Finite Group Databasebeta

← #10695749

Subgroups — #10695749

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
1C2×#4739C212yesmaximal
2#73361C211yesmaximal, characteristic
3C2×#4467C212yesmaximal
4#69424C211yesmaximal, characteristic
5C3×#1280C211yesmaximal, characteristic
6#9967—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×#924C2212yes—
9C22×Dic12—22no—
10C2×C3×C22⋊C8C2212yes—
11C2×C3×#95C2211yescharacteristic
12#5212—22no—
13C2×C3⋊#95—22no—
14C2×C3⋊#95C2211yescharacteristic
15C2×#816—22no—
16#4739—24no—
17#4739—24no—
18#4661—22no—
19#4656—22no—
20#4467—24no—
21C2×#451—32no—
22#2659—31no—
23C2×#328—32no—
24#1446—31no—
25#1280D311yescharacteristic
26C23×C3×C4D412yes—
27C23×C3×C4C2311yescharacteristic
28C23×Dic3—21no—
29C23×Dic3—21no—
30C22×Dic6—22no—
31C22×Dic6—22no—
32C22×Dic6—21no—
33C22×Dic6—44no—
34C22×Dic6D412yes—
35C22×Dic6D412yes—
36C22×C3×C8—22no—
37C2×#217—22no—
38C2×C4⋊Dic3—22no—
39C2×C4⋊Dic3—22no—
40C2×C4⋊Dic3D412yes—
41#924—22no—
42#924—22no—
43#924—22no—
44#924—22no—
45C2×Dic3⋊C4—22no—
46C2×Dic12—28no—
47C2×Dic12—48no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8D414yes—
50C3×#95—22no—
51C3⋊#95—24no—
52C3⋊#95—22no—
53C3⋊#95—44no—
54#816—28no—
55C23×Q8—31no—
56C22×Q16—62no—
57C2×C22⋊Q8—32no—
58#451—64no—
59#451—64no—
60C2×#103—62no—
61C2×C22⋊C8C2×D312yes—
62#398—62no—
63#395—62no—
64#380—62no—
65C2×#95—31no—
66C2×#95—62no—
67C2×#95C2×D311yescharacteristic
68#328—64no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—22no—
71C22×C3×C4—22no—
72C22×C3×C4—42no—
73C22×C3×C4—42no—
74C22×C3×C4—42no—
75C22×C3×C4Q1614yescomplemented
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D412yes—
78C22×C3×C4C2×D411yescharacteristic
79C22×C3×C4#4111yescharacteristic
80C22×Dic3—24no—
81C22×Dic3—22no—
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3—42no—
86C22×Dic3—41no—
87C22×Dic3—41no—
88C22×Dic3—42no—
89C22×Dic3—41no—
90C22×Dic3—41no—
91C22×Dic3—42no—
92C2×Dic6—28no—
93C2×Dic6—24no—
94C2×Dic6—22no—
95C2×Dic6—22no—
96C2×Dic6—44no—
97C2×Dic6—48no—
98C2×Dic6—44no—
99C2×Dic6—44no—
100C2×Dic6—42no—
101C2×Dic6—44no—
102C2×C3×C8—24no—
103C2×C3×C8—44no—
104#217—42no—
105#217—42no—
106C4⋊Dic3—22no—
107C4⋊Dic3—22no—
108C4⋊Dic3—82no—
109C4⋊Dic3—82no—
110Dic3⋊C4—42no—
111Dic3⋊C4—42no—
112Dic12—416no—
113C22×Q8—32no—
114C22×Q8—32no—
115C22×Q8—61no—
116C22×Q8—62no—
117C22×Q8—62no—
118C22×Q8—124no—
119C23×C4—61no—
120C23×C4—61no—
121C23×C4D1212yes—
122C23×C4C22×D311yescharacteristic
123C2×Q16—68no—
124C2×Q16—128no—
125C22×C8—22no—
126C22⋊Q8—62no—
127C22⋊Q8—62no—
128C22⋊Q8—62no—
129C22⋊Q8—62no—
130C2×C4⋊C4—32no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4—62no—
133C2×C4⋊C4—62no—
134C2×C22⋊C4—62no—
135#103—68no—
136C22⋊C8—22no—
137C22⋊C8D1214yes—
138#95—22no—
139#95—64no—
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—
150C2×C3×C4—24no—
151C2×C3×C4—24no—
152C2×C3×C4—42no—
153C2×C3×C4—41no—
