Finite Group Databasebeta

← #10693076

Subgroups — #10693076

Order 768.

458 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 99 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×#2921C211yesmaximal, characteristic
2#82258C211yesmaximal, characteristic
3#71572C211yesmaximal, characteristic
4#69857C212yesmaximal, complemented
5#67666C212yesmaximal, complemented
6#9493—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×C4×Dic6C2211yescharacteristic
9C3×#433—24no—
10C3×#423—21no—
11C3×#407—21no—
12C3×C4×M16C2212yes—
13#5210—21no—
14#4981C411yescharacteristic
15#4979C411yescharacteristic
16#4978—21no—
17#4953C411yescharacteristic
18#4947C411yescharacteristic
19#4944—21no—
20#4937C2211yescharacteristic
21#4933—21no—
22#4926C2212yes—
23#4528—21no—
24#4525—21no—
25#4456—24no—
26#2921D311yescharacteristic
27#2599—31no—
28#1491—32no—
29#1464—31no—
30#1288—32no—
31C23×C3×C4D411yescharacteristic
32C22×Dic6C2×C411yescharacteristic
33C2×C3×M16—22no—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42C2×#217—21no—
43C2×#217—21no—
44C2×C4⋊Dic3C2×C411yescharacteristic
45C2×Dic3⋊C4—21no—
46C2×Dic3⋊C4—21no—
47C2×Dic3⋊C4—21no—
48C2×C4×Dic3—21no—
49C4×Dic6—22no—
50C4×Dic6—22no—
51C3×C4⋊C8—24no—
52C3×C22⋊C8—24no—
53C3×#97—22no—
54C3×C4×C8D414yes—
55C3⋊#95—21no—
56C3⋊#95—21no—
57C3⋊#95—21no—
58C3⋊#95—21no—
59C3⋊#95—21no—
60C3⋊#95—21no—
61C3⋊#95C2×C411yescharacteristic
62C3⋊#95C2×C411yescharacteristic
63#830—22no—
64C4⋊(C3⋊C8)—22no—
65#803—22no—
66C2×C4×Q8—31no—
67C2×#117C2×D311yescharacteristic
68#433—24no—
69#432—32no—
70#423—21no—
71#407—21no—
72C4×M16C2×D312yes—
73#398—31no—
74#397—31no—
75#395—31no—
76#393—31no—
77#388—61no—
78#387—61no—
79#387—61no—
80#385—61no—
81#383—31no—
82Q8⋊C8—64no—
83#324—61no—
84#323—61no—
85C24×C3C22⋊C411yescharacteristic
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—41no—
89C22×C3×C4—41no—
90C22×C3×C4C22⋊C411yescharacteristic
91C22×C3×C4C22⋊C411yescharacteristic
92C22×C3×C4C22⋊C411yescharacteristic
93C22×C3×C4C22×C411yescharacteristic
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×Dic3—21no—
97C22×Dic3—21no—
98C22×Dic3—21no—
99C22×Dic3—21no—
100C2×Dic6—22no—
101C2×Dic6—42no—
102C3×M16—42no—
103C3×M16—42no—
104C2×C3×C8—24no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C42—21no—
110C3×C42—21no—
111C3×C42C2×D412yes—
112C3×C42#4112yes—
113#217—42no—
114#217—42no—
115C4⋊Dic3—22no—
116Dic3⋊C4—42no—
117Dic3⋊C4—42no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—42no—
121C2×C3⋊C8—42no—
122C22×Q8—31no—
123C23×C4#6711yescharacteristic
124C2×M16—22no—
125C4×Q8—62no—
126C4×Q8—62no—
127#117—22no—
128#117—22no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—61no—
132C2×C4⋊C4—61no—
133C2×C4⋊C4C4×D311yescharacteristic
134C2×C4⋊C4D1211yescharacteristic
135C2×C22⋊C4—61no—
136C2×C22⋊C4—61no—
137C2×C22⋊C4C4×D311yescharacteristic
138C2×C22⋊C4D1211yescharacteristic
139C2×C42—61no—
140C2×C42#6711yescharacteristic
141C2×C42C22×D311yescharacteristic
142C4⋊C8—24no—
143C4⋊C8—62no—
144C22⋊C8—24no—
145C22⋊C8—62no—
146#97—22no—
147#97—62no—
148C4×C8#6714yes—
149#95—31no—
150#95—31no—
151#95—61no—
152#95—61no—
153#95—61no—
154#95—61no—
155#95—61no—
156#95—61no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—41no—
