Finite Group Databasebeta

← #10900941

Subgroups — #10900941

Order 768.

1230 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4⋊Q8C211yesmaximal, characteristic
2C2×#4971C212yesmaximal
3#82075C211yesmaximal, characteristic
4#71794C214yesmaximal, complemented
5#71783C214yesmaximal, complemented
6#71553C212yesmaximal
7#71531C211yesmaximal, characteristic
8#29641—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#4971—28no—
22C2×#810C2214yes—
23C2×#808C2214yes—
24#4933—22no—
25#4929C2212yes—
26#4928C2212yes—
27#4529C2218yes—
28#4524C2214yes—
29C22×C4⋊Q8D311yescharacteristic
30C2×#483—32no—
31#2855—31no—
32#2058—34no—
33#2024—34no—
34#1823—32no—
35#1817—31no—
36C22×C3×Q8—28no—
37C22×C3×Q8—24no—
38C22×C3×Q8—44no—
39C22×C3×Q8—44no—
40C23×C3×C4—22no—
41C23×C3×C4D411yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C22×Dic6C2311yescharacteristic
45C3×C4⋊Q8—216no—
46C3×C4⋊Q8—216no—
47C2×C3×C4⋊C4—28no—
48C2×C3×C4⋊C4—24no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4—24no—
51C2×C3×C4⋊C4D418yes—
52C2×C3×C4⋊C4C2314yes—
53C2×C3×C42—24no—
54C2×C3×C42D412yes—
55C2×C3×C42C2312yes—
56C2×#217—21no—
57C2×#217—21no—
58C2×C4⋊Dic3C2311yescharacteristic
59C2×C4⋊Dic3C2312yes—
60#924—24no—
61C2×Dic3⋊C4—22no—
62C2×Dic3⋊C4—21no—
63C22×C3⋊C8C2314yes—
64C3⋊#95—22no—
65C3⋊#95—22no—
66C3⋊#95—22no—
67C3⋊#95—21no—
68#830—24no—
69#810—28no—
70#808—28no—
71C23×Q8—21no—
72C2×C4⋊Q8—28no—
73C2×C4⋊Q8—22no—
74C2×C4⋊Q8C2×D314yes—
75C2×C22⋊Q8—31no—
76C22×C4⋊C4—21no—
77C22×C4⋊C4C2×D312yes—
78C22×C42C2×D311yescharacteristic
79#483—68no—
80C2×#106—34no—
81C2×#103—34no—
82C2×C22⋊C8—32no—
83#391—32no—
84#389—32no—
85#387—62no—
86#385—61no—
87#381—61no—
88#380—31no—
89#340—38no—
90#336—34no—
91C24×C3C2×D411yescharacteristic
92C2×C3×Q8—216no—
93C2×C3×Q8—48no—
94C2×C3×Q8—432no—
95C2×C3×Q8—48no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—24no—
100C22×C3×C4—24no—
101C22×C3×C4—28no—
102C22×C3×C4—24no—
103C22×C3×C4—44no—
104C22×C3×C4—44no—
105C22×C3×C4SD1614yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D414yes—
108C22×C3×C4C2×D412yes—
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2411yescharacteristic
111C22×Dic3—22no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C2×Dic6—22no—
115C2×Dic6—24no—
116C3×C4⋊C4—216no—
117C3×C4⋊C4—28no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4C2×D418yes—
120C3×C42—24no—
121C3×C42—24no—
122#217—42no—
123#217—42no—
124C4⋊Dic3—22no—
125C4⋊Dic3—24no—
126Dic3⋊C4—42no—
127Dic3⋊C4—44no—
128C2×C3⋊C8—24no—
129C2×C3⋊C8—28no—
130C22×Q8—28no—
131C22×Q8—24no—
132C22×Q8—31no—
133C22×Q8—44no—
134C22×Q8—44no—
135C23×C4—22no—
136C23×C4#6711yescharacteristic
137C23×C4#6711yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—34no—
140C4⋊Q8—216no—
141C4⋊Q8—216no—
142C22⋊Q8—64no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—32no—
149C2×C4⋊C4—61no—
150C2×C4⋊C4—62no—
151C2×C4⋊C4#6718yes—
152C2×C4⋊C4C22×D314yes—
153C2×C22⋊C4—61no—
154C2×C22⋊C4—61no—
155C2×C42—24no—
156C2×C42#6712yes—
157C2×C42C22×D312yes—
158#106—68no—
159#103—68no—
160C22⋊C8—64no—
161#95—61no—
162#95—62no—
