Finite Group Databasebeta

← #10948981

Subgroups — #10948981

Order 768.

1124 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 211 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
1C23×C3×Q16C211yesmaximal, characteristic, complemented
2C2×#5179C212yesmaximal, complemented
3#83742C211yesmaximal, characteristic
4#72064C214yesmaximal, complemented
5#72063C214yesmaximal, complemented
6#72061C212yesmaximal, complemented
7#71963C211yesmaximal, characteristic, complemented
8#33518—31nomaximal, Sylow
9C23×C3×Q8—21no—
10C22×C3×Q16—28no—
11C22×C3×Q16—21no—
12C22×C3×Q16—22no—
13C22×C3×Q16C2214yes—
14C22×C3×Q16C2212yes—
15C23×C3×C8C2211yescharacteristic
16#5791—21no—
17C2×#904C2211yescharacteristic
18C2×#829C2214yescomplemented
19#5179—28no—
20C2×#828C2214yescomplemented
21#5116C2212yes—
22#5115C2212yes—
23#5114—22no—
24#5113—21no—
25#5106C2211yescharacteristic
26C2×#812C2212yes—
27#4564C2218yescomplemented
28#4549C2214yes—
29C23×Q16D311yescharacteristic, complemented
30C2×#510—32no—
31#3050—31no—
32#2188—34no—
33#2185—34no—
34#2141—32no—
35#2112—31no—
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—22no—
39C22×C3×Q8—44no—
40C22×C3×Q8—42no—
41C22×C3×Q8—41no—
42C23×C3×C4—41no—
43C23×C3×C4D411yescharacteristic
44C2×#235—21no—
45C2×C3×Q16—28no—
46C2×C3×Q16—216no—
47C2×C3×Q16—216no—
48C2×C3×Q16—24no—
49C2×C3×Q16—22no—
50C2×C3×Q16—24no—
51C2×C3×Q16D418yescomplemented
52C2×C3×Q16C2314yes—
53C22×C3×C8—24no—
54C22×C3×C8D411yescharacteristic
55C22×C3×C8D412yes—
56C22×C3×C8C2312yes—
57C22×C3×C8C2311yescharacteristic
58#953—41no—
59#930—22no—
60#929—41no—
61C4×#67—42no—
62#926—21no—
63#924—22no—
64C2×Dic12C2311yescharacteristic
65#904—24no—
66C2×D24C2311yescharacteristic
67C2×#204—21no—
68#829—28no—
69#828—28no—
70#821—22no—
71#818C2312yes—
72#817—21no—
73#816—22no—
74#812—24no—
75C2×C3⋊C16C2314yes—
76C23×Q8—21no—
77C2×#135—31no—
78C22×Q16—28no—
79C22×Q16—21no—
80C22×Q16—22no—
81C22×Q16C2×D314yes—
82C22×Q16C2×D312yes—
83C23×C8C2×D311yescharacteristic
84#525—61no—
85#510—68no—
86C2×Q32—34no—
87C2×SD32—34no—
88C2×M32—32no—
89#467—32no—
90#466—32no—
91#465—62no—
92#427—31no—
93#416—61no—
94#358—38no—
95C22⋊C16—34no—
96C24×C3D811yescharacteristic
97C2×C3×Q8—216no—
98C2×C3×Q8—28no—
99C2×C3×Q8—24no—
100C2×C3×Q8—48no—
101C2×C3×Q8—48no—
102C2×C3×Q8—416no—
103C2×C3×Q8—44no—
104C2×C3×Q8—42no—
105C2×C3×Q8—44no—
106C22×C3×C4—22no—
107C22×C3×C4—22no—
108C22×C3×C4—48no—
109C22×C3×C4—44no—
110C22×C3×C4—42no—
111C22×C3×C4D812yes—
112C22×C3×C4D811yescharacteristic
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D411yescharacteristic
115C2×#67—42no—
116C2×#67—41no—
117#235—42no—
118#235—42no—
119C2×D12—21no—
120C2×C4×D3—41no—
121C2×Dic6—21no—
122C3×Q16—216no—
123C3×Q16—28no—
124C3×Q16—24no—
125C3×Q16C2×D418yes—
126C2×C3×C8—22no—
127C2×C3×C8—24no—
128C2×C3×C8—24no—
129C2×C3×C8C2×D412yes—
130C2×C3×C8C2×D411yescharacteristic
131C2×C3×C8C2×D414yes—
132C2×C3×C8C2411yescharacteristic
133#217—42no—
134#217—41no—
135(C2×D3)⋊C4—42no—
136C4⋊Dic3—22no—
137Dic3⋊C4—42no—
138C4×Dic3—41no—
139Dic12—22no—
140D24—22no—
141#204—42no—
142C3⋊C16—24no—
143C2×#41—61no—
144C22×Q8—28no—
145C22×Q8—24no—
146C22×Q8—22no—
147C22×Q8—44no—
148C22×Q8—42no—
149C22×Q8—41no—
150C23×C4—41no—
151C23×C4#6711yescharacteristic
152#135—64no—
153C2×Q16—28no—
154C2×Q16—216no—
155C2×Q16—216no—
156C2×Q16—24no—
157C2×Q16—22no—
158C2×Q16—24no—
159C2×Q16—31no—
160C2×Q16#6718yescomplemented
161C2×Q16C22×D314yes—
162C2×SD16—61no—
163C2×D8—31no—
164C22×C8—24no—
165C22×C8#6711yescharacteristic
166C22×C8#6712yes—
167C22×C8C22×D312yes—
168C22×C8C22×D311yescharacteristic
169#123—121no—
170C22⋊Q8—62no—
171#121—62no—
172#120—121no—
173C4×D4—122no—
174#117—61no—
175Q32—68no—
176SD32—68no—
177M32—64no—
178C2×C16—34no—
179#107—32no—
