Finite Group Databasebeta

← #78709

Subgroups — #78709

Order 384.

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

normal subgroups only

This group has 698 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×#1007C211yesmaximal, characteristic, complemented
2#5816C211yesmaximal, characteristic, complemented
3#5748C212yesmaximal, complemented
4C2×C3×#101C211yesmaximal, characteristic, complemented
5#5139C212yesmaximal, complemented
6#4753C214yesmaximal, complemented
7#4752C214yesmaximal, complemented
8#3020—31nomaximal, Sylow
9C22×C3×Q8C2211yescharacteristic
10#1007—22no—
11#1007—24no—
12#1007C2214yes—
13C2×#239—22no—
14C2×D3×Q8—22no—
15C2×#235—22no—
16C2×#235C2212yes—
17C2×#235C2211yescharacteristic
18C2×C3×M16C2212yes—
19#947C2212yescomplemented
20#946—22no—
21#944C2212yescomplemented
22#929—22no—
23#926C2211yescharacteristic
24#909—22no—
25#909C2214yescomplemented
26#908—22no—
27#908C2214yescomplemented
28#904—24no—
29C3×#101C2214yescomplemented
30#825C2218yescomplemented
31C2×2−1+4—31no—
32#575—32no—
33#536—31no—
34#456—34no—
35#454—34no—
36#421—32no—
37C2×#101D311yescharacteristic, complemented
38C2×C3×Q8—22no—
39C2×C3×Q8—22no—
40C2×C3×Q8D412yes—
41C2×C3×Q8C2312yes—
42C22×C3×C4D412yes—
43C22×C3×C4C2311yescharacteristic
44C2×#67—22no—
45C2×#67—21no—
46#239—28no—
47#239—28no—
48D3×Q8—28no—
49D3×Q8—28no—
50#235—28no—
51#235—28no—
52#235—22no—
53#235—22no—
54#235—22no—
55#235—22no—
56#235D414yes—
57#235C2314yes—
58C2×D12—22no—
59C2×D12D412yes—
60C2×D12C2311yescharacteristic
61C2×C4×D3—24no—
62C2×C4×D3—22no—
63C2×C4×D3—21no—
64C2×Dic6—22no—
65C2×Dic6D412yes—
66C2×Dic6C2311yescharacteristic
67C3×M16—22no—
68C3×M16C2314yes—
69C2×C3×C8—22no—
70#217—21no—
71(C2×D3)⋊C4—24no—
72C4⋊Dic3—21no—
73Dic3⋊C4—24no—
74C4×Dic3C2312yes—
75Dic12—24no—
76D24—24no—
77#204—24no—
78#204—24no—
792−1+4—34no—
802−1+4—64no—
812−1+4—62no—
82C2×#41—32no—
83C2×#41—31no—
84C2×#41—62no—
85C2×#41—62no—
86C22×Q8—62no—
87C22×Q8C2×D311yescharacteristic
88#137—34no—
89#137—62no—
90AGL(1,ℤ/8)—34no—
91AGL(1,ℤ/8)—62no—
92#135—64no—
93C2×M16C2×D312yes—
94C4⋊Q8—32no—
95#124—32no—
96#123—62no—
97C22⋊Q8—62no—
98#117—31no—
99#104—38no—
100#101C2×D314yescomplemented
101C23×C3C2×D411yescharacteristic
102C22×D3—22no—
103C22×D3—21no—
104C3×Q8—24no—
105C3×Q8—24no—
106C2×C3×C4—22no—
107C2×C3×C4—22no—
108C2×C3×C4C2×D414yes—
109C2×C3×C4C2×D411yescharacteristic
110C2×C3×C4C2×D412yes—
111C2×C3×C4C2×D412yes—
112C2×C3×C4C2411yescharacteristic
113#67—24no—
114#67—24no—
115#67—41no—
116#67—41no—
117C2×Dic3—22no—
118C2×Dic3—22no—
119C2×Dic3—21no—
120D12—28no—
121D12—22no—
122D12—22no—
123D12C2×D414yes—
124C4×D3—216no—
125C4×D3—24no—
126C4×D3—24no—
127C4×D3—42no—
128Dic6—28no—
129Dic6—22no—
