Finite Group Databasebeta

← #81072

Subgroups — #81072

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#5557C211yesmaximal, characteristic
3C3×#476C211yesmaximal, characteristic, complemented
4#5091C211yesmaximal, characteristic, complemented
5#5019C211yesmaximal, characteristic, complemented
6#4867C212yesmaximal, complemented
7#4857C212yesmaximal, complemented
8#4849C211yesmaximal, characteristic, complemented
9#4837C211yesmaximal, characteristic, complemented
10#4780C212yesmaximal
11#4767C212yesmaximal, complemented
12#3252—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4×D4C2211yescharacteristic
15#946—21no—
16Q8×Dic3—21no—
17#938C2211yescharacteristic
18#936C2211yescharacteristic
19C2×#214C2212yes—
20C4×#67—21no—
21#926—21no—
22C2×Dic12C2212yes—
23C2×ΓL(1,𝔽25)C2212yes—
24#897C2211yescharacteristic
25#897C2212yes—
26#892—21no—
27#890—21no—
28#890C2211yescharacteristic
29#887C2212yes—
30#883—21no—
31#879—21no—
32C42×D3C2211yescharacteristic
33#869C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35C3×#106C2212yes—
36C3×C4⋊C8C2211yescharacteristic
37C3×#102C2212yes—
38#816C2212yes—
39Dic3⋊C8C2212yes—
40#810C2212yes—
41#810C2212yes—
42#807C2212yes—
43C4⋊(C3⋊C8)C2211yescharacteristic
44#533—31no—
45#529—31no—
46#479—32no—
47#477—31no—
48#476D311yescharacteristic, complemented
49#470—32no—
50#468—32no—
51#464—32no—
52#462—31no—
53#461—31no—
54#423—31no—
55C2×C3×Q8—21no—
56C2×C3×D4C2311yescharacteristic
57C22×C3×C4—21no—
58C2×#67—21no—
59C2×C4×D3D411yescharacteristic
60C2×C4×D3D411yescharacteristic
61C2×C4×D3C2311yescharacteristic
62C2×Dic6C2311yescharacteristic
63C2×Dic6C2312yes—
64C2×C3×C8C2312yes—
65C3×C4⋊C4—21no—
66C3×C4⋊C4C2312yes—
67C3×C4⋊C4C2311yescharacteristic
68C3×C22⋊C4—21no—
69C3×C42C2311yescharacteristic
70#217—21no—
71#217—21no—
72#217—21no—
73#214—42no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3—22no—
81C4⋊Dic3C2311yescharacteristic
82Dic3⋊C4—21no—
83Dic3⋊C4—21no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C4×Dic3—22no—
87C4×Dic3D411yescharacteristic
88C4×Dic3D411yescharacteristic
89C4×Dic3C2311yescharacteristic
90C2×C3⋊C8C2312yes—
91Dic12—42no—
92ΓL(1,𝔽25)—42no—
93C2×Q16—32no—
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—31no—
109C4×Q8—61no—
110C4×D4—61no—
111C4×D4C2×D311yescharacteristic
112#117—61no—
113#117—61no—
114C2×C42—31no—
115#107—32no—
116#106C2×D312yes—
117C4⋊C8—32no—
118C4⋊C8—31no—
119C4⋊C8C2×D311yescharacteristic
120#103—32no—
121#103—32no—
122#103—32no—
123#102C2×D312yes—
124C23×C3—21no—
125C22×D3C2×D411yescharacteristic
126C3×Q8—42no—
127C3×D4—21no—
128C3×D4—21no—
129C2×C3×C4—22no—
130C2×C3×C4—21no—
131C2×C3×C4—42no—
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2411yescharacteristic
135#67—42no—
136C2×Dic3—21no—
137C2×Dic3—22no—
138C2×Dic3—21no—
139C2×Dic3C2×D411yescharacteristic
140C2×Dic3C2×D411yescharacteristic
141C2×Dic3C2×D411yescharacteristic
142C4×D3—22no—
143C4×D3—21no—
144C4×D3—21no—
145C4×D3—41no—
146Dic6—21no—
147Dic6—22no—
148Dic6—21no—
149Dic6—22no—
150C3×C8—22no—
151C3⋊C8—22no—
152C2×Q8—21no—
153C2×Q8—31no—
154C2×Q8—32no—
155C2×D4—61no—
156C2×D4C22×D311yescharacteristic
157C22×C4—21no—
158C22×C4—31no—
159C22×C4—31no—
160C22×C4—31no—
161Q16—122no—
162SD16—122no—
163M16—122no—
164C2×C8—32no—
165C2×C8C22×D312yes—
166C4⋊C4—21no—
167C4⋊C4—31no—
168C4⋊C4—61no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4C22×D312yes—
177C4⋊C4C22×D311yescharacteristic
178C22⋊C4—21no—
179C22⋊C4—62no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C42—31no—
185C42—31no—
186C42—31no—
187C42—62no—
188C42—61no—
189C42C22×D311yescharacteristic
190C22×C3—41no—
191C22×C3—41no—
192C22×C3—41no—
193C22×C3C22×D411yescharacteristic
194C2×D3—21no—
195C2×D3—21no—
196C2×D3—21no—
197C3×C4—21no—
198C3×C4—41no—
199C3×C4—42no—
200C3×C4#13712yes—
201C3×C4H(5,𝔽2)11yescharacteristic
202C3×C42−1+411yescharacteristic
203Dic3—21no—
204Dic3—21no—
205Dic3—21no—
206Dic3—41no—
207Dic3—42no—
208Dic3—41no—
209C23—21no—
210C23—31no—
211Q8—42no—
212Q8—61no—
213Q8—62no—
214Q8—62no—
215Q8—61no—
216D4—21no—
217D4—21no—
218D4—122no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—31no—
222C2×C4—31no—
223C2×C4—31no—
224C2×C4—42no—
225C2×C4—62no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—62no—
231C2×C4—121no—
232C2×C4D3×D411yescharacteristic
233C2×C4D3×D411yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—22no—
236C8—62no—
237C2×C3—81no—
238C2×C3#53711yescharacteristic
239C2×C3C2×#13711yescharacteristic
240C2×C3#57511yescharacteristic
241D3—41no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22—61no—
246C22—61no—
247C22—61no—
248C22C2×D3×D411yescharacteristic
249C4—21no—
250C4—41no—
251C4—42no—
252C4—61no—
253C4—61no—
254C4—61no—
255C4—121no—
256C4—121no—
257C4—122no—
258C4#91512yes—
259C4#100711yescharacteristic
260C4#100911yescharacteristic
261C3#325211yescharacteristic, complemented, Sylow
262C2—81no—
263C2—121no—
264C2#562911yescharacteristic
265C2C2×#91511yescharacteristic
266C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#476C2
socleC22×C3C22×D4
prime radical#938C22
hypercenter#476D3
hypocenterC3#3252
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
1. #5728C2semidirect
3. C3×#476C2semidirect
4. #5091C2semidirect
5. #5019C2semidirect
6. #4867C2semidirect
7. #4857C2semidirect
8. #4849C2semidirect
9. #4837C2semidirect
11. #4767C2semidirect
48. #476D3semidirect
261. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#57282
#55572
C3×#4762
#50912
#50192
#48672
#48572
#48492
#48372
#47802
#47672
#32523