Finite Group Databasebeta

← #10696523

Subgroups — #10696523

Order 768.

500 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 97 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×#2822C211yesmaximal, characteristic
2C2×#5064C212yesmaximal
3C2×#4543C212yesmaximal
4#69552C211yesmaximal, characteristic
5#67630C211yesmaximal, characteristic
6#10087—31nomaximal, Sylow
7C2×C3×#125C2212yes—
8C2×#869C2211yescharacteristic
9C3×#400—22no—
10C3×#398—22no—
11C3×#398—22no—
12C3×#382C2211yescharacteristic
13#5064—24no—
14#5064—24no—
15C2×#810—22no—
16C2×#808—22no—
17C2×#807—22no—
18C2×#803C2211yescharacteristic
19C2×#803C2212yes—
20#4543—24no—
21C2×#501—32no—
22#2822D311yescharacteristic
23#1960—31no—
24C2×#330—32no—
25#1295—31no—
26C22×Dic6—21no—
27C3×#125—22no—
28C3×#125—22no—
29C3×#125—22no—
30C3×#125—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4D412yes—
35C2×C3×C4⋊C4D412yes—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×C4⋊Dic3—22no—
39C22×C3⋊C8—22no—
40C22×C3⋊C8—21no—
41#869—22no—
42#869—21no—
43#869D412yes—
44C3×#95—24no—
45C3×#95—22no—
46C3×#95—22no—
47#810—44no—
48#808—28no—
49#807—28no—
50#803—22no—
51#803—22no—
52#803D414yes—
53C2×C4⋊Q8—31no—
54C2×#125C2×D312yes—
55#501—64no—
56#501—64no—
57C2×#107—62no—
58C2×#106—62no—
59C2×#103—62no—
60C2×#97—32no—
61C2×#97—31no—
62#400—22no—
63#398—22no—
64#398—22no—
65#382C2×D311yescharacteristic
66#330—64no—
67C22×C3×C4—22no—
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4C2×D412yes—
71C22×C3×C4C2×D411yescharacteristic
72C22×Dic3—41no—
73C2×Dic6—22no—
74C2×Dic6—42no—
75C2×Dic6—42no—
76C2×Dic6—41no—
77C3×C4⋊C4—22no—
78C3×C4⋊C4—22no—
79C3×C4⋊C4—22no—
80C3×C4⋊C4—22no—
81C3×C4⋊C4—42no—
82C3×C4⋊C4—42no—
83C3×C4⋊C4—42no—
84C3×C4⋊C4—42no—
85C3×C4⋊C4—42no—
86C3×C4⋊C4—82no—
87C3×C42C2×D411yescharacteristic
88C3×C42C2×D412yes—
89C3×C42#4111yescharacteristic
90C4⋊Dic3—24no—
91C4⋊Dic3—42no—
92C2×C3⋊C8—28no—
93C2×C3⋊C8—24no—
94C2×C3⋊C8—22no—
95C2×C3⋊C8—41no—
96C2×C3⋊C8—41no—
97C22×Q8—61no—
98C22×C8—62no—
99C22×C8—61no—
100C4⋊Q8—32no—
101C4⋊Q8—62no—
102C4⋊Q8—61no—
103#125—22no—
104#125—22no—
105#125—22no—
106#125—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—62no—
111C2×C4⋊C4#6712yes—
112C2×C4⋊C4#6712yes—
113C2×C4⋊C4#6712yes—
114C2×C42C22×D311yescharacteristic
115#107—68no—
116#106—68no—
117#103—124no—
118#97—34no—
119#97—62no—
120#97—62no—
121#95—24no—
122#95—22no—
123#95—22no—
124C23×C3#12011yescharacteristic
125C2×C3×C4—21no—
126C2×C3×C4—21no—
127C2×C3×C4—24no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—44no—
134C2×C3×C4—44no—
135C2×C3×C4#12012yes—
136C2×C3×C4#12111yescharacteristic
137C2×C3×C4#12112yes—
138C2×C3×C4#12711yescharacteristic
139C2×Dic3—42no—
140C2×Dic3—81no—
141C2×Dic3—81no—
142Dic6—44no—
143Dic6—82no—
144C3⋊C8—28no—
145C3⋊C8—42no—
146C2×Q8—62no—
147C2×Q8—122no—
148C2×Q8—122no—
149C2×Q8—121no—
150C22×C4—22no—
151C22×C4—22no—
152C22×C4—22no—
153C22×C4—121no—
154C22×C4C2×#6712yes—
155C22×C4C2×#6711yescharacteristic
156C2×C8—68no—
157C2×C8—64no—
158C2×C8—62no—
159C2×C8—121no—
160C2×C8—121no—
161C4⋊C4—22no—
162C4⋊C4—22no—
163C4⋊C4—22no—
164C4⋊C4—22no—
165C4⋊C4—42no—
166C4⋊C4—42no—
167C4⋊C4—42no—
168C4⋊C4—42no—
169C4⋊C4—42no—
170C4⋊C4—64no—
171C4⋊C4—82no—
172C4⋊C4—122no—
173C42D3×D411yescharacteristic
174C42D3×D412yes—
175C42#23711yescharacteristic
176C22×C3#39211yescharacteristic
177C22×C3#45311yescharacteristic
178C22×C3#45411yescharacteristic
179C22×C3#45614yes—
180C3×C4—22no—
181C3×C4—22no—
182C3×C4—21no—
183C3×C4—21no—
184C3×C4—82no—
185C3×C4—82no—
186C3×C4—82no—
187Dic3—82no—
188C23#95311yescharacteristic
189Q8—124no—
190Q8—242no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—24no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—42no—
199C2×C4—44no—
200C2×C4—44no—
201C2×C4—122no—
202C2×C4—241no—
203C2×C4—241no—
204C2×C4#93712yes—
205C2×C4#93811yescharacteristic
206C2×C4#93912yes—
207C2×C4#94011yescharacteristic
208C8—68no—
209C8—122no—
210C2×C3#166012yes—
211C2×C3#166312yes—
212C2×C3#200512yes—
213C2×C3#200611yescharacteristic
214C22#522311yescharacteristic
215C22#524111yescharacteristic
216C22#524511yescharacteristic
217C22#524614yes—
218C4—22no—
219C4—22no—
220C4—21no—
221C4—21no—
222C4—82no—
223C4—82no—
224C4—82no—
225C4—242no—
226C3#1008711yescharacteristic, complemented, Sylow
227C2#7186812yes—
228C2#7187911yescharacteristic
229C2#7195612yes—
230C2#7196112yes—
subgroupG/H
centerC23#953
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2822C2
socle, prime radicalC23×C3#120
hypercenter#2822D3
hypocenterC3#10087
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2822

Complements

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

NH ≅ G/Ntype
226. C3#10087semidirect

Maximal subgroups

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

subgroupindex
C3×#28222
C2×#50642
C2×#45432
#695522
#676302
#100873