Finite Group Databasebeta

← #10692557

Subgroups — #10692557

Order 768.

506 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 506 conjugacy classes of subgroups — too many to draw. Only its 135 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×#2982C211yesmaximal, characteristic
2#82475C211yesmaximal, characteristic
3#71637C211yesmaximal, characteristic
4#70778C212yesmaximal
5#67746C212yesmaximal
6#9394—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#944—21no—
9C2×#869C2211yescharacteristic
10C3×C8×Q8—24no—
11C3×#431—22no—
12C3×#431C2212yes—
13C2×#836—21no—
14#4981C414yes—
15#4973—21no—
16C4×C4⋊Dic3C2211yescharacteristic
17#4928—22no—
18#4924C2212yes—
19#4530—21no—
20#4486—24no—
21#2982D311yescharacteristic
22#2546—31no—
23#1801—31no—
24#1546—32no—
25#1303—32no—
26C22×C3×Q8D411yescharacteristic
27C22×Dic6—21no—
28C3×C4×Q8—22no—
29C3×C4×Q8—22no—
30C2×C3×C4⋊C4C2×C412yes—
31C2×C3×C4⋊C4D411yescharacteristic
32C2×C3×C42D412yes—
33C2×C3×C42C2311yescharacteristic
34#944—42no—
35#944—42no—
36C2×C4⋊Dic3—21no—
37C2×C4⋊Dic3—21no—
38C2×C4⋊Dic3C2×C412yes—
39C2×Dic3⋊C4—24no—
40C2×C4×Dic3—21no—
41C2×#208—22no—
42#869—22no—
43#869—22no—
44C3×C4⋊C8—24no—
45C3×C4⋊C8D414yes—
46C3×C22⋊C8—22no—
47C3×C4×C8—24no—
48C3×C4×C8—22no—
49#836—42no—
50C3⋊#95—24no—
51C3⋊#95C2×C412yes—
52C4⋊(C3⋊C8)—22no—
53C4⋊(C3⋊C8)—22no—
54C2×C4⋊Q8—31no—
55C2×C4⋊Q8—61no—
56C2×C4×Q8C2×D311yescharacteristic
57C8×Q8—24no—
58#431—22no—
59#431C2×D312yes—
60#423—32no—
61C2×#101—61no—
62#398—34no—
63#391—61no—
64#389—62no—
65C4×C4⋊C4—31no—
66#343—61no—
67#332—64no—
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8D812yes—
71C2×C3×Q8SD1612yes—
72C22×C3×C4C22⋊C412yes—
73C22×C3×C4C22⋊C412yes—
74C22×C3×C4C22×C411yescharacteristic
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×Dic3—22no—
78C22×Dic3—22no—
79C2×Dic6—42no—
80C2×Dic6—41no—
81C2×Dic6—41no—
82C2×Dic6—42no—
83C2×C3×C8—24no—
84C3×C4⋊C4—24no—
85C3×C4⋊C4—22no—
86C3×C42—22no—
87C3×C42—22no—
88C3×C42C2×D412yes—
89C3×C42#4112yes—
90C4⋊Dic3—24no—
91C4⋊Dic3—24no—
92C4⋊Dic3—41no—
93C4⋊Dic3—41no—
94Dic3⋊C4—48no—
95C4×Dic3—41no—
96C4×Dic3—41no—
97#208—42no—
98#208—42no—
99C2×C3⋊C8—42no—
100C22×Q8—61no—
101C22×Q8D1211yescharacteristic
102C2×M16—62no—
103C4⋊Q8—62no—
104C4⋊Q8—62no—
105C4⋊Q8—122no—
106C4⋊Q8—122no—
107C4×Q8—22no—
108C4×Q8—22no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—64no—
112C2×C4⋊C4—61no—
113C2×C4⋊C4C4×D312yes—
114C2×C4⋊C4D1211yescharacteristic
115C2×C42—61no—
116C2×C42#6712yes—
117C2×C42C22×D311yescharacteristic
118C4⋊C8—24no—
119C4⋊C8—62no—
120C4⋊C8—62no—
121C4⋊C8#6714yes—
122#101—122no—
123C22⋊C8—22no—
124C4×C8—24no—
125C4×C8—22no—
126#95—32no—
127#95—64no—
128C23×C3C2×C22⋊C411yescharacteristic
129C3×Q8—24no—
130C3×Q8—24no—
