Finite Group Databasebeta

← #10692578

Subgroups — #10692578

Order 768.

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

normal subgroups only

This group has 574 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×#2548C211yesmaximal, characteristic
2#82435C211yesmaximal, characteristic, complemented
3#71759C212yesmaximal
4#71637C211yesmaximal, characteristic, complemented
5#67620C212yesmaximal, complemented
6#9398—31nomaximal, Sylow
7C2×C3×#127C2211yescharacteristic
8C2×C3×#124—21no—
9C2×C3×C4×Q8C2211yescharacteristic
10C3×#394C414yes—
11C3×#390—21no—
12C3×#390—22no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#836—21no—
15Q8×C3⋊C8—24no—
16#4925—22no—
17#4925C2212yes—
18#4924C2212yes—
19#4544—24no—
20#4530—21no—
21#2982—31no—
22#2548D311yescharacteristic
23#1801—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C3×#127—22no—
31C3×#127—22no—
32C3×#124—42no—
33C3×#124—42no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D411yescharacteristic
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—24no—
41C2×C3×C42—21no—
42C2×C3×C42D412yes—
43C2×C3×C42C2311yescharacteristic
44C2×#208—22no—
45C3×#95C2×C412yes—
46#836—42no—
47#830—22no—
48C4⋊(C3⋊C8)—24no—
49C4⋊(C3⋊C8)—22no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)D414yes—
52C4×C3⋊C8—24no—
53C4×C3⋊C8—22no—
54C2×#127C2×D311yescharacteristic
55C2×#124—21no—
56C2×C4×Q8C2×D311yescharacteristic
57C8×Q8—64no—
58#431—32no—
59#431—62no—
60#423—32no—
61C2×#101—61no—
62#394Dic314yes—
63#390—21no—
64#390—22no—
65C4×C4⋊C4C2×D311yescharacteristic
66#343—61no—
67#331—64no—
68C24×C3—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8D812yes—
72C2×C3×Q8SD1612yes—
73C2×C3×D4—24no—
74C2×C3×D4—44no—
75C2×C3×D4—41no—
76C2×C3×D4—42no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—42no—
81C2×C3×D4—82no—
82C22×C3×C4—22no—
83C22×C3×C4C22⋊C412yes—
84C22×C3×C4C22⋊C412yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C3×C4⋊C4—24no—
89C3×C4⋊C4—22no—
90C3×C4⋊C4—42no—
91C3×C22⋊C4—48no—
92C3×C22⋊C4—84no—
93C3×C42—22no—
94C3×C42—22no—
95C3×C42—41no—
96C3×C42—41no—
97C3×C42C2×D412yes—
98C3×C42#4112yes—
99#208—42no—
100#208—42no—
101C2×C3⋊C8—24no—
102C2×C3⋊C8—42no—
103C22×Q8#6711yescharacteristic
104C22×D4—21no—
105C22×D4—21no—
106C22×D4C2×Dic312yes—
107C2×M16—62no—
108#127—22no—
109#127—22no—
110#124—42no—
111#124—42no—
112C4×Q8—22no—
113C4×Q8—22no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4C2×Dic312yes—
116C2×C4⋊C4#6711yescharacteristic
117C2×C22⋊C4—24no—
118C2×C22⋊C4—24no—
119C2×C42—21no—
120C2×C42#6712yes—
121C2×C42C22×D311yescharacteristic
122C4⋊C8—34no—
123C4⋊C8—64no—
124C4⋊C8—62no—
125C4⋊C8—62no—
126#101—122no—
127C22⋊C8—62no—
128C4×C8—64no—
129C4×C8—62no—
130#95C2×Dic312yes—
131C23×C3—44no—
132C23×C3—44no—
133C23×C3—42no—
134C23×C3—42no—
135C23×C3—42no—
136C23×C3C2×C22⋊C411yescharacteristic
137C3×Q8—24no—
138C3×Q8—24no—
139C3×D4—84no—
140C3×D4—82no—
141C3×D4—82no—
142C3×D4—82no—
143C3×D4—82no—
