Finite Group Databasebeta

← #80388

Subgroups — #80388

Order 384.

400 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 113 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#5716C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5173C212yesmaximal, complemented
5#5168C212yesmaximal, complemented
6#5072C211yesmaximal, characteristic, complemented
7#5031C211yesmaximal, characteristic, complemented
8#4841C211yesmaximal, characteristic, complemented
9#4833C211yesmaximal, characteristic, complemented
10#4781C212yesmaximal, complemented
11#4768C212yesmaximal, complemented
12#3212—31nomaximal, Sylow
13C2×#237C2212yes—
14C2×C3×SD16C2212yes—
15C3×#127C2211yescharacteristic
16C3×C4×Q8C2211yescharacteristic
17#947C2211yescharacteristic
18#946C2211yescharacteristic
19C2×#216C2212yes—
20#938—21no—
21#938—21no—
22#936C2212yes—
23D4×Dic3—21no—
24D4×Dic3—21no—
25C2×#213C2212yes—
26C2×#204C2212yes—
27#898—21no—
28#895C2211yescharacteristic
29#894—21no—
30#890C2211yescharacteristic
31C4×D12—21no—
32C4×D12C2211yescharacteristic
33#872—21no—
34C42×D3C2211yescharacteristic
35#869C2211yescharacteristic
36C3×C4⋊C8C2211yescharacteristic
37C3×#102C2212yes—
38#832C2212yes—
39(C2×D3)⋊C8C2212yes—
40#816C2212yes—
41C8×Dic3C2212yes—
42#810C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#532—31no—
45#529—31no—
46#495—32no—
47#488—32no—
48#462—31no—
49#461—31no—
50#460D311yescharacteristic, complemented
51#459—31no—
52#452—32no—
53#449—32no—
54#431—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×D4—21no—
57C2×C3×D4—21no—
58C2×C3×D4C2312yes—
59C2×#67—22no—
60C22×Dic3—22no—
61#237—42no—
62#237—42no—
63C2×D12—21no—
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×C4×D3D411yescharacteristic
67C2×C4×D3D411yescharacteristic
68C2×C4×D3C2311yescharacteristic
69C2×Dic6C2312yes—
70C3×SD16—42no—
71C2×C3×C8C2312yes—
72C3×C4⋊C4—21no—
73C3×C4⋊C4C2311yescharacteristic
74C3×C42—21no—
75C3×C42C2311yescharacteristic
76#217—22no—
77#217—22no—
78#216—42no—
79#213—42no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84(C2×D3)⋊C4—21no—
85C4⋊Dic3—21no—
86C4⋊Dic3—21no—
87C4⋊Dic3—21no—
88C4⋊Dic3C2311yescharacteristic
89Dic3⋊C4—21no—
90Dic3⋊C4—21no—
91C4×Dic3D411yescharacteristic
92C4×Dic3D411yescharacteristic
93C4×Dic3C2311yescharacteristic
94C2×C3⋊C8C2312yes—
95#204—42no—
96C2×#41—32no—
97C2×Q16—32no—
98C2×SD16—32no—
99C2×SD16C2×D312yes—
100C2×D8—32no—
101C4⋊Q8—31no—
102#127C2×D311yescharacteristic
103#126—61no—
104#125—31no—
105#124—31no—
106#124—32no—
107#123—61no—
108C22⋊Q8—31no—
109#121—31no—
110#121—61no—
111#121—61no—
112C4×Q8C2×D311yescharacteristic
113C4×D4—31no—
114C4×D4—61no—
115C4×D4—61no—
116C4×D4—61no—
117#117—61no—
118C2×C42—31no—
119C4⋊C8—31no—
120C4⋊C8C2×D311yescharacteristic
121#103—32no—
122#103—32no—
123#102—32no—
124#102C2×D312yes—
125C22⋊C8—32no—
126C4×C8—32no—
127C23×C3—22no—
128C22×D3—21no—
129C22×D3C2×D411yescharacteristic
130C3×Q8—21no—
131C3×Q8—21no—
132C3×D4—22no—
133C3×D4—22no—
134C3×D4—42no—
135C3×D4—42no—
136C2×C3×C4—21no—
137C2×C3×C4—21no—
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2411yescharacteristic
141#67—44no—
142C2×Dic3—21no—
