Finite Group Databasebeta

← #10692553

Subgroups — #10692553

Order 768.

558 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 139 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×#2547C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#71773C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic
5#67620C212yesmaximal, complemented
6#9393—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×C3×#124—21no—
9C2×C3×C4×D4C2211yescharacteristic
10C3×#397C414yes—
11C3×#391—21no—
12C3×#380—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#833—21no—
15C2×#830—22no—
16D4×C3⋊C8—24no—
17#4925C2212yes—
18#4924C2212yes—
19#4547—24no—
20#4530—21no—
21#2972—31no—
22#2547D311yescharacteristic
23#1800—31no—
24#1546—32no—
25#1303—32no—
26C22×C3×Q8—21no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4D412yes—
29C3×C4⋊Q8—22no—
30C3×C4⋊Q8—22no—
31C3×#124—42no—
32C3×#124—42no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4C2×C412yes—
41C2×C3×C42—21no—
42C2×C3×C42C2311yescharacteristic
43C2×#208—22no—
44C22×C3⋊C8—24no—
45C3×#95—24no—
46C3×#95C2×C412yes—
47#833—42no—
48#830—24no—
49#830—22no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)D414yes—
53C4×C3⋊C8—22no—
54C2×C4⋊Q8C2×D311yescharacteristic
55C2×#124—21no—
56C2×C4×D4C2×D311yescharacteristic
57C8×D4—64no—
58#431—32no—
59#423—32no—
60C2×#100—61no—
61C2×C22⋊C8—62no—
62#397Dic314yes—
63#391—21no—
64#380—22no—
65C4×C4⋊C4C2×D311yescharacteristic
66#343—61no—
67#332—64no—
68C24×C3C22⋊C412yes—
69C2×C3×Q8—42no—
70C2×C3×Q8—41no—
71C2×C3×Q8—41no—
72C2×C3×Q8—42no—
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4D812yes—
76C2×C3×D4SD1612yes—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—22no—
81C22×C3×C4—44no—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4C22×C411yescharacteristic
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C3×C4⋊C4—22no—
87C3×C4⋊C4—24no—
88C3×C4⋊C4—42no—
89C3×C4⋊C4—41no—
90C3×C4⋊C4—41no—
91C3×C22⋊C4—24no—
92C3×C22⋊C4—48no—
93C3×C42—41no—
94C3×C42—41no—
95C3×C42C2×D412yes—
96C3×C42#4112yes—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—24no—
100C2×C3⋊C8—42no—
101C2×C3⋊C8—84no—
102C22×Q8—21no—
103C22×D4#6711yescharacteristic
104C23×C4#6712yes—
105C2×M16—62no—
106C22×C8—64no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109#124—42no—
110#124—42no—
111C4×D4—22no—
112C4×D4—22no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2×Dic312yes—
116C2×C4⋊C4#6711yescharacteristic
117C2×C22⋊C4—24no—
118C2×C22⋊C4C2×Dic312yes—
119C2×C42—21no—
120C2×C42C22×D311yescharacteristic
121C4⋊C8—34no—
122C4⋊C8—62no—
123C4⋊C8—62no—
124#100—122no—
125C22⋊C8—64no—
126C22⋊C8—62no—
127C4×C8—62no—
128#95—24no—
129#95C2×Dic312yes—
130C23×C3—22no—
131C23×C3—22no—
132C23×C3—42no—
133C23×C3—42no—
134C23×C3#10214yes—
135C23×C3C2×C22⋊C411yescharacteristic
136C3×Q8—82no—
137C3×Q8—82no—
138C3×D4—24no—
139C3×D4—24no—
140C2×C3×C4—22no—
141C2×C3×C4—22no—
142C2×C3×C4—41no—
143C2×C3×C4—44no—
144C2×C3×C4—42no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—82no—
152C2×C3×C4C4×D412yes—
153C2×C3×C4#12011yescharacteristic
154C2×C3×C4#12112yes—
155C2×C3×C4#12311yescharacteristic
156C2×C3×C4C2×D811yescharacteristic
157C2×C3×C4C2×SD1611yescharacteristic
158C2×C3×C4#13512yes—
159C3⋊C8—44no—
160C3⋊C8—82no—
161C24#21712yes—
162C2×Q8—41no—
163C2×Q8—41no—
164C2×Q8—42no—
165C2×Q8—42no—
166C2×D4—22no—
167C2×D4—22no—
168C2×D4#21312yes—
169C2×D4#21412yes—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—44no—
175C22×C4#21712yes—
176C22×C4C22×Dic311yescharacteristic
177C22×C4C2×#6711yescharacteristic
178C22×C4C2×#6711yescharacteristic
179M16—122no—
180M16—122no—
181C2×C8—64no—
182C2×C8—122no—
183C2×C8—244no—
184C4⋊C4—22no—
185C4⋊C4—24no—
186C4⋊C4—42no—
187C4⋊C4—41no—
188C4⋊C4—41no—
189C22⋊C4—24no—
190C22⋊C4—48no—
191C42—41no—
192C42—41no—
193C42D3×D412yes—
194C42#23712yes—
195C22×C3—21no—
196C22×C3—21no—
197C22×C3—24no—
198C22×C3—44no—
199C22×C3—42no—
200C22×C3—42no—
201C22×C3—82no—
202C22×C3#38611yescharacteristic
203C22×C3C2×#10211yescharacteristic
204C22×C3#41611yescharacteristic
205C3×C4—21no—
206C3×C4—22no—
207C3×C4—42no—
208C3×C4—82no—
209C3×C4—82no—
210C3×C4#45512yes—
211C3×C4#46512yes—
212C3×C4#46612yes—
213C23—22no—
214C23—22no—
215C23—42no—
216C23—42no—
217C23#83214yes—
218C23C2×#21711yescharacteristic
219Q8—82no—
220Q8—82no—
221D4—24no—
222D4—24no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—41no—
226C2×C4—44no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—82no—
235C2×C4C2×#21311yescharacteristic
236C2×C4C2×#21411yescharacteristic
237C2×C4D4×Dic312yes—
238C2×C4#93511yescharacteristic
239C2×C4#93711yescharacteristic
240C2×C4#93912yes—
241C2×C4#95012yes—
242C8—124no—
243C8—242no—
244C2×C3—21no—
245C2×C3—21no—
246C2×C3—44no—
247C2×C3#188511yescharacteristic
248C2×C3#189311yescharacteristic
249C2×C3#191511yescharacteristic
250C22—21no—
251C22—21no—
252C22—24no—
253C22—44no—
254C22—42no—
255C22—42no—
256C22—82no—
257C22C2×#83211yescharacteristic
258C22#522011yescharacteristic
259C22#523311yescharacteristic
260C4—21no—
261C4—22no—
262C4—42no—
263C4—82no—
264C4—82no—
265C4#503812yes—
266C4#504112yes—
267C4#506112yes—
268C3#939311yescharacteristic, complemented, Sylow
269C2—21no—
270C2—21no—
271C2—44no—
272C2#7168111yescharacteristic
273C2#7172311yescharacteristic
274C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2547C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenter#2547D3
hypocenterC3#9393
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
3. #71773C2semidirect
5. #67620C2semidirect
268. C3#9393semidirect

Maximal subgroups

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

subgroupindex
C3×#25472
#823012
#717732
#716172
#676202
#93933