Finite Group Databasebeta

← #80352

Subgroups — #80352

Order 384.

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

normal subgroups only

This group has 494 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
1#5717C211yesmaximal, characteristic, complemented
2#5551C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5157C212yesmaximal, complemented
5#5156C212yesmaximal, complemented
6#5070C211yesmaximal, characteristic, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#4846C211yesmaximal, characteristic, complemented
9#4834C211yesmaximal, characteristic, complemented
10#4774C212yesmaximal, complemented
11D3×#102C212yesmaximal, complemented
12#3294—31nomaximal, Sylow
13C2×D3×D4—21no—
14C2×D3×D4C2212yes—
15C2×#235—21no—
16C2×C3×D8C2212yescomplemented
17C3×#127C2211yescharacteristic
18C3×C4×D4C2211yescharacteristic, complemented
19#938—21no—
20#938C2212yes—
21#938C2212yescomplemented
22#937—22no—
23C2×#213C2211yescharacteristic
24#924—21no—
25C2×D3×C8C2212yescomplemented
26D3×C4⋊C4C2212yescomplemented
27#883—22no—
28C4×D12C2211yescharacteristic
29C4×D12C2211yescharacteristic
30#869C2211yescharacteristic, complemented
31C3×C4⋊C8C2211yescharacteristic, complemented
32C3×#102C2212yescomplemented
33#832C2212yescomplemented
34#832C2211yescharacteristic, complemented
35#832C2212yescomplemented
36#821C2212yes—
37(C2×D3)⋊C8C2212yescomplemented
38#818C2212yescomplemented
39#809C2211yescharacteristic
40#807C2211yescharacteristic, complemented
41C4×C3⋊C8C2211yescharacteristic, complemented
42#547—31no—
43D42—31no—
44#486—31no—
45#480—32no—
46#474—31no—
47#466—32no—
48#459D311yescharacteristic, complemented
49#447—32no—
50C4×D8—31no—
51C8×D4—31no—
52C2×#102—32no—
53C23×D3—22no—
54C2×C3×D4—21no—
55C2×C3×D4—21no—
56C2×C3×D4C2312yes—
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—21no—
59C2×#67—22no—
60C2×#67—22no—
61C2×#67—22no—
62D3×D4—24no—
63D3×D4—42no—
64D3×D4—44no—
65#235—44no—
66C2×D12D411yescharacteristic
67C2×D12C2311yescharacteristic
68C2×C4×D3—22no—
69C2×C4×D3—21no—
70C2×C4×D3C2312yes—
71C2×Dic6—21no—
72C3×D8—42no—
73C2×C3×C8C2312yes—
74C3×C4⋊C4C2311yescharacteristic
75C3×C22⋊C4—21no—
76C3×C42C2311yescharacteristic
77#217—22no—
78#217—22no—
79#213—22no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4D412yes—
82C4⋊Dic3—21no—
83C4⋊Dic3D411yescharacteristic
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2312yes—
86Dic3⋊C4—22no—
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2311yescharacteristic
89D3×C8—42no—
90C2×#41—61no—
91C22×D4—32no—
92C22×D4—61no—
93C2×D8—31no—
94C2×D8C2×D312yescomplemented
95C22×C8—32no—
96C4⋊Q8—31no—
97#127C2×D311yescharacteristic
98#123—62no—
99C22⋊Q8—61no—
100#121—32no—
101#121—32no—
102#121—61no—
103#120—62no—
104C4×D4—31no—
105C4×D4—31no—
106C4×D4C2×D311yescharacteristic, complemented
107C2×C4⋊C4—32no—
108#107—32no—
109#107—31no—
110C4⋊C8C2×D311yescharacteristic, complemented
111#102—32no—
112#102—31no—
113#102—32no—
114#102—32no—
115#102—31no—
116#102C2×D312yescomplemented
117C22⋊C8—32no—
118C4×C8—31no—
119C23×C3—21no—
120C23×C3—22no—
121C22×D3—21no—
122C22×D3—44no—
123C22×D3—44no—
124C22×D3—82no—
125C22×D3C2×D412yes—
126C3×D4—21no—
127C3×D4—22no—
128C3×D4—21no—
129C3×D4—22no—
130C3×D4—42no—
131C3×D4—81no—
132C2×C3×C4—21no—
133C2×C3×C4—42no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2411yescharacteristic
137#67—44no—
138#67—44no—
139#67—44no—
140C2×Dic3—22no—
141C2×Dic3—21no—
142C2×Dic3C2×D412yes—
143D12—21no—
