Finite Group Databasebeta

← #10698431

Subgroups — #10698431

Order 768.

406 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 77 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×#2608C211yesmaximal, characteristic
2#69534C211yesmaximal, characteristic
3#69442C212yesmaximal
4#67629C211yesmaximal, characteristic
5#67616C212yesmaximal
6#10389—31nomaximal, Sylow
7C2×#926C2211yescharacteristic
8C3×#396—22no—
9C3×#389—22no—
10C3×#387—22no—
11C3×#381C2211yescharacteristic
12C3×C4×C22⋊C4C2212yes—
13#5211C2211yescharacteristic
14#5207C2212yes—
15#4527—22no—
16#4523—22no—
17#4523—22no—
18#2608D311yescharacteristic
19#1826—32no—
20#1820—31no—
21#1306—31no—
22#1284—32no—
23C23×C3×C4C2311yescharacteristic
24C2×C3×C4⋊C4—22no—
25C2×C3×C4⋊C4—22no—
26C2×C3×C4⋊C4—22no—
27C2×C3×C22⋊C4—22no—
28C2×C3×C22⋊C4D412yes—
29C2×C3×C42D412yes—
30C2×C3×C42D412yes—
31C2×#217—21no—
32C2×#217—21no—
33#926—22no—
34#926—22no—
35C2×C4⋊Dic3Q812yes—
36C2×C4×Dic3—21no—
37C2×#208—22no—
38C3×#95—22no—
39C3×#95—22no—
40C3×#95—22no—
41C3⋊#95—22no—
42#830—22no—
43#830—22no—
44C2×#117—31no—
45#407—32no—
46#396—22no—
47#389—22no—
48#387—22no—
49#381—31no—
50#381C2×D311yescharacteristic
51C4×C22⋊C4C2×D312yes—
52#342—62no—
53#339—62no—
54#339—62no—
55C24×C3C2×D411yescharacteristic
56C22×C3×C4—22no—
57C22×C3×C4—22no—
58C22×C3×C4—22no—
59C22×C3×C4—42no—
60C22×C3×C4C2×D412yes—
61C22×C3×C4C2×Q811yescharacteristic
62C22×C3×C4#4112yes—
63C22×C3×C4#4111yescharacteristic
64C22×Dic3—22no—
65C22×Dic3—41no—
66C3×C4⋊C4—44no—
67C3×C4⋊C4—44no—
68C3×C4⋊C4—82no—
69C3×C22⋊C4—44no—
70C3×C22⋊C4—42no—
71C3×C42—42no—
72C3×C42—42no—
73#217—41no—
74#217—41no—
75#217—42no—
76C4⋊Dic3—24no—
77C4×Dic3—42no—
78#208—42no—
79#208—42no—
80C2×C3⋊C8—42no—
81C23×C4C22×D311yescharacteristic
82C2×M16—62no—
83#117—62no—
84#117—62no—
85C2×C4⋊C4—22no—
86C2×C4⋊C4—22no—
87C2×C4⋊C4—22no—
88C2×C4⋊C4—32no—
89C2×C22⋊C4—22no—
90C2×C22⋊C4—61no—
91C2×C22⋊C4—61no—
92C2×C22⋊C4#6712yes—
93C2×C42—61no—
94C2×C42#6712yes—
95C2×C42#6712yes—
96C22⋊C8—62no—
97C22⋊C8—62no—
98#95—22no—
99#95—22no—
100#95—22no—
101#95—62no—
102C23×C3—22no—
103C23×C3—42no—
104C23×C3#12011yescharacteristic
105C23×C3C22⋊Q811yescharacteristic
106C23×C3#12411yescharacteristic
107C2×C3×C4—22no—
108C2×C3×C4—22no—
109C2×C3×C4—24no—
110C2×C3×C4—41no—
111C2×C3×C4—41no—
112C2×C3×C4—42no—
113C2×C3×C4—42no—
114C2×C3×C4—42no—
115C2×C3×C4—42no—
116C2×C3×C4—42no—
117C2×C3×C4—42no—
118C2×C3×C4—42no—
119C2×C3×C4—42no—
120C2×C3×C4—42no—
121C2×C3×C4C22⋊Q812yes—
122C2×C3×C4#12412yes—
123C2×Dic3—42no—
124C2×Dic3—82no—
125C2×Dic3—81no—
126C2×Dic3—82no—
127C3⋊C8—82no—
128C24C2×#6711yescharacteristic
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—22no—
132C22×C4—42no—
133C22×C4—62no—
134C22×C4—121no—
135C22×C4D3×Q811yescharacteristic
136C22×C4#23912yes—
137C22×C4#23911yescharacteristic
138C22×C4C2×#6712yes—
139M16—122no—
140M16—122no—
141C2×C8—122no—
142C4⋊C4—44no—
143C4⋊C4—44no—
144C4⋊C4—64no—
145C4⋊C4—82no—
146C22⋊C4—44no—
147C22⋊C4—42no—
148C22⋊C4—121no—
149C22⋊C4—121no—
150C22⋊C4—122no—
151C42—42no—
152C42—42no—
153C42—122no—
154C22×C3—22no—
155C22×C3—21no—
156C22×C3—21no—
157C22×C3—22no—
158C22×C3—22no—
159C22×C3—82no—
160C22×C3#39311yescharacteristic
161C22×C3#45812yes—
162C3×C4—22no—
163C3×C4—22no—
164C3×C4—44no—
165C3×C4—82no—
166C3×C4—82no—
167Dic3—82no—
168Dic3—161no—
169C23—22no—
170C23—42no—
171C23#94611yescharacteristic
172C23#94711yescharacteristic
173C23#95311yescharacteristic
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—24no—
177C2×C4—41no—
178C2×C4—41no—
179C2×C4—42no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—122no—
189C2×C4—242no—
190C2×C4—241no—
191C2×C4—242no—
192C2×C4#94612yes—
193C2×C4#94712yes—
194C8—242no—
195C2×C3—22no—
196C2×C3—41no—
197C2×C3—41no—
198C2×C3#203011yescharacteristic
199C2×C3#203112yes—
200C22—22no—
201C22—22no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—82no—
206C22#523111yescharacteristic
207C22#524312yes—
208C4—22no—
209C4—22no—
210C4—44no—
211C4—82no—
212C4—82no—
213C4—242no—
214C4—481no—
215C3#1038911yescharacteristic, complemented, Sylow
216C2—22no—
217C2—41no—
218C2—41no—
219C2#7184212yes—
220C2#7189411yescharacteristic
subgroupG/H
centerC22#5231
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2608C2
socleC22×C3#393
prime radicalC24×C3C2×D4
hypercenter#2608D3
hypocenterC3#10389
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2608

Complements

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

NH ≅ G/Ntype
215. C3#10389semidirect

Maximal subgroups

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

subgroupindex
C3×#26082
#695342
#694422
#676292
#676162
#103893