Finite Group Databasebeta

← #10708472

Subgroups — #10708472

Order 768.

310 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 71 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×#3249C211yesmaximal, characteristic
2#71079C212yesmaximal
3#70463C211yesmaximal, characteristic
4#67776C212yesmaximal
5#67709C211yesmaximal, characteristic
6#12292—31nomaximal, Sylow
7C3×#533—21no—
8C3×#477—22no—
9C3×#467—21no—
10C3×#467C2212yes—
11C3×#462—22no—
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×#818C2211yescharacteristic
14#4928—21no—
15#4564—22no—
16#4551—21no—
17#4551C2211yescharacteristic
18#4549C2212yes—
19#4489—22no—
20#3249D311yescharacteristic
21#2236—32no—
22#2087—31no—
23#1348—32no—
24#1322—31no—
25C2×C3×Q16—21no—
26C2×C3×Q16D412yes—
27C22×C3×C8D411yescharacteristic
28C22×C3×C8C2311yescharacteristic
29C3×C4⋊Q8—21no—
30C3×C4⋊Q8—21no—
31C3×#125—41no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—41no—
34C3×C4×Q8—22no—
35C3×C4×Q8—41no—
36C3×#117—41no—
37C2×C3×C42D411yescharacteristic
38C2×C4⋊Dic3—21no—
39C2×C4⋊Dic3—21no—
40C2×Dic3⋊C4—41no—
41C3×#107—21no—
42C3×#107D412yes—
43C3×C4⋊C8—22no—
44C3×#103—22no—
45C3×#103—22no—
46C3⋊#95—41no—
47#818—22no—
48C2×C3⋊C16—22no—
49#533—21no—
50#477—22no—
51#467—21no—
52#467C2×D312yes—
53#462—22no—
54C2×#107—31no—
55C2×C4⋊C8C2×D311yescharacteristic
56#389—61no—
57#366—62no—
58#358—62no—
59C22⋊C16—32no—
60#340—31no—
61#340—61no—
62C2×C3×Q8—22no—
63C2×C3×Q8—41no—
64C22×C3×C4D811yescharacteristic
65C22×C3×C4D811yescharacteristic
66C22×C3×C4C2×D411yescharacteristic
67C22×Dic3—41no—
68C22×Dic3—41no—
69C3×Q16—42no—
70C3×Q16—81no—
71C2×C3×C8—21no—
72C2×C3×C8—41no—
73C2×C3×C8—41no—
74C2×C3×C8C2×D412yes—
75C3×C4⋊C4—22no—
76C3×C4⋊C4—41no—
77C3×C4⋊C4—41no—
78C3×C4⋊C4—42no—
79C3×C4⋊C4—41no—
80C3×C4⋊C4—42no—
81C3×C22⋊C4—42no—
82C3×C42—21no—
83C3×C42—21no—
84C3×C42—42no—
85C4⋊Dic3—22no—
86C4⋊Dic3—41no—
87C4⋊Dic3—41no—
88C4⋊Dic3—41no—
89Dic3⋊C4—82no—
90C3⋊C16—42no—
91C2×Q16—21no—
92C2×Q16#6712yes—
93C22×C8#6711yescharacteristic
94C22×C8C22×D311yescharacteristic
95C4⋊Q8—21no—
96C4⋊Q8—21no—
97#125—41no—
98C22⋊Q8—22no—
99C22⋊Q8—41no—
100C4×Q8—22no—
101C4×Q8—41no—
102#117—41no—
103C2×C4⋊C4—61no—
104C2×C4⋊C4—61no—
105C2×C4⋊C4—121no—
106C2×C42#6711yescharacteristic
107C2×C16—62no—
108#107—21no—
109#107—62no—
110#107#6712yes—
111C4⋊C8—22no—
112#103—22no—
113#103—22no—
114#95—121no—
115C23×C3C2×D811yescharacteristic
116C3×Q8—42no—
117C3×Q8—42no—
118C3×Q8—82no—
119C2×C3×C4—21no—
120C2×C3×C4—21no—
121C2×C3×C4—21no—
122C2×C3×C4—21no—
123C2×C3×C4—21no—
124C2×C3×C4—41no—
125C2×C3×C4—42no—
126C2×C3×C4—42no—
127C2×C3×C4Q3212yes—
128C2×C3×C4#12011yescharacteristic
129C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
130C2×Dic3—42no—
131C2×Dic3—81no—
132C2×Dic3—81no—
133C2×Dic3—81no—
134C2×Dic3—81no—
135C2×Dic3—81no—
136C3×C8—22no—
137C3×C8—41no—
138C2×Q8—22no—
139C2×Q8—41no—
140C22×C4—121no—
141C22×C4—121no—
142C22×C4#21311yescharacteristic
143C22×C4#21311yescharacteristic
144C22×C4C2×#6711yescharacteristic
145Q16—42no—
146Q16—81no—
147C2×C8—21no—
148C2×C8—41no—
149C2×C8—41no—
150C2×C8C2×#6712yes—
151C4⋊C4—22no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—42no—
155C4⋊C4—41no—
156C4⋊C4—42no—
157C4⋊C4—62no—
158C4⋊C4—121no—
159C4⋊C4—121no—
160C4⋊C4—121no—
161C4⋊C4—242no—
162C22⋊C4—42no—
163C42—21no—
164C42—21no—
165C42—42no—
166C16—122no—
167C22×C3—21no—
168C22×C3—21no—
169C22×C3#44711yescharacteristic
170C22×C3C2×Q3211yescharacteristic
171C22×C3#51011yescharacteristic
172C3×C4—21no—
173C3×C4—22no—
174C3×C4—41no—
175C3×C4—82no—
176C3×C4—82no—
177C3×C4#45611yescharacteristic
178C3×C4#45811yescharacteristic
179Dic3—82no—
180Dic3—161no—
181C23C2×#21311yescharacteristic
182Q8—42no—
183Q8—42no—
184Q8—82no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—122no—
194C2×C4—241no—
195C2×C4—241no—
196C2×C4—241no—
197C2×C4—241no—
198C2×C4—241no—
199C2×C4#82912yes—
200C2×C4#93211yescharacteristic
201C2×C4#95311yescharacteristic
202C8—22no—
203C8—41no—
204C2×C3—21no—
205C2×C3—21no—
206C2×C3#161011yescharacteristic
207C2×C3#218811yescharacteristic
208C2×C3#219011yescharacteristic
209C22—21no—
210C22—21no—
211C22#517911yescharacteristic
212C22C2×#82911yescharacteristic
213C22#521711yescharacteristic
214C4—22no—
215C4—21no—
216C4—41no—
217C4—82no—
218C4—82no—
219C4—242no—
220C4—481no—
221C4#524311yescharacteristic
222C4#524611yescharacteristic
223C3#1229211yescharacteristic, complemented, Sylow
224C2—21no—
225C2—21no—
226C2#7187411yescharacteristic
227C2#7206411yescharacteristic
228C2#7211211yescharacteristic
subgroupG/H
centerC22#5217
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#3249C2
socleC22×C3#447
prime radicalC23×C3C2×D8
hypercenter#3249D3
hypocenterC3#12292
LCSG ▷ C22×C3×C8 ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C8 ◁ #3249

Complements

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

NH ≅ G/Ntype
223. C3#12292semidirect

Maximal subgroups

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

subgroupindex
C3×#32492
#710792
#704632
#677762
#677092
#122923