Finite Group Databasebeta

← #10695746

Subgroups — #10695746

Order 768.

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

normal subgroups only

This group has 804 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×#5050C212yesmaximal
3C2×#4538C212yesmaximal
4#69424C211yesmaximal, characteristic
5#67613C211yesmaximal, characteristic
6#9967—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×C22⋊Q8C2212yes—
9C22×#216—22no—
10C3×#398—22no—
11C3×#395—22no—
12C3×#380—22no—
13C2×C3×#95C2211yescharacteristic
14C2×C3⋊#95—22no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5050—24no—
18#5050—24no—
19C2×#810—22no—
20#4538—24no—
21C2×#451—32no—
22#2659D311yescharacteristic
23C2×#328—32no—
24#1446—31no—
25#1280—31no—
26C22×C3×Q8D412yes—
27C23×C3×C4D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×Dic6—22no—
32C22×Dic6—22no—
33C22×Dic6—21no—
34C22×Dic6—44no—
35C22×Dic6D412yes—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4—22no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C22⋊C4—22no—
45C2×#216—28no—
46C2×#216—48no—
47C22×C3⋊C8—22no—
48C3×#95—24no—
49C3×#95—22no—
50C3⋊#95—22no—
51C3⋊#95—44no—
52#830—22no—
53#830D414yes—
54#810—28no—
55C23×Q8—31no—
56C22×Q16—62no—
57C2×C22⋊Q8C2×D312yes—
58#451—64no—
59#451—64no—
60C2×#103—62no—
61C2×C22⋊C8—32no—
62#398—22no—
63#395—22no—
64#380—22no—
65C2×#95—31no—
66C2×#95—62no—
67C2×#95C2×D311yescharacteristic
68#328—64no—
69C24×C3C2×D411yescharacteristic
70C2×C3×Q8—22no—
71C2×C3×Q8—22no—
72C2×C3×Q8—44no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4Q1614yescomplemented
81C22×C3×C4C2×D411yescharacteristic
82C22×C3×C4C2×D412yes—
83C22×C3×C4C2×D412yes—
84C22×C3×C4#4111yescharacteristic
85C22×Dic3—24no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—42no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3—42no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—42no—
95C2×Dic6—28no—
96C2×Dic6—24no—
97C2×Dic6—44no—
98C2×Dic6—48no—
99C2×Dic6—44no—
100C2×Dic6—44no—
101C2×Dic6—42no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—42no—
105C3×C4⋊C4—42no—
106C3×C4⋊C4—82no—
107C3×C4⋊C4—82no—
108C3×C22⋊C4—42no—
109C3×C22⋊C4—42no—
110#216—416no—
111C2×C3⋊C8—24no—
112C2×C3⋊C8—44no—
113C22×Q8—32no—
114C22×Q8—61no—
115C22×Q8—62no—
116C22×Q8—62no—
117C22×Q8—124no—
118C22×Q8#6712yes—
119C23×C4—61no—
120C23×C4—61no—
121C23×C4#6712yes—
122C23×C4C22×D311yescharacteristic
123C2×Q16—68no—
124C2×Q16—128no—
125C22×C8—62no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4#6712yes—
134C2×C22⋊C4—22no—
135#103—68no—
136C22⋊C8—34no—
137C22⋊C8—62no—
138#95—24no—
139#95—22no—
140#95—62no—
141#95—124no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—22no—
145C23×C3#12011yescharacteristic
146C23×C3#12111yescharacteristic
147C23×C3#12711yescharacteristic
148C23×C3C2×Q1612yes—
149C23×C3#13712yes—
150C3×Q8—44no—
151C3×Q8—82no—
152C2×C3×C4—24no—
153C2×C3×C4—24no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—44no—
160C2×C3×C4—42no—
161C2×C3×C4—42no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—44no—
165C2×C3×C4—82no—
