Finite Group Databasebeta

← #83415

Subgroups — #83415

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
2C3×#536C211yesmaximal, characteristic, complemented
3#5822C212yesmaximal, complemented
4C2×#836C211yesmaximal, characteristic, complemented
5#5095C214yesmaximal, complemented
6#5062C214yesmaximal, complemented
7#5005C212yesmaximal, complemented
8#3020—31nomaximal, Sylow
9C2×C3×#41C2211yescharacteristic
10C22×C3×Q8C2211yescharacteristic
11#1007—22no—
12#1007—24no—
13#1007C2214yes—
14C2×#239—22no—
15C2×D3×Q8—22no—
16C2×#235—22no—
17C2×#235C2212yes—
18C3×C4⋊Q8C2212yescomplemented
19C3×#124C2212yescomplemented
20C3×#123—22no—
21C3×C22⋊Q8—22no—
22C3×#117C2211yescharacteristic
23#951—22no—
24#950—24no—
25#949—22no—
26#942C2214yescomplemented
27#932C2214yescomplemented
28C2×#208C2212yes—
29#836C2214yescomplemented
30#805C2218yescomplemented
31C2×2−1+4—31no—
32#575—32no—
33#536D311yescharacteristic, complemented
34#456—34no—
35#454—34no—
36#421—32no—
37C2×#101—31no—
38C3×#41—22no—
39C3×#41—22no—
40C2×C3×Q8—22no—
41C2×C3×Q8—22no—
42C2×C3×Q8D412yes—
43C2×C3×Q8C2311yescharacteristic
44C2×C3×Q8C2312yes—
45C2×C3×D4—21no—
46C2×C3×D4C2311yescharacteristic
47C22×C3×C4—21no—
48C22×C3×C4D412yes—
49C22×C3×C4C2311yescharacteristic
50C2×#67—22no—
51#239—28no—
52#239—28no—
53D3×Q8—28no—
54D3×Q8—28no—
55#235—28no—
56#235—28no—
57#235—22no—
58#235—22no—
59#235D414yes—
60#235C2314yes—
61C2×D12—22no—
62C2×D12D412yes—
63C2×C4×D3—24no—
64C2×C4×D3—22no—
65C2×Dic6—22no—
66C2×Dic6D412yes—
67C3×C4⋊C4—21no—
68C3×C4⋊C4—24no—
69C3×C22⋊C4—21no—
70C3×C22⋊C4—24no—
71C3×C42C2312yes—
72#216—24no—
73#215—24no—
74#214—24no—
75#213—24no—
76#208—22no—
77#208C2314yes—
78C2×C3⋊C8—22no—
792−1+4—34no—
802−1+4—64no—
812−1+4—62no—
82C2×#41—32no—
83C2×#41—62no—
84C2×#41—62no—
85C2×#41C2×D311yescharacteristic
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×M16—32no—
94C4⋊Q8C2×D312yescomplemented
95#124C2×D312yescomplemented
96#123—22no—
97C22⋊Q8—22no—
98#117C2×D311yescharacteristic
99#104—38no—
100#101—34no—
101C23×C3—21no—
102C23×C3C2×D411yescharacteristic
103C22×D3—22no—
104C3×Q8—24no—
105C3×Q8—24no—
106C3×Q8—22no—
107C3×D4—22no—
108C3×D4—41no—
109C3×D4—41no—
110C2×C3×C4—22no—
111C2×C3×C4—22no—
112C2×C3×C4—22no—
113C2×C3×C4—21no—
114C2×C3×C4—42no—
115C2×C3×C4C2×D412yes—
116C2×C3×C4C2×D412yes—
117C2×C3×C4C2×D414yes—
118C2×C3×C4C2×D411yescharacteristic
119C2×C3×C4C2411yescharacteristic
120#67—24no—
121#67—24no—
122C2×Dic3—22no—
123D12—28no—
124D12—22no—
125D12C2×D414yes—
126C4×D3—216no—
127C4×D3—24no—
128C4×D3—24no—
129Dic6—28no—
130Dic6—22no—
131Dic6C2×D414yes—
132C3⋊C8—24no—
133#41—22no—
134#41—22no—
135#41—34no—
136#41—34no—
