Finite Group Databasebeta

← #10692864

Subgroups — #10692864

Order 768.

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

normal subgroups only

This group has 518 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#82382C211yesmaximal, characteristic
2#82302C211yesmaximal, characteristic
3C3×#1466C211yesmaximal, characteristic
4#70824C212yesmaximal, complemented
5#67749C212yesmaximal, complemented
6#9451—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#936C412yes—
9C2×#869C2211yescharacteristic
10C3×#432C2212yes—
11#5212—21no—
12#5212—21no—
13#5207—21no—
14C2×#830—21no—
15#5011—24no—
16#4973C412yes—
17#4953—21no—
18#4953—21no—
19C4×C4⋊Dic3C2211yescharacteristic
20#4926C2212yes—
21C3×#323—21no—
22C3×#323—21no—
23#4481—24no—
24#2973—31no—
25#2547—31no—
26#1558—32no—
27#1466D311yescharacteristic
28#1285—32no—
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C22×Dic6C2×C412yes—
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—
41#936—44no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—21no—
44C2×C4⋊Dic3—21no—
45C2×C4×Dic3C2×C412yes—
46C2×#208—22no—
47C22×C3⋊C8—22no—
48#869—22no—
49#869—22no—
50C3×C4⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×#97—22no—
53C3⋊#95—22no—
54C3⋊#95—22no—
55C3⋊#95—21no—
56#830—22no—
57#830—22no—
58#830—22no—
59C4⋊(C3⋊C8)D414yes—
60#803—22no—
61C2×C4⋊Q8—31no—
62C2×#124—32no—
63C2×C4×D4C2×D311yescharacteristic
64#435—64no—
65#432—32no—
66#432C2×D312yes—
67#407—61no—
68C2×C22⋊C8—61no—
69#397—61no—
70#397—61no—
71#391—32no—
72#380—61no—
73#380—61no—
74C4×C4⋊C4—31no—
75#333—64no—
76#323—21no—
77#323—21no—
78C24×C3C22⋊C411yescharacteristic
79C24×C3C22⋊C411yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—41no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4C22⋊C411yescharacteristic
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C22×Dic3—22no—
95C22×Dic3—22no—
96C2×Dic6—24no—
97C2×Dic6—42no—
98C2×Dic6—42no—
99C2×C3×C8—42no—
100C3×C4⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C42C2×D412yes—
104C3×C42#4112yes—
105#217—44no—
106#217—44no—
107C4⋊Dic3—42no—
108C4⋊Dic3—41no—
109C4⋊Dic3—42no—
110C4⋊Dic3—41no—
111C4×Dic3—42no—
112#208—82no—
113C2×C3⋊C8—24no—
114C2×C3⋊C8—82no—
115C22×Q8—32no—
116C22×D4C4×D311yescharacteristic
117C23×C4D1211yescharacteristic
118C23×C4D1211yescharacteristic
119C2×M16—62no—
120C22×C8—62no—
121C4⋊Q8—62no—
122C4⋊Q8—62no—
123#124—124no—
124C4×D4—22no—
125C4×D4—22no—
126C2×C4⋊C4—61no—
127C2×C4⋊C4—61no—
128C2×C4⋊C4—61no—
129C2×C4⋊C4C4×D311yescharacteristic
130C2×C22⋊C4—62no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4#6711yescharacteristic
133C2×C22⋊C4#6711yescharacteristic
134C2×C42—32no—
135C2×C42C22×D311yescharacteristic
136C4⋊C8—22no—
137C4⋊C8—34no—
138C22⋊C8—22no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141C22⋊C8—62no—
142#97—22no—
143#97—62no—
144#95—62no—
145#95—62no—
146#95—61no—
147C23×C3—21no—
148C23×C3—21no—
149C23×C3—21no—
150C23×C3—21no—
151C23×C3—21no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—41no—
156C23×C3#10012yes—
157C23×C3C2×C22⋊C411yescharacteristic
158C3×D4—44no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—81no—
170C2×C3×C4—81no—
171C2×C3×C4C4×D412yes—
172C2×C3×C4#12011yescharacteristic
173C2×C3×C4#12112yes—
174C2×C3×C4#12311yescharacteristic
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181Dic6—84no—
182C3×C8—82no—
183C3⋊C8—44no—
184C24(C2×D3)⋊C411yescharacteristic
185C24(C2×D3)⋊C411yescharacteristic
186C2×Q8—64no—
187C2×Q8—122no—
188C2×Q8—122no—
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—41no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4C2×C4×D311yescharacteristic
203C22×C4C2×D1211yescharacteristic
204C22×C4C2×#6711yescharacteristic
205M16—242no—
206C2×C8—42no—
207C2×C8—64no—
208C2×C8—242no—
209C4⋊C4—22no—
210C4⋊C4—122no—
211C4⋊C4—121no—
212C4⋊C4—121no—
213C4⋊C4—122no—
214C22⋊C4—22no—
215C22⋊C4—22no—
216C22⋊C4—124no—
217C22⋊C4—124no—
218C42—122no—
219C42D3×D412yes—
220C42#23712yes—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3—22no—
224C22×C3—42no—
225C22×C3—42no—
226C22×C3—41no—
227C22×C3—41no—
228C22×C3—41no—
229C22×C3—41no—
230C22×C3—41no—
231C22×C3—81no—
232C22×C3—81no—
233C22×C3#38611yescharacteristic
234C22×C3C2×#10011yescharacteristic
235C22×C3#41311yescharacteristic
236C3×C4—21no—
237C3×C4—22no—
238C3×C4—22no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4#45512yes—
242Dic3—82no—
243Dic3—82no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23#82312yes—
254C23C2×(C2×D3)⋊C411yescharacteristic
255Q8—244no—
256D4—44no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—122no—
275C2×C4#67⋊C412yes—
276C2×C4#88211yescharacteristic
277C2×C4#88611yescharacteristic
278C2×C4#93912yes—
279C8—82no—
280C8—124no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—42no—
284C2×C3—81no—
285C2×C3#188511yescharacteristic
286C2×C3#190011yescharacteristic
287C2×C3#190911yescharacteristic
288C22—21no—
289C22—21no—
290C22—22no—
291C22—42no—
292C22—42no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—81no—
299C22—81no—
300C22#495811yescharacteristic
301C22C2×#82311yescharacteristic
302C22#513211yescharacteristic
303C4—22no—
304C4—22no—
305C4—21no—
306C4—41no—
307C4—41no—
308C4—242no—
309C4—242no—
310C4#474612yes—
311C3#945111yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#6998511yescharacteristic
317C2#7004611yescharacteristic
318C2#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#9451
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. #70824C2semidirect
5. #67749C2semidirect
311. C3#9451semidirect

Maximal subgroups

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

subgroupindex
#823822
#823022
C3×#14662
#708242
#677492
#94513