Finite Group Databasebeta

← #78692

Subgroups — #78692

Order 384.

716 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 145 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#5964C211yesmaximal, characteristic
2#5825C211yesmaximal, characteristic, complemented
3#5526C212yesmaximal, complemented
4C3×#410C211yesmaximal, characteristic, complemented
5#5006C212yesmaximal, complemented
6#4742C218yesmaximal, complemented
7#3028—31nomaximal, Sylow
8C2×C3×#41C2211yescharacteristic
9#1010—22no—
10#1009—22no—
11D3×#41—24no—
12#1007—21no—
13#1004—21no—
14#1004C2214yes—
15C2×#235—22no—
16C2×#235C2212yes—
17C22×Dic6C2211yescharacteristic, complemented
18C3×#117C2212yescomplemented
19#944—21no—
20#936—21no—
21#935C2214yescomplemented
22#926C2211yescharacteristic, complemented
23#924—22no—
24#897—24no—
25#886C2218yescomplemented
26#875—22no—
27#869—22no—
28C3×#99C2214yescomplemented
29#806C2218yescomplemented
30#585—31no—
31#536—31no—
32#536—32no—
33#458—38no—
34#410—32no—
35#410D311yescharacteristic, complemented
36C3×#41—22no—
37C3×#41—22no—
38C2×C3×Q8D411yescharacteristic
39C2×C3×D4D411yescharacteristic
40C2×C3×D4C2312yes—
41C22×C3×C4D412yes—
42C22×C3×C4C2311yescharacteristic
43C2×#67—24no—
44C2×#67C2314yes—
45C22×Dic3C2312yes—
46#239—24no—
47D3×Q8—24no—
48#237—24no—
49#237—28no—
50D3×D4—24no—
51D3×D4—28no—
52#235—24no—
53#235—24no—
54#235—22no—
55#235—24no—
56#235—22no—
57#235—22no—
58#235—22no—
59C2×D12—22no—
60C2×D12D412yes—
61C2×C4×D3—24no—
62C2×C4×D3D414yes—
63C2×Dic6—22no—
64C2×Dic6—22no—
65C2×Dic6—22no—
66C2×Dic6—21no—
67C2×Dic6—21no—
68C2×Dic6D412yes—
69C3×C4⋊C4—22no—
70C3×C22⋊C4C2314yes—
71C3×C42—22no—
72#217—24no—
73#217C2312yes—
74(C2×D3)⋊C4—28no—
75C4⋊Dic3—21no—
76C4⋊Dic3—28no—
77Dic3⋊C4—24no—
78C4×Dic3—21no—
792−1+4—62no—
802−1+4—61no—
81H(5,𝔽2)—34no—
82H(5,𝔽2)—62no—
83H(5,𝔽2)—61no—
84C2×#41—32no—
85C2×#41—64no—
86C2×#41—62no—
87C2×#41C2×D311yescharacteristic
88C22×Q8—31no—
89C4⋊Q8—62no—
90C4⋊Q8—61no—
91#124—62no—
92#124—61no—
93#123—34no—
94#123—38no—
95C22⋊Q8—64no—
96C22⋊Q8—62no—
97#117—31no—
98#117C2×D312yescomplemented
99#99—38no—
100#99C2×D314yescomplemented
101C23×C3C2×D412yes—
102C23×C3C2411yescharacteristic
103C22×D3—24no—
104C22×D3C2×D414yes—
105C3×Q8—22no—
106C3×D4—24no—
107C3×D4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—22no—
110C2×C3×C4—24no—
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2×D411yescharacteristic
113C2×C3×C4C2×D411yescharacteristic
114C2×C3×C4C2×D412yes—
115C2×C3×C4C2×D412yes—
116#67—28no—
117#67—24no—
118#67—24no—
119C2×Dic3—24no—
120C2×Dic3—22no—
121C2×Dic3—22no—
122C2×Dic3—24no—
123C2×Dic3C2×D414yes—
124D12—24no—
125D12—22no—
126D12—22no—
127C4×D3—28no—
128C4×D3—24no—
129C4×D3—24no—
130Dic6—24no—
131Dic6—22no—
132Dic6—22no—
133Dic6—41no—
134Dic6—42no—
135Dic6—41no—
