Finite Group Databasebeta

← #10695115

Subgroups — #10695115

Order 768.

422 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 127 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×#3170C211yesmaximal, characteristic
2#71839C212yesmaximal
3#71826C212yesmaximal
4#69750C211yesmaximal, characteristic
5#69730C211yesmaximal, characteristic
6#9853—31nomaximal, Sylow
7C3×#533C2212yes—
8C3×#500C411yescharacteristic, complemented
9C3×#494C411yescharacteristic, complemented
10C3×#487C412yescomplemented
11C3×#467—22no—
12C3×C4×Q16—22no—
13C2×C3×C4×C8C2211yescharacteristic
14C42×Dic3C2211yescharacteristic
15#4924C2212yes—
16C2×C4×C3⋊C8C2211yescharacteristic
17#4550—22no—
18#4535—24no—
19#4532—24no—
20#4457—22no—
21#3170D311yescharacteristic
22#1994—32no—
23#1744—31no—
24#1566—31no—
25#1493—32no—
26C2×C3×Q16—22no—
27C22×C3×C8D412yes—
28C3×C4⋊Q8C2×C412yes—
29C3×C4⋊Q8C2×C412yes—
30C3×#125—22no—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×#117—22no—
36C2×C3×C42C2311yescharacteristic
37C2×C4×Dic3—21no—
38C2×C4×Dic3—21no—
39C2×C4×Dic3D412yes—
40C2×#208—22no—
41C22×C3⋊C8—21no—
42C3×#107—22no—
43C3×#103—24no—
44C3×C4×C8C2×C411yescharacteristic
45C3×C4×C8C2×C411yescharacteristic
46C3×C4×C8D412yes—
47C4⋊(C3⋊C8)—22no—
48C4⋊(C3⋊C8)—22no—
49C4×C3⋊C8—21no—
50C4×C3⋊C8D412yes—
51#533C2×D312yes—
52#500Dic311yescharacteristic, complemented
53#494Dic311yescharacteristic, complemented
54#487Dic312yescomplemented
55#467—22no—
56C4×Q16—22no—
57#423—32no—
58C2×C4×C8—31no—
59C2×C4×C8C2×D311yescharacteristic
60C43—31no—
61#339—64no—
62#336—62no—
63#334—62no—
64Q8⋊C8—64no—
65C2×C3×Q8—22no—
66C2×C3×Q8—22no—
67C22×C3×C4C2×D411yescharacteristic
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4#4111yescharacteristic
70C22×Dic3—22no—
71C3×Q16—44no—
72C2×C3×C8—42no—
73C2×C3×C8C22⋊C412yes—
74C2×C3×C8C22⋊C412yes—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—22no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—42no—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—42no—
82C3×C42—42no—
83C3×C42C22×C411yescharacteristic
84C3×C42C2×D411yescharacteristic
85C3×C42C2×D411yescharacteristic
86C3×C42#4111yescharacteristic
87C4×Dic3—24no—
88C4×Dic3—24no—
89C4×Dic3—42no—
90#208—44no—
91C2×C3⋊C8—22no—
92C2×C3⋊C8—42no—
93C2×C3⋊C8—42no—
94C2×Q16—22no—
95C2×M16—62no—
96C22×C8—61no—
97C22×C8#6712yes—
98C4⋊Q8C2×Dic312yes—
99C4⋊Q8C2×Dic312yes—
100#125—22no—
101C22⋊Q8—22no—
102C22⋊Q8—22no—
103C4×Q8—22no—
104C4×Q8—22no—
105#117—22no—
106C2×C42—32no—
107C2×C42—61no—
108C2×C42—61no—
109C2×C42C22×D311yescharacteristic
110#107—22no—
111C4⋊C8—62no—
112C4⋊C8—62no—
113#103—24no—
114C4×C8—32no—
115C4×C8—61no—
116C4×C8C2×Dic311yescharacteristic
117C4×C8C2×Dic311yescharacteristic
118C4×C8#6712yes—
119C23×C3#12111yescharacteristic
120C3×Q8—44no—
121C3×Q8—44no—
