Finite Group Databasebeta

← #83293

Subgroups — #83293

Order 384.

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

normal subgroups only

This group has 410 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×#532C211yesmaximal, characteristic, complemented
2#5479C211yesmaximal, characteristic, complemented
3#5089C211yesmaximal, characteristic, complemented
4#5081C211yesmaximal, characteristic, complemented
5#5074C211yesmaximal, characteristic, complemented
6#5071C211yesmaximal, characteristic, complemented
7#5041C212yesmaximal, complemented
8#5038C212yesmaximal, complemented
9#5018C212yesmaximal, complemented
10#5014C212yesmaximal, complemented
11C2×#803C211yesmaximal, characteristic, complemented
12#3180—31nomaximal, Sylow
13C2×C3×#41—21no—
14C3×C4⋊Q8C2211yescharacteristic
15C3×#127C2211yescharacteristic
16C3×#124—21no—
17C3×#121—21no—
18C3×#121C2212yes—
19C3×C4×D4—21no—
20C3×C4×D4C2212yes—
21C2×C3×C42C2211yescharacteristic
22C2×#214C2212yes—
23C2×#213C2212yes—
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#832C2212yes—
36#832C2212yes—
37#810C2212yes—
38#809C2212yes—
39#808C2212yes—
40#807C2212yes—
41#803C2212yes—
42#803C2211yescharacteristic
43#803C2211yescharacteristic
44#532D311yescharacteristic, complemented
45#529—31no—
46#501—31no—
47#496—31no—
48#490—31no—
49#489—31no—
50#466—32no—
51#465—32no—
52#442—32no—
53#440—32no—
54C2×#97—31no—
55C3×#41—44no—
56C2×C3×Q8—21no—
57C2×C3×D4—22no—
58C2×C3×D4—21no—
59C2×C3×D4—21no—
60C2×C3×D4C2312yes—
61C22×C3×C4—22no—
62C22×C3×C4D411yescharacteristic
63C22×C3×C4D411yescharacteristic
64C22×C3×C4C2311yescharacteristic
65C2×#67—21no—
66C2×D12C2311yescharacteristic
67C2×C4×D3—21no—
68C2×Dic6C2311yescharacteristic
69C3×C4⋊C4—21no—
70C3×C4⋊C4C2312yes—
71C3×C22⋊C4—22no—
72C3×C22⋊C4—22no—
73C3×C42D411yescharacteristic
74C3×C42D411yescharacteristic
75C3×C42C2311yescharacteristic
76C3×C42C2311yescharacteristic
77#217—21no—
78#214—24no—
79#213—24no—
80(C2×D3)⋊C4—21no—
81(C2×D3)⋊C4—21no—
82(C2×D3)⋊C4—21no—
83(C2×D3)⋊C4—21no—
84C4⋊Dic3C2311yescharacteristic
85C4⋊Dic3C2311yescharacteristic
86Dic3⋊C4—21no—
87Dic3⋊C4—21no—
88Dic3⋊C4—21no—
89Dic3⋊C4—21no—
90C4×Dic3—21no—
91C2×C3⋊C8—24no—
92C2×C3⋊C8C2312yes—
93C2×C3⋊C8C2312yes—
94C2×#41—21no—
95C2×SD16—32no—
96C2×D8—32no—
97C22×C8—32no—
98C4⋊Q8C2×D311yescharacteristic
99#127C2×D311yescharacteristic
100#126—61no—
101#125—31no—
102#124—21no—
103#124—31no—
104#123—61no—
105C22⋊Q8—31no—
106#121—21no—
107#121—31no—
108#121C2×D312yes—
109C4×Q8—31no—
110C4×D4—21no—
111C4×D4—31no—
112C4×D4—61no—
113C4×D4C2×D312yes—
114#117—61no—
115C2×C42C2×D311yescharacteristic
116#107—32no—
117#106—32no—
118#103—32no—
119#102—32no—
120#102—32no—
121#102—32no—
122#97—32no—
123#97—31no—
124#97—31no—
125C23×C3—22no—
126C23×C3C2×D411yescharacteristic
127C22×D3—21no—
128C3×Q8—42no—
129C3×D4—22no—
130C3×D4—22no—
131C3×D4—42no—
132C3×D4—42no—
133C3×D4—44no—
134C2×C3×C4—22no—
135C2×C3×C4—22no—
136C2×C3×C4—22no—
137C2×C3×C4—41no—
138C2×C3×C4—44no—
139C2×C3×C4C2×D411yescharacteristic
140C2×C3×C4C2×D411yescharacteristic
141C2×C3×C4C2×D411yescharacteristic
