Finite Group Databasebeta

← #10692874

Subgroups — #10692874

Order 768.

498 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#82384C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3C3×#1466C211yesmaximal, characteristic
4#70823C212yesmaximal, complemented
5#67749C212yesmaximal, complemented
6#9453—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#869C2211yescharacteristic
9C3×#432C2212yes—
10#5221C412yes—
11#5212—21no—
12#5212—21no—
13#5207—21no—
14C2×#830—21no—
15#5011—24no—
16#4977C412yes—
17#4947—21no—
18#4947—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2212yes—
21C3×#323—21no—
22C3×#323—21no—
23#4482—24no—
24#2973—31no—
25#2563—31no—
26#1556—32no—
27#1466D311yescharacteristic
28#1285—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6—21no—
33C3×C4×D4—22no—
34C3×C4×D4—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C22⋊C4D411yescharacteristic
37C2×C3×C22⋊C4D411yescharacteristic
38C2×C3×C42C2311yescharacteristic
39C2×#217—22no—
40C2×#217—22no—
41C2×C4⋊Dic3—21no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3C2×C412yes—
44C2×C4×Dic3C2×C412yes—
45C2×#208—22no—
46C22×C3⋊C8—22no—
47#869—22no—
48#869—22no—
49C3×C4⋊C8—22no—
50C3×C22⋊C8—22no—
51C3×#97—22no—
52C3⋊#95—22no—
53C3⋊#95—22no—
54C3⋊#95—21no—
55#830—22no—
56#830—22no—
57#830—22no—
58C4⋊(C3⋊C8)D414yes—
59#803—22no—
60C2×C4⋊Q8—31no—
61C2×C4×D4C2×D311yescharacteristic
62#435—64no—
63#432—32no—
64#432C2×D312yes—
65#407—61no—
66C2×C22⋊C8—61no—
67#393—61no—
68#393—61no—
69#390—32no—
70#382—32no—
71#380—61no—
72#380—61no—
73C4×C4⋊C4—31no—
74#332—64no—
75#323—21no—
76#323—21no—
77C24×C3C22⋊C411yescharacteristic
78C24×C3C22⋊C411yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—41no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22×C411yescharacteristic
91C22×C3×C4C2×D411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×Dic3—22no—
94C22×Dic3—22no—
95C2×Dic6—42no—
96C2×Dic6—82no—
97C2×C3×C8—42no—
98C3×C4⋊C4—22no—
99C3×C22⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C42C2×D412yes—
102C3×C42#4112yes—
103#217—82no—
104#217—82no—
105C4⋊Dic3—24no—
106C4⋊Dic3—42no—
107C4⋊Dic3—41no—
108C4⋊Dic3—41no—
109C4×Dic3—42no—
110#208—82no—
111C2×C3⋊C8—24no—
112C2×C3⋊C8—82no—
113C22×Q8—61no—
114C22×D4C4×D311yescharacteristic
115C23×C4D1211yescharacteristic
116C23×C4D1211yescharacteristic
117C2×M16—62no—
118C22×C8—62no—
119C4⋊Q8—62no—
120C4⋊Q8—62no—
121C4×D4—22no—
122C4×D4—22no—
123C2×C4⋊C4—32no—
124C2×C4⋊C4—61no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4C4×D311yescharacteristic
127C2×C22⋊C4—62no—
128C2×C22⋊C4—62no—
129C2×C22⋊C4#6711yescharacteristic
130C2×C22⋊C4#6711yescharacteristic
131C2×C42—32no—
132C2×C42C22×D311yescharacteristic
133C4⋊C8—22no—
134C4⋊C8—34no—
135C22⋊C8—22no—
136C22⋊C8—62no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139#97—22no—
140#97—62no—
141#95—62no—
142#95—62no—
143#95—61no—
144C23×C3—21no—
145C23×C3—21no—
146C23×C3—21no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—41no—
150C23×C3—41no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3#10112yes—
154C23×C3C2×C22⋊C411yescharacteristic
155C3×D4—44no—
156C2×C3×C4—21no—