144C2×C3×C4—22no—
145C2×C3×C4—22no—
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—41no—
149C2×C3×C4—42no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4#10214yes—
154C2×C3×C4C4×D412yes—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12311yescharacteristic
158C2×C3×C4C2×D811yescharacteristic
159C2×C3×C4C2×SD1611yescharacteristic
160C2×C3×C4#13512yes—
161C3⋊C8—44no—
162C3⋊C8—82no—
163C24—22no—
164C2×Q8—22no—
165C2×Q8—22no—
166C2×Q8#21312yes—
167C2×Q8#21412yes—
168C2×D4—24no—
169C2×D4—44no—
170C2×D4—41no—
171C2×D4—41no—
172C2×D4—41no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—82no—
177C22×C4—22no—
178C22×C4#21712yes—
179C22×C4#21712yes—
180C22×C4C22×Dic311yescharacteristic
181C22×C4C2×#6711yescharacteristic
182C22×C4C2×#6711yescharacteristic
183M16—122no—
184M16—122no—
185C2×C8—64no—
186C2×C8—122no—
187C4⋊C4—24no—
188C4⋊C4—22no—
189C4⋊C4—42no—
190C22⋊C4—48no—
191C22⋊C4—84no—
192C42—22no—
193C42—22no—
194C42—41no—
195C42—41no—
196C42D3×D412yes—
197C42#23712yes—
198C22×C3—21no—
199C22×C3—21no—
200C22×C3—84no—
201C22×C3—84no—
202C22×C3—82no—
203C22×C3—82no—
204C22×C3—82no—
205C22×C3#38611yescharacteristic
206C22×C3C2×#10211yescharacteristic
207C22×C3#41611yescharacteristic
208C3×C4—21no—
209C3×C4—24no—
210C3×C4—22no—
211C3×C4—42no—
212C3×C4—82no—
213C3×C4#45412yes—
214C3×C4#46512yes—
215C3×C4#46612yes—
216C23—44no—
217C23—44no—
218C23—42no—
219C23—42no—
220C23—42no—
221C23C2×#21711yescharacteristic
222Q8—24no—
223Q8—24no—
224D4—84no—
225D4—82no—
226D4—82no—
227D4—82no—
228D4—82no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—22no—
233C2×C4—41no—
234C2×C4—42no—
235C2×C4—42no—
236C2×C4—42no—
237C2×C4—42no—
238C2×C4#83214yes—
239C2×C4C2×#21311yescharacteristic
240C2×C4C2×#21411yescharacteristic
241C2×C4D4×Dic312yes—
242C2×C4#93511yescharacteristic
243C2×C4#93711yescharacteristic
244C2×C4#93912yes—
245C2×C4#95012yes—
246C8—124no—
247C8—242no—
248C2×C3—21no—
249C2×C3—21no—
250C2×C3—162no—
251C2×C3#188811yescharacteristic
252C2×C3#189311yescharacteristic
253C2×C3#191411yescharacteristic
254C22—21no—
255C22—21no—
256C22—84no—
257C22—84no—
258C22—82no—
259C22—82no—
260C22—82no—
261C22C2×#83211yescharacteristic
262C22#522011yescharacteristic
263C22#523311yescharacteristic
264C4—24no—
265C4—21no—
266C4—22no—
267C4—42no—
268C4—82no—
269C4#503812yes—
270C4#504112yes—
271C4#506212yes—
272C3#939811yescharacteristic, complemented, Sylow
273C2—21no—
274C2—21no—
275C2—162no—
276C2#7168111yescharacteristic
277C2#7172411yescharacteristic
278C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2548C2
socleC22×C3#386
prime radicalC2×C3×#127C22
cosocleC2×C3×C42C23
hypercenter#2548D3
hypocenterC3#9398
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. #82435C2semidirect
4. #71637C2semidirect
5. #67620C2semidirect
272. C3#9398semidirect

Maximal subgroups

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

subgroupindex
C3×#25482
#824352
#717592
#716372
#676202
#93983