154C2×C3×C4—41no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—44no—
158C2×C3×C4—82no—
159C2×C3×C4#12012yes—
160C2×C3×C4#12112yes—
161C2×Dic3—24no—
162C2×Dic3—42no—
163C2×Dic3—44no—
164C2×Dic3—42no—
165C2×Dic3—44no—
166C2×Dic3—44no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—42no—
170C2×Dic3—42no—
171C2×Dic3—42no—
172C2×Dic3—44no—
173C2×Dic3—82no—
174C2×Dic3—81no—
175C2×Dic3—81no—
176Dic6—28no—
177Dic6—44no—
178Dic6—48no—
179Dic6—44no—
180Dic6—82no—
181C3×C8—44no—
182C24C2×D1211yescharacteristic
183C2×Q8—68no—
184C2×Q8—64no—
185C2×Q8—62no—
186C2×Q8—62no—
187C2×Q8—124no—
188C2×Q8—124no—
189C2×Q8—124no—
190C2×Q8—124no—
191C2×Q8—128no—
192C2×Q8—122no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—64no—
199C22×C4—62no—
200C22×C4—62no—
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×C4Dic1214yescomplemented
211C22×C4C2×D1212yes—
212C22×C4D3×D412yes—
213C22×C4D3×D411yescharacteristic
214C22×C4#23911yescharacteristic
215Q16—1216no—
216C2×C8—24no—
217C2×C8—44no—
218C4⋊C4—62no—
219C4⋊C4—62no—
220C4⋊C4—122no—
221C4⋊C4—122no—
222C4⋊C4—242no—
223C4⋊C4—242no—
224C22⋊C4—122no—
225C22⋊C4—122no—
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—
245Dic3—44no—
246Dic3—82no—
247Dic3—82no—
248Dic3—82no—
249C23—22no—
250C23—21no—
251C23—21no—
252C23#87411yescharacteristic
253C23#88211yescharacteristic
254C23#89511yescharacteristic
255C23C2×Dic1212yes—
256C23#90912yes—
257Q8—68no—
258Q8—124no—
259Q8—128no—
260Q8—124no—
261Q8—242no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—44no—
270C2×C4—64no—
271C2×C4—82no—
272C2×C4—122no—
273C2×C4—124no—
274C2×C4—122no—
275C2×C4—124no—
276C2×C4—124no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—124no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—242no—
284C2×C4—241no—
285C2×C4—241no—
286C2×C4#88212yes—
287C2×C4#89512yes—
288C8—44no—
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—22no—
298C22—22no—
299C22—22no—
300C22—22no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—21no—
305C22#467111yescharacteristic
306C22#469911yescharacteristic
307C22#471511yescharacteristic
308C22#473912yes—
309C22#474112yes—
310C22#474611yescharacteristic
311C22#475211yescharacteristic
312C22#484912yes—
313C4—22no—
314C4—22no—
315C4—44no—
316C4—124no—
317C4—242no—
318C4—242no—
319C4—242no—
320C3#996711yescharacteristic, complemented, Sylow
321C2—21no—
322C2—21no—
323C2—22no—
324C2#6811512yes—
325C2#6813912yes—
326C2#6868811yescharacteristic
327C2#6896511yescharacteristic
328C2#6900911yescharacteristic
subgroupG/H
centerC23#882
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1280C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenter#1280D3
hypocenterC3#9967
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
75. C22×C3×C4Q16semidirect
210. C22×C4Dic12semidirect
320. C3#9967semidirect

Maximal subgroups

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

subgroupindex
C2×#47392
#733612
C2×#44672
#694242
C3×#12802
#99673