Finite Group Databasebeta

← #10714874

Subgroups — #10714874

Order 768.

384 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2809C211yesmaximal, characteristic
2#69672C212yesmaximal
3#69672C211yesmaximal, characteristic
4#67644C211yesmaximal, characteristic
5#67644C212yesmaximal
6#13465—31nomaximal, Sylow
7#5812C2211yescharacteristic
8C3×#400—21no—
9C3×#398—22no—
10C3×#391C2211yescharacteristic
11C3×#391C2212yes—
12C3×#389—21no—
13C3×#387—22no—
14C2×#836C2212yes—
15C2×#836C2211yescharacteristic
16#4964—22no—
17#4964—21no—
18#4523—22no—
19#4523—21no—
20#2809D311yescharacteristic
21#1857—32no—
22#1857—31no—
23#1394—31no—
24#1394—32no—
25C22×C3×Q8C2311yescharacteristic
26C2×C3×C4⋊C4—22no—
27C2×C3×C4⋊C4—21no—
28C2×C3×C4⋊C4—21no—
29C2×C3×C4⋊C4—21no—
30C2×C3×C4⋊C4D412yes—
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C42D412yes—
33C2×C3×C42D411yescharacteristic
34Q8×Dic3—41no—
35Q8×Dic3—41no—
36#926—22no—
37#926—21no—
38C2×#208—22no—
39C2×#208—21no—
40C3×#95—21no—
41C3×#95—22no—
42C3×#95—22no—
43C3×#95—22no—
44C3×#95—22no—
45#836—22no—
46#836—21no—
47#836Q814yes—
48#836Q812yes—
49#808—42no—
50#808—41no—
51#807—42no—
52#807—41no—
53#519—31no—
54#428—62no—
55#428—61no—
56C2×#101—32no—
57C2×#101—31no—
58#400—21no—
59#398—22no—
60#391C2×D311yescharacteristic
61#391C2×D312yes—
62#389—21no—
63#387—22no—
64#339—62no—
65#339—61no—
66C2×C3×Q8—41no—
67C2×C3×Q8—41no—
68C2×C3×Q8C2×Q812yes—
69C2×C3×Q8C2×Q811yescharacteristic
70C2×C3×Q8#4111yescharacteristic
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C3×C4⋊C4—42no—
78C3×C4⋊C4—41no—
79C3×C4⋊C4—44no—
80C3×C4⋊C4—42no—
81C3×C4⋊C4—81no—
82C3×C4⋊C4—81no—
83C3×C42—41no—
84C3×C42—42no—
85#217—41no—
86C4⋊Dic3—21no—
87C4⋊Dic3—22no—
88C4⋊Dic3—22no—
89C4⋊Dic3—21no—
90C4×Dic3—42no—
91C4×Dic3—41no—
92#208—24no—
93#208—22no—
94#208—41no—
95#208—42no—
96C2×C3⋊C8—42no—
97C2×C3⋊C8—41no—
98C22×Q8C22×D311yescharacteristic
99C2×M16—62no—
100C2×M16—61no—
101C4×Q8—121no—
102C4×Q8—121no—
103#117—62no—
104#117—61no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4#6712yes—
110C2×C4⋊C4#6711yescharacteristic
111C2×C42#6712yes—
112C2×C42#6711yescharacteristic
113#107—122no—
114#107—121no—
115#106—122no—
116#106—121no—
117#101—34no—
118#101—32no—
119#101—62no—
120#101—61no—
121#95—21no—
122#95—22no—
123#95—22no—
124#95—22no—
125#95—22no—
126C23×C3#12011yescharacteristic
127C3×Q8—42no—
128C3×Q8—41no—
129C3×Q8—41no—
130C2×C3×C4—22no—
131C2×C3×C4—21no—
132C2×C3×C4—21no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—41no—
136C2×C3×C4—41no—
137C2×C3×C4—42no—
138C2×C3×C4—42no—
139C2×C3×C4—42no—
140C2×C3×C4—42no—
141C2×C3×C4—42no—
142C2×C3×C4—41no—
143C2×C3×C4—42no—
144C2×C3×C4—42no—
145C2×C3×C4C22⋊Q812yes—
146C2×C3×C4C22⋊Q811yescharacteristic
147C2×C3×C4C4⋊Q812yes—
148C2×C3×C4C4⋊Q811yescharacteristic
149C2×Dic3—41no—
150C2×Dic3—41no—
151C3⋊C8—44no—
152C3⋊C8—42no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D3×Q811yescharacteristic
156C2×Q8D3×Q812yes—
157C2×Q8#23911yescharacteristic
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—22no—
161C22×C4—22no—
162C22×C4C2×#6712yes—
163C22×C4C2×#6711yescharacteristic
164M16—64no—
165M16—62no—
166M16—121no—
167M16—122no—
168C2×C8—122no—
169C2×C8—121no—
170C4⋊C4—42no—
171C4⋊C4—41no—
172C4⋊C4—44no—
173C4⋊C4—42no—
174C4⋊C4—62no—
175C4⋊C4—61no—
176C4⋊C4—61no—
177C4⋊C4—62no—
178C4⋊C4—81no—
179C4⋊C4—81no—
180C22⋊C4—121no—
181C42—41no—
182C42—42no—
183C42—122no—
184C42—121no—
185C22×C3—21no—
186C22×C3—21no—
187C22×C3#39511yescharacteristic
188C22×C3B(4,𝔽2)11yescharacteristic
189C22×C3#45811yescharacteristic
190C3×C4—22no—
191C3×C4—21no—
192C3×C4—22no—
193C3×C4—21no—
194C3×C4—42no—
195C3×C4—82no—
196C3×C4—81no—
197C3×C4—82no—
198Dic3—81no—
199Dic3—81no—
200C23#95311yescharacteristic
201Q8—41no—
202Q8—42no—
203Q8—41no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—41no—
218C2×C4—42no—
219C2×C4—121no—
220C2×C4—121no—
221C2×C4#94412yes—
222C2×C4#94411yescharacteristic
223C2×C4#94612yes—
224C2×C4#94611yescharacteristic
225C8—122no—
226C8—124no—
227C2×C3—21no—
228C2×C3—21no—
229C2×C3#206211yescharacteristic
230C2×C3#220311yescharacteristic
231C2×C3#220411yescharacteristic
232C22—21no—
233C22—21no—
234C22#523011yescharacteristic
235C22#524311yescharacteristic
236C22#524411yescharacteristic
237C4—21no—
238C4—22no—
239C4—21no—
240C4—22no—
241C4—42no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—241no—
246C4—241no—
247C3#1346511yescharacteristic, complemented, Sylow
248C2—21no—
249C2—21no—
250C2#7189511yescharacteristic
251C2#7193011yescharacteristic
252C2#7195911yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC22×C3×Q8C23
FrattiniC22×Q8C22×D3
FittingC3×#2809C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2809D3
hypocenterC3#13465
LCSG ▷ C22×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #2809

Complements

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

NH ≅ G/Ntype
247. C3#13465semidirect

Maximal subgroups

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

subgroupindex
C3×#28092
#696722
#676442
#134653