Finite Group Databasebeta

← #83521

Subgroups — #83521

Order 384.

386 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 123 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×#547C211yesmaximal, characteristic, complemented
2#5563C211yesmaximal, characteristic, complemented
3#5087C211yesmaximal, characteristic, complemented
4#5080C211yesmaximal, characteristic, complemented
5#5039C212yesmaximal, complemented
6#5035C212yesmaximal, complemented
7C4×#213C211yesmaximal, characteristic, complemented
8#5012C211yesmaximal, characteristic, complemented
9#5008C211yesmaximal, characteristic, complemented
10#4968C212yesmaximal, complemented
11#4964C212yesmaximal, complemented
12#3354—31nomaximal, Sylow
13C2×C3×#41—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#123—22no—
16C3×C22⋊Q8—21no—
17C3×#121C2212yes—
18C3×C4×D4C2211yescharacteristic
19C3×C4×D4C2211yescharacteristic
20C2×C3×C4⋊C4C2212yes—
21#938—21no—
22C2×#213C2211yescharacteristic
23#930C2212yes—
24#926C2212yes—
25C2×#208C2212yes—
26#892—21no—
27#885—22no—
28C4×D12C2211yescharacteristic
29#869C2211yescharacteristic
30#832C2212yes—
31#832C2211yescharacteristic
32#830C2212yes—
33#809C2211yescharacteristic
34#809C2212yes—
35#808C2212yes—
36#808C2212yes—
37#808C2212yes—
38#807C2211yescharacteristic
39#807C2212yes—
40C4⋊(C3⋊C8)C2211yescharacteristic
41C4×C3⋊C8C2211yescharacteristic
42#555—31no—
43#547D311yescharacteristic, complemented
44#498—31no—
45#482—32no—
46#481—32no—
47#476—31no—
48#474—31no—
49#469—32no—
50C4×D8—31no—
51#436—31no—
52#428—32no—
53C3×#41—44no—
54C2×C3×Q8—21no—
55C2×C3×D4—22no—
56C2×C3×D4D411yescharacteristic
57C2×C3×D4C2311yescharacteristic
58C22×C3×C4—22no—
59C22×C3×C4—21no—
60C22×C3×C4C2312yes—
61C2×#67—22no—
62C2×D12C2311yescharacteristic
63C2×C4×D3—21no—
64C2×Dic6—21no—
65C3×C4⋊C4—22no—
66C3×C4⋊C4—21no—
67C3×C4⋊C4D411yescharacteristic
68C3×C4⋊C4C2311yescharacteristic
69C3×C4⋊C4C2312yes—
70C3×C22⋊C4—21no—
71C3×C22⋊C4—22no—
72C3×C22⋊C4D412yes—
73C3×C42C2311yescharacteristic
74#217—22no—
75#213—22no—
76(C2×D3)⋊C4—21no—
77(C2×D3)⋊C4—22no—
78C4⋊Dic3—21no—
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2312yes—
81C4×Dic3—22no—
82#208—24no—
83C2×C3⋊C8C2311yescharacteristic
84C2×C3⋊C8C2312yes—
85C2×C3⋊C8C2311yescharacteristic
86C2×#41—21no—
87C2×D8—31no—
88C2×M16—32no—
89C4⋊Q8—31no—
90C4⋊Q8C2×D311yescharacteristic
91#124—62no—
92#123—22no—
93C22⋊Q8—21no—
94#121—32no—
95#121—61no—
96#121C2×D312yes—
97C4×D4—31no—
98C4×D4C2×D311yescharacteristic
99C4×D4C2×D311yescharacteristic
100#117—32no—
101#117—61no—
102C2×C4⋊C4C2×D312yes—
103#107—31no—
104#107—32no—
105#106—32no—
106#106—32no—
107#106—32no—
108C4⋊C8—31no—
109#102—31no—
110#102—32no—
111#102—31no—
112#102—32no—
113C22⋊C8—32no—
114C4×C8—31no—
115C23×C3—21no—
116C23×C3C2×D412yes—
117C22×D3—21no—
118C3×Q8—42no—
119C3×D4—22no—
120C3×D4—21no—
121C3×D4—21no—
122C3×D4—44no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—22no—
127C2×C3×C4—44no—
