Finite Group Databasebeta

← #82445

Subgroups — #82445

Order 384.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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×C3×C4×Q8C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic
3#5228C211yesmaximal, characteristic, complemented
4#5227C211yesmaximal, characteristic
5#5030C212yesmaximal, complemented
6#5028C212yesmaximal, complemented
7#5026C212yesmaximal, complemented
8#5022C212yesmaximal, complemented
9#4970C211yesmaximal, characteristic, complemented
10#4968C211yesmaximal, characteristic
11#4926C211yesmaximal, characteristic, complemented
12#3117—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C3×C4×Q8—21no—
15C3×C4×Q8—21no—
16C3×C4×Q8C2212yes—
17C3×C4×Q8C2212yes—
18C2×C3×C4⋊C4—21no—
19C2×C3×C4⋊C4C2211yescharacteristic
20C2×C3×C42—21no—
21C2×C3×C42C2211yescharacteristic
22C2×#216C2212yes—
23C2×#215C2212yes—
24#930C2211yescharacteristic
25#929—21no—
26C4×#67—21no—
27#924C2211yescharacteristic
28#876—21no—
29#875C2211yescharacteristic
30C4×D12C2211yescharacteristic
31#872—21no—
32#870C2211yescharacteristic
33C4×Dic6C2211yescharacteristic
34#835C2212yes—
35#835C2212yes—
36#830C2211yescharacteristic
37#830C2211yescharacteristic
38#810C2212yes—
39#809C2212yes—
40#808C2212yes—
41#807C2212yes—
42C4⋊(C3⋊C8)C2212yes—
43#803C2212yes—
44#529—31no—
45C2×C4×Q8D311yescharacteristic, complemented
46#484—31no—
47#481—31no—
48#479—32no—
49#464—32no—
50#451—31no—
51#448—31no—
52#441—32no—
53#440—32no—
54#432—31no—
55C2×C3×Q8—24no—
56C2×C3×Q8—21no—
57C2×C3×Q8C2312yes—
58C22×C3×C4—21no—
59C22×C3×C4—21no—
60C22×C3×C4D411yescharacteristic
61C22×C3×C4D411yescharacteristic
62C22×C3×C4C2311yescharacteristic
63C2×#67—21no—
64C2×D12C2311yescharacteristic
65C2×C4×D3—21no—
66C2×Dic6C2311yescharacteristic
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C42—22no—
72C3×C42—22no—
73C3×C42D411yescharacteristic
74C3×C42D411yescharacteristic
75C3×C42C2311yescharacteristic
76C3×C42C2311yescharacteristic
77#217—21no—
78#216—24no—
79#215—24no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—21no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
93C22×Q8C2×D311yescharacteristic
94C2×Q16—32no—
95C2×SD16—32no—
96#126—61no—
97#125—31no—
98#124—31no—
99#123—61no—
100C22⋊Q8—31no—
101#121—31no—
102C4×Q8—21no—
103C4×Q8—21no—
104C4×Q8—31no—
105C4×Q8C2×D312yes—
106C4×Q8C2×D312yes—
107C4×D4—31no—
108C4×D4—61no—
109#117—61no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4C2×D311yescharacteristic
112C2×C42—21no—
113C2×C42C2×D311yescharacteristic
114#107—32no—
115#106—32no—
116C4⋊C8—32no—
117#103—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121C22⋊C8—31no—
122C22⋊C8—31no—
123#97—32no—
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C3×Q8—24no—
127C3×Q8—22no—
128C3×Q8#4114yes—
129C2×C3×C4—24no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—41no—
135C2×C3×C4—42no—
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2411yescharacteristic
142#67—42no—
143C2×Dic3—21no—
144C2×Dic3—21no—
145C2×Dic3—21no—
146D12—22no—
147C4×D3—42no—
148Dic6—22no—
149C3⋊C8—22no—
150C3⋊C8—22no—
151C2×Q8—24no—
152C2×Q8—21no—
153C2×Q8—31no—
154C2×Q8C22×D312yes—
155C2×D4—31no—
156C2×D4—61no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—61no—
160C22×C4#6711yescharacteristic
161C22×C4#6711yescharacteristic
162C22×C4C22×D311yescharacteristic
163Q16—64no—
164SD16—64no—
165C2×C8—32no—
166C2×C8—32no—
167C4⋊C4—22no—
168C4⋊C4—22no—
169C4⋊C4—21no—
170C4⋊C4—31no—
171C4⋊C4—31no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4—61no—
175C4⋊C4—61no—
176C4⋊C4C22×D312yes—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C22⋊C4—61no—
182C42—22no—
183C42—22no—
184C42—61no—
185C42#6711yescharacteristic
186C42#6711yescharacteristic
187C42C22×D311yescharacteristic
188C42C22×D311yescharacteristic
189C22×C3—21no—
190C22×C3—21no—
191C22×C3#13712yes—
192C22×C3C22×D411yescharacteristic
193C2×D3—41no—
194C2×D3—41no—
195C2×D3—41no—
196C3×C4—24no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—21no—
200C3×C4—21no—
201C3×C4—21no—
202C3×C4—42no—
203C3×C4C2×#4112yes—
204Dic3—41no—
205Dic3—41no—
206Dic3—41no—
207C23—61no—
208C23C2×#6711yescharacteristic
209Q8—24no—
210Q8—22no—
211Q8—62no—
212Q8#23514yes—
213D4—62no—
214D4—122no—
215C2×C4—24no—
216C2×C4—22no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—41no—
221C2×C4—42no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—122no—
226C2×C4C2×#6711yescharacteristic
227C2×C4C2×#6711yescharacteristic
228C2×C4C2×#6711yescharacteristic
229C2×C4C2×#6711yescharacteristic
230C2×C4C2×#6711yescharacteristic
231C2×C4C23×D311yescharacteristic
232C8—62no—
233C8—62no—
234C2×C3—22no—
235C2×C3#52511yescharacteristic
236C2×C3C2×#13711yescharacteristic
237C2×C3#57511yescharacteristic
238D3—81no—
239C22—21no—
240C22—21no—
241C22—121no—
242C22—121no—
243C22—121no—
244C22#94212yes—
245C22C22×#6711yescharacteristic
246C4—24no—
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—42no—
253C4—121no—
254C4—121no—
255C4—121no—
256C4C2×#23512yes—
257C3#311711yescharacteristic, complemented, Sylow
258C2—22no—
259C2—241no—
260C2#579111yescharacteristic
261C2C2×#94211yescharacteristic
262C2#582211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×C4×Q8C2
socleC22×C3C22×D4
prime radical#930C22
hypercenterC2×C4×Q8D3
hypocenterC3#3117
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×Q8C2semidirect
3. #5228C2semidirect
5. #5030C2semidirect
6. #5028C2semidirect
7. #5026C2semidirect
8. #5022C2semidirect
9. #4970C2semidirect
11. #4926C2semidirect
45. C2×C4×Q8D3semidirect
257. C3#3117semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Q82
#54792
#52282
#52272
#50302
#50282
#50262
#50222
#49702
#49682
#49262
#31173