Finite Group Databasebeta

← #80498

Subgroups — #80498

Order 384.

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

normal subgroups only

This group has 348 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#5728C211yesmaximal, characteristic, complemented
2#5580C211yesmaximal, characteristic
3C3×#462C211yesmaximal, characteristic, complemented
4#5189C212yesmaximal, complemented
5#5185C212yesmaximal, complemented
6#5083C211yesmaximal, characteristic, complemented
7#5021C211yesmaximal, characteristic, complemented
8#4842C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4812C212yesmaximal
11#4799C212yesmaximal, complemented
12#3252—31nomaximal, Sylow
13C2×C3×Q16C2212yes—
14C3×C4⋊Q8C2211yescharacteristic
15C3×C4×Q8C2211yescharacteristic
16#947C2211yescharacteristic
17#946C2211yescharacteristic
18#946C2212yes—
19Q8×Dic3C2212yes—
20C2×#215C2212yes—
21C2×ΓL(1,𝔽25)C2212yes—
22#898—21no—
23#897—21no—
24#895C2211yescharacteristic
25#894—21no—
26#892—21no—
27#890—21no—
28#890C2211yescharacteristic
29#887—21no—
30C4×D12—21no—
31C4×D12C2211yescharacteristic
32#872—21no—
33C42×D3C2211yescharacteristic
34#869C2211yescharacteristic
35C3×C4⋊C8C2211yescharacteristic
36C3×#103C2212yes—
37#835C2212yes—
38#835C2212yes—
39#821C2212yes—
40#817C2212yes—
41Dic3⋊C8C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#533—31no—
45#529—31no—
46#479—32no—
47#477—31no—
48#476—31no—
49#470—32no—
50#468—32no—
51#464—32no—
52#462D311yescharacteristic, complemented
53#461—31no—
54#423—31no—
55C2×C3×Q8C2311yescharacteristic
56C2×C3×Q8C2312yes—
57C2×D12—21no—
58C2×D12C2311yescharacteristic
59C2×C4×D3—21no—
60C2×C4×D3D411yescharacteristic
61C2×C4×D3D411yescharacteristic
62C2×C4×D3C2311yescharacteristic
63C2×Dic6—21no—
64C3×Q16—42no—
65C2×C3×C8C2312yes—
66C3×C4⋊C4—21no—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—21no—
69C3×C4⋊C4C2311yescharacteristic
70C3×C42—21no—
71C3×C42C2311yescharacteristic
72#215—42no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—21no—
78(C2×D3)⋊C4—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3—21no—
81C4⋊Dic3—22no—
82C4⋊Dic3C2312yes—
83C4⋊Dic3C2311yescharacteristic
84Dic3⋊C4—21no—
85Dic3⋊C4—21no—
86Dic3⋊C4—22no—
87C4×Dic3—22no—
88C4×Dic3D411yescharacteristic
89C4×Dic3D411yescharacteristic
90C4×Dic3C2311yescharacteristic
91C2×C3⋊C8C2312yes—
92ΓL(1,𝔽25)—42no—
93C2×Q16C2×D312yes—
94C2×SD16—32no—
95C2×M16—32no—
96C4⋊Q8—31no—
97C4⋊Q8C2×D311yescharacteristic
98#126—61no—
99#125—31no—
100#125—61no—
101#124—31no—
102#123—61no—
103C22⋊Q8—31no—
104C22⋊Q8—32no—
105C22⋊Q8—61no—
106#121—31no—
107C4×Q8—32no—
108C4×Q8—61no—
109C4×Q8C2×D311yescharacteristic
110C4×D4—31no—
111C4×D4—61no—
112#117—61no—
113#117—61no—
114C2×C42—31no—
115#107—32no—
116#106—32no—
117C4⋊C8—32no—
118C4⋊C8—31no—
119C4⋊C8C2×D311yescharacteristic
120#103—32no—
121#103—32no—
122#103C2×D312yes—
123#102—32no—
124C22×D3—21no—
125C22×D3C2×D411yescharacteristic
126C3×Q8—22no—
127C3×Q8—21no—
128C3×Q8—21no—
129C3×Q8—22no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136C2×Dic3—22no—
137C2×Dic3—21no—
138C2×Dic3C2×D411yescharacteristic
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141D12—21no—
142D12—21no—
143D12—42no—
144C4×D3—22no—
145C4×D3—21no—
146C4×D3—21no—
147C4×D3—41no—
148C4×D3—42no—
149Dic6—42no—
150C3×C8—22no—
151C3⋊C8—22no—
152C2×Q8—61no—
153C2×Q8C22×D311yescharacteristic
154C2×Q8C22×D312yes—
155C2×D4—31no—
156C2×D4—61no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—61no—
161Q16—42no—
162SD16—122no—
163M16—122no—
164C2×C8—32no—
165C2×C8C22×D312yes—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—32no—
170C4⋊C4—31no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—62no—
175C4⋊C4—62no—
176C4⋊C4—61no—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—62no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C42—21no—
185C42—31no—
186C42—31no—
187C42—31no—
188C42—62no—
189C42C22×D311yescharacteristic
190C22×C3C22×D411yescharacteristic
191C2×D3—21no—
192C2×D3—21no—
193C2×D3—21no—
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—41no—
197C3×C4—21no—
198C3×C4—42no—
199C3×C4—41no—
200C3×C4—41no—
201C3×C4#13712yes—
202C3×C4H(5,𝔽2)11yescharacteristic
203C3×C42−1+411yescharacteristic
204Dic3—21no—
205Dic3—21no—
206Dic3—21no—
207Dic3—41no—
208Dic3—42no—
209C23—31no—
210C23—61no—
211Q8—22no—
212Q8—21no—
213Q8—21no—
214Q8—22no—
215Q8—122no—
216D4—61no—
217D4—61no—
218D4—122no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—31no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—62no—
229C2×C4—61no—
230C2×C4—121no—
231C2×C4—122no—
232C2×C4D3×D411yescharacteristic
233C2×C4D3×D411yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—22no—
236C8—62no—
237C2×C3#53711yescharacteristic
238C2×C3C2×#13711yescharacteristic
239C2×C3#57511yescharacteristic
240D3—41no—
241D3—81no—
242C22—61no—
243C22—61no—
244C22—61no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—41no—
252C4—42no—
253C4—61no—
254C4—61no—
255C4—61no—
256C4—121no—
257C4—122no—
258C4#91812yes—
259C4#100411yescharacteristic
260C4#101011yescharacteristic
261C3#325211yescharacteristic, complemented, Sylow
262C2—121no—
263C2—241no—
264C2#560811yescharacteristic
265C2C2×#91811yescharacteristic
266C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#462C2
socleC22×C3C22×D4
prime radical#895C22
hypercenter#462D3
hypocenterC3#3252
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#462C2semidirect
4. #5189C2semidirect
5. #5185C2semidirect
6. #5083C2semidirect
7. #5021C2semidirect
8. #4842C2semidirect
9. #4837C2semidirect
11. #4799C2semidirect
52. #462D3semidirect
261. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#57282
#55802
C3×#4622
#51892
#51852
#50832
#50212
#48422
#48372
#48122
#47992
#32523