Finite Group Databasebeta

← #83346

Subgroups — #83346

Order 384.

366 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C3×#533C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic
3#5089C211yesmaximal, characteristic, complemented
4#5086C211yesmaximal, characteristic, complemented
5#5082C211yesmaximal, characteristic, complemented
6#5081C211yesmaximal, characteristic, complemented
7#5051C212yesmaximal, complemented
8#5048C212yesmaximal
9#5030C212yesmaximal, complemented
10#5026C212yesmaximal, complemented
11C2×#803C211yesmaximal, characteristic, complemented
12#3181—31nomaximal, Sylow
13C3×C4⋊Q8C2211yescharacteristic
14C3×C4⋊Q8C2211yescharacteristic
15C3×#125—21no—
16C3×C22⋊Q8—21no—
17C3×C22⋊Q8C2212yes—
18C3×C4×Q8—21no—
19C3×C4×Q8C2212yes—
20C3×#117—21no—
21C2×C3×C42C2211yescharacteristic
22C2×#216C2212yes—
23C2×#215C2212yes—
24#930C2211yescharacteristic
25#929—21no—
26C4×#67—21no—
27#924C2211yescharacteristic
28C22×C3⋊C8C2212yes—
29#876—21no—
30#875C2211yescharacteristic
31C4×D12C2211yescharacteristic
32#872—21no—
33#870C2211yescharacteristic
34C4×Dic6C2211yescharacteristic
35#835C2212yes—
36#835C2212yes—
37#810C2212yes—
38#809C2212yes—
39#808C2212yes—
40#807C2212yes—
41#803C2212yes—
42#803C2211yescharacteristic
43#803C2211yescharacteristic
44#533D311yescharacteristic, complemented
45#529—31no—
46#501—31no—
47#497—31no—
48#491—31no—
49#489—31no—
50#467—32no—
51#465—32no—
52#441—32no—
53#440—32no—
54C2×#97—31no—
55C2×C3×Q8—21no—
56C2×C3×Q8C2312yes—
57C22×C3×C4D411yescharacteristic
58C22×C3×C4D411yescharacteristic
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61C2×D12C2311yescharacteristic
62C2×C4×D3—21no—
63C2×Dic6C2311yescharacteristic
64C3×C4⋊C4—21no—
65C3×C4⋊C4—21no—
66C3×C4⋊C4—21no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4—22no—
69C3×C4⋊C4C2312yes—
70C3×C22⋊C4—22no—
71C3×C42—22no—
72C3×C42D411yescharacteristic
73C3×C42D411yescharacteristic
74C3×C42C2311yescharacteristic
75C3×C42C2311yescharacteristic
76#217—21no—
77#216—24no—
78#215—24no—
79(C2×D3)⋊C4—21no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83C4⋊Dic3C2311yescharacteristic
84C4⋊Dic3C2311yescharacteristic
85Dic3⋊C4—21no—
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89C4×Dic3—21no—
90C2×C3⋊C8—24no—
91C2×C3⋊C8C2312yes—
92C2×C3⋊C8C2312yes—
93C2×Q16—32no—
94C2×SD16—32no—
95C22×C8—32no—
96C4⋊Q8C2×D311yescharacteristic
97C4⋊Q8C2×D311yescharacteristic
98#126—61no—
99#125—21no—
100#125—31no—
101#124—31no—
102#123—61no—
103C22⋊Q8—21no—
104C22⋊Q8—31no—
105C22⋊Q8C2×D312yes—
106#121—31no—
107C4×Q8—21no—
108C4×Q8—31no—
109C4×Q8C2×D312yes—
110C4×D4—31no—
111C4×D4—61no—
112#117—21no—
113#117—61no—
114C2×C42C2×D311yescharacteristic
115#107—32no—
116#106—32no—
117#103—32no—
118#103—32no—
119#103—32no—
120#102—32no—
121#97—32no—
122#97—31no—
123#97—31no—
124C23×C3C2×D411yescharacteristic
125C22×D3—21no—
126C3×Q8—22no—
127C3×Q8—22no—
128C3×Q8—42no—
129C2×C3×C4—22no—