131C2×C3×C4—22no—
132C2×C3×C4—22no—
133C2×C3×C4—22no—
134C2×C3×C4—22no—
135C2×C3×C4—41no—
136C2×C3×C4—42no—
137C2×C3×C4#10214yes—
138C2×C3×C4C4×D412yes—
139C2×C3×C4#12011yescharacteristic
140C2×C3×C4#12112yes—
141C2×C3×C4#12311yescharacteristic
142C2×C3×C4C2×D811yescharacteristic
143C2×C3×C4C2×SD1611yescharacteristic
144C2×C3×C4#13512yes—
145C2×Dic3—24no—
146C2×Dic3—42no—
147C2×Dic3—42no—
148C2×Dic3—42no—
149C2×Dic3—42no—
150C2×Dic3—42no—
151Dic6—82no—
152Dic6—82no—
153C3×C8—44no—
154C3⋊C8—82no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—121no—
158C2×Q8—121no—
159C2×Q8—122no—
160C2×Q8—122no—
161C2×Q8#20412yes—
162C2×Q8D2412yes—
163C22×C4—62no—
164C22×C4—62no—
165C22×C4(C2×D3)⋊C412yes—
166C22×C4(C2×D3)⋊C412yes—
167C22×C4C2×C4×D311yescharacteristic
168C22×C4C2×D1211yescharacteristic
169C22×C4C2×#6711yescharacteristic
170M16—122no—
171M16—122no—
172C2×C8—24no—
173C2×C8—122no—
174C4⋊C4—24no—
175C4⋊C4—22no—
176C4⋊C4—64no—
177C4⋊C4—64no—
178C4⋊C4—128no—
179C4⋊C4—121no—
180C4⋊C4—121no—
181C42—22no—
182C42—22no—
183C42—121no—
184C42—121no—
185C42#23512yes—
186C42C2×#6712yes—
187C22×C3—21no—
188C22×C3—21no—
189C22×C3#38611yescharacteristic
190C22×C3C2×#10211yescharacteristic
191C22×C3#41611yescharacteristic
192C3×C4—21no—
193C3×C4—24no—
194C3×C4—22no—
195C3×C4—42no—
196C3×C4#45612yes—
197C3×C4#46512yes—
198C3×C4#46612yes—
199Dic3—44no—
200Dic3—82no—
201C23C2×(C2×D3)⋊C411yescharacteristic
202Q8—24no—
203Q8—24no—
204Q8—242no—
205Q8—242no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4—64no—
213C2×C4—122no—
214C2×C4—122no—
215C2×C4—122no—
216C2×C4—122no—
217C2×C4—122no—
218C2×C4#82114yes—
219C2×C4C2×#20411yescharacteristic
220C2×C4C2×D2411yescharacteristic
221C2×C4#90412yes—
222C2×C4C4×#6712yes—
223C2×C4#92911yescharacteristic
224C2×C4#93012yes—
225C2×C4#95311yescharacteristic
226C8—44no—
227C8—242no—
228C2×C3—21no—
229C2×C3—21no—
230C2×C3#188611yescharacteristic
231C2×C3#189311yescharacteristic
232C2×C3#191511yescharacteristic
233C22—21no—
234C22—21no—
235C22C2×#82111yescharacteristic
236C22#511311yescharacteristic
237C22#521311yescharacteristic
238C4—24no—
239C4—22no—
240C4—21no—
241C4—42no—
242C4—124no—
243C4—242no—
244C4#511412yes—
245C4#511512yes—
246C4#524612yes—
247C3#939411yescharacteristic, complemented, Sylow
248C2—21no—
249C2—21no—
250C2#7166011yescharacteristic
251C2#7197311yescharacteristic
252C2#7203111yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2982C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2982D3
hypocenterC3#9394
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
247. C3#9394semidirect

Maximal subgroups

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

subgroupindex
C3×#29822
#824752
#716372
#707782
#677462
#93943