163#95—62no—
164#95—62no—
165C23×C3—22no—
166C23×C3—22no—
167C23×C3#12012yes—
168C23×C3C2×SD1612yes—
169C23×C3AGL(1,ℤ/8)12yes—
170C23×C3C22×D411yescharacteristic
171C3×Q8—432no—
172C2×C3×C4—28no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—28no—
179C2×C3×C4—28no—
180C2×C3×C4—28no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—416no—
184C2×C3×C4#12012yes—
185C2×C3×C4C2×SD1614yes—
186C2×C3×C4#13714yes—
187C2×C3×C4C22×D412yes—
188C2×C3×C42−1+414yes—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194C2×Dic3—42no—
195C2×Dic3—42no—
196Dic6—44no—
197C3⋊C8—44no—
198C24C2×#6711yescharacteristic
199C2×Q8—216no—
200C2×Q8—432no—
201C2×Q8—48no—
202C2×Q8—48no—
203C2×Q8—64no—
204C2×Q8—62no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—24no—
212C22×C4—44no—
213C22×C4—44no—
214C22×C4—62no—
215C22×C4—61no—
216C22×C4—61no—
217C22×C4#21414yes—
218C22×C4C2×#6712yes—
219C22×C4C2×#6712yes—
220C22×C4C2×#6711yescharacteristic
221C22×C4C2×#6714yes—
222C22×C4C23×D311yescharacteristic
223C2×C8—68no—
224C2×C8—64no—
225C4⋊C4—216no—
226C4⋊C4—28no—
227C4⋊C4—24no—
228C4⋊C4—64no—
229C4⋊C4—62no—
230C4⋊C4—124no—
231C4⋊C4—122no—
232C4⋊C4C2×#6718yes—
233C22⋊C4—122no—
234C22⋊C4—122no—
235C42—24no—
236C42—24no—
237C22×C3—22no—
238C22×C3—24no—
239C22×C3—24no—
240C22×C3—22no—
241C22×C3#45014yes—
242C22×C3C2×#12011yescharacteristic
243C22×C3#53612yes—
244C22×C3C22×SD1611yescharacteristic
245C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
246C22×C3C2×#13712yes—
247C3×C4—22no—
248C3×C4—24no—
249C3×C4—24no—
250C3×C4—48no—
251C3×C4#53614yes—
252Dic3—81no—
253Dic3—82no—
254Dic3—81no—
255C23—22no—
256C23—22no—
257C23#93212yes—
258C23C2×#21412yes—
259C23#95312yes—
260C23C22×#6711yescharacteristic
261Q8—432no—
262Q8—124no—
263C2×C4—28no—
264C2×C4—28no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—28no—
271C2×C4—28no—
272C2×C4—416no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—122no—
276C2×C4—122no—
277C2×C4—122no—
278C2×C4—122no—
279C2×C4—122no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4C2×#21414yes—
283C2×C4#94214yes—
284C2×C4#95312yes—
285C2×C4#100714yes—
286C2×C4C22×#6712yes—
287C8—124no—
288C2×C3—24no—
289C2×C3#261411yescharacteristic
290C2×C3C2×#45011yescharacteristic
291C2×C3#300411yescharacteristic
292C2×C3#322912yes—
293C2×C3#323812yes—
294C22—24no—
295C22—24no—
296C22—22no—
297C22—22no—
298C22#521814yes—
299C22C2×#93211yescharacteristic
300C22C22×#21411yescharacteristic
301C22C2×#94212yes—
302C22#581612yes—
303C22C2×#95311yescharacteristic
304C4—24no—
305C4—24no—
306C4—22no—
307C4—48no—
308C4—242no—
309C4—241no—
310C4—241no—
311C4#581614yes—
312C3#2964111yescharacteristic, complemented, Sylow
313C2—24no—
314C2#8321712yes—
315C2#8322312yes—
316C2C2×#521811yescharacteristic
317C2#8405911yescharacteristic
318C2#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#29641
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. #71783C2semidirect
312. C3#29641semidirect

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×#49712
#820752
#717942
#717832
#715532
#715312
#296413