144D12D814yescomplemented
145D12#4112yescomplemented
146C4×D3—22no—
147C4×D3—22no—
148C4×D3—44no—
149C4×D3—42no—
150Dic6—42no—
151C3×C8—22no—
152C3⋊C8—21no—
153C3⋊C8—21no—
154C24—62no—
155#41—124no—
156C2×Q8—61no—
157C2×D4—21no—
158C2×D4—21no—
159C2×D4—31no—
160C2×D4—31no—
161C2×D4—62no—
162C2×D4—64no—
163C2×D4—62no—
164C2×D4—62no—
165C2×D4—122no—
166C2×D4—124no—
167C2×D4C22×D312yes—
168C2×D4C22×D311yescharacteristic
169C22×C4—21no—
170C22×C4—32no—
171C22×C4—62no—
172C22×C4—61no—
173D8—42no—
174D8—62no—
175C2×C8—31no—
176C2×C8—31no—
177C2×C8—122no—
178C2×C8C22×D312yes—
179C4⋊C4—31no—
180C4⋊C4—32no—
181C4⋊C4—31no—
182C4⋊C4—62no—
183C4⋊C4—61no—
184C4⋊C4C22×D311yescharacteristic
185C22⋊C4—21no—
186C22⋊C4—32no—
187C22⋊C4—62no—
188C22⋊C4—62no—
189C22⋊C4—61no—
190C42C22×D311yescharacteristic
191C22×C3—42no—
192C22×C3—41no—
193C22×C3—41no—
194C22×C3—42no—
195C22×C3—42no—
196C22×C3—41no—
197C22×C3C22×D411yescharacteristic
198C2×D3—22no—
199C2×D3—22no—
200C2×D3—22no—
201C2×D3—41no—
202C2×D3—41no—
203C2×D3—88no—
204C2×D3C2×D814yes—
205C3×C4—21no—
206C3×C4—41no—
207C3×C4C2×D812yes—
208C3×C4C2×#4111yescharacteristic
209C3×C4H(5,𝔽2)11yescharacteristic
210Dic3—22no—
211Dic3—42no—
212Dic3—41no—
213C23—21no—
214C23—22no—
215C23—32no—
216C23—61no—
217C23—124no—
218C23—124no—
219C23—242no—
220Q8—122no—
221D4—21no—
222D4—22no—
223D4—21no—
224D4—22no—
225D4—34no—
226D4—32no—
227D4—42no—
228D4—61no—
229D4—81no—
230D4—124no—
231D4—124no—
232D4—124no—
233C2×C4—21no—
234C2×C4—32no—
235C2×C4—42no—
236C2×C4—62no—
237C2×C4—62no—
238C2×C4—61no—
239C2×C4—62no—
240C2×C4—124no—
241C2×C4—122no—
242C2×C4D3×D411yescharacteristic
243C2×C4D3×D411yescharacteristic
244C2×C4C23×D311yescharacteristic
245C8—22no—
246C8—61no—
247C8—61no—
248C2×C3—82no—
249C2×C3—81no—
250C2×C3#54611yescharacteristic
251C2×C3C22×D811yescharacteristic
252C2×C3#57711yescharacteristic
253D3—24no—
254D3—42no—
255C22—34no—
256C22—41no—
257C22—42no—
258C22—42no—
259C22—41no—
260C22—42no—
261C22—41no—
262C22—62no—
263C22—62no—
264C22—62no—
265C22—121no—
266C22—121no—
267C22—248no—
268C22C2×D3×D411yescharacteristic
269C4—21no—
270C4—41no—
271C4—62no—
272C4—121no—
273C4—122no—
274C4D3×D812yes—
275C4#100411yescharacteristic
276C4D3×#4111yescharacteristic
277C3#329411yescharacteristic, complemented, Sylow
278C2—64no—
279C2—82no—
280C2—81no—
281C2—122no—
282C2#561211yescharacteristic
283C2C2×D3×D811yescharacteristic
284C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3294
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5717C2semidirect
2. #5551C2semidirect
3. C3×#459C2semidirect
4. #5157C2semidirect
5. #5156C2semidirect
6. #5070C2semidirect
7. C4×#213C2semidirect
8. #4846C2semidirect
9. #4834C2semidirect
10. #4774C2semidirect
11. D3×#102C2semidirect
16. C2×C3×D8C22semidirect
18. C3×C4×D4C22semidirect
21. #938C22semidirect
25. C2×D3×C8C22semidirect
26. D3×C4⋊C4C22semidirect
30. #869C22semidirect
31. C3×C4⋊C8C22semidirect
32. C3×#102C22semidirect
33. #832C22semidirect
34. #832C22semidirect
35. #832C22semidirect
37. (C2×D3)⋊C8C22semidirect
38. #818C22semidirect
40. #807C22semidirect
41. C4×C3⋊C8C22semidirect
48. #459D3semidirect
94. C2×D8C2×D3semidirect
106. C4×D4C2×D3semidirect
110. C4⋊C8C2×D3semidirect
116. #102C2×D3semidirect
144. D12D8semidirect
145. D12#41semidirect
277. C3#3294semidirect

Maximal subgroups

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

subgroupindex
#57172
#55512
C3×#4592
#51572
#51562
#50702
C4×#2132
#48462
#48342
#47742
D3×#1022
#32943