Finite Group Databasebeta

← #10826102

Subgroups — #10826102

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#888C212yesmaximal
2C3×C4×#106C211yesmaximal, characteristic
3#70567C212yesmaximal
4#70565C212yesmaximal
5C4×#817C211yesmaximal, characteristic
6#69750C211yesmaximal, characteristic
7C4×#808C212yesmaximal
8#68835C212yesmaximal
9#68827C212yesmaximal
10#23837—31nomaximal, Sylow
11C2×#888C2212yes—
12C2×C4×Dic6—22no—
13C2×C3×#106C2212yes—
14C2×C3×C4×C8C2211yescharacteristic
15C3×C4×C4⋊C4C2212yes—
16C2×#817C2212yes—
17#4984C2212yes—
18#4975C2212yes—
19#4972C2212yes—
20C2×#808C2212yes—
21C2×#808C2212yes—
22C4×C4⋊Dic3C2212yes—
23C4×Dic3⋊C4—22no—
24C42×Dic3C2211yescharacteristic
25C2×C4×C3⋊C8C2211yescharacteristic
26#4647—22no—
27#4646—22no—
28C3×#340C2211yescharacteristic
29C3×#340C2211yescharacteristic
30#4551C2211yescharacteristic
31#4551C2211yescharacteristic
32#4550C2212yes—
33#4550C2212yes—
34#4529C2212yes—
35#4529C2212yes—
36C4×C4⋊Q8—32no—
37#2084—32no—
38#2084—32no—
39#2058—32no—
40#2058—32no—
41C4×#106—31no—
42C4×#106—32no—
43C4×#106D311yescharacteristic
44#1744—31no—
45C22×Dic6—22no—
46C22×C3×C8C2311yescharacteristic
47C22×C3×C8C2311yescharacteristic
48C2×C3×C4⋊C4C2312yes—
49C2×C3×C4⋊C4C2312yes—
50C2×C3×C42—22no—
51C2×C3×C42C2311yescharacteristic
52C2×C4⋊Dic3—22no—
53C2×C4⋊Dic3C2312yes—
54C2×C4⋊Dic3C2312yes—
55C2×Dic3⋊C4—22no—
56C2×Dic3⋊C4—22no—
57C2×Dic3⋊C4—22no—
58C2×Dic3⋊C4—22no—
59C2×C4×Dic3—22no—
60C2×C4×Dic3D411yescharacteristic
61C2×C4×Dic3D411yescharacteristic
62C2×C4×Dic3D412yes—
63C2×C4×Dic3C2312yes—
64C22×C3⋊C8C2312yes—
65#888—24no—
66#888—24no—
67C4×Dic6—48no—
68C3×#106—24no—
69C3×C4×C8Q814yescomplemented
70C3×#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74#817—24no—
75#808—24no—
76#808—24no—
77C4×C3⋊C8Q814yescomplemented
78C2×C4⋊Q8—32no—
79C2×C4×Q8—62no—
80C2×#106—32no—
81C2×#106—32no—
82C2×#106—32no—
83C2×#106C2×D312yes—
84C2×C4×C8—31no—
85C2×C4×C8C2×D311yescharacteristic
86#391—62no—
87#389—32no—
88#389—32no—
89#389—62no—
90#384—32no—
91C4×C4⋊C4—32no—
92C4×C4⋊C4—62no—
93C4×C4⋊C4C2×D312yes—
94C43—31no—
95#340—31no—
96#340—31no—
97#340—32no—
98#340—32no—
99#340C2×D311yescharacteristic
100#340C2×D311yescharacteristic
101#336—32no—
102#336—32no—
103C22×C3×C4—22no—
104C22×C3×C4—22no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2411yescharacteristic
108C22×Dic3—22no—
109C22×Dic3—22no—
110C22×Dic3C2×D412yes—
111C22×Dic3C2×D412yes—
112C2×Dic6—28no—
113C2×Dic6—44no—
114C2×Dic6—44no—
115C2×C3×C8—22no—
116C2×C3×C8—22no—
117C2×C3×C8C2×Q812yes—
118C2×C3×C8C2×Q812yes—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4#4114yes—
123C3×C42—44no—
124C3×C42C2×Q812yes—
125C3×C42C2×Q812yes—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—44no—
130C4⋊Dic3#4114yes—
131Dic3⋊C4—28no—
132Dic3⋊C4—44no—
133Dic3⋊C4—44no—
134Dic3⋊C4—44no—
135C4×Dic3—22no—
136C4×Dic3—22no—
137C4×Dic3—41no—
138C4×Dic3—41no—
139C4×Dic3—44no—
140C4×Dic3SD1618yes—
141C2×C3⋊C8—22no—
142C2×C3⋊C8—22no—
143C2×C3⋊C8C2×Q814yes—
144C22×Q8—62no—
145C22×C8—32no—
146C22×C8C22×D311yescharacteristic
147C22×C8C22×D311yescharacteristic
148C4⋊Q8—64no—
149C4⋊Q8—64no—
150C4×Q8—128no—
151C2×C4⋊C4—32no—
152C2×C4⋊C4—32no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4—62no—
155C2×C4⋊C4—62no—
156C2×C4⋊C4—62no—
157C2×C4⋊C4—62no—
158C2×C4⋊C4C22×D312yes—
159C2×C4⋊C4C22×D312yes—
160C2×C42—22no—
161C2×C42—32no—
162C2×C42—32no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—62no—
166C2×C42C22×D311yescharacteristic
167#106—24no—
168#106—64no—
169#106—64no—
170#106—64no—
171C4×C8—34no—