180#106—61no—
181#103—62no—
182#102—62no—
183C23×C3—22no—
184C23×C3—24no—
185C23×C3C2×D811yescharacteristic
186C23×C3C2×D812yes—
187C22×D3—41no—
188C3×Q8—28no—
189C3×Q8—416no—
190C3×Q8—44no—
191C3×Q8—48no—
192C2×C3×C4—22no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—24no—
196C2×C3×C4—416no—
197C2×C3×C4—44no—
198C2×C3×C4—48no—
199C2×C3×C4#12012yes—
200C2×C3×C4C2×D812yes—
201C2×C3×C4C2×D811yescharacteristic
202C2×C3×C4AGL(1,ℤ/8)12yes—
203C2×C3×C4C22×D411yescharacteristic
204#67—82no—
205#67—84no—
206C2×Dic3—42no—
207C2×Dic3—41no—
208D12—41no—
209D12—41no—
210C4×D3—82no—
211Dic6—41no—
212Dic6—41no—
213C3×C8—22no—
214C3×C8#12012yes—
215C3×C8C22×D412yes—
216C24#21311yescharacteristic
217#41—122no—
218#41—122no—
219C2×Q8—216no—
220C2×Q8—28no—
221C2×Q8—24no—
222C2×Q8—48no—
223C2×Q8—416no—
224C2×Q8—48no—
225C2×Q8—44no—
226C2×Q8—42no—
227C2×Q8—44no—
228C2×Q8—61no—
229C2×D4—61no—
230C2×D4—122no—
231C2×D4—121no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—48no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—121no—
238C22×C4#21312yes—
239C22×C4#21311yescharacteristic
240C22×C4C2×#6712yes—
241C22×C4C2×#6711yescharacteristic
242Q16—216no—
243Q16—28no—
244Q16—24no—
245Q16—62no—
246Q16C2×#6718yes—
247SD16—122no—
248D8—62no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8C2×#6712yes—
253C2×C8C2×#6711yescharacteristic
254C2×C8C2×#6714yes—
255C2×C8C23×D311yescharacteristic
256C4⋊C4—62no—
257C4⋊C4—122no—
258C22⋊C4—121no—
259C22⋊C4—122no—
260C22⋊C4—122no—
261C42—121no—
262C16—64no—
263C22×C3—28no—
264C22×C3—22no—
265C22×C3—22no—
266C22×C3—22no—
267C22×C3#44712yes—
268C22×C3#51014yescomplemented
269C22×C3C22×D811yescharacteristic
270C2×D3—82no—
271C2×D3—81no—
272C3×C4—22no—
273C3×C4—48no—
274C3×C4#44712yes—
275C3×C4C2×#12011yescharacteristic
276C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
277Dic3—82no—
278Dic3—81no—
279C23—24no—
280C23—22no—
281C23—121no—
282C23C2×#21312yes—
283C23C2×#21311yescharacteristic
284Q8—28no—
285Q8—416no—
286Q8—48no—
287Q8—44no—
288Q8—121no—
289Q8—121no—
290D4—121no—
291D4—121no—
292D4—244no—
293D4—242no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—416no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—122no—
302C2×C4—121no—
303C2×C4—242no—
304C2×C4C2×#21312yes—
305C2×C4C2×#21311yescharacteristic
306C2×C4#93212yes—
307C2×C4#95312yes—
308C2×C4C22×#6711yescharacteristic
309C8—22no—
310C8#95312yes—
311C8C22×#6712yes—
312C2×C3—22no—
313C2×C3—24no—
314C2×C3C2×#44711yescharacteristic
315C2×C3C2×#51012yes—
316D3—161no—
317C22—28no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—241no—
322C22—242no—
323C22#517914yescomplemented
324C22#521712yes—
325C22C22×#21311yescharacteristic
326C4—22no—
327C4—48no—
328C4—241no—
329C4—242no—
330C4#521712yes—
331C4C2×#93211yescharacteristic
332C4C2×#95311yescharacteristic
333C3#3351811yescharacteristic, complemented, Sylow
334C2—24no—
335C2—22no—
336C2—481no—
337C2C2×#517912yes—
338C2C2×#521711yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC23×C3×Q16C2
socleC22×C3C22×D8
prime radical#83742C2
hypercenterC23×Q16D3
hypocenterC3#33518
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×Q16

Complements

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

NH ≅ G/Ntype
1. C23×C3×Q16C2semidirect
2. C2×#5179C2semidirect
4. #72064C2semidirect
5. #72063C2semidirect
6. #72061C2semidirect
7. #71963C2semidirect
18. C2×#829C22semidirect
20. C2×#828C22semidirect
27. #4564C22semidirect
29. C23×Q16D3semidirect
51. C2×C3×Q16D4semidirect
160. C2×Q16#67semidirect
268. C22×C3#510semidirect
323. C22#5179semidirect
333. C3#33518semidirect

Maximal subgroups

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

subgroupindex
C23×C3×Q162
C2×#51792
#837422
#720642
#720632
#720612
#719632
#335183