143C2×Dic3—22no—
144C2×Dic3—44no—
145C2×Dic3C2×D411yescharacteristic
146C2×Dic3C2×D411yescharacteristic
147C2×Dic3C2×D411yescharacteristic
148D12—21no—
149D12—21no—
150D12—42no—
151C4×D3—22no—
152C4×D3—21no—
153C4×D3—21no—
154C4×D3—41no—
155C4×D3—42no—
156Dic6—22no—
157Dic6—22no—
158C3×C8—22no—
159C3⋊C8—22no—
160#41—122no—
161#41—122no—
162C2×Q8—32no—
163C2×Q8C22×D311yescharacteristic
164C2×D4—21no—
165C2×D4—21no—
166C2×D4—31no—
167C2×D4—61no—
168C2×D4—62no—
169C2×D4C22×D312yes—
170C22×C4—31no—
171C22×C4—31no—
172C22×C4—31no—
173C22×C4—62no—
174C22×C4—61no—
175Q16—122no—
176SD16—42no—
177SD16—122no—
178D8—122no—
179C2×C8—32no—
180C2×C8C22×D312yes—
181C4⋊C4—21no—
182C4⋊C4—31no—
183C4⋊C4—61no—
184C4⋊C4—61no—
185C4⋊C4—61no—
186C4⋊C4—61no—
187C4⋊C4—61no—
188C4⋊C4C22×D311yescharacteristic
189C22⋊C4—61no—
190C22⋊C4—61no—
191C22⋊C4—61no—
192C22⋊C4—61no—
193C22⋊C4—62no—
194C22⋊C4—62no—
195C22⋊C4—61no—
196C42—21no—
197C42—31no—
198C42—31no—
199C42—31no—
200C42C22×D311yescharacteristic
201C22×C3—42no—
202C22×C3—42no—
203C22×C3—42no—
204C22×C3C22×D411yescharacteristic
205C2×D3—21no—
206C2×D3—21no—
207C2×D3—21no—
208C2×D3—41no—
209C2×D3—41no—
210C2×D3—41no—
211C3×C4—21no—
212C3×C4—41no—
213C3×C4—41no—
214C3×C4#13512yes—
215C3×C4H(5,𝔽2)11yescharacteristic
216C3×C4H(5,𝔽2)11yescharacteristic
217Dic3—21no—
218Dic3—21no—
219Dic3—21no—
220Dic3—42no—
221Dic3—41no—
222C23—22no—
223C23—31no—
224C23—61no—
225Q8—21no—
226Q8—21no—
227Q8—62no—
228Q8—62no—
229D4—22no—
230D4—22no—
231D4—42no—
232D4—42no—
233D4—61no—
234D4—61no—
235D4—122no—
236D4—124no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—31no—
240C2×C4—31no—
241C2×C4—31no—
242C2×C4—62no—
243C2×C4—61no—
244C2×C4—61no—
245C2×C4—61no—
246C2×C4—62no—
247C2×C4—121no—
248C2×C4—124no—
249C2×C4—122no—
250C2×C4D3×D411yescharacteristic
251C2×C4D3×D411yescharacteristic
252C2×C4C23×D311yescharacteristic
253C8—22no—
254C8—62no—
255C2×C3—82no—
256C2×C3#53511yescharacteristic
257C2×C3C2×#13511yescharacteristic
258C2×C3#57511yescharacteristic
259D3—41no—
260D3—81no—
261C22—42no—
262C22—42no—
263C22—42no—
264C22—61no—
265C22—61no—
266C22—61no—
267C22—121no—
268C22—121no—
269C22—121no—
270C22C2×D3×D411yescharacteristic
271C4—21no—
272C4—41no—
273C4—41no—
274C4—61no—
275C4—61no—
276C4—61no—
277C4—121no—
278C4—122no—
279C4#91612yes—
280C4#100411yescharacteristic
281C4#100911yescharacteristic
282C3#321211yescharacteristic, complemented, Sylow
283C2—82no—
284C2—121no—
285C2—241no—
286C2#560711yescharacteristic
287C2C2×#91611yescharacteristic
288C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
hypercenter#460D3
hypocenterC3#3212
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
1. #5716C2semidirect
2. #5580C2semidirect
3. C3×#460C2semidirect
4. #5173C2semidirect
5. #5168C2semidirect
6. #5072C2semidirect
7. #5031C2semidirect
8. #4841C2semidirect
9. #4833C2semidirect
10. #4781C2semidirect
11. #4768C2semidirect
50. #460D3semidirect
282. C3#3212semidirect

Maximal subgroups

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

subgroupindex
#57162
#55802
C3×#4602
#51732
#51682
#50722
#50312
#48412
#48332
#47812
#47682
#32123