122C2×C3×C4—22no—
123C2×C3×C4—22no—
124C2×C3×C4—42no—
125C2×C3×C4—42no—
126C2×C3×C4C2×C22⋊C411yescharacteristic
127C2×C3×C4C4×D411yescharacteristic
128C2×C3×C4C4×D411yescharacteristic
129C2×C3×C4#12111yescharacteristic
130C2×C3×C4#12411yescharacteristic
131C2×C3×C4#12711yescharacteristic
132C2×C3×C4#13714yes—
133C2×Dic3—24no—
134C2×Dic3—42no—
135C2×Dic3—42no—
136C3×C8—22no—
137C3×C8—22no—
138C3⋊C8—42no—
139C3⋊C8—82no—
140C2×Q8—22no—
141C2×Q8—22no—
142C22×C4—62no—
143C22×C4D3×D411yescharacteristic
144C22×C4#23711yescharacteristic
145C22×C4C2×#6711yescharacteristic
146Q16—44no—
147M16—124no—
148C2×C8—42no—
149C2×C8—62no—
150C2×C8—122no—
151C2×C8—122no—
152C2×C8#21712yes—
153C2×C8#21712yes—
154C4⋊C4—22no—
155C4⋊C4—22no—
156C4⋊C4—22no—
157C4⋊C4—22no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C22⋊C4—42no—
161C42—42no—
162C42—64no—
163C42—64no—
164C42—122no—
165C42D3×D411yescharacteristic
166C42#23711yescharacteristic
167C42C22×Dic311yescharacteristic
168C42C2×#6711yescharacteristic
169C22×C3—21no—
170C22×C3—21no—
171C22×C3#39011yescharacteristic
172C22×C3#47012yes—
173C3×C4—21no—
174C3×C4—21no—
175C3×C4—82no—
176C3×C4—82no—
177C3×C4#42112yes—
178C3×C4#44112yes—
179C3×C4#44312yes—
180C3×C4#49712yes—
181Dic3—44no—
182C23#93811yescharacteristic
183Q8—44no—
184Q8—44no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—42no—
188C2×C4—42no—
189C2×C4—64no—
190C2×C4—122no—
191C2×C4—122no—
192C2×C4#91814yes—
193C2×C4D4×Dic311yescharacteristic
194C2×C4D4×Dic311yescharacteristic
195C2×C4#93611yescharacteristic
196C2×C4#93811yescharacteristic
197C2×C4#94011yescharacteristic
198C2×C4C2×#21711yescharacteristic
199C8—22no—
200C8—22no—
201C8—122no—
202C8—242no—
203C2×C3—22no—
204C2×C3#196311yescharacteristic
205C2×C3#197111yescharacteristic
206C2×C3#197411yescharacteristic
207C22—21no—
208C22—21no—
209C22#518912yes—
210C22#522111yescharacteristic
211C4—21no—
212C4—21no—
213C4—82no—
214C4—82no—
215C4—124no—
216C4#518412yes—
217C4#519012yes—
218C4#519612yes—
219C4#523612yes—
220C3#985311yescharacteristic, complemented, Sylow
221C2—22no—
222C2#7181611yescharacteristic
223C2#7206711yescharacteristic
224C2#7208411yescharacteristic
subgroupG/H
centerC2×C4#940
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#3170C2
socleC22×C3#390
prime radicalC23×C3#121
cosocleC2×C3×C42C23
hypercenter#3170D3
hypocenterC3#9853
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3170

Complements

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

NH ≅ G/Ntype
8. C3×#500C4semidirect
9. C3×#494C4semidirect
10. C3×#487C4semidirect
52. #500Dic3semidirect
53. #494Dic3semidirect
54. #487Dic3semidirect
220. C3#9853semidirect

Maximal subgroups

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

subgroupindex
C3×#31702
#718392
#718262
#697502
#697302
#98533