Finite Group Databasebeta

← #10826088

Subgroups — #10826088

Order 768.

778 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×C4⋊Q8C211yesmaximal, characteristic
2C4×#869C211yesmaximal, characteristic
3#71782C212yesmaximal
4#69607C212yesmaximal
5#69604C212yesmaximal
6#69569C212yesmaximal
7C4×#807C214yesmaximal
8#69388C211yesmaximal, characteristic
9#23835—31nomaximal, Sylow
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×C3×C4×Q8—21no—
12C2×#869C2211yescharacteristic
13C2×C4×Dic6—21no—
14C3×#391—21no—
15C3×#389—21no—
16C3×#389C2212yes—
17C3×#384C2211yescharacteristic
18C3×C4×C4⋊C4—21no—
19C3×C4×C4⋊C4C2212yes—
20C3×C43C2211yescharacteristic
21C2×#807C2214yes—
22C2×#807C2214yes—
23#4935C2211yescharacteristic
24C4×C4⋊Dic3—21no—
25C4×C4⋊Dic3C2212yes—
26#4929—21no—
27#4928—21no—
28#4928C2212yes—
29C2×C4×C3⋊C8C2212yes—
30#4529C2214yes—
31#4529C2214yes—
32#4524C2212yes—
33#4524C2212yes—
34C4×C4⋊Q8—31no—
35C4×C4⋊Q8D311yescharacteristic
36#2083—32no—
37#2083—32no—
38#2057—32no—
39#2057—32no—
40C4×#107—34no—
41#1744—31no—
42C22×C3×Q8—21no—
43C22×Dic6—21no—
44C3×C4⋊Q8—24no—
45C3×C4×Q8—44no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—22no—
50C2×C3×C4⋊C4C2312yes—
51C2×C3×C4⋊C4C2312yes—
52C2×C3×C42—22no—
53C2×C3×C42D411yescharacteristic
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C2×C3×C42C2311yescharacteristic
58C2×C3×C42C2311yescharacteristic
59C2×C3×C42C2311yescharacteristic
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3C2312yes—
64C2×C4⋊Dic3C2312yes—
65C2×Dic3⋊C4—22no—
66C2×C4×Dic3—22no—
67C22×C3⋊C8C2314yes—
68#869—24no—
69C4×Dic6—44no—
70C3×#95—22no—
71C3×#95—22no—
72C3⋊#95—22no—
73C3⋊#95—22no—
74#807—28no—
75#807—28no—
76C4×C3⋊C8Q818yescomplemented
77C2×C4⋊Q8—31no—
78C2×C4⋊Q8C2×D311yescharacteristic
79C2×C4×Q8—21no—
80C2×C4×Q8—61no—
81C2×#107—34no—
82C2×#107—34no—
83C2×C4×C8—32no—
84#391—21no—
85#391—61no—
86#389—21no—
87#389—32no—
88#389—61no—
89#389C2×D312yes—
90#384—31no—
91#384C2×D311yescharacteristic
92C4×C4⋊C4—21no—
93C4×C4⋊C4—32no—
94C4×C4⋊C4—61no—
95C4×C4⋊C4C2×D312yes—
96C43C2×D311yescharacteristic
97#340—34no—
98#340—34no—
99#336—32no—
100#336—32no—
101C2×C3×Q8—24no—
102C2×C3×Q8—42no—
103C2×C3×Q8—42no—
104C22×C3×C4—22no—
105C22×C3×C4—22no—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4C2×D411yescharacteristic
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2411yescharacteristic
113C22×Dic3—22no—
114C22×Dic3—22no—
115C2×Dic6—24no—
116C2×Dic6—42no—
117C2×Dic6—42no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—42no—
122C3×C4⋊C4—44no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4#4114yes—
125C3×C42—22no—
126C3×C42—22no—
127C3×C42—41no—
128C3×C42—41no—
129C3×C42—44no—
130C3×C42D814yes—
131C3×C42Q1614yes—
132C3×C42C2×Q814yes—
133C4⋊Dic3—24no—
134C4⋊Dic3—22no—
135C4⋊Dic3—24no—
136C4⋊Dic3—42no—
137C4⋊Dic3—42no—
138C4⋊Dic3#4114yes—
139Dic3⋊C4—44no—
140C4×Dic3—44no—
141C2×C3⋊C8—24no—
142C2×C3⋊C8—24no—
143C2×C3⋊C8C2×Q818yes—
144C22×Q8—21no—
145C22×Q8—61no—
146C22×C8—34no—
147C4⋊Q8—24no—
148C4⋊Q8—64no—
149C4×Q8—44no—
150C4×Q8—124no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—32no—
156C2×C4⋊C4—32no—
157C2×C4⋊C4—61no—
158C2×C4⋊C4—61no—
159C2×C4⋊C4—62no—
160C2×C4⋊C4—61no—
161C2×C4⋊C4C22×D312yes—
162C2×C4⋊C4C22×D312yes—
163C2×C42—22no—
164C2×C42—62no—
165C2×C42#6711yescharacteristic
166C2×C42#6711yescharacteristic
167C2×C42#6711yescharacteristic
168C2×C42#6711yescharacteristic
169C2×C42C22×D311yescharacteristic
170C2×C42C22×D311yescharacteristic
171C2×C42C22×D311yescharacteristic
172#107—68no—
173#107—68no—
174C4×C8—38no—
175#95—22no—
176#95—22no—
177#95—62no—
178#95—62no—
179C23×C3C22×D411yescharacteristic