166C2×C3×C4#12012yes—
167C2×C3×C4#12112yes—
168C2×Dic3—24no—
169C2×Dic3—42no—
170C2×Dic3—44no—
171C2×Dic3—42no—
172C2×Dic3—44no—
173C2×Dic3—44no—
174C2×Dic3—42no—
175C2×Dic3—82no—
176C2×Dic3—81no—
177C2×Dic3—81no—
178Dic6—28no—
179Dic6—44no—
180Dic6—48no—
181C3⋊C8—44no—
182C24C2×#6711yescharacteristic
183C2×Q8—22no—
184C2×Q8—22no—
185C2×Q8—44no—
186C2×Q8—68no—
187C2×Q8—64no—
188C2×Q8—124no—
189C2×Q8—124no—
190C2×Q8—124no—
191C2×Q8—128no—
192C2×Q8—122no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—64no—
201C22×C4—62no—
202C22×C4—62no—
203C22×C4—122no—
204C22×C4—121no—
205C22×C4—121no—
206C22×C4—122no—
207C22×C4—121no—
208C22×C4—121no—
209C22×C4—122no—
210C22×C4#21614yescomplemented
211C22×C4D3×D411yescharacteristic
212C22×C4D3×D412yes—
213C22×C4#23711yescharacteristic
214C22×C4C2×#6712yes—
215Q16—1216no—
216C2×C8—64no—
217C2×C8—124no—
218C4⋊C4—22no—
219C4⋊C4—22no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C4⋊C4—82no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22×C3—22no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3#39211yescharacteristic
235C22×C3#45112yes—
236C22×C3#45511yescharacteristic
237C22×C3#45611yescharacteristic
238C22×C3B(4,𝔽2)12yes—
239C22×C3#46212yes—
240C22×C3#49411yescharacteristic
241C22×C3#49711yescharacteristic
242C3×C4—22no—
243C3×C4—22no—
244C3×C4—44no—
245C3×C4—82no—
246C3×C4—82no—
247Dic3—44no—
248Dic3—82no—
249C23—21no—
250C23—22no—
251C23—21no—
252C23#93811yescharacteristic
253C23#94011yescharacteristic
254C23#94212yes—
255C23C2×#21612yes—
256C23#95311yescharacteristic
257Q8—44no—
258Q8—68no—
259Q8—82no—
260Q8—124no—
261Q8—128no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—64no—
276C2×C4—82no—
277C2×C4—122no—
278C2×C4—124no—
279C2×C4—122no—
280C2×C4—124no—
281C2×C4—122no—
282C2×C4—124no—
283C2×C4—241no—
284C2×C4—242no—
285C2×C4—241no—
286C2×C4#93712yes—
287C2×C4#93912yes—
288C8—124no—
289C2×C3—21no—
290C2×C3—22no—
291C2×C3—21no—
292C2×C3#160212yes—
293C2×C3#160512yes—
294C2×C3#200711yescharacteristic
295C2×C3#201611yescharacteristic
296C2×C3#202111yescharacteristic
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22#508611yescharacteristic
306C22#509012yes—
307C22#509211yescharacteristic
308C22#522311yescharacteristic
309C22#522812yes—
310C22#524211yescharacteristic
311C22#524412yes—
312C22#524611yescharacteristic
313C4—22no—
314C4—22no—
315C4—44no—
316C4—82no—
317C4—82no—
318C4—124no—
319C4—242no—
320C3#996711yescharacteristic, complemented, Sylow
321C2—21no—
322C2—22no—
323C2—21no—
324C2#7179311yescharacteristic
325C2#7185111yescharacteristic
326C2#7188111yescharacteristic
327C2#7189312yes—
328C2#7190412yes—
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#2659C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#2659D3
hypocenterC3#9967
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
80. C22×C3×C4Q16semidirect
210. C22×C4#216semidirect
320. C3#9967semidirect

Maximal subgroups

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

subgroupindex
C3×#26592
C2×#50502
C2×#45382
#694242
#676132
#99673