Finite Group Databasebeta

← #80138

Subgroups — #80138

Order 384.

488 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 123 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×#940C211yesmaximal, characteristic
2#5622C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5175C212yesmaximal, complemented
6#5163C212yesmaximal, complemented
7#5046C211yesmaximal, characteristic, complemented
8#4762C212yesmaximal, complemented
9#4760C212yesmaximal, complemented
10#4731C211yesmaximal, characteristic, complemented
11#4718C211yesmaximal, characteristic, complemented
12#3208—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×#67—21no—
15C22×#67—21no—
16C22×D12C2211yescharacteristic
17C2×C3×SD16C2212yescomplemented
18C3×C22⋊Q8C2211yescharacteristic
19Q8×Dic3C2211yescharacteristic, complemented
20#944C2211yescharacteristic, complemented
21C2×#215C2212yescomplemented
22#940—21no—
23#940—21no—
24#940C2212yes—
25#940C2212yes—
26C2×#214C2212yescomplemented
27#924C2211yescharacteristic
28C2×C4×Dic3C2211yescharacteristic
29C2×#204C2212yescomplemented
30#889—21no—
31#888C2211yescharacteristic, complemented
32#887—21no—
33#887C2211yescharacteristic, complemented
34#878—21no—
35#877—21no—
36C3×#102C2212yescomplemented
37C3×C22⋊C8C2211yescharacteristic
38#832C2212yescomplemented
39#830C2211yescharacteristic
40#821C2212yescomplemented
41Dic3⋊C8C2212yescomplemented
42C8×Dic3C2212yescomplemented
43#809C2212yescomplemented
44#533—31no—
45C2×#127—31no—
46#492—32no—
47#486—32no—
48#460—32no—
49#460—32no—
50#450—31no—
51#450—31no—
52#450—31no—
53#450D311yescharacteristic, complemented
54#431—31no—
55C24×C3—21no—
56C23×D3—21no—
57C2×C3×Q8C2311yescharacteristic
58C2×C3×D4—21no—
59C2×C3×D4—42no—
60C2×C3×D4C2312yes—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×#67—22no—
64C2×#67—22no—
65C2×#67—22no—
66C2×#67—44no—
67C2×#67—44no—
68C22×Dic3D411yescharacteristic
69C22×Dic3D411yescharacteristic
70C2×D12—21no—
71C2×D12—42no—
72C2×D12C2312yes—
73C2×Dic6—21no—
74C2×Dic6C2311yescharacteristic
75C3×SD16—42no—
76C2×C3×C8C2312yes—
77C3×C4⋊C4—21no—
78C3×C4⋊C4C2311yescharacteristic
79C3×C22⋊C4—21no—
80#217—21no—
81#215—42no—
82#214—42no—
83C4⋊Dic3—21no—
84C4⋊Dic3C2311yescharacteristic
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—21no—
91C4×Dic3—21no—
92C4×Dic3D411yescharacteristic
93C4×Dic3D411yescharacteristic
94C4×Dic3C2311yescharacteristic
95C4×Dic3C2311yescharacteristic
96C2×C3⋊C8C2312yes—
97#204—42no—
98C22×D4—31no—
99C22×D4—61no—
100C22×D4—61no—
101C22×D4C2×D311yescharacteristic
102C2×SD16—32no—
103C2×SD16—32no—
104C2×SD16—32no—
105C2×SD16C2×D312yescomplemented
106C4⋊Q8—31no—
107C4⋊Q8—31no—
108#127—32no—
109#127—32no—
110#127—61no—
111#127—61no—
112#125—61no—
113C22⋊Q8—31no—
114C22⋊Q8—61no—
115C22⋊Q8C2×D311yescharacteristic
116C4×Q8—31no—
117C4×Q8—31no—
118C4×Q8—61no—
119#117—61no—
120C2×C42—31no—
121C4⋊C8—32no—
122#102—32no—
123#102—32no—
124#102—32no—
125#102C2×D312yescomplemented
126C22⋊C8—31no—
127C22⋊C8C2×D311yescharacteristic
128C4×C8—32no—
129C23×C3—22no—
130C23×C3—42no—
131C23×C3—42no—
132C23×C3—42no—
133C23×C3C2×D411yescharacteristic
134C22×D3—22no—
135C22×D3—42no—
136C22×D3—42no—
137C22×D3—42no—
138C3×Q8—22no—
139C3×D4—22no—
140C3×D4—22no—
141C3×D4—42no—
142C2×C3×C4—22no—
143C2×C3×C4—21no—
144C2×C3×C4—21no—
145C2×C3×C4C2×D411yescharacteristic
146C2×C3×C4C2411yescharacteristic
147#67—44no—
148#67—44no—
149#67—44no—
150#67—44no—
151C2×Dic3—22no—
152C2×Dic3—21no—
153C2×Dic3—21no—
154C2×Dic3SD1614yescomplemented
155C2×Dic3C2×D411yescharacteristic
156C2×Dic3C2×D411yescharacteristic
