Finite Group Databasebeta

← #10695783

Subgroups — #10695783

Order 768.

632 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 121 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
1C2×#4721C212yesmaximal
2#73361C211yesmaximal, characteristic
3C2×#4467C212yesmaximal
4#69422C211yesmaximal, characteristic
5C3×#1280C211yesmaximal, characteristic
6#9973—31nomaximal, Sylow
7C22×C4⋊Dic3C2211yescharacteristic
8C2×#924C2212yes—
9C2×C3×C22⋊C8C2212yes—
10C2×C3×#95C2211yescharacteristic
11#5212—22no—
12C2×C3⋊#95—22no—
13C2×C3⋊#95C2211yescharacteristic
14C2×#817—22no—
15C2×#816—22no—
16#4721—24no—
17#4721—24no—
18#4661—22no—
19#4656—22no—
20#4467—24no—
21C2×#483—32no—
22#2659—31no—
23C2×#328—32no—
24#1445—31no—
25#1280D311yescharacteristic
26C23×C3×C4D412yes—
27C23×C3×C4C2311yescharacteristic
28C23×Dic3—21no—
29C23×Dic3—21no—
30C22×Dic6D412yes—
31C22×C3×C8—22no—
32C2×#217—22no—
33C2×C4⋊Dic3—22no—
34C2×C4⋊Dic3—22no—
35C2×C4⋊Dic3—21no—
36C2×C4⋊Dic3—22no—
37C2×C4⋊Dic3—22no—
38C2×C4⋊Dic3D412yes—
39C2×C4⋊Dic3D412yes—
40#924—22no—
41#924—22no—
42#924—22no—
43#924—22no—
44C2×Dic3⋊C4—22no—
45C3×C22⋊C8—22no—
46C3×C22⋊C8D414yes—
47C3×#95—22no—
48C3⋊#95—24no—
49C3⋊#95—22no—
50C3⋊#95—44no—
51#817—28no—
52#816—28no—
53C2×C22⋊Q8—32no—
54C22×C4⋊C4—31no—
55#483—64no—
56#483—64no—
57C2×#106—62no—
58C2×#103—62no—
59C2×C22⋊C8C2×D312yes—
60#398—62no—
61#395—62no—
62#380—62no—
63C2×#95—31no—
64C2×#95—62no—
65C2×#95C2×D311yescharacteristic
66#328—64no—
67C24×C3C2×D411yescharacteristic
68C22×C3×C4—22no—
69C22×C3×C4—22no—
70C22×C3×C4—42no—
71C22×C3×C4—42no—
72C22×C3×C4—42no—
73C22×C3×C4SD1614yes—
74C22×C3×C4C2×D412yes—
75C22×C3×C4C2×D412yes—
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4#4111yescharacteristic
78C22×Dic3—22no—
79C22×Dic3—22no—
80C22×Dic3—22no—
81C22×Dic3—22no—
82C22×Dic3—41no—
83C22×Dic3—42no—
84C22×Dic3—41no—
85C22×Dic3—41no—
86C22×Dic3—41no—
87C22×Dic3—42no—
88C22×Dic3—42no—
89C22×Dic3—42no—
90C2×Dic6—22no—
91C2×Dic6—22no—
92C2×Dic6—44no—
93C2×C3×C8—24no—
94C2×C3×C8—44no—
95#217—42no—
96#217—42no—
97C4⋊Dic3—24no—
98C4⋊Dic3—22no—
99C4⋊Dic3—22no—
100C4⋊Dic3—42no—
101C4⋊Dic3—82no—
102C4⋊Dic3—82no—
103Dic3⋊C4—42no—
104Dic3⋊C4—42no—
105C22×Q8—32no—
106C23×C4—61no—
107C23×C4—61no—
108C23×C4D1212yes—
109C23×C4C22×D311yescharacteristic
110C22×C8—22no—
111C22⋊Q8—62no—
112C22⋊Q8—62no—
113C22⋊Q8—62no—
114C22⋊Q8—62no—
115C2×C4⋊C4—32no—
116C2×C4⋊C4—32no—
117C2×C4⋊C4—62no—
118C2×C4⋊C4—62no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—61no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—62no—
123C2×C22⋊C4—62no—
124#106—68no—
125#103—68no—
126C22⋊C8—22no—
127C22⋊C8D1214yes—
128#95—22no—
129#95—64no—
130#95—62no—
131#95—124no—
132C23×C3—21no—
133C23×C3—21no—
134C23×C3—22no—
135C23×C3#12011yescharacteristic
136C23×C3#12111yescharacteristic
137C23×C3#12711yescharacteristic
138C23×C3C2×SD1612yes—
139C23×C3#13712yes—
140C2×C3×C4—24no—
141C2×C3×C4—24no—
142C2×C3×C4—42no—
143C2×C3×C4—41no—
144C2×C3×C4—41no—
145C2×C3×C4—42no—
146C2×C3×C4—42no—
147C2×C3×C4—44no—
148C2×C3×C4—82no—
149C2×C3×C4#12012yes—
150C2×C3×C4#12112yes—
151C2×Dic3—44no—
