Finite Group Databasebeta

← #78952

Subgroups — #78952

Order 384.

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

normal subgroups only

This group has 372 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
1#5816C211yesmaximal, characteristic
2#5519C211yesmaximal, characteristic, complemented
3C3×#419C211yesmaximal, characteristic, complemented
4C2×#835C211yesmaximal, characteristic, complemented
5#5133C211yesmaximal, characteristic
6#5122C211yesmaximal, characteristic, complemented
7#5002C211yesmaximal, characteristic
8#4800C212yesmaximal, complemented
9#4797C212yesmaximal, complemented
10#4794C212yesmaximal, complemented
11#4792C212yesmaximal, complemented
12#3137—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C2×C3×M16C2211yescharacteristic
16C3×#117C2211yescharacteristic
17#947C2212yes—
18#946—21no—
19#946—21no—
20#944C2212yes—
21#929—22no—
22#926C2211yescharacteristic
23C2×C4⋊Dic3C2211yescharacteristic
24C22×C3⋊C8C2211yescharacteristic
25#890C2212yes—
26#888C2212yes—
27#878—22no—
28#870—21no—
29#869—21no—
30C3×#103C2212yes—
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#821C2212yes—
35#816C2212yes—
36#815C2212yes—
37C8×Dic3C2212yes—
38#810C2211yescharacteristic
39#810C2211yescharacteristic
40#809C2211yescharacteristic
41#809C2211yescharacteristic
42#544—31no—
43#536—31no—
44#491—32no—
45#489—32no—
46#488—32no—
47#487—32no—
48#419D311yescharacteristic, complemented
49#417—31no—
50#416—31no—
51C2×#103—31no—
52#405—31no—
53C2×C3×Q8—21no—
54C2×C3×Q8—22no—
55C2×C3×Q8—22no—
56C2×C3×Q8C2312yes—
57C22×C3×C4—21no—
58C22×C3×C4C2311yescharacteristic
59C2×#67—21no—
60C22×Dic3—21no—
61#235—22no—
62#235—22no—
63C2×D12C2311yescharacteristic
64C2×C4×D3—21no—
65C2×Dic6—22no—
66C2×Dic6C2311yescharacteristic
67C3×M16—22no—
68C2×C3×C8C2312yes—
69C3×C4⋊C4C2311yescharacteristic
70C3×C4⋊C4C2311yescharacteristic
71C3×C22⋊C4—21no—
72C3×C42—21no—
73#217D412yes—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—21no—
77C4⋊Dic3—22no—
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3D411yescharacteristic
80C4⋊Dic3C2312yes—
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C4×Dic3C2312yes—
86C2×C3⋊C8—22no—
87C2×C3⋊C8C2311yescharacteristic
88C2×C3⋊C8C2311yescharacteristic
89C2×#41—31no—
90C22×Q8C2×D311yescharacteristic
91C2×M16C2×D311yescharacteristic
92C22×C8—31no—
93C4⋊Q8—32no—
94C4⋊Q8—32no—
95C4⋊Q8—61no—
96#125—32no—
97#125—61no—
98#124—32no—
99#123—62no—
100C22⋊Q8—62no—
101C22⋊Q8—61no—
102C22⋊Q8—61no—
103#117—31no—
104#117C2×D311yescharacteristic
105C2×C4⋊C4—31no—
106#103—32no—
107#103—32no—
108#103—32no—
109#103—31no—
110#103—31no—
111#103C2×D312yes—
112#103C2×D312yes—
113#102—32no—
114#102—31no—
115#102—31no—
116#97—32no—
117C4×C8—32no—
118C23×C3C2×D411yescharacteristic
119C22×D3—21no—
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—42no—
123C2×C3×C4—21no—
124C2×C3×C4—22no—
125C2×C3×C4—21no—
126C2×C3×C4—42no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4#4112yes—
129C2×C3×C4#4112yes—
130C2×C3×C4C2411yescharacteristic
131#67—42no—
132C2×Dic3—21no—
133C2×Dic3—22no—
134C2×Dic3—81no—
135C2×Dic3C2×D412yes—
136C2×Dic3C2×D412yes—
137D12—21no—
138D12—21no—
139C4×D3—42no—
140Dic6—21no—
141Dic6—21no—
142Dic6—82no—
143C3×C8—22no—
144C3⋊C8—21no—
145C3⋊C8—21no—
146#41—62no—
147#41—62no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—31no—
152C2×Q8—62no—
153C2×Q8C22×D312yes—
154C2×D4—31no—
155C2×D4—61no—
156C22×C4—21no—
157C22×C4—61no—
158C22×C4—61no—
159C22×C4C22×D311yescharacteristic
160M16—22no—
161C2×C8—31no—
162C2×C8—31no—
163C2×C8—62no—
164C2×C8C22×D312yes—
165C4⋊C4—31no—
166C4⋊C4—31no—
167C4⋊C4—32no—
168C4⋊C4—62no—
169C4⋊C4—62no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—61no—
174C4⋊C4C22×D311yescharacteristic
175C4⋊C4C22×D311yescharacteristic
176C22⋊C4—21no—
177C22⋊C4—32no—
178C22⋊C4—62no—
179C22⋊C4—62no—
180C42—21no—
181C42—32no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3#12412yes—
185C22×C3C22×D411yescharacteristic
186C2×D3—41no—
187C2×D3—41no—
188C2×D3—41no—
189C3×C4—42no—
190C3×C4—41no—
191C3×C4—41no—
192C3×C4#12412yes—
193C3×C4C2×#4111yescharacteristic
194C3×C4C2×#4111yescharacteristic
195Dic3—22no—
196Dic3—22no—
197Dic3—41no—
198Dic3—42no—
199C23—61no—
200C23D3×D411yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—42no—
204Q8—61no—
205Q8—61no—
206Q8—242no—
207D4—61no—
208D4—61no—
209D4—122no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—32no—
214C2×C4—32no—
215C2×C4—42no—
216C2×C4—62no—
217C2×C4—61no—
218C2×C4—122no—
219C2×C4—241no—
220C2×C4#23512yes—
221C2×C4D3×D411yescharacteristic
222C2×C4#23712yes—
223C2×C4C23×D311yescharacteristic
224C8—22no—
225C8—61no—
226C8—61no—
227C2×C3—22no—
228C2×C3C2×#12411yescharacteristic
229C2×C3#57711yescharacteristic
230C2×C3#57811yescharacteristic
231D3—81no—
232C22—21no—
233C22—21no—
234C22—121no—
235C22—121no—
236C22—121no—
237C22#88512yes—
238C22C2×D3×D411yescharacteristic
239C4—42no—
240C4—41no—
241C4—41no—
242C4—62no—
243C4—62no—
244C4—121no—
245C4—122no—
246C4#88512yes—
247C4C2×#23511yescharacteristic
248C4C2×#23711yescharacteristic
249C3#313711yescharacteristic, complemented, Sylow
250C2—22no—
251C2—241no—
252C2C2×#88511yescharacteristic
253C2#577811yescharacteristic
254C2#577911yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#419C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenter#419D3
hypocenterC3#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
2. #5519C2semidirect
3. C3×#419C2semidirect
4. C2×#835C2semidirect
6. #5122C2semidirect
8. #4800C2semidirect
9. #4797C2semidirect
10. #4794C2semidirect
11. #4792C2semidirect
48. #419D3semidirect
249. C3#3137semidirect

Maximal subgroups

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

subgroupindex
#58162
#55192
C3×#4192
C2×#8352
#51332
#51222
#50022
#48002
#47972
#47942
#47922
#31373