136#41—22no—
137#41—22no—
138#41—64no—
139#41—64no—
140#41—68no—
141#41—64no—
142#41—64no—
143#41—62no—
144#41—62no—
145#41—64no—
146#41—62no—
147#41—62no—
148C2×Q8—32no—
149C2×Q8—62no—
150C2×Q8—64no—
151C2×Q8—61no—
152C2×Q8—61no—
153C2×Q8—62no—
154C2×Q8—62no—
155C2×Q8D1211yescharacteristic
156C2×D4—34no—
157C2×D4—32no—
158C2×D4—62no—
159C2×D4—68no—
160C2×D4—64no—
161C2×D4—64no—
162C2×D4D1211yescharacteristic
163C2×D4C22×D312yes—
164C22×C4—34no—
165C22×C4—32no—
166C22×C4—64no—
167C22×C4D1212yes—
168C22×C4C22×D311yescharacteristic
169C4⋊C4—22no—
170C4⋊C4—68no—
171C4⋊C4—61no—
172C4⋊C4—64no—
173C22⋊C4—32no—
174C22⋊C4—68no—
175C22⋊C4—64no—
176C22⋊C4C22×D314yes—
177C42—22no—
178C42—61no—
179C22×C3—22no—
180C22×C3—41no—
181C22×C3—42no—
182C22×C3C22×D411yescharacteristic
183C22×C3C22×D412yes—
184C2×D3—24no—
185C2×D3—44no—
186C2×D3—84no—
187C3×C4—22no—
188C3×C4—44no—
189C3×C4#12011yescharacteristic
190C3×C4#12012yes—
191C3×C4#12011yescharacteristic
192Dic3—24no—
193Dic3—42no—
194Dic3—42no—
195C23—34no—
196C23—64no—
197C23C2×D1212yes—
198C23C23×D311yescharacteristic
199Q8—22no—
200Q8—64no—
201Q8—62no—
202Q8—62no—
203Q8—121no—
204Q8—121no—
205Q8—122no—
206D4—24no—
207D4—22no—
208D4—64no—
209D4—68no—
210D4—64no—
211D4—64no—
212D4—62no—
213D4—62no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—24no—
217C2×C4—34no—
218C2×C4—68no—
219C2×C4—64no—
220C2×C4—64no—
221C2×C4—64no—
222C2×C4—62no—
223C2×C4—62no—
224C2×C4—64no—
225C2×C4C2×D1212yes—
226C2×C4C2×D1211yescharacteristic
227C2×C4C2×D1211yescharacteristic
228C2×C4D3×D412yes—
229C2×C4D3×D412yes—
230C2×C3—21no—
231C2×C3—22no—
232C2×C3—42no—
233C2×C3C2×#12011yescharacteristic
234D3—44no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—64no—
239C22—124no—
240C22—244no—
241C22C22×D1211yescharacteristic
242C22C2×D3×D412yes—
243C4—22no—
244C4—44no—
245C4—64no—
246C4—122no—
247C4—122no—
248C4#88211yescharacteristic
249C4#88211yescharacteristic
250C4#88212yes—
251C3#302811yescharacteristic, complemented, Sylow
252C2—22no—
253C2—21no—
254C2—42no—
255C2—124no—
256C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC23×C3C24
FrattiniC23C23×D3
FittingC3×#410C2
socleC2×C3C2×#120
prime radical#5964C2
hypercenter#410D3
hypocenterC3#3028
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #5825C2semidirect
3. #5526C2semidirect
4. C3×#410C2semidirect
5. #5006C2semidirect
6. #4742C2semidirect
17. C22×Dic6C22semidirect
18. C3×#117C22semidirect
21. #935C22semidirect
22. #926C22semidirect
25. #886C22semidirect
28. C3×#99C22semidirect
29. #806C22semidirect
35. #410D3semidirect
98. #117C2×D3semidirect
100. #99C2×D3semidirect
251. C3#3028semidirect

Maximal subgroups

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

subgroupindex
#59642
#58252
#55262
C3×#4102
#50062
#47422
#30283