157C2×Dic3C2×D411yescharacteristic
158C2×Dic3C2×D411yescharacteristic
159D12—22no—
160D12—22no—
161D12—42no—
162Dic6—22no—
163Dic6—42no—
164C3×C8—22no—
165C3⋊C8—22no—
166C24—21no—
167C24—61no—
168C2×Q8—31no—
169C2×Q8—61no—
170C2×Q8C22×D311yescharacteristic
171C2×D4—21no—
172C2×D4—32no—
173C2×D4—42no—
174C2×D4—62no—
175C2×D4—62no—
176C2×D4—62no—
177C2×D4—62no—
178C2×D4—61no—
179C2×D4—122no—
180C2×D4—124no—
181C2×D4—124no—
182C2×D4C22×D312yes—
183C22×C4—31no—
184C22×C4—31no—
185C22×C4C22×D311yescharacteristic
186SD16—42no—
187SD16—122no—
188SD16—122no—
189SD16—122no—
190C2×C8—32no—
191C2×C8C22×D312yes—
192C4⋊C4—21no—
193C4⋊C4—31no—
194C4⋊C4—61no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4—61no—
198C4⋊C4—61no—
199C4⋊C4—61no—
200C4⋊C4C22×D311yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—61no—
203C42—31no—
204C42—31no—
205C42—31no—
206C42—31no—
207C42—61no—
208C42—61no—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—42no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—84no—
215C22×C3C2×SD1612yes—
216C22×C3C22×D411yescharacteristic
217C2×D3—42no—
218C2×D3—42no—
219C2×D3—42no—
220C2×D3—84no—
221C3×C4—21no—
222C3×C4—41no—
223C3×C4—41no—
224C3×C4H(5,𝔽2)12yes—
225Dic3—21no—
226Dic3—41no—
227Dic3—41no—
228Dic3C2×SD1612yes—
229Dic3C2×SD1612yes—
230Dic3AGL(1,ℤ/8)12yes—
231C23—22no—
232C23—42no—
233C23—42no—
234C23—42no—
235C23—62no—
236C23—122no—
237C23—122no—
238C23—122no—
239C23D3×D411yescharacteristic
240Q8—22no—
241Q8—62no—
242Q8—122no—
243D4—22no—
244D4—22no—
245D4—42no—
246D4—62no—
247D4—62no—
248D4—122no—
249D4—124no—
250D4—124no—
251D4—124no—
252D4—124no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—34no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—62no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4D3×D411yescharacteristic
265C2×C4C23×D311yescharacteristic
266C8—22no—
267C8—62no—
268C2×C3—22no—
269C2×C3—82no—
270C2×C3#53511yescharacteristic
271C2×C3C22×SD1611yescharacteristic
272C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
273D3—82no—
274C22—21no—
275C22—21no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—84no—
280C22—122no—
281C22—122no—
282C22—122no—
283C22—244no—
284C22D3×SD1612yes—
285C22C2×D3×D411yescharacteristic
286C4—21no—
287C4—32no—
288C4—32no—
289C4—32no—
290C4—41no—
291C4—41no—
292C4—61no—
293C4—121no—
294C4—121no—
295C4#100412yes—
296C3#320811yescharacteristic, complemented, Sylow
297C2—22no—
298C2—82no—
299C2—242no—
300C2#559211yescharacteristic
301C2C2×D3×SD1611yescharacteristic
302C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radicalC2×#940C2
hypercenter#450D3
hypocenterC3#3208
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
2. #5622C2semidirect
3. C3×#450C2semidirect
4. #5218C2semidirect
5. #5175C2semidirect
6. #5163C2semidirect
7. #5046C2semidirect
8. #4762C2semidirect
9. #4760C2semidirect
10. #4731C2semidirect
11. #4718C2semidirect
17. C2×C3×SD16C22semidirect
19. Q8×Dic3C22semidirect
20. #944C22semidirect
21. C2×#215C22semidirect
26. C2×#214C22semidirect
29. C2×#204C22semidirect
31. #888C22semidirect
33. #887C22semidirect
36. C3×#102C22semidirect
38. #832C22semidirect
40. #821C22semidirect
41. Dic3⋊C8C22semidirect
42. C8×Dic3C22semidirect
43. #809C22semidirect
53. #450D3semidirect
105. C2×SD16C2×D3semidirect
125. #102C2×D3semidirect
154. C2×Dic3SD16semidirect
296. C3#3208semidirect

Maximal subgroups

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

subgroupindex
C2×#9402
#56222
C3×#4502
#52182
#51752
#51632
#50462
#47622
#47602
#47312
#47182
#32083