Finite Group Databasebeta

← #78785

Subgroups — #78785

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#5500C211yesmaximal, characteristic, complemented
3C2×C3×#103C211yesmaximal, characteristic, complemented
4#5225C211yesmaximal, characteristic, complemented
5#5113C211yesmaximal, characteristic
6#5101C211yesmaximal, characteristic, complemented
7#4966C211yesmaximal, characteristic
8#4800C212yesmaximal, complemented
9#4797C212yesmaximal, complemented
10#4794C212yesmaximal, complemented
11#4792C212yesmaximal, complemented
12#3137—31nomaximal, Sylow
13C22×C3×Q8C2211yescharacteristic
14C2×#235C2211yescharacteristic
15C22×C3×C8C2211yescharacteristic
16C2×C3×C4⋊C4C2211yescharacteristic
17#947C2212yes—
18#946—21no—
19#946—21no—
20#944C2212yes—
21#929—22no—
22#926C2211yescharacteristic
23#926C2211yescharacteristic
24#924—22no—
25C2×#208C2211yescharacteristic
26#890—21no—
27#890C2212yes—
28#888—21no—
29#888C2212yes—
30C3×#103C2212yes—
31C3×#103C2212yes—
32#835C2212yes—
33#835C2212yes—
34#821C2211yescharacteristic
35#821C2211yescharacteristic
36#816C2211yescharacteristic
37#816C2211yescharacteristic
38#815C2212yes—
39C8×Dic3C2212yes—
40#810C2212yes—
41#809C2212yes—
42#544—31no—
43#536—31no—
44#491—32no—
45#489—32no—
46#488—32no—
47#487—32no—
48#419—31no—
49#417—31no—
50#416—31no—
51C2×#103D311yescharacteristic, complemented
52#405—31no—
53C2×C3×Q8—21no—
54C2×C3×Q8—22no—
55C2×C3×Q8—22no—
56C2×C3×Q8C2312yes—
57C22×C3×C4—21no—
58C22×C3×C4—21no—
59C22×C3×C4C2311yescharacteristic
60C2×#67—21no—
61#235—22no—
62#235—22no—
63C2×D12C2311yescharacteristic
64C2×C4×D3—21no—
65C2×Dic6—22no—
66C2×Dic6C2311yescharacteristic
67C2×C3×C8—22no—
68C2×C3×C8C2311yescharacteristic
69C2×C3×C8C2311yescharacteristic
70C3×C4⋊C4—21no—
71C3×C4⋊C4C2312yes—
72#217—21no—
73#217D412yes—
74(C2×D3)⋊C4—22no—
75(C2×D3)⋊C4—22no—
76C4⋊Dic3—22no—
77C4⋊Dic3D411yescharacteristic
78C4⋊Dic3D411yescharacteristic
79C4⋊Dic3C2311yescharacteristic
80C4⋊Dic3C2311yescharacteristic
81Dic3⋊C4—22no—
82Dic3⋊C4—22no—
83Dic3⋊C4—22no—
84Dic3⋊C4—22no—
85C4×Dic3—21no—
86C4×Dic3C2312yes—
87#208—22no—
88C2×C3⋊C8C2312yes—
89C2×#41—31no—
90C22×Q8C2×D311yescharacteristic
91C2×M16—31no—
92C22×C8C2×D311yescharacteristic
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#117—31no—
105C2×C4⋊C4C2×D311yescharacteristic
106#103—32no—
107#103—32no—
108#103—31no—
109#103—31no—
110#103—32no—
111#103C2×D312yes—
112#103C2×D312yes—
113#102—31no—
114#102—31no—
115#102—32no—
116#97—32no—
117C4×C8—32no—
118C23×C3C2×D411yescharacteristic
119C22×D3—21no—
120C3×Q8—22no—
121C3×Q8—22no—
122C3×Q8—42no—
123C2×C3×C4—22no—
124C2×C3×C4—22no—
125C2×C3×C4—42no—
126C2×C3×C4—81no—
127C2×C3×C4C2×D411yescharacteristic
128C2×C3×C4#4112yes—
129C2×C3×C4#4112yes—
130C2×C3×C4C2411yescharacteristic
131#67—42no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3C2×D412yes—
136C2×Dic3C2×D412yes—
137D12—21no—
138D12—21no—
139C4×D3—42no—
140Dic6—21no—
141Dic6—21no—
142Dic6—82no—
143C3×C8—21no—
144C3×C8—21no—
145C3⋊C8—22no—
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—21no—
158C22×C4—61no—
159C22×C4C22×D311yescharacteristic
160M16—62no—
161C2×C8—22no—
162C2×C8—32no—
163C2×C8C22×D311yescharacteristic
164C2×C8C22×D311yescharacteristic
165C4⋊C4—21no—
166C4⋊C4—31no—
167C4⋊C4—31no—
168C4⋊C4—31no—
169C4⋊C4—31no—
170C4⋊C4—62no—
171C4⋊C4—62no—
172C4⋊C4—62no—
173C4⋊C4—62no—
174C4⋊C4—62no—
175C4⋊C4C22×D312yes—
176C22⋊C4—32no—
177C22⋊C4—62no—
178C22⋊C4—62no—
179C22⋊C4—61no—
180C42—32no—
181C42—61no—
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—42no—
191C3×C4#12412yes—
192C3×C4C2×#4111yescharacteristic
193C3×C4C2×#4111yescharacteristic
194Dic3—22no—
195Dic3—22no—
196Dic3—41no—
197Dic3—41no—
198Dic3—41no—
199C23—61no—
200C23D3×D411yescharacteristic
201Q8—22no—
202Q8—22no—
203Q8—42no—
204Q8—61no—
205Q8—61no—
206Q8—242no—
207D4—61no—
208D4—61no—
209D4—122no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—32no—
213C2×C4—32no—
214C2×C4—42no—
215C2×C4—61no—
216C2×C4—61no—
217C2×C4—61no—
218C2×C4—81no—
219C2×C4—122no—
220C2×C4#23512yes—
221C2×C4D3×D411yescharacteristic
222C2×C4#23712yes—
223C2×C4C23×D311yescharacteristic
224C8—21no—
225C8—21no—
226C8—62no—
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—42no—
241C4—62no—
242C4—62no—
243C4—121no—
244C4—121no—
245C4—121no—
246C4#88512yes—
247C4C2×#23511yescharacteristic
248C4C2×#23711yescharacteristic
249C3#313711yescharacteristic, complemented, Sylow
250C2—22no—
251C2—241no—
252C2C2×#88511yescharacteristic
253C2#576211yescharacteristic
254C2#576611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#103C2
socleC22×C3C22×D4
prime radicalC2×#235C22
hypercenterC2×#103D3
hypocenterC3#3137
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
2. #5500C2semidirect
3. C2×C3×#103C2semidirect
4. #5225C2semidirect
6. #5101C2semidirect
8. #4800C2semidirect
9. #4797C2semidirect
10. #4794C2semidirect
11. #4792C2semidirect
51. C2×#103D3semidirect
249. C3#3137semidirect

Maximal subgroups

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

subgroupindex
#58162
#55002
C2×C3×#1032
#52252
#51132
#51012
#49662
#48002
#47972
#47942
#47922
#31373