
177C4×D3—22no—
178C4×D3—22no—
179C4×D3C2×D414yes—
180Dic6—24no—
181Dic6—22no—
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—31no—
197C2×Q8—64no—
198C2×Q8—61no—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—62no—
203C2×D4—62no—
204C2×D4—68no—
205C2×D4—61no—
206C2×D4—61no—
207C2×D4C22×D311yescharacteristic
208C2×D4C22×D312yes—
209C22×C4—31no—
210C22×C4—31no—
211C22×C4—31no—
212C22×C4—31no—
213C22×C4—31no—
214C22×C4—64no—
215C22×C4—62no—
216C22×C4—62no—
217C22×C4—61no—
218C22×C4—68no—
219C22×C4—61no—
220C22×C4C22×D311yescharacteristic
221Q16—64no—
222SD16—64no—
223SD16—64no—
224D8—24no—
225C2×C8—32no—
226C2×C8—64no—
227C2×C8C22×D312yes—
228C4⋊C4—31no—
229C4⋊C4—61no—
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—41no—
245C22×C3—41no—
246C22×C3—41no—
247C22×C3#13512yescomplemented
248C22×C3C22×D411yescharacteristic
249C2×D3—21no—
250C2×D3—21no—
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—21no—
254C2×D3—44no—
255C2×D3#12012yes—
256C3×C4—21no—
257C3×C4—41no—
258C3×C4C22×D412yes—
259Dic3—24no—
260Dic3—21no—
261Dic3—41no—
262Dic3—41no—
263Dic3#12012yes—
264C23—24no—
265C23—22no—
266C23—21no—
267C23—31no—
268C23—31no—
269C23—42no—
270C23—42no—
271C23—62no—
272C23—62no—
273C23—121no—
274C23D3×D411yescharacteristic
275Q8—34no—
276Q8—64no—
277Q8—62no—
278Q8—62no—
279D4—24no—
280D4—22no—
281D4—22no—
282D4—42no—
283D4—68no—
284D4—68no—
285D4—122no—
286D4—122no—
287D4D3×D414yes—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—31no—
291C2×C4—34no—
292C2×C4—31no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4—62no—
296C2×C4—68no—
297C2×C4—64no—
298C2×C4—68no—
299C2×C4—62no—
300C2×C4—62no—
301C2×C4—61no—
302C2×C4—61no—
303C2×C4—122no—
304C2×C4D3×D411yescharacteristic
305C2×C4C23×D311yescharacteristic
306C8—22no—
307C8—62no—
308C2×C3—22no—
309C2×C3—44no—
310C2×C3—81no—
311C2×C3C2×#12011yescharacteristic
312C2×C3C2×#13511yescharacteristic
313C2×C3#57511yescharacteristic
314D3—22no—
315D3—41no—
316C22—21no—
317C22—21no—
318C22—24no—
319C22—32no—
320C22—48no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—61no—
327C22—61no—
328C22—61no—
329C22—61no—
330C22—61no—
331C22—124no—
332C22#91212yescomplemented
333C22C2×D3×D411yescharacteristic
334C4—21no—
335C4—32no—
336C4—41no—
337C4—61no—
338C4—64no—
339C4—121no—
340C4—121no—
341C4C2×D3×D412yes—
342C3#300211yescharacteristic, complemented, Sylow
343C2—22no—
344C2—44no—
345C2—62no—
346C2—81no—
347C2—121no—
348C2D3×#12011yescharacteristic
349C2C2×#91211yescharacteristic
350C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3002
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. C22×#237C2semidirect
2. C2×#912C2semidirect
3. #5606C2semidirect
4. C3×#447C2semidirect
5. #5218C2semidirect
6. #5158C2semidirect
7. #5040C2semidirect
8. #4783C2semidirect
9. #4772C2semidirect
10. #4739C2semidirect
11. D3×C22⋊C8C2semidirect
21. C2×#237C22semidirect
24. C2×C3×D8C22semidirect
26. #938C22semidirect
29. C2×#214C22semidirect
30. C2×#214C22semidirect
33. C2×Dic12C22semidirect
34. C2×D3×C8C22semidirect
35. #897C22semidirect
36. #892C22semidirect
39. C3×#102C22semidirect
41. #832C22semidirect
43. (C2×D3)⋊C8C22semidirect
44. #816C22semidirect
45. #810C22semidirect
54. #447D3semidirect
82. #237D4semidirect
124. C2×D8C2×D3semidirect
138. #102C2×D3semidirect
247. C22×C3#135semidirect
332. C22#912semidirect
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×#9122
#56062
C3×#4472
#52182
#51582
#50402
#47832
#47722
#47392
D3×C22⋊C82
#30023