142C2×C3×C4C2×D411yescharacteristic
143C2×C3×C4C2×D411yescharacteristic
144C2×C3×C4C2411yescharacteristic
145#67—42no—
146C2×Dic3—21no—
147C2×Dic3—21no—
148C2×Dic3—21no—
149D12—22no—
150C4×D3—42no—
151Dic6—22no—
152C3⋊C8—22no—
153C3⋊C8#4114yescomplemented
154#41—44no—
155C2×Q8—21no—
156C2×Q8—31no—
157C2×D4—22no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—31no—
161C2×D4—61no—
162C2×D4C22×D312yes—
163C22×C4—22no—
164C22×C4—61no—
165C22×C4#6711yescharacteristic
166C22×C4#6711yescharacteristic
167C22×C4C22×D311yescharacteristic
168SD16—64no—
169D8—64no—
170C2×C8—32no—
171C2×C8—32no—
172C2×C8—64no—
173C4⋊C4—21no—
174C4⋊C4—31no—
175C4⋊C4—31no—
176C4⋊C4—61no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—61no—
180C4⋊C4C22×D312yes—
181C22⋊C4—22no—
182C22⋊C4—22no—
183C22⋊C4—61no—
184C22⋊C4—61no—
185C22⋊C4—61no—
186C22⋊C4—61no—
187C22⋊C4—61no—
188C42—61no—
189C42#6711yescharacteristic
190C42#6711yescharacteristic
191C42C22×D311yescharacteristic
192C42C22×D311yescharacteristic
193C22×C3—21no—
194C22×C3—21no—
195C22×C3—42no—
196C22×C3—42no—
197C22×C3—42no—
198C22×C3AGL(1,ℤ/8)12yes—
199C22×C3C22×D411yescharacteristic
200C2×D3—41no—
201C2×D3—41no—
202C2×D3—41no—
203C3×C4—21no—
204C3×C4—21no—
205C3×C4—21no—
206C3×C4—21no—
207C3×C4—21no—
208C3×C4—42no—
209C3×C4C2×#4112yes—
210Dic3—41no—
211Dic3—41no—
212Dic3—41no—
213C23—22no—
214C23—61no—
215C23C2×#6711yescharacteristic
216Q8—42no—
217Q8—62no—
218D4—22no—
219D4—22no—
220D4—42no—
221D4—42no—
222D4—44no—
223D4—62no—
224D4—122no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—44no—
230C2×C4—61no—
231C2×C4—61no—
232C2×C4—61no—
233C2×C4—122no—
234C2×C4C2×#6711yescharacteristic
235C2×C4C2×#6711yescharacteristic
236C2×C4C2×#6711yescharacteristic
237C2×C4C2×#6711yescharacteristic
238C2×C4C2×#6711yescharacteristic
239C2×C4C23×D311yescharacteristic
240C8—34no—
241C8—62no—
242C2×C3—22no—
243C2×C3—82no—
244C2×C3#53211yescharacteristic
245C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
246C2×C3#57511yescharacteristic
247D3—81no—
248C22—21no—
249C22—21no—
250C22—42no—
251C22—42no—
252C22—42no—
253C22—121no—
254C22—121no—
255C22—121no—
256C22#93212yes—
257C22C22×#6711yescharacteristic
258C4—21no—
259C4—21no—
260C4—21no—
261C4—21no—
262C4—21no—
263C4—42no—
264C4—121no—
265C4—121no—
266C4—121no—
267C4D3×#4112yes—
268C3#318011yescharacteristic, complemented, Sylow
269C2—22no—
270C2—82no—
271C2—241no—
272C2C2×#93211yescharacteristic
273C2#582211yescharacteristic
274C2#583211yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#532C2
socleC22×C3C22×D4
hypercenter#532D3
hypocenterC3#3180
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5479C2semidirect
3. #5089C2semidirect
4. #5081C2semidirect
5. #5074C2semidirect
6. #5071C2semidirect
7. #5041C2semidirect
8. #5038C2semidirect
9. #5018C2semidirect
10. #5014C2semidirect
11. C2×#803C2semidirect
44. #532D3semidirect
153. C3⋊C8#41semidirect
268. C3#3180semidirect

Maximal subgroups

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

subgroupindex
C3×#5322
#54792
#50892
#50812
#50742
#50712
#50412
#50382
#50182
#50142
C2×#8032
#31803