Finite Group Databasebeta

← #83394

Subgroups — #83394

Order 384.

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

normal subgroups only

This group has 762 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
1C2×#1004C211yesmaximal, characteristic, complemented
2C3×#535C211yesmaximal, characteristic, complemented
3#5822C212yesmaximal, complemented
4C2×#833C211yesmaximal, characteristic, complemented
5#5077C214yesmaximal, complemented
6#5061C214yesmaximal, complemented
7#5005C212yesmaximal, complemented
8#3019—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic, complemented
10C22×C3×D4C2211yescharacteristic
11C22×#67—22no—
12#1004—22no—
13#1004—24no—
14#1004C2214yes—
15C2×#237—22no—
16C2×D3×D4—22no—
17C2×#235C2212yes—
18C3×#127C2212yescomplemented
19C3×#124C2212yescomplemented
20C3×#121—22no—
21C3×#120—22no—
22C3×#117C2211yescharacteristic, complemented
23#951—22no—
24#950—24no—
25#949—22no—
26#942C2214yescomplemented
27#932C2214yescomplemented
28C2×#208C2212yescomplemented
29#833C2214yescomplemented
30#805C2218yescomplemented
31C2×H(5,𝔽2)—31no—
32#575—32no—
33#535D311yescharacteristic, complemented
34#455—34no—
35#453—34no—
36#421—32no—
37C2×#100—31no—
38C24×C3D412yes—
39C23×D3—22no—
40C3×#41—22no—
41C3×#41—22no—
42C2×C3×Q8C2311yescharacteristic, complemented
43C2×C3×D4—21no—
44C2×C3×D4—24no—
45C2×C3×D4—22no—
46C2×C3×D4—22no—
47C2×C3×D4D412yes—
48C2×C3×D4C2312yes—
49C2×C3×D4C2311yescharacteristic, complemented
50C22×C3×C4—21no—
51C22×C3×C4C2311yescharacteristic
52C2×#67—24no—
53C2×#67—28no—
54C2×#67—22no—
55C2×#67—28no—
56C22×Dic3—22no—
57#237—28no—
58#237—28no—
59D3×D4—28no—
60D3×D4—28no—
61#235—22no—
62#235—22no—
63#235D414yescomplemented
64#235C2314yes—
65C2×D12D412yescomplemented
66C2×C4×D3—22no—
67C2×Dic6D412yescomplemented
68C3×C4⋊C4—21no—
69C3×C22⋊C4—21no—
70C3×C22⋊C4—24no—
71C3×C42C2312yescomplemented
72#216—24no—
73#215—24no—
74#214—24no—
75#213—24no—
76#208—22no—
77#208C2314yescomplemented
78C2×C3⋊C8—22no—
79H(5,𝔽2)—34no—
80H(5,𝔽2)—64no—
81H(5,𝔽2)—62no—
82C2×#41—32no—
83C2×#41—62no—
84C2×#41C2×D311yescharacteristic, complemented
85C22×D4—62no—
86C22×D4—62no—
87C22×D4C2×D311yescharacteristic
88#137—34no—
89#137—62no—
90AGL(1,ℤ/8)—34no—
91AGL(1,ℤ/8)—62no—
92#135—64no—
93C2×M16—32no—
94#127C2×D312yescomplemented
95#124C2×D312yescomplemented
96#121—22no—
97#120—22no—
98#117C2×D311yescharacteristic, complemented
99#104—38no—
100#100—34no—
101C23×C3—22no—
102C23×C3—22no—
103C23×C3—21no—
104C23×C3—82no—
105C23×C3C2×D414yes—
106C23×C3C2×D411yescharacteristic
107C22×D3—22no—
108C22×D3—28no—
109C22×D3—84no—
110C3×Q8—22no—
111C3×D4—24no—
112C3×D4—24no—
113C3×D4—22no—
114C3×D4—41no—
115C3×D4—41no—
116C3×D4—84no—
117C2×C3×C4—22no—
118C2×C3×C4—21no—
119C2×C3×C4—42no—
120C2×C3×C4C2×D412yes—
121C2×C3×C4C2×D412yes—
122C2×C3×C4C2×D411yescharacteristic
123C2×C3×C4C2411yescharacteristic
124#67—216no—
125#67—24no—
126#67—24no—
127C2×Dic3—28no—
128C2×Dic3—22no—
129D12—22no—
130D12C2×D414yes—
131C4×D3—24no—
132C4×D3—24no—
133Dic6—22no—
134Dic6C2×D414yes—
135C3⋊C8—24no—
136C24—62no—
137C24#6712yes—
138#41—22no—
139#41—22no—
140#41—34no—
141#41—34no—
142#41—68no—