130C2×C3×C4—22no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—41no—
134C2×C3×C4C2×D411yescharacteristic
135C2×C3×C4C2×D411yescharacteristic
136C2×C3×C4C2×D411yescharacteristic
137C2×C3×C4C2×D411yescharacteristic
138C2×C3×C4C2×D411yescharacteristic
139C2×C3×C4C2411yescharacteristic
140#67—42no—
141C2×Dic3—21no—
142C2×Dic3—21no—
143C2×Dic3—21no—
144D12—22no—
145C4×D3—42no—
146Dic6—22no—
147C3⋊C8—22no—
148C3⋊C8#4114yes—
149C2×Q8—21no—
150C2×Q8—31no—
151C2×Q8C22×D312yes—
152C2×D4—31no—
153C2×D4—61no—
154C22×C4—61no—
155C22×C4#6711yescharacteristic
156C22×C4#6711yescharacteristic
157C22×C4C22×D311yescharacteristic
158Q16—64no—
159SD16—64no—
160C2×C8—32no—
161C2×C8—32no—
162C2×C8—64no—
163C4⋊C4—21no—
164C4⋊C4—21no—
165C4⋊C4—21no—
166C4⋊C4—22no—
167C4⋊C4—22no—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—61no—
171C4⋊C4—61no—
172C4⋊C4—61no—
173C4⋊C4—61no—
174C4⋊C4C22×D312yes—
175C22⋊C4—22no—
176C22⋊C4—61no—
177C22⋊C4—61no—
178C22⋊C4—61no—
179C22⋊C4—61no—
180C22⋊C4—61no—
181C42—22no—
182C42—61no—
183C42#6711yescharacteristic
184C42#6711yescharacteristic
185C42C22×D311yescharacteristic
186C42C22×D311yescharacteristic
187C22×C3—21no—
188C22×C3—21no—
189C22×C3#13712yes—
190C22×C3C22×D411yescharacteristic
191C2×D3—41no—
192C2×D3—41no—
193C2×D3—41no—
194C3×C4—21no—
195C3×C4—21no—
196C3×C4—21no—
197C3×C4—21no—
198C3×C4—21no—
199C3×C4—42no—
200C3×C4—42no—
201C3×C4C2×#4112yes—
202Dic3—41no—
203Dic3—41no—
204Dic3—41no—
205C23—61no—
206C23C2×#6711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—62no—
211D4—62no—
212D4—122no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—61no—
219C2×C4—61no—
220C2×C4—61no—
221C2×C4—122no—
222C2×C4C2×#6711yescharacteristic
223C2×C4C2×#6711yescharacteristic
224C2×C4C2×#6711yescharacteristic
225C2×C4C2×#6711yescharacteristic
226C2×C4C2×#6711yescharacteristic
227C2×C4C23×D311yescharacteristic
228C8—34no—
229C8—62no—
230C2×C3—22no—
231C2×C3#53211yescharacteristic
232C2×C3C2×#13711yescharacteristic
233C2×C3#57511yescharacteristic
234D3—81no—
235C22—21no—
236C22—21no—
237C22—121no—
238C22—121no—
239C22—121no—
240C22#94212yes—
241C22C22×#6711yescharacteristic
242C4—21no—
243C4—21no—
244C4—21no—
245C4—21no—
246C4—21no—
247C4—42no—
248C4—42no—
249C4—121no—
250C4—121no—
251C4—121no—
252C4D3×#4112yes—
253C3#318111yescharacteristic, complemented, Sylow
254C2—22no—
255C2—241no—
256C2C2×#94211yescharacteristic
257C2#582211yescharacteristic
258C2#583211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#533C2
socleC22×C3C22×D4
prime radical#930C22
hypercenter#533D3
hypocenterC3#3181
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
3. #5089C2semidirect
4. #5086C2semidirect
5. #5082C2semidirect
6. #5081C2semidirect
7. #5051C2semidirect
9. #5030C2semidirect
10. #5026C2semidirect
11. C2×#803C2semidirect
44. #533D3semidirect
253. C3#3181semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#54792
#50892
#50862
#50822
#50812
#50512
#50482
#50302
#50262
C2×#8032
#31813