152C2×Dic3—42no—
153C2×Dic3—42no—
154C2×Dic3—42no—
155C2×Dic3—42no—
156C2×Dic3—42no—
157C2×Dic3—42no—
158C2×Dic3—44no—
159C2×Dic3—44no—
160C2×Dic3—82no—
161C2×Dic3—81no—
162C2×Dic3—82no—
163C2×Dic3—81no—
164C2×Dic3—81no—
165C2×Dic3—81no—
166Dic6—44no—
167Dic6—82no—
168C3×C8—44no—
169C24C2×D1211yescharacteristic
170C2×Q8—62no—
171C2×Q8—62no—
172C2×Q8—124no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—42no—
176C22×C4—42no—
177C22×C4—42no—
178C22×C4—62no—
179C22×C4—62no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—122no—
183C22×C4—121no—
184C22×C4—121no—
185C22×C4—121no—
186C22×C4—121no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4#20414yes—
191C22×C4C2×D1212yes—
192C22×C4D3×D412yes—
193C22×C4D3×D411yescharacteristic
194C22×C4#23911yescharacteristic
195C2×C8—24no—
196C2×C8—44no—
197C4⋊C4—64no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—122no—
201C4⋊C4—122no—
202C4⋊C4—122no—
203C4⋊C4—242no—
204C4⋊C4—242no—
205C22⋊C4—122no—
206C22⋊C4—122no—
207C22×C3—22no—
208C22×C3—22no—
209C22×C3—22no—
210C22×C3—22no—
211C22×C3—21no—
212C22×C3—21no—
213C22×C3—21no—
214C22×C3—21no—
215C22×C3#39211yescharacteristic
216C22×C3#44812yes—
217C22×C3#45511yescharacteristic
218C22×C3#45611yescharacteristic
219C22×C3#45812yes—
220C22×C3#46112yes—
221C22×C3#49211yescharacteristic
222C22×C3#49711yescharacteristic
223C3×C4—22no—
224C3×C4—22no—
225C3×C4—44no—
226Dic3—82no—
227Dic3—82no—
228Dic3—82no—
229Dic3—82no—
230C23—22no—
231C23—21no—
232C23—21no—
233C23#87411yescharacteristic
234C23#88211yescharacteristic
235C23#89511yescharacteristic
236C23C2×#20412yes—
237C23#90912yes—
238Q8—124no—
239Q8—242no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—82no—
249C2×C4—124no—
250C2×C4—122no—
251C2×C4—122no—
252C2×C4—122no—
253C2×C4—122no—
254C2×C4—124no—
255C2×C4—122no—
256C2×C4—122no—
257C2×C4—124no—
258C2×C4—242no—
259C2×C4—241no—
260C2×C4—241no—
261C2×C4—241no—
262C2×C4—241no—
263C2×C4—242no—
264C2×C4#88212yes—
265C2×C4#89512yes—
266C8—44no—
267C2×C3—21no—
268C2×C3—22no—
269C2×C3—21no—
270C2×C3#160812yes—
271C2×C3#161112yes—
272C2×C3#200711yescharacteristic
273C2×C3#201311yescharacteristic
274C2×C3#202211yescharacteristic
275C22—22no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22#467111yescharacteristic
284C22#469311yescharacteristic
285C22#471511yescharacteristic
286C22#473812yes—
287C22#474212yes—
288C22#474611yescharacteristic
289C22#475211yescharacteristic
290C22#484112yes—
291C4—22no—
292C4—22no—
293C4—44no—
294C4—242no—
295C4—242no—
296C4—242no—
297C4—242no—
298C3#997311yescharacteristic, complemented, Sylow
299C2—21no—
300C2—21no—
301C2—22no—
302C2#6811612yes—
303C2#6814012yes—
304C2#6868711yescharacteristic
305C2#6896611yescharacteristic
306C2#6900911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#1280D3
hypocenterC3#9973
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1280

Complements

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

NH ≅ G/Ntype
298. C3#9973semidirect

Maximal subgroups

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

subgroupindex
C2×#47212
#733612
C2×#44672
#694222
C3×#12802
#99733