Finite Group Databasebeta

← #10697413

Subgroups — #10697413

Order 768.

468 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 79 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
1#83274C212yesmaximal
2#79069C211yesmaximal, characteristic, complemented
3C3×#1963C211yesmaximal, characteristic
4#69433C212yesmaximal, complemented
5#67716C211yesmaximal, characteristic, complemented
6#10216—31nomaximal, Sylow
7C3×#532C2211yescharacteristic
8#5470C2212yes—
9C3×#423C2211yescharacteristic
10C2×C3×#104—22no—
11C3×#376C2211yescharacteristic
12#5137—22no—
13#5081—22no—
14#5071—22no—
15#5035—24no—
16#5008—22no—
17#5007—22no—
18#4926C2212yes—
19#4851—22no—
20#4849—22no—
21#4452—24no—
22#3255—31no—
23#3235—32no—
24#1963D311yescharacteristic
25#1506—32no—
26#1307—31no—
27C2×C3×#41—21no—
28C2×C3×M16D412yes—
29C3×C4⋊Q8D411yescharacteristic
30C3×#127D411yescharacteristic
31C3×#124—21no—
32C3×#121—22no—
33C3×C4×D4—22no—
34C2×C3×C42—21no—
35C2×C3×C42C2311yescharacteristic
36#926—22no—
37#924—22no—
38#924—22no—
39C2×Dic12—22no—
40#870—22no—
41#869D412yes—
42#869D412yes—
43C4×Dic6—22no—
44C4×Dic6—22no—
45C3×C4⋊C8—22no—
46C3×#104—44no—
47C3×#95—22no—
48#832—24no—
49#832—24no—
50#830D414yes—
51#817—22no—
52#816—24no—
53#808—24no—
54#807—24no—
55C4⋊(C3⋊C8)—22no—
56#803—22no—
57#533—32no—
58#532C2×D311yescharacteristic
59#501—62no—
60#490—62no—
61#482—64no—
62#476—62no—
63#475—62no—
64#474—62no—
65#470—62no—
66#462—62no—
67#432—32no—
68#423C2×D311yescharacteristic
69C2×#104—22no—
70#376C2×D311yescharacteristic
71#328—64no—
72C3×#41—42no—
73C3×#41—42no—
74C2×C3×Q8—21no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—41no—
78C22×C3×C4—41no—
79C22×C3×C4—41no—
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D411yescharacteristic
82C2×Dic6—22no—
83C2×Dic6—22no—
84C3×M16—42no—
85C2×C3×C8—22no—
86C3×C4⋊C4—22no—
87C3×C22⋊C4—42no—
88C3×C42—22no—
89C3×C42—41no—
90C3×C42C2×D411yescharacteristic
91C3×C42C2×D412yes—
92C3×C42#4111yescharacteristic
93#217—42no—
94C4⋊Dic3—22no—
95C4⋊Dic3—22no—
96C4⋊Dic3—22no—
97C4⋊Dic3—22no—
98Dic3⋊C4—42no—
99Dic3⋊C4—42no—
100C4×Dic3—42no—
101C2×C3⋊C8—24no—
102Dic12—82no—
103C2×#41—21no—
104C2×Q16—62no—
105C2×M16#6712yes—
106C4⋊Q8—32no—
107C4⋊Q8—32no—
108C4⋊Q8D1211yescharacteristic
109#127D1211yescharacteristic
110#125—62no—
111#124—21no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114#121—22no—
115C4×Q8—62no—
116C4×Q8—62no—
117C4×D4—22no—
118#117—62no—
119C2×C42—21no—
120C2×C42C22×D311yescharacteristic
121#107—64no—
122#106—64no—
123#106—62no—
124C4⋊C8—22no—
125C4⋊C8—62no—
126#104—44no—
127#103—64no—
128#102—64no—
129#102—64no—
130C22⋊C8—34no—
131#97—62no—
132#95—22no—
133C23×C3—41no—
134C23×C3#12711yescharacteristic
135C3×Q8—42no—
136C3×D4—44no—
137C3×D4—42no—
138C3×D4—82no—
139C2×C3×C4—21no—
140C2×C3×C4—21no—
141C2×C3×C4—42no—
142C2×C3×C4—42no—
143C2×C3×C4—41no—
144C2×C3×C4—82no—
145C2×C3×C4—81no—
146C2×C3×C4—81no—
147C2×C3×C4#12012yes—
148C2×C3×C4#12011yescharacteristic
149C2×C3×C4#12112yes—
150C2×C3×C4#12111yescharacteristic
151C2×Dic3—42no—
152C2×Dic3—42no—
153Dic6—42no—
154Dic6—42no—
155Dic6—82no—
156C3×C8—42no—
157C3⋊C8—44no—
158#41—42no—
159#41—42no—
160C2×Q8—21no—
161C2×Q8—62no—
162C2×Q8—62no—
163C2×D4—22no—
164C2×D4—21no—
165C2×D4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4D3×D412yes—
169C22×C4C2×#6711yescharacteristic
170Q16—242no—
171M16—42no—
172C2×C8—22no—
173C2×C8—64no—
174C4⋊C4—22no—
175C4⋊C4—62no—
176C4⋊C4—62no—
177C4⋊C4—62no—
178C4⋊C4—62no—
179C4⋊C4—122no—
180C4⋊C4—122no—
181C22⋊C4—42no—
182C22⋊C4—122no—
183C42—22no—
184C42—41no—
185C42—122no—
186C42C2×D1211yescharacteristic
187C42#23511yescharacteristic
188C42D3×D412yes—
189C22×C3—21no—
190C22×C3—22no—
191C22×C3—81no—
192C22×C3—82no—
193C22×C3#39211yescharacteristic
194C3×C4—21no—
195C3×C4—22no—
196C3×C4—22no—
197C3×C4—82no—
198C3×C4—81no—
199C3×C4#45311yescharacteristic
200C3×C4#45411yescharacteristic
201Dic3—82no—
202Dic3—82no—
203C23—41no—
204C23#94011yescharacteristic
205Q8—42no—
206Q8—122no—
207Q8—122no—
208Q8—242no—
209D4—42no—
210D4—44no—
211D4—82no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—41no—
217C2×C4—82no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—122no—
221C2×C4—122no—
222C2×C4#88212yes—
223C2×C4#88412yes—
224C2×C4#93011yescharacteristic
225C2×C4#93711yescharacteristic
226C8—42no—
227C8—124no—
228C2×C3—41no—
229C2×C3—161no—
230C2×C3#200611yescharacteristic
231C2×C3#200912yes—
232C22—22no—
233C22—21no—
234C22—82no—
235C22—81no—
236C22#496011yescharacteristic
237C4—22no—
238C4—22no—
239C4—21no—
240C4—82no—
241C4—81no—
242C4—242no—
243C4—242no—
244C4#506211yescharacteristic
245C4#507711yescharacteristic
246C3#1021611yescharacteristic, complemented, Sylow
247C2—41no—
248C2—161no—
249C2#6966411yescharacteristic
250C2#7041212yes—
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1963C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#1963D3
hypocenterC3#10216
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1963

Complements

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

NH ≅ G/Ntype
2. #79069C2semidirect
4. #69433C2semidirect
5. #67716C2semidirect
246. C3#10216semidirect

Maximal subgroups

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

subgroupindex
#832742
#790692
C3×#19632
#694332
#677162
#102163