128C2×C3×C4—42no—
129C2×C3×C4C2×D412yes—
130C2×C3×C4C2×D411yescharacteristic
131C2×C3×C4C2×D411yescharacteristic
132C2×C3×C4C2411yescharacteristic
133#67—82no—
134C2×Dic3—21no—
135C2×Dic3—22no—
136D12—21no—
137D12—21no—
138C4×D3—42no—
139Dic6—81no—
140C3⋊C8—22no—
141C3⋊C8—21no—
142C3⋊C8#4112yescomplemented
143#41—44no—
144C2×Q8—21no—
145C2×Q8—61no—
146C2×D4—22no—
147C2×D4—31no—
148C2×D4—62no—
149C2×D4#6711yescharacteristic
150C2×D4C22×D311yescharacteristic
151C22×C4—22no—
152C22×C4—21no—
153C22×C4—61no—
154C22×C4C22×D312yes—
155D8—62no—
156M16—64no—
157C2×C8—31no—
158C2×C8—31no—
159C2×C8—32no—
160C4⋊C4—22no—
161C4⋊C4—21no—
162C4⋊C4—32no—
163C4⋊C4—31no—
164C4⋊C4—61no—
165C4⋊C4#6711yescharacteristic
166C4⋊C4C22×D311yescharacteristic
167C4⋊C4C22×D312yes—
168C22⋊C4—21no—
169C22⋊C4—22no—
170C22⋊C4—62no—
171C22⋊C4—62no—
172C22⋊C4—61no—
173C22⋊C4#6712yes—
174C42—62no—
175C42C22×D311yescharacteristic
176C22×C3—22no—
177C22×C3—22no—
178C22×C3—22no—
179C22×C3—41no—
180C22×C3—41no—
181C22×C3—41no—
182C22×C3C22×D411yescharacteristic
183C2×D3—41no—
184C2×D3—41no—
185C2×D3—41no—
186C3×C4—22no—
187C3×C4—21no—
188C3×C4—41no—
189C3×C4—42no—
190C3×C4AGL(1,ℤ/8)12yes—
191C3×C4C2×#4111yescharacteristic
192C3×C42−1+411yescharacteristic
193Dic3—41no—
194Dic3—42no—
195C23—21no—
196C23—61no—
197C23C2×#6712yes—
198Q8—42no—
199Q8—241no—
200D4—22no—
201D4—21no—
202D4—21no—
203D4—44no—
204D4—61no—
205D4—61no—
206D4—242no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—21no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—42no—
213C2×C4—61no—
214C2×C4—62no—
215C2×C4—122no—
216C2×C4C2×#6712yes—
217C2×C4C2×#6711yescharacteristic
218C2×C4C2×#6711yescharacteristic
219C2×C4C23×D311yescharacteristic
220C8—32no—
221C8—62no—
222C8—61no—
223C2×C3—42no—
224C2×C3—81no—
225C2×C3#54711yescharacteristic
226C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
227C2×C3#57711yescharacteristic
228D3—81no—
229C22—22no—
230C22—22no—
231C22—22no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—121no—
236C22—121no—
237C22—121no—
238C22C22×#6711yescharacteristic
239C4—22no—
240C4—21no—
241C4—41no—
242C4—42no—
243C4—121no—
244C4—122no—
245C4#93212yes—
246C4D3×#4111yescharacteristic
247C4#101011yescharacteristic
248C3#335411yescharacteristic, complemented, Sylow
249C2—42no—
250C2—81no—
251C2—241no—
252C2C2×#93211yescharacteristic
253C2#582411yescharacteristic
254C2#583711yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#547C2
socleC22×C3C22×D4
hypercenter#547D3
hypocenterC3#3354
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5563C2semidirect
3. #5087C2semidirect
4. #5080C2semidirect
5. #5039C2semidirect
6. #5035C2semidirect
7. C4×#213C2semidirect
8. #5012C2semidirect
9. #5008C2semidirect
10. #4968C2semidirect
11. #4964C2semidirect
43. #547D3semidirect
142. C3⋊C8#41semidirect
248. C3#3354semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#55632
#50872
#50802
#50392
#50352
C4×#2132
#50122
#50082
#49682
#49642
#33543