Finite Group Databasebeta

← #10695780

Subgroups — #10695780

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
1C3×#2659C211yesmaximal, characteristic
2C2×#5043C212yesmaximal
3C2×#4538C212yesmaximal
4#69422C211yesmaximal, characteristic
5#67613C211yesmaximal, characteristic
6#9973—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C22×C4⋊Dic3C2211yescharacteristic
9C3×#398—22no—
10C3×#395—22no—
11C3×#380—22no—
12C2×C3×#95C2211yescharacteristic
13C2×#835—22no—
14C2×C3⋊#95—22no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5043—24no—
18#5043—24no—
19C2×#808—22no—
20#4538—24no—
21C2×#483—32no—
22#2659D311yescharacteristic
23C2×#328—32no—
24#1445—31no—
25#1280—31no—
26C22×C3×Q8D412yes—
27C23×C3×C4D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—22no—
40C2×C4⋊Dic3—22no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3D412yes—
44C22×C3⋊C8—22no—
45C3×#95—24no—
46C3×#95—22no—
47#835—28no—
48C3⋊#95—22no—
49C3⋊#95—44no—
50#830—22no—
51#830D414yes—
52#808—28no—
53C2×C22⋊Q8C2×D312yes—
54C22×C4⋊C4—31no—
55#483—64no—
56#483—64no—
57C2×#106—62no—
58C2×#103—62no—
59C2×C22⋊C8—32no—
60#398—22no—
61#395—22no—
62#380—22no—
63C2×#95—31no—
64C2×#95—62no—
65C2×#95C2×D311yescharacteristic
66#328—64no—
67C24×C3C2×D411yescharacteristic
68C2×C3×Q8—22no—
69C2×C3×Q8—22no—
70C2×C3×Q8—44no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—42no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4SD1614yes—
79C22×C3×C4C2×D411yescharacteristic
80C22×C3×C4C2×D412yes—
81C22×C3×C4C2×D412yes—
82C22×C3×C4#4111yescharacteristic
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3—41no—
86C22×Dic3—42no—
87C22×Dic3—41no—
88C22×Dic3—41no—
89C22×Dic3—41no—
90C22×Dic3—42no—
91C22×Dic3—42no—
92C22×Dic3—42no—
93C3×C4⋊C4—22no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—42no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—82no—
98C3×C4⋊C4—82no—
99C3×C22⋊C4—42no—
100C3×C22⋊C4—42no—
101C4⋊Dic3—24no—
102C4⋊Dic3—42no—
103C2×C3⋊C8—24no—
104C2×C3⋊C8—44no—
105C22×Q8#6712yes—
106C23×C4—61no—
107C23×C4—61no—
108C23×C4#6712yes—
109C23×C4C22×D311yescharacteristic
110C22×C8—62no—
111C22⋊Q8—22no—
112C22⋊Q8—22no—
113C22⋊Q8—22no—
114C22⋊Q8—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—62no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—61no—
122C2×C4⋊C4#6712yes—
123C2×C22⋊C4—22no—
124#106—68no—
125#103—68no—
126C22⋊C8—34no—
127C22⋊C8—62no—
128#95—24no—
129#95—22no—
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—
140C3×Q8—44no—
141C3×Q8—82no—
142C2×C3×C4—24no—
143C2×C3×C4—24no—
144C2×C3×C4—41no—
145C2×C3×C4—42no—
146C2×C3×C4—41no—
147C2×C3×C4—42no—
148C2×C3×C4—42no—
149C2×C3×C4—44no—
150C2×C3×C4—42no—
151C2×C3×C4—42no—
152C2×C3×C4—42no—
153C2×C3×C4—42no—
154C2×C3×C4—44no—
155C2×C3×C4—82no—
156C2×C3×C4#12012yes—
157C2×C3×C4#12112yes—
158C2×Dic3—44no—
159C2×Dic3—42no—
160C2×Dic3—42no—
161C2×Dic3—44no—
162C2×Dic3—82no—
163C2×Dic3—81no—
164C2×Dic3—82no—
165C2×Dic3—81no—
166C2×Dic3—81no—
167C2×Dic3—81no—
168C3⋊C8—44no—
169C24C2×#6711yescharacteristic
170C2×Q8—22no—
171C2×Q8—22no—
172C2×Q8—44no—
173C22×C4—22no—
174C22×C4—22no—
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—42no—
178C22×C4—42no—
179C22×C4—42no—
180C22×C4—62no—
181C22×C4—62no—
182C22×C4—121no—
183C22×C4—122no—
184C22×C4—121no—
185C22×C4—121no—
186C22×C4—122no—
187C22×C4—122no—
188C22×C4—122no—
189C22×C4—121no—
190C22×C4#21514yes—
191C22×C4D3×D411yescharacteristic
192C22×C4D3×D412yes—
193C22×C4#23711yescharacteristic
194C22×C4C2×#6712yes—
195C2×C8—64no—
196C2×C8—124no—
197C4⋊C4—22no—
198C4⋊C4—22no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—64no—
202C4⋊C4—82no—
203C4⋊C4—82no—
204C4⋊C4—122no—
205C22⋊C4—42no—
206C22⋊C4—42no—
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—
226C3×C4—82no—
227C3×C4—82no—
228Dic3—82no—
229Dic3—82no—
230C23—21no—
231C23—22no—
232C23—21no—
233C23#93811yescharacteristic
234C23#94011yescharacteristic
235C23C2×#21512yes—
236C23#94212yes—
237C23#95311yescharacteristic
238Q8—44no—
239Q8—82no—
240C2×C4—24no—
241C2×C4—24no—
242C2×C4—41no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—44no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—44no—
252C2×C4—42no—
253C2×C4—82no—
254C2×C4—122no—
255C2×C4—124no—
256C2×C4—122no—
257C2×C4—124no—
258C2×C4—241no—
259C2×C4—242no—
260C2×C4—242no—
261C2×C4—241no—
262C2×C4—241no—
263C2×C4—241no—
264C2×C4#93712yes—
265C2×C4#93912yes—
266C8—124no—
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—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—22no—
280C22—22no—
281C22—22no—
282C22—22no—
283C22#508312yes—
284C22#508511yescharacteristic
285C22#508611yescharacteristic
286C22#522311yescharacteristic
287C22#522712yes—
288C22#524211yescharacteristic
289C22#524312yes—
290C22#524611yescharacteristic
291C4—22no—
292C4—22no—
293C4—44no—
294C4—82no—
295C4—82no—
296C4—242no—
297C4—242no—
298C3#997311yescharacteristic, complemented, Sylow
299C2—21no—
300C2—22no—
301C2—21no—
302C2#7179111yescharacteristic
303C2#7184911yescharacteristic
304C2#7188111yescharacteristic
305C2#7189012yes—
306C2#7189812yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2659C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#2659D3
hypocenterC3#9973
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2659

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
C3×#26592
C2×#50432
C2×#45382
#694222
#676132
#99733