180C3×Q8—48no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—24no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4C4⋊Q812yes—
198C2×C3×C4C4⋊Q812yes—
199C2×C3×C4C2×D812yes—
200C2×C3×C4C2×D812yes—
201C2×C3×C4C2×D812yes—
202C2×C3×C4C2×Q1612yes—
203C2×C3×C4C2×Q1612yes—
204C2×C3×C4C2×Q1612yes—
205C2×C3×C4C22×Q812yes—
206C2×C3×C4C2×#4112yes—
207C2×C3×C4C2×#4112yes—
208C2×Dic3—24no—
209C2×Dic3—42no—
210C2×Dic3—42no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3—42no—
214Dic6—48no—
215C3⋊C8—28no—
216C2×Q8—24no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—64no—
220C2×Q8—122no—
221C2×Q8—122no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—62no—
225C22×C4—62no—
226C22×C4C2×#6711yescharacteristic
227C22×C4C2×#6711yescharacteristic
228C22×C4C2×#6711yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C22×C4C2×#6711yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C22×C4C23×D311yescharacteristic
233C2×C8—38no—
234C2×C8—64no—
235C2×C8—64no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—24no—
239C4⋊C4—34no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C4⋊C4—64no—
244C4⋊C4—62no—
245C4⋊C4—64no—
246C4⋊C4—122no—
247C4⋊C4—124no—
248C4⋊C4—122no—
249C4⋊C4#23514yes—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—44no—
255C42—124no—
256C42#21314yes—
257C42#21614yes—
258C42D3×Q814yes—
259C22×C3#52511yescharacteristic
260C22×C3#52511yescharacteristic
261C22×C3C2×C4⋊Q811yescharacteristic
262C22×C3C22×D811yescharacteristic
263C22×C3C22×Q1611yescharacteristic
264C22×C3#57511yescharacteristic
265C22×C3#57511yescharacteristic
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—22no—
271C3×C4—44no—
272C3×C4—82no—
273C3×C4#50014yes—
274C3×C4#53314yes—
275Dic3—44no—
276Dic3—82no—
277C23C22×#6711yescharacteristic
278Q8—48no—
279Q8—128no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—24no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—64no—
297C2×C4—122no—
298C2×C4—122no—
299C2×C4—122no—
300C2×C4—122no—
301C2×C4—122no—
302C2×C4C2×#21312yes—
303C2×C4C2×#21312yes—
304C2×C4C2×#21312yes—
305C2×C4C2×#21612yes—
306C2×C4C2×#21612yes—
307C2×C4C2×#21612yes—
308C2×C4#94412yes—
309C2×C4#94412yes—
310C2×C4C2×#23512yes—
311C2×C4C2×D3×Q812yes—
312C2×C4D3×#4112yes—
313C8—68no—
314C2×C3#239911yescharacteristic
315C2×C3#310011yescharacteristic
316C2×C3#310011yescharacteristic
317C2×C3#310411yescharacteristic
318C2×C3#310411yescharacteristic
319C2×C3C2×#50011yescharacteristic
320C2×C3#316211yescharacteristic
321C22#579111yescharacteristic
322C22C22×#21311yescharacteristic
323C22C22×#21611yescharacteristic
324C22C2×#94411yescharacteristic
325C22#582211yescharacteristic
326C22#582211yescharacteristic
327C22#582811yescharacteristic
328C4—22no—
329C4—22no—
330C4—22no—
331C4—22no—
332C4—22no—
333C4—44no—
334C4—82no—
335C4—124no—
336C4—242no—
337C4#507814yes—
338C4#557814yes—
339C3#2383511yescharacteristic, complemented, Sylow
340C2#8230811yescharacteristic
341C2#8244711yescharacteristic
342C2#8246711yescharacteristic
343C2C2×#507811yescharacteristic
344C2#8329811yescharacteristic
345C2#8334011yescharacteristic
346C2#8336811yescharacteristic
subgroupG/H
centerC22×C4C2×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×C4⋊Q8C2
socle, prime radicalC23×C3C22×D4
hypercenterC4×C4⋊Q8D3
hypocenterC3#23835
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C43 ◁ C4×C4⋊Q8

Complements

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

NH ≅ G/Ntype
76. C4×C3⋊C8Q8semidirect
339. C3#23835semidirect

Maximal subgroups

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

subgroupindex
C3×C4×C4⋊Q82
C4×#8692
#717822
#696072
#696042
#695692
C4×#8072
#693882
#238353