Finite Group Databasebeta

← #10900948

Subgroups — #10900948

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×#4970C212yesmaximal
3#82075C211yesmaximal, characteristic
4#71794C214yesmaximal, complemented
5#71782C214yesmaximal, complemented
6#71554C212yesmaximal
7#71531C211yesmaximal, characteristic
8#29642—31nomaximal, Sylow
9C23×C3×Q8—21no—
10C2×C3×C4⋊Q8—28no—
11C2×C3×C4⋊Q8—22no—
12C2×C3×C4⋊Q8C2214yes—
13C22×C3×C4⋊C4—21no—
14C22×C3×C4⋊C4C2212yes—
15C22×C3×C42C2211yescharacteristic
16C2×#924C2211yescharacteristic
17#5212C2211yescharacteristic
18#5211—21no—
19#5210—21no—
20C2×#830C2212yes—
21#4970—28no—
22C2×#810C2214yes—
23C2×#807C2214yes—
24#4933—22no—
25#4929C2212yes—
26#4928C2212yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8D311yescharacteristic
30C2×#484—32no—
31#2855—31no—
32#2057—34no—
33#2024—34no—
34#1824—32no—
35#1817—31no—
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—48no—
39C23×C3×C4—22no—
40C23×C3×C4D411yescharacteristic
41C23×C3×C4D411yescharacteristic
42C23×C3×C4C2311yescharacteristic
43C22×Dic6C2311yescharacteristic
44C3×C4⋊Q8—216no—
45C3×C4⋊Q8—216no—
46C2×C3×C4⋊C4—28no—
47C2×C3×C4⋊C4—24no—
48C2×C3×C4⋊C4—22no—
49C2×C3×C4⋊C4—24no—
50C2×C3×C4⋊C4D418yes—
51C2×C3×C4⋊C4C2314yes—
52C2×C3×C42—24no—
53C2×C3×C42D412yes—
54C2×C3×C42C2312yes—
55C2×#217—21no—
56C2×#217—21no—
57C2×C4⋊Dic3C2311yescharacteristic
58C2×C4⋊Dic3C2312yes—
59#924—24no—
60C2×Dic3⋊C4—22no—
61C2×Dic3⋊C4—21no—
62C22×C3⋊C8C2314yes—
63C3⋊#95—22no—
64C3⋊#95—22no—
65C3⋊#95—22no—
66C3⋊#95—21no—
67#830—24no—
68#810—28no—
69#807—28no—
70C23×Q8—21no—
71C2×C4⋊Q8—28no—
72C2×C4⋊Q8—22no—
73C2×C4⋊Q8C2×D314yes—
74C2×C22⋊Q8—31no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4C2×D312yes—
77C22×C42C2×D311yescharacteristic
78#484—68no—
79C2×#107—34no—
80C2×#103—34no—
81C2×C22⋊C8—32no—
82#391—32no—
83#389—32no—
84#387—62no—
85#385—61no—
86#381—61no—
87#380—31no—
88#340—38no—
89#336—34no—
90C24×C3C2×D411yescharacteristic
91C2×C3×Q8—216no—
92C2×C3×Q8—432no—
93C2×C3×Q8—48no—
94C2×C3×Q8—48no—
95C22×C3×C4—22no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—24no—
99C22×C3×C4—24no—
100C22×C3×C4—28no—
101C22×C3×C4—24no—
102C22×C3×C4—48no—
103C22×C3×C4Q1614yes—
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D414yes—
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D412yes—
108C22×C3×C4C2411yescharacteristic
109C22×Dic3—22no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C2×Dic6—22no—
113C2×Dic6—24no—
114C3×C4⋊C4—216no—
115C3×C4⋊C4—28no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4C2×D418yes—
118C3×C42—24no—
119C3×C42—24no—
120#217—42no—
121#217—42no—
122C4⋊Dic3—22no—
123C4⋊Dic3—24no—
124Dic3⋊C4—42no—
125Dic3⋊C4—44no—
126C2×C3⋊C8—24no—
127C2×C3⋊C8—28no—
128C22×Q8—28no—
129C22×Q8—24no—
130C22×Q8—31no—
131C22×Q8—48no—
132C23×C4—22no—
133C23×C4#6711yescharacteristic
134C23×C4#6711yescharacteristic
135C23×C4C22×D311yescharacteristic
136C22×C8—34no—
137C4⋊Q8—216no—
138C4⋊Q8—216no—
139C22⋊Q8—64no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—32no—
146C2×C4⋊C4—61no—
147C2×C4⋊C4—62no—
148C2×C4⋊C4#6718yes—
149C2×C4⋊C4C22×D314yes—
150C2×C22⋊C4—61no—
151C2×C22⋊C4—61no—
152C2×C42—24no—
153C2×C42#6712yes—
154C2×C42C22×D312yes—
