Finite Group Databasebeta

← #83409

Subgroups — #83409

Order 384.

388 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 117 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×#536C211yesmaximal, characteristic, complemented
2#5535C211yesmaximal, characteristic, complemented
3#5238C212yesmaximal, complemented
4#5227C212yesmaximal, complemented
5#5085C211yesmaximal, characteristic, complemented
6#5083C212yesmaximal, complemented
7#5082C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5057C212yesmaximal, complemented
10#5056C211yesmaximal, characteristic, complemented
11#5000C211yesmaximal, characteristic, complemented
12#3221—31nomaximal, Sylow
13C2×C3×#41C2211yescharacteristic
14C22×C3×Q8C2211yescharacteristic
15C3×C4⋊Q8C2211yescharacteristic
16C3×C4⋊Q8C2211yescharacteristic
17C3×#124C2211yescharacteristic
18C3×#124C2211yescharacteristic
19C3×#123—22no—
20C3×C22⋊Q8—22no—
21C3×#117C2211yescharacteristic
22C2×#215C2212yes—
23C2×#215C2212yes—
24C2×#215C2212yes—
25C2×#213C2212yes—
26#930C2212yes—
27#894—22no—
28#884—22no—
29#874C2211yescharacteristic
30C4×D12C2212yes—
31#870C2211yescharacteristic
32#835C2212yes—
33#835C2212yes—
34#835C2212yes—
35#832C2212yes—
36#830C2212yes—
37C4⋊(C3⋊C8)C2212yes—
38#803C2211yescharacteristic
39C4×C3⋊C8C2211yescharacteristic
40#540—31no—
41#536D311yescharacteristic, complemented
42#496—31no—
43#492—31no—
44#491—31no—
45#488—31no—
46#463—32no—
47#461—32no—
48#449—32no—
49#448—32no—
50#433—31no—
51C3×#41—42no—
52C2×C3×Q8—21no—
53C2×C3×Q8—42no—
54C2×C3×Q8C2311yescharacteristic
55C2×C3×Q8C2311yescharacteristic
56C2×C3×Q8C2311yescharacteristic
57C2×C3×D4—21no—
58C2×C3×D4C2311yescharacteristic
59C22×C3×C4—21no—
60C22×C3×C4—21no—
61C22×C3×C4C2311yescharacteristic
62C2×#67—22no—
63C2×D12—22no—
64C2×D12C2312yes—
65C2×C4×D3—22no—
66C3×C4⋊C4—22no—
67C3×C4⋊C4—22no—
68C3×C4⋊C4D412yes—
69C3×C22⋊C4—22no—
70C3×C22⋊C4—22no—
71C3×C22⋊C4D412yes—
72C3×C42C2311yescharacteristic
73C3×C42C2311yescharacteristic
74#215—42no—
75#215—42no—
76#215—42no—
77#213—42no—
78(C2×D3)⋊C4—22no—
79(C2×D3)⋊C4—22no—
80C4⋊Dic3C2312yes—
81Dic3⋊C4—22no—
82C2×C3⋊C8C2312yes—
83C2×C3⋊C8C2312yes—
84C2×#41C2×D311yescharacteristic
85C22×Q8C2×D311yescharacteristic
86C2×SD16—32no—
87C2×SD16—32no—
88C2×SD16—32no—
89C2×D8—32no—
90C4⋊Q8C2×D311yescharacteristic
91C4⋊Q8C2×D311yescharacteristic
92#127—31no—
93#125—31no—
94#124C2×D311yescharacteristic
95#124C2×D311yescharacteristic
96#123—22no—
97#123—62no—
98C22⋊Q8—22no—
99#121—32no—
100#121—62no—
101C4×D4—32no—
102#117C2×D311yescharacteristic
103C4⋊C8—32no—
104#103—32no—
105#103—32no—
106#103—32no—
107#102—32no—
108C22⋊C8—32no—
109#97—31no—
110C4×C8—31no—
111C23×C3—21no—
112C23×C3C2×D411yescharacteristic
113C22×D3—22no—
114C3×Q8—22no—
115C3×Q8—22no—
116C3×Q8—22no—