161C23×C3—41no—
162C23×C3C2×C22⋊C411yescharacteristic
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—41no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4C4×D412yes—
177C2×C3×C4#12011yescharacteristic
178C2×C3×C4#12112yes—
179C2×C3×C4#12311yescharacteristic
180C2×Dic3—22no—
181C2×Dic3—41no—
182C2×Dic3—41no—
183C2×Dic3—41no—
184C2×Dic3—41no—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—41no—
188C2×Dic3—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—41no—
192Dic6—24no—
193Dic6—42no—
194C3×C8—44no—
195C3⋊C8—82no—
196C24(C2×D3)⋊C411yescharacteristic
197C2×Q8—62no—
198C2×Q8—122no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4—61no—
204C22×C4—61no—
205C22×C4—61no—
206C22×C4—61no—
207C22×C4(C2×D3)⋊C411yescharacteristic
208C22×C4(C2×D3)⋊C411yescharacteristic
209C22×C4(C2×D3)⋊C411yescharacteristic
210C22×C4C2×C4×D311yescharacteristic
211C22×C4C2×D1211yescharacteristic
212C22×C4C2×#6711yescharacteristic
213M16—42no—
214M16—42no—
215C2×C8—24no—
216C2×C8—122no—
217C4⋊C4—22no—
218C4⋊C4—22no—
219C4⋊C4—62no—
220C4⋊C4—122no—
221C4⋊C4—122no—
222C4⋊C4—121no—
223C4⋊C4—121no—
224C22⋊C4—22no—
225C22⋊C4—22no—
226C22⋊C4—122no—
227C22⋊C4—122no—
228C42—21no—
229C42—21no—
230C42—122no—
231C42#23512yes—
232C42C2×#6712yes—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—42no—
236C22×C3—41no—
237C22×C3—41no—
238C22×C3—41no—
239C22×C3—81no—
240C22×C3#38611yescharacteristic
241C22×C3#42111yescharacteristic
242C22×C3#42111yescharacteristic
243C3×C4—22no—
244C3×C4—21no—
245C3×C4—21no—
246C3×C4—22no—
247C3×C4—41no—
248C3×C4—41no—
249C3×C4—41no—
250Dic3—42no—
251Dic3—81no—
252Dic3—81no—
253Dic3—81no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23C2×(C2×D3)⋊C411yescharacteristic
260Q8—64no—
261Q8—122no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—62no—
276C2×C4—121no—
277C2×C4—121no—
278C2×C4—121no—
279C2×C4—121no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4—121no—
283C2×C4—121no—
284C2×C4—121no—
285C2×C4—121no—
286C2×C4—121no—
287C2×C4C4×#6712yes—
288C2×C4#92911yescharacteristic
289C2×C4#93012yes—
290C2×C4#95311yescharacteristic
291C8—44no—
292C8—242no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—81no—
296C2×C3#189111yescharacteristic
297C2×C3#189711yescharacteristic
298C2×C3#190911yescharacteristic
299C22—21no—
300C22—21no—
301C22—42no—
302C22—41no—
303C22—41no—
304C22—41no—
305C22—81no—
306C22#500511yescharacteristic
307C22#500511yescharacteristic
308C22#521311yescharacteristic
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—41no—
314C4—41no—
315C4—41no—
316C4—122no—
317C4—241no—
318C4—241no—
319C4—241no—
320C3#949311yescharacteristic, complemented, Sylow
321C2—21no—
322C2—21no—
323C2—81no—
324C2#7159111yescharacteristic
325C2#7202411yescharacteristic
326C2#7202911yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2921C2
socleC22×C3#386
prime radicalC24×C3C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2921D3
hypocenterC3#9493
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
4. #69857C2semidirect
5. #67666C2semidirect
320. C3#9493semidirect

Maximal subgroups

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

subgroupindex
C3×#29212
#822582
#715722
#698572
#676662
#94933