155#107—68no—
156#103—68no—
157C22⋊C8—64no—
158#95—61no—
159#95—62no—
160#95—62no—
161#95—62no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3#12012yes—
165C23×C3C2×Q1612yes—
166C23×C3#13712yes—
167C23×C3C22×D411yescharacteristic
168C3×Q8—432no—
169C2×C3×C4—28no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—28no—
176C2×C3×C4—28no—
177C2×C3×C4—28no—
178C2×C3×C4—416no—
179C2×C3×C4—48no—
180C2×C3×C4#12012yes—
181C2×C3×C4C2×Q1614yes—
182C2×C3×C4AGL(1,ℤ/8)14yes—
183C2×C3×C4C22×D412yes—
184C2×C3×C42−1+414yes—
185C2×Dic3—41no—
186C2×Dic3—41no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—42no—
191C2×Dic3—42no—
192Dic6—44no—
193C3⋊C8—44no—
194C24C2×#6711yescharacteristic
195C2×Q8—216no—
196C2×Q8—432no—
197C2×Q8—48no—
198C2×Q8—48no—
199C2×Q8—64no—
200C2×Q8—62no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—28no—
207C22×C4—24no—
208C22×C4—48no—
209C22×C4—62no—
210C22×C4—61no—
211C22×C4—61no—
212C22×C4#21614yes—
213C22×C4C2×#6712yes—
214C22×C4C2×#6712yes—
215C22×C4C2×#6711yescharacteristic
216C22×C4C2×#6714yes—
217C22×C4C23×D311yescharacteristic
218C2×C8—68no—
219C2×C8—64no—
220C4⋊C4—216no—
221C4⋊C4—28no—
222C4⋊C4—24no—
223C4⋊C4—64no—
224C4⋊C4—62no—
225C4⋊C4—124no—
226C4⋊C4—122no—
227C4⋊C4C2×#6718yes—
228C22⋊C4—122no—
229C22⋊C4—122no—
230C42—24no—
231C42—24no—
232C22×C3—22no—
233C22×C3—24no—
234C22×C3—24no—
235C22×C3—22no—
236C22×C3#45114yes—
237C22×C3C2×#12011yescharacteristic
238C22×C3#53612yes—
239C22×C3C22×Q1611yescharacteristic
240C22×C3C2×AGL(1,ℤ/8)12yes—
241C22×C3C2×#13711yescharacteristic
242C3×C4—22no—
243C3×C4—24no—
244C3×C4—24no—
245C3×C4—48no—
246C3×C4#53614yes—
247Dic3—81no—
248Dic3—82no—
249Dic3—81no—
250C23—22no—
251C23—22no—
252C23#94212yes—
253C23C2×#21612yes—
254C23#95312yes—
255C23C22×#6711yescharacteristic
256Q8—432no—
257Q8—124no—
258C2×C4—28no—
259C2×C4—28no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—28no—
266C2×C4—28no—
267C2×C4—416no—
268C2×C4—48no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4—122no—
273C2×C4—122no—
274C2×C4—121no—
275C2×C4—121no—
276C2×C4#93214yes—
277C2×C4C2×#21614yes—
278C2×C4#95312yes—
279C2×C4#100714yes—
280C2×C4C22×#6712yes—
281C8—124no—
282C2×C3—24no—
283C2×C3#261411yescharacteristic
284C2×C3C2×#45111yescharacteristic
285C2×C3#300311yescharacteristic
286C2×C3#323012yes—
287C2×C3#323912yes—
288C22—24no—
289C22—24no—
290C22—22no—
291C22—22no—
292C22#522814yes—
293C22C2×#93212yes—
294C22C2×#94211yescharacteristic
295C22C22×#21611yescharacteristic
296C22#581612yes—
297C22C2×#95311yescharacteristic
298C4—24no—
299C4—24no—
300C4—22no—
301C4—48no—
302C4—242no—
303C4—241no—
304C4—241no—
305C4#581614yes—
306C3#2964211yescharacteristic, complemented, Sylow
307C2—24no—
308C2#8321812yes—
309C2#8322412yes—
310C2#8403811yescharacteristic
311C2C2×#522811yescharacteristic
312C2#8406911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC22×C3×C4⋊Q8C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC22×C4⋊Q8D3
hypocenterC3#29642
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #71794C2semidirect
5. #71782C2semidirect
306. C3#29642semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊Q82
C2×#49702
#820752
#717942
#717822
#715542
#715312
#296423