143#41—68no—
144#41—62no—
145#41—62no—
146C2×Q8—32no—
147C2×Q8C22×D311yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—24no—
150C2×D4—22no—
151C2×D4—22no—
152C2×D4—32no—
153C2×D4—68no—
154C2×D4—68no—
155C2×D4—64no—
156C2×D4—68no—
157C2×D4—62no—
158C2×D4—68no—
159C2×D4#6712yes—
160C2×D4C22×D312yes—
161C2×D4C22×D311yescharacteristic, complemented
162C22×C4—21no—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4C22×D311yescharacteristic
166Q16—64no—
167SD16—64no—
168SD16—64no—
169D8—64no—
170M16—34no—
171M16—62no—
172C2×C8—62no—
173C4⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—24no—
176C42C22×D312yescomplemented
177C22×C3—24no—
178C22×C3—41no—
179C22×C3—44no—
180C22×C3—44no—
181C22×C3—42no—
182C22×C3—41no—
183C22×C3—82no—
184C22×C3#12011yescharacteristic
185C22×C3#12011yescharacteristic
186C22×C3C22×D411yescharacteristic
187C2×D3—24no—
188C2×D3—44no—
189C2×D3—88no—
190C3×C4—42no—
191C3×C4—41no—
192C3×C4#12012yes—
193C3×C4C22×D412yes—
194Dic3—24no—
195C23—22no—
196C23—22no—
197C23—21no—
198C23—68no—
199C23—62no—
200C23—82no—
201C23—244no—
202C23C2×#6714yes—
203C23C2×#6711yescharacteristic
204Q8—22no—
205Q8—34no—
206Q8—62no—
207D4—24no—
208D4—24no—
209D4—22no—
210D4—34no—
211D4—41no—
212D4—41no—
213D4—616no—
214D4—64no—
215D4—64no—
216D4—62no—
217D4—84no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—42no—
221C2×C4—68no—
222C2×C4—62no—
223C2×C4—64no—
224C2×C4—64no—
225C2×C4D3×D412yes—
226C2×C4D3×D412yes—
227C2×C4C2×#6711yescharacteristic
228C2×C4C23×D311yescharacteristic
229C8—64no—
230C2×C3—21no—
231C2×C3—21no—
232C2×C3—21no—
233C2×C3—44no—
234C2×C3—81no—
235C2×C3C2×#12011yescharacteristic
236D3—44no—
237C22—24no—
238C22—41no—
239C22—44no—
240C22—44no—
241C22—42no—
242C22—41no—
243C22—64no—
244C22—82no—
245C22—124no—
246C22—248no—
247C22#93711yescharacteristic
248C22#93711yescharacteristic
249C22C22×#6711yescharacteristic
250C4—42no—
251C4—41no—
252C4—64no—
253C4#93712yes—
254C4C2×D3×D412yes—
255C3#301911yescharacteristic, complemented, Sylow
256C2—21no—
257C2—21no—
258C2—21no—
259C2—44no—
260C2—81no—
261C2—124no—
262C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC2×C3C2×#120
hypercenter#535D3
hypocenterC3#3019
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#1004C2semidirect
2. C3×#535C2semidirect
3. #5822C2semidirect
4. C2×#833C2semidirect
5. #5077C2semidirect
6. #5061C2semidirect
7. #5005C2semidirect
9. C2×C3×#41C22semidirect
18. C3×#127C22semidirect
19. C3×#124C22semidirect
22. C3×#117C22semidirect
26. #942C22semidirect
27. #932C22semidirect
28. C2×#208C22semidirect
29. #833C22semidirect
30. #805C22semidirect
33. #535D3semidirect
42. C2×C3×Q8C23semidirect
49. C2×C3×D4C23semidirect
63. #235D4semidirect
65. C2×D12D4semidirect
67. C2×Dic6D4semidirect
71. C3×C42C23semidirect
77. #208C23semidirect
84. C2×#41C2×D3semidirect
94. #127C2×D3semidirect
95. #124C2×D3semidirect
98. #117C2×D3semidirect
147. C2×Q8C22×D3semidirect
161. C2×D4C22×D3semidirect
176. C42C22×D3semidirect
255. C3#3019semidirect

Maximal subgroups

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

subgroupindex
C2×#10042
C3×#5352
#58222
C2×#8332
#50772
#50612
#50052
#30193