130Dic6—22no—
131Dic6C2×D414yes—
132C3×C8—24no—
133#41—34no—
134#41—34no—
135#41—68no—
136#41—68no—
137#41—68no—
138#41—62no—
139#41—68no—
140#41—62no—
141#41—62no—
142#41—62no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—32no—
146C2×Q8—31no—
147C2×Q8—62no—
148C2×Q8—68no—
149C2×Q8—68no—
150C2×Q8D1212yes—
151C2×Q8C22×D312yes—
152C2×D4—32no—
153C2×D4—31no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4—61no—
157C22×C4—64no—
158C22×C4—62no—
159C22×C4—61no—
160C22×C4D1212yes—
161C22×C4C22×D311yescharacteristic
162Q16—64no—
163SD16—64no—
164SD16—64no—
165D8—64no—
166M16—22no—
167M16C22×D314yes—
168C2×C8—22no—
169C4⋊C4—61no—
170C4⋊C4—64no—
171C22⋊C4—61no—
172C22⋊C4—64no—
173C42—32no—
174C22×C3—41no—
175C22×C3#12011yescharacteristic
176C22×C3#12011yescharacteristic
177C22×C3C22×D411yescharacteristic
178C2×D3—24no—
179C2×D3—44no—
180C2×D3—42no—
181C2×D3—41no—
182C3×C4—24no—
183C3×C4#12012yes—
184C3×C4C22×D412yes—
185Dic3—24no—
186Dic3—42no—
187Dic3—41no—
188C23—62no—
189C23—61no—
190C23C2×D1211yescharacteristic
191Q8—24no—
192Q8—24no—
193Q8—34no—
194Q8—68no—
195Q8—62no—
196Q8—62no—
197D4—34no—
198D4—68no—
199D4—64no—
200D4—64no—
201D4—62no—
202D4—62no—
203D4—121no—
204D4—121no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—616no—
208C2×C4—62no—
209C2×C4—64no—
210C2×C4—64no—
211C2×C4—62no—
212C2×C4—61no—
213C2×C4—122no—
214C2×C4C2×D1214yes—
215C2×C4C2×D1211yescharacteristic
216C2×C4D3×D412yes—
217C2×C4D3×D412yes—
218C2×C4C23×D311yescharacteristic
219C8—24no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3C2×#12011yescharacteristic
224D3—44no—
225D3—81no—
226C22—41no—
227C22—64no—
228C22—124no—
229C22—122no—
230C22—121no—
231C22#88211yescharacteristic
232C22#88211yescharacteristic
233C22C22×D1211yescharacteristic
234C4—24no—
235C4—64no—
236C4—122no—
237C4—121no—
238C4#88212yes—
239C4C2×D3×D412yes—
240C3#302011yescharacteristic, complemented, Sylow
241C2—21no—
242C2—21no—
243C2—21no—
244C2—124no—
245C2—241no—
246C2C2×#88211yescharacteristic
subgroupG/H
centerC2C2×#882
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#101C2
socleC2×C3C2×#120
hypercenterC2×#101D3
hypocenterC3#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
1. C2×#1007C2semidirect
2. #5816C2semidirect
3. #5748C2semidirect
4. C2×C3×#101C2semidirect
5. #5139C2semidirect
6. #4753C2semidirect
7. #4752C2semidirect
19. #947C22semidirect
21. #944C22semidirect
25. #909C22semidirect
27. #908C22semidirect
29. C3×#101C22semidirect
30. #825C22semidirect
37. C2×#101D3semidirect
100. #101C2×D3semidirect
240. C3#3020semidirect

Maximal subgroups

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

subgroupindex
C2×#10072
#58162
#57482
C2×C3×#1012
#51392
#47532
#47522
#30203