157C2×C3×C4—21no—
158C2×C3×C4—22no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—41no—
162C2×C3×C4—42no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—41no—
166C2×C3×C4—81no—
167C2×C3×C4—81no—
168C2×C3×C4C4×D412yes—
169C2×C3×C4#12011yescharacteristic
170C2×C3×C4#12112yes—
171C2×C3×C4#12311yescharacteristic
172C2×Dic3—42no—
173C2×Dic3—42no—
174C2×Dic3—42no—
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178Dic6—84no—
179C3×C8—82no—
180C3⋊C8—44no—
181C24(C2×D3)⋊C411yescharacteristic
182C24(C2×D3)⋊C411yescharacteristic
183C2×Q8—122no—
184C2×Q8—242no—
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—41no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4—62no—
195C22×C4—62no—
196C22×C4(C2×D3)⋊C411yescharacteristic
197C22×C4(C2×D3)⋊C411yescharacteristic
198C22×C4C2×C4×D311yescharacteristic
199C22×C4C2×D1211yescharacteristic
200C22×C4C2×#6711yescharacteristic
201M16—242no—
202C2×C8—42no—
203C2×C8—64no—
204C2×C8—242no—
205C4⋊C4—22no—
206C4⋊C4—64no—
207C4⋊C4—122no—
208C4⋊C4—121no—
209C4⋊C4—121no—
210C22⋊C4—22no—
211C22⋊C4—22no—
212C22⋊C4—242no—
213C22⋊C4—242no—
214C42—122no—
215C42D3×D412yes—
216C42#23712yes—
217C22×C3—21no—
218C22×C3—21no—
219C22×C3—22no—
220C22×C3—42no—
221C22×C3—42no—
222C22×C3—41no—
223C22×C3—41no—
224C22×C3—41no—
225C22×C3—41no—
226C22×C3—41no—
227C22×C3—81no—
228C22×C3—81no—
229C22×C3#38611yescharacteristic
230C22×C3C2×#10111yescharacteristic
231C22×C3#41311yescharacteristic
232C3×C4—21no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—41no—
236C3×C4—41no—
237C3×C4#45512yes—
238Dic3—82no—
239Dic3—82no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23—41no—
249C23#82412yes—
250C23C2×(C2×D3)⋊C411yescharacteristic
251Q8—244no—
252D4—44no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—22no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—81no—
264C2×C4—81no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4#67⋊C412yes—
272C2×C4#88211yescharacteristic
273C2×C4#88611yescharacteristic
274C2×C4#93912yes—
275C8—82no—
276C8—124no—
277C2×C3—21no—
278C2×C3—21no—
279C2×C3—42no—
280C2×C3—81no—
281C2×C3#188511yescharacteristic
282C2×C3#190111yescharacteristic
283C2×C3#191011yescharacteristic
284C22—21no—
285C22—21no—
286C22—22no—
287C22—42no—
288C22—42no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—81no—
295C22—81no—
296C22#495811yescharacteristic
297C22#513211yescharacteristic
298C22C2×#82411yescharacteristic
299C4—22no—
300C4—22no—
301C4—21no—
302C4—41no—
303C4—41no—
304C4—242no—
305C4—242no—
306C4#474612yes—
307C3#945311yescharacteristic, complemented, Sylow
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#6998611yescharacteristic
313C2#7004711yescharacteristic
314C2#7006911yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1466C2
socleC22×C3#386
prime radicalC22×C3×D4C2×C4
cosocleC2×C3×C42C23
hypercenter#1466D3
hypocenterC3#9453
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
4. #70823C2semidirect
5. #67749C2semidirect
307. C3#9453semidirect

Maximal subgroups

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

subgroupindex
#823842
#823022
C3×#14662
#708232
#677492
#94533