172C4×C8Dic614yescomplemented
173#95—22no—
174#95—62no—
175#95—62no—
176#95—62no—
177C23×C3C22×D411yescharacteristic
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—24no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4C4⋊Q812yes—
187C2×C3×C4C4⋊Q812yes—
188C2×C3×C4C2×SD1614yes—
189C2×C3×C4C22×Q811yescharacteristic
190C2×C3×C4C22×Q811yescharacteristic
191C2×C3×C4C2×#4112yes—
192C2×C3×C4C2×#4112yes—
193C2×Dic3—22no—
194C2×Dic3—22no—
195C2×Dic3—22no—
196C2×Dic3—22no—
197C2×Dic3—24no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—42no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3C2×SD1614yes—
204C2×Dic3C2×SD1614yes—
205Dic6—416no—
206C3×C8—22no—
207C3×C8—22no—
208C3⋊C8—24no—
209C2×Q8—68no—
210C2×Q8—124no—
211C2×Q8—124no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—32no—
215C22×C4—32no—
216C22×C4—62no—
217C22×C4—62no—
218C22×C4D3×D411yescharacteristic
219C22×C4D3×D411yescharacteristic
220C22×C4C23×D311yescharacteristic
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—34no—
224C2×C8—62no—
225C2×C8—62no—
226C2×C8C2×Dic612yes—
227C2×C8C2×Dic612yes—
228C4⋊C4—22no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—34no—
232C4⋊C4—68no—
233C4⋊C4—62no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C4⋊C4—124no—
237C4⋊C4—124no—
238C4⋊C4—124no—
239C4⋊C4—124no—
240C4⋊C4#23514yes—
241C42—38no—
242C42—44no—
243C42—62no—
244C42—62no—
245C42—121no—
246C42—121no—
247C42—124no—
248C42C2×Dic612yes—
249C42D3×Q812yes—
250C22×C3#52512yes—
251C22×C3C2×C4⋊Q811yescharacteristic
252C22×C3C22×SD1612yes—
253C22×C3#57511yescharacteristic
254C22×C3#57511yescharacteristic
255C3×C4—22no—
256C3×C4—44no—
257C3×C4—82no—
258C3×C4#49814yes—
259C3×C4#53311yescharacteristic
260C3×C4#53311yescharacteristic
261C3×C4#53312yes—
262Dic3—24no—
263Dic3—24no—
264Dic3—44no—
265Dic3—82no—
266C23C2×D3×D411yescharacteristic
267Q8—1216no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—24no—
271C2×C4—34no—
272C2×C4—34no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—62no—
279C2×C4—62no—
280C2×C4—62no—
281C2×C4—62no—
282C2×C4—64no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4#88812yes—
289C2×C4#88812yes—
290C2×C4D3×SD1614yes—
291C2×C4C22×Dic611yescharacteristic
292C2×C4C2×#23512yes—
293C2×C4C2×D3×Q811yescharacteristic
294C2×C4D3×#4112yes—
295C8—22no—
296C8—22no—
297C8—64no—
298C2×C3#239911yescharacteristic
299C2×C3#310112yes—
300C2×C3#310312yes—
301C2×C3C2×#49811yescharacteristic
302C2×C3#316211yescharacteristic
303C22C2×#88811yescharacteristic
304C22#554712yes—
305C22C2×D3×SD1612yes—
306C22#576811yescharacteristic
307C22#576811yescharacteristic
308C4—22no—
309C4—44no—
310C4—64no—
311C4—64no—
312C4—82no—
313C4—124no—
314C4—242no—
315C4#485514yes—
316C4#547011yescharacteristic
317C4#551711yescharacteristic
318C4#557812yes—
319C3#2383711yescharacteristic, complemented, Sylow
320C2#7366911yescharacteristic
321C2C2×#485511yescharacteristic
322C2#7917111yescharacteristic
323C2#7965112yes—
324C2#7965312yes—
subgroupG/H
centerC22×C4D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#106C2
socle, prime radicalC23×C3C22×D4
hypercenterC4×#106D3
hypocenterC3#23837
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ C4×#106

Complements

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

NH ≅ G/Ntype
69. C3×C4×C8Q8semidirect
77. C4×C3⋊C8Q8semidirect
172. C4×C8Dic6semidirect
319. C3#23837semidirect

Maximal subgroups

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

subgroupindex
C4×#8882
C3×C4×#1062
#705672
#705652
C4×#8172
#697502
C4×#8082
#688352
#688272
#238373