Finite Group Databasebeta

← #10708209

Subgroups — #10708209

Order 768.

342 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 63 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
2#69713C212yesmaximal
3#69543C211yesmaximal, characteristic
4#67650C212yesmaximal
5#67632C211yesmaximal, characteristic
6#12251—31nomaximal, Sylow
7C2×C3×#125—21no—
8#5500C2211yescharacteristic
9C3×#400—22no—
10C3×#398—21no—
11C3×#398—22no—
12C3×#398C2212yes—
13C3×#382C2211yescharacteristic
14#4971—22no—
15#4964—21no—
16#4530C2211yescharacteristic
17#4528C2212yes—
18#4523—21no—
19#4452—22no—
20#2822D311yescharacteristic
21#2061—31no—
22#1611—32no—
23#1346—32no—
24#1324—31no—
25C3×#125—42no—
26C3×#125—42no—
27C2×C3×C4⋊C4—21no—
28C2×C3×C4⋊C4—22no—
29C2×C3×C4⋊C4—21no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—21no—
32C2×C3×C4⋊C4D411yescharacteristic
33C2×C3×C4⋊C4D412yes—
34C2×C3×C4⋊C4C2311yescharacteristic
35C2×C3×C42D411yescharacteristic
36#926—21no—
37#924—22no—
38C2×#208—21no—
39#890—41no—
40#888—41no—
41C3×#95—22no—
42C3×#95—21no—
43C3×#95—22no—
44C3×#95—22no—
45C3×#95D412yes—
46#830—22no—
47#810—42no—
48#808—41no—
49#808—42no—
50#807—41no—
51#544—31no—
52C2×#125—21no—
53#483—62no—
54#428—61no—
55#400—22no—
56#398—21no—
57#398—22no—
58#398C2×D312yes—
59#382C2×D311yescharacteristic
60#343—31no—
61#339—61no—
62#328—62no—
63#324—32no—
64C22×C3×C4—21no—
65C22×C3×C4—21no—
66C22×C3×C4—22no—
67C22×C3×C4—22no—
68C22×C3×C4C2×D411yescharacteristic
69C22×C3×C4C2×D412yes—
70C2×Dic6—22no—
71C3×C4⋊C4—41no—
72C3×C4⋊C4—41no—
73C3×C4⋊C4—42no—
74C3×C4⋊C4—42no—
75C3×C4⋊C4—44no—
76C3×C4⋊C4—42no—
77C3×C4⋊C4—42no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—81no—
80C3×C42—21no—
81C3×C42SD1612yes—
82#217—41no—
83C4⋊Dic3—21no—
84C4⋊Dic3—21no—
85C4⋊Dic3—22no—
86Dic3⋊C4—42no—
87C4×Dic3—41no—
88#208—22no—
89#208—41no—
90C2×C3⋊C8—41no—
91C2×C3⋊C8—42no—
92C2×M16—61no—
93C4⋊Q8—121no—
94#125—42no—
95#125—42no—
96#125—121no—
97C22⋊Q8—62no—
98#117—61no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—22no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4#6711yescharacteristic
105C2×C4⋊C4#6712yes—
106C2×C4⋊C4C22×D311yescharacteristic
107C2×C42#6711yescharacteristic
108#107—121no—
109#106—121no—
110#106—122no—
111#103—122no—
112C22⋊C8—62no—
113#95—22no—
114#95—21no—
115#95—22no—
116#95—22no—
117#95#6712yes—
118C23×C3#12011yescharacteristic
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—22no—
122C2×C3×C4—22no—
123C2×C3×C4—41no—
124C2×C3×C4—41no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4—42no—
128C2×C3×C4—42no—
129C2×C3×C4—42no—
130C2×C3×C4—42no—
131C2×C3×C4—42no—
132C2×C3×C4—42no—
133C2×C3×C4—41no—
134C2×C3×C4C2×SD1611yescharacteristic
135C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
136C2×Dic3—41no—
137C2×Dic3—41no—
138Dic6—82no—
139C3⋊C8—42no—
140C3⋊C8—82no—
141C2×Q8—62no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4C2×#6711yescharacteristic
147C22×C4C2×#6712yes—
148M16—62no—
149M16—121no—
150C2×C8—121no—
151C2×C8—122no—
152C4⋊C4—41no—
153C4⋊C4—42no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—42no—
160C4⋊C4—61no—
161C4⋊C4—62no—
162C4⋊C4—61no—
163C4⋊C4—81no—
164C4⋊C4—122no—
165C22⋊C4—121no—
166C42—21no—
167C42—121no—
168C42#21412yes—
169C22×C3—21no—
170C22×C3—21no—
171C22×C3#45011yescharacteristic
172C22×C3#45611yescharacteristic
173C22×C3B(4,𝔽2)11yescharacteristic
174C3×C4—21no—
175C3×C4—21no—
176C3×C4—42no—
177C3×C4—42no—
178C3×C4—82no—
179C3×C4—82no—
180C3×C4—81no—
181Dic3—81no—
182Dic3—81no—
183C23#95311yescharacteristic
184Q8—242no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—42no—
199C2×C4—42no—
200C2×C4—121no—
201C2×C4—121no—
202C2×C4#93211yescharacteristic
203C2×C4C2×#21411yescharacteristic
204C8—122no—
205C8—242no—
206C2×C3—21no—
207C2×C3—21no—
208C2×C3#160411yescharacteristic
209C2×C3#220111yescharacteristic
210C2×C3#220311yescharacteristic
211C22—21no—
212C22—21no—
213C22#521811yescharacteristic
214C22#524411yescharacteristic
215C22#524611yescharacteristic
216C4—21no—
217C4—21no—
218C4—42no—
219C4—42no—
220C4—82no—
221C4—81no—
222C4—82no—
223C4—241no—
224C4—241no—
225C3#1225111yescharacteristic, complemented, Sylow
226C2—21no—
227C2—21no—
228C2#7188011yescharacteristic
229C2#7195811yescharacteristic
230C2#7195911yescharacteristic
subgroupG/H
centerC22#5244
commutator, cosocleC2×C3×C4⋊C4C23
FrattiniC2×C4⋊C4C22×D3
FittingC3×#2822C2
socleC22×C3B(4,𝔽2)
prime radicalC23×C3#120
hypercenter#2822D3
hypocenterC3#12251
LCSG ▷ C2×C3×C4⋊C4 ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2822

Complements

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

NH ≅ G/Ntype
225. C3#12251semidirect

Maximal subgroups

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

subgroupindex
C3×#28222
#697132
#695432
#676502
#676322
#122513