137#41—68no—
138#41—68no—
139#41—68no—
140#41—62no—
141#41—62no—
142#41—68no—
143C2×Q8—22no—
144C2×Q8—22no—
145C2×Q8—32no—
146C2×Q8—62no—
147C2×Q8—68no—
148C2×Q8—68no—
149C2×Q8#6712yes—
150C2×Q8C22×D311yescharacteristic
151C2×Q8C22×D312yes—
152C2×D4—21no—
153C2×D4—32no—
154C2×D4—62no—
155C2×D4—62no—
156C2×D4C22×D311yescharacteristic
157C22×C4—21no—
158C22×C4—64no—
159C22×C4—62no—
160C22×C4#6712yes—
161C22×C4C22×D311yescharacteristic
162Q16—64no—
163SD16—64no—
164SD16—64no—
165D8—64no—
166M16—34no—
167M16—62no—
168C2×C8—62no—
169C4⋊C4—21no—
170C4⋊C4—24no—
171C22⋊C4—21no—
172C22⋊C4—24no—
173C42C22×D312yes—
174C22×C3—41no—
175C22×C3—42no—
176C22×C3—41no—
177C22×C3#12011yescharacteristic
178C22×C3#12011yescharacteristic
179C22×C3C22×D411yescharacteristic
180C2×D3—24no—
181C2×D3—44no—
182C3×C4—24no—
183C3×C4—42no—
184C3×C4—41no—
185C3×C4#12012yes—
186C3×C4C22×D412yes—
187Dic3—24no—
188C23—21no—
189C23—62no—
190C23C2×#6711yescharacteristic
191Q8—24no—
192Q8—24no—
193Q8—22no—
194Q8—34no—
195Q8—68no—
196Q8—62no—
197D4—22no—
198D4—34no—
199D4—41no—
200D4—41no—
201D4—68no—
202D4—64no—
203D4—64no—
204D4—62no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—21no—
209C2×C4—42no—
210C2×C4—616no—
211C2×C4—62no—
212C2×C4—64no—
213C2×C4—64no—
214C2×C4D3×D412yes—
215C2×C4D3×D412yes—
216C2×C4C2×#6714yes—
217C2×C4C2×#6711yescharacteristic
218C2×C4C23×D311yescharacteristic
219C8—64no—
220C2×C3—21no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—81no—
224C2×C3C2×#12011yescharacteristic
225D3—44no—
226C22—41no—
227C22—42no—
228C22—41no—
229C22—64no—
230C22—124no—
231C22#93711yescharacteristic
232C22#93711yescharacteristic
233C22C22×#6711yescharacteristic
234C4—24no—
235C4—42no—
236C4—41no—
237C4—64no—
238C4#93712yes—
239C4C2×D3×D412yes—
240C3#302011yescharacteristic, complemented, Sylow
241C2—21no—
242C2—21no—
243C2—21no—
244C2—81no—
245C2—124no—
246C2C2×#93711yescharacteristic
subgroupG/H
centerC2C2×#937
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#536C2
socleC2×C3C2×#120
hypercenter#536D3
hypocenterC3#3020
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #536

Complements

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

NH ≅ G/Ntype
1. C2×#1007C2semidirect
2. C3×#536C2semidirect
3. #5822C2semidirect
4. C2×#836C2semidirect
5. #5095C2semidirect
6. #5062C2semidirect
7. #5005C2semidirect
18. C3×C4⋊Q8C22semidirect
19. C3×#124C22semidirect
26. #942C22semidirect
27. #932C22semidirect
29. #836C22semidirect
30. #805C22semidirect
33. #536D3semidirect
94. C4⋊Q8C2×D3semidirect
95. #124C2×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
C3×#5362
#58222
C2×#8362
#50952
#50622
#50052
#30203