Finite Group Databasebeta

← #79063

Subgroups — #79063

Order 384.

348 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 121 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#5479C211yesmaximal, characteristic
2#5470C211yesmaximal, characteristic, complemented
3C3×#423C211yesmaximal, characteristic, complemented
4#5137C212yesmaximal, complemented
5#5134C212yesmaximal
6#4850C211yesmaximal, characteristic, complemented
7#4849C211yesmaximal, characteristic, complemented
8#4845C212yesmaximal, complemented
9#4842C211yesmaximal, characteristic, complemented
10#4841C211yesmaximal, characteristic, complemented
11#4829C212yesmaximal, complemented
12#3252—31nomaximal, Sylow
13C2×C3×M16C2212yes—
14C2×C3×C42C2211yescharacteristic
15#930C2211yescharacteristic
16#929—21no—
17C4×#67—21no—
18#926—21no—
19#924—21no—
20#924C2212yes—
21#924C2211yescharacteristic
22C2×Dic12C2212yes—
23C2×#204C2212yes—
24#876—21no—
25#875C2211yescharacteristic
26C4×D12C2211yescharacteristic
27#872—21no—
28#870—21no—
29#870C2211yescharacteristic
30#869C2211yescharacteristic
31#869C2211yescharacteristic
32C4×Dic6—21no—
33C4×Dic6C2212yes—
34C4×Dic6C2211yescharacteristic
35C3×C4⋊C8C2212yes—
36C3×C4⋊C8C2211yescharacteristic
37C3×C4⋊C8C2211yescharacteristic
38#821C2212yes—
39#818C2212yes—
40#817C2212yes—
41#816C2212yes—
42#816C2212yes—
43#816C2212yes—
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#462—31no—
53#461—31no—
54#423D311yescharacteristic, complemented
55C22×C3×C4D411yescharacteristic
56C22×C3×C4D411yescharacteristic
57C22×C3×C4C2311yescharacteristic
58C2×#67—21no—
59C2×D12C2311yescharacteristic
60C2×C4×D3—21no—
61C2×Dic6—21no—
62C2×Dic6C2312yes—
63C2×Dic6C2311yescharacteristic
64C3×M16—42no—
65C2×C3×C8C2312yes—
66C2×C3×C8C2312yes—
67C3×C42D411yescharacteristic
68C3×C42D411yescharacteristic
69C3×C42C2311yescharacteristic
70C3×C42C2311yescharacteristic
71#217—22no—
72#217—21no—
73(C2×D3)⋊C4—21no—
74(C2×D3)⋊C4—21no—
75(C2×D3)⋊C4—21no—
76(C2×D3)⋊C4—21no—
77C4⋊Dic3—21no—
78C4⋊Dic3—21no—
79C4⋊Dic3—21no—
80C4⋊Dic3C2311yescharacteristic
81C4⋊Dic3C2311yescharacteristic
82C4⋊Dic3C2312yes—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89C4×Dic3—22no—
90C4×Dic3—21no—
91Dic12—42no—
92#204—42no—
93C2×Q16—32no—
94C2×SD16—32no—
95C2×M16C2×D312yes—
96C4⋊Q8—31no—
97C4⋊Q8—31no—
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—31no—
108C4×Q8—32no—
109C4×Q8—61no—
110C4×D4—31no—
111C4×D4—61no—
112#117—61no—
113#117—61no—
114C2×C42C2×D311yescharacteristic
115#107—32no—
116#106—32no—
117C4⋊C8C2×D311yescharacteristic
118C4⋊C8C2×D311yescharacteristic
119C4⋊C8C2×D312yes—
120#103—32no—
121#103—32no—
122#103—32no—
123#102—32no—
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C2×C3×C4—22no—
127C2×C3×C4—21no—
128C2×C3×C4—21no—
129C2×C3×C4—41no—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2×D411yescharacteristic
133C2×C3×C4C2×D411yescharacteristic
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2411yescharacteristic
136#67—42no—
137C2×Dic3—21no—
138C2×Dic3—22no—
139C2×Dic3—22no—
140C2×Dic3—21no—
141C2×Dic3—21no—
142D12—21no—
143D12—21no—
144C4×D3—42no—
145Dic6—22no—
146Dic6—21no—
147Dic6—22no—
148Dic6—21no—
149Dic6—42no—
150C3×C8—22no—
151C3×C8—22no—
152C2×Q8—31no—
153C2×Q8—32no—
154C2×Q8—61no—
155C2×D4—31no—
156C2×D4—61no—
157C22×C4—61no—
158C22×C4D1211yescharacteristic
159C22×C4D1211yescharacteristic
160C22×C4C22×D311yescharacteristic
161Q16—122no—
162SD16—122no—
163M16—42no—
164C2×C8C22×D312yes—
165C2×C8C22×D312yes—
166C4⋊C4—32no—
167C4⋊C4—31no—
168C4⋊C4—31no—
169C4⋊C4—61no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—62no—
177C4⋊C4—61no—
178C22⋊C4—62no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C22⋊C4—61no—
183C22⋊C4—61no—
184C42—61no—
185C42—62no—
186C42D1211yescharacteristic
187C42D1211yescharacteristic
188C42C22×D311yescharacteristic
189C42C22×D311yescharacteristic
190C22×C3—21no—
191C22×C3—21no—
192C22×C3—21no—
193C22×C3C22×D411yescharacteristic
194C2×D3—41no—
195C2×D3—41no—
196C2×D3—41no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4#13712yes—
202C3×C4H(5,𝔽2)11yescharacteristic
203C3×C42−1+411yescharacteristic
204Dic3—42no—
205Dic3—41no—
206Dic3—41no—
207Dic3—42no—
208Dic3—41no—
209C23—61no—
210C23C2×D1211yescharacteristic
211Q8—62no—
212Q8—61no—
213Q8—61no—
214Q8—62no—
215Q8—122no—
216D4—61no—
217D4—61no—
218D4—122no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—41no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—62no—
226C2×C4—62no—
227C2×C4—61no—
228C2×C4—122no—
229C2×C4C2×D1211yescharacteristic
230C2×C4C2×D1211yescharacteristic
231C2×C4C2×D1211yescharacteristic
232C2×C4C2×D1211yescharacteristic
233C2×C4C2×D1211yescharacteristic
234C2×C4C23×D311yescharacteristic
235C8—22no—
236C8—22no—
237C2×C3—41no—
238C2×C3#53711yescharacteristic
239C2×C3C2×#13711yescharacteristic
240C2×C3#57511yescharacteristic
241D3—81no—
242C22—21no—
243C22—21no—
244C22—21no—
245C22—121no—
246C22—121no—
247C22—121no—
248C22C22×D1211yescharacteristic
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—121no—
254C4—121no—
255C4—122no—
256C4—121no—
257C4—122no—
258C4#90912yes—
259C4#100411yescharacteristic
260C4#100711yescharacteristic
261C3#325211yescharacteristic, complemented, Sylow
262C2—41no—
263C2—241no—
264C2#551011yescharacteristic
265C2C2×#90911yescharacteristic
266C2#574811yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#423C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#423D3
hypocenterC3#3252
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
2. #5470C2semidirect
3. C3×#423C2semidirect
4. #5137C2semidirect
6. #4850C2semidirect
7. #4849C2semidirect
8. #4845C2semidirect
9. #4842C2semidirect
10. #4841C2semidirect
11. #4829C2semidirect
54. #423D3semidirect
261. C3#3252semidirect

Maximal subgroups

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

subgroupindex
#54792
#54702
C3×#4232
#51372
#51342
#48502
#48492
#48452
#48422
#48412
#48292
#32523