Finite Group Databasebeta

← #10876218

Subgroups — #10876218

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2314C211yesmaximal, characteristic
2#82230C211yesmaximal, characteristic
3#71584C211yesmaximal, characteristic
4#69589C212yesmaximal, complemented
5#69561C212yesmaximal, complemented
6#69525C212yesmaximal
7#69506C212yesmaximal
8#69465C212yesmaximal, complemented
9#69464C212yesmaximal, complemented
10#27592—31nomaximal, Sylow
11C23×Dic6C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×#926C2211yescharacteristic
14C3×#388—22no—
15C3×#387—22no—
16C3×#384C2211yescharacteristic
17C3×#382C2211yescharacteristic
18C3×#380C2212yes—
19C3×C4×C4⋊C4C2212yes—
20C2×#830C2212yes—
21#4973—22no—
22C2×#810C2212yes—
23C2×#810C2212yes—
24C2×#810C2212yes—
25C2×#810C2212yes—
26C2×#808C2212yes—
27C2×#807C2212yes—
28#4943—22no—
29#4929C2212yes—
30#4929C2212yes—
31#4529C2212yes—
32#4529C2212yes—
33#4524C2211yescharacteristic
34#4524C2211yescharacteristic
35#4524C2212yes—
36#2314D311yescharacteristic
37#2310—31no—
38#2097—32no—
39#2072—32no—
40#1928—32no—
41#1927—32no—
42#1854—32no—
43#1818—31no—
44#1787—32no—
45C23×C3×C4C2311yescharacteristic
46C23×Dic3—21no—
47C22×Dic6—21no—
48C22×Dic6—24no—
49C22×Dic6—24no—
50C22×Dic6—41no—
51C22×Dic6—41no—
52C22×Dic6C2312yes—
53C3×#117—24no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4D412yes—
56C2×C3×C4⋊C4C2312yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—22no—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C42—22no—
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—21no—
64#926—41no—
65#926—41no—
66C2×C4⋊Dic3C2312yes—
67C2×C4×Dic3—21no—
68C22×C3⋊C8C2312yes—
69C22×C3⋊C8C2312yes—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—22no—
73C3⋊#95—24no—
74C3⋊#95—24no—
75#830—24no—
76#810—24no—
77#810—24no—
78#810—42no—
79#810—42no—
80#808—42no—
81#807—42no—
82C23×Q8—31no—
83C2×#117—31no—
84C2×#117C2×D311yescharacteristic
85C2×#107—32no—
86C2×#106—32no—
87C2×#103—32no—
88C2×#103—32no—
89C2×#103—32no—
90C2×#103—32no—
91C2×C22⋊C8—32no—
92#391—32no—
93#391—32no—
94#391—62no—
95#388—22no—
96#387—22no—
97#384C2×D311yescharacteristic
98#382C2×D311yescharacteristic
99#381—62no—
100#380C2×D312yes—
101C4×C4⋊C4C2×D312yes—
102#340—32no—
103#340—32no—
104#336—32no—
105#336—31no—
106#336—31no—
107C24×C3C2×D411yescharacteristic
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—22no—
112C22×C3×C4—22no—
113C22×C3×C4C2×D412yes—
114C22×C3×C4C2×D412yes—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4#4114yes—
117C22×C3×C4C2411yescharacteristic
118C22×Dic3—24no—
119C22×Dic3—22no—
120C22×Dic3—22no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C2×Dic6—22no—
124C2×Dic6—22no—
125C2×Dic6—22no—
126C2×Dic6—22no—
127C2×Dic6—24no—
128C2×Dic6—22no—
129C2×Dic6—48no—
130C2×Dic6—48no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6#4114yes—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—82no—
140C3×C22⋊C4—24no—
141C3×C22⋊C4—82no—
142C3×C42—22no—
143C3×C42—22no—
144C3×C42—82no—
145#217—42no—
146C4⋊Dic3—22no—
147C4⋊Dic3—22no—
148C4×Dic3—42no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—22no—
151C2×C3⋊C8—22no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—22no—
155C22×Q8—32no—
156C22×Q8—64no—
157C22×Q8—61no—
158C22×Q8—64no—
159C22×Q8—121no—
160C22×Q8—121no—
161C23×C4—61no—
162C23×C4C22×D311yescharacteristic
163C22×C8—32no—
164C22×C8—32no—
165#117—24no—
166#117—121no—
167#117—121no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4#6712yes—
171C2×C4⋊C4C22×D312yes—
172C2×C4⋊C4C22×D312yes—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—61no—
175C2×C22⋊C4#6712yes—
176C2×C42—22no—
177C2×C42—61no—
178C2×C42C22×D311yescharacteristic
179C2×C42C22×D311yescharacteristic
180#107—122no—
181#106—122no—
182#103—64no—
183#103—64no—
184#103—122no—
185#103—122no—
186C22⋊C8—64no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—64no—
191#95—64no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3#12312yes—
197C23×C3C22×D411yescharacteristic
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—24no—
201C2×C3×C4—24no—
202C2×C3×C4—22no—
203C2×C3×C4—22no—
204C2×C3×C4—22no—
205C2×C3×C4—22no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—42no—
209C2×C3×C4—42no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4#12312yes—
215C2×C3×C4C2×#4112yes—
216C2×C3×C4C2×#4112yes—
217C2×C3×C4H(5,𝔽2)11yescharacteristic
218C2×C3×C4H(5,𝔽2)11yescharacteristic
219C2×Dic3—24no—
220C2×Dic3—48no—
221C2×Dic3—42no—
222C2×Dic3—42no—
223C2×Dic3—42no—
224C2×Dic3—42no—
225C2×Dic3—42no—
226Dic6—24no—
227Dic6—24no—
228Dic6—48no—
229Dic6—42no—
230Dic6—42no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233C24C2×#6711yescharacteristic
234C2×Q8—34no—
235C2×Q8—64no—
236C2×Q8—62no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—62no—
240C2×Q8—62no—
241C2×Q8—128no—
242C2×Q8—128no—
243C2×Q8—122no—
244C2×Q8—122no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—64no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—122no—
254C22×C4—122no—
255C22×C4#23514yes—
256C22×C4C2×#6712yes—
257C22×C4C2×#6712yes—
258C22×C4C2×#6711yescharacteristic
259C22×C4C23×D311yescharacteristic
260C2×C8—62no—
261C2×C8—62no—
262C2×C8—62no—
263C2×C8—62no—
264C2×C8—62no—
265C2×C8—62no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—82no—
274C22⋊C4—24no—
275C22⋊C4—82no—
276C22⋊C4—122no—
277C42—22no—
278C42—22no—
279C42—82no—
280C42—122no—
281C22×C3—24no—
282C22×C3—22no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—42no—
286C22×C3C2×#12311yescharacteristic
287C22×C3#52511yescharacteristic
288C22×C3#53511yescharacteristic
289C22×C3#57712yes—
290C22×C3#57812yes—
291C3×C4—22no—
292C3×C4—42no—
293C3×C4—42no—
294C3×C4—82no—
295C3×C4—82no—
296C3×C4#55112yes—
297C3×C4#55112yes—
298Dic3—44no—
299Dic3—82no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23#92912yes—
305C23C22×#6711yescharacteristic
306Q8—64no—
307Q8—64no—
308