117C3×Q8—42no—
118C3×D4—22no—
119C3×D4—42no—
120C2×C3×C4—22no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4C2×D412yes—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4C2×D412yes—
129C2×C3×C4C2411yescharacteristic
130#67—82no—
131C2×Dic3—22no—
132D12—22no—
133D12—22no—
134D12—82no—
135C4×D3—82no—
136C3⋊C8—22no—
137C3⋊C8—22no—
138#41—42no—
139C2×Q8—21no—
140C2×Q8—42no—
141C2×Q8C22×D311yescharacteristic
142C2×Q8C22×D311yescharacteristic
143C2×Q8C22×D311yescharacteristic
144C2×D4—21no—
145C2×D4—32no—
146C2×D4—62no—
147C2×D4—62no—
148C2×D4C22×D311yescharacteristic
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—62no—
152C22×C4C22×D311yescharacteristic
153SD16—122no—
154SD16—122no—
155SD16—122no—
156D8—122no—
157C2×C8—32no—
158C2×C8—32no—
159C4⋊C4—22no—
160C4⋊C4—22no—
161C4⋊C4—32no—
162C4⋊C4—62no—
163C4⋊C4#6712yes—
164C22⋊C4—22no—
165C22⋊C4—22no—
166C22⋊C4—62no—
167C22⋊C4—62no—
168C22⋊C4#6712yes—
169C42C22×D311yescharacteristic
170C42C22×D311yescharacteristic
171C22×C3—21no—
172C22×C3—22no—
173C22×C3—41no—
174C22×C3—42no—
175C22×C3C22×D411yescharacteristic
176C2×D3—42no—
177C2×D3—42no—
178C2×D3—42no—
179C3×C4—21no—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—41no—
183C3×C4—41no—
184C3×C4—41no—
185C3×C4H(5,𝔽2)12yes—
186Dic3—42no—
187C23—21no—
188C23—62no—
189C23C2×#6711yescharacteristic
190Q8—22no—
191Q8—22no—
192Q8—22no—
193Q8—42no—
194D4—22no—
195D4—42no—
196D4—62no—
197D4—62no—
198D4—242no—
199D4—242no—
200C2×C4—22no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—62no—
207C2×C4—242no—
208C2×C4C2×#6712yes—
209C2×C4C2×#6712yes—
210C2×C4C2×#6711yescharacteristic
211C2×C4C23×D311yescharacteristic
212C8—62no—
213C8—62no—
214C2×C3—41no—
215C2×C3—81no—
216C2×C3#53511yescharacteristic
217C2×C3#57712yes—
218D3—82no—
219C22—21no—
220C22—22no—
221C22—41no—
222C22—42no—
223C22—122no—
224C22—122no—
225C22—122no—
226C22C22×#6711yescharacteristic
227C4—21no—
228C4—22no—
229C4—22no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—122no—
234C4#100412yes—
235C3#322111yescharacteristic, complemented, Sylow
236C2—41no—
237C2—81no—
238C2—242no—
239C2#580511yescharacteristic
240C2#583712yes—
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC22×C3C22×D4
hypercenter#536D3
hypocenterC3#3221
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C3×#536C2semidirect
2. #5535C2semidirect
3. #5238C2semidirect
4. #5227C2semidirect
5. #5085C2semidirect
6. #5083C2semidirect
7. #5082C2semidirect
8. #5058C2semidirect
9. #5057C2semidirect
10. #5056C2semidirect
11. #5000C2semidirect
41. #536D3semidirect
235. C3#3221semidirect

Maximal subgroups

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

subgroupindex
C3×#5362
#55352
#52382
#52272
#50852
#50832
#50822
#50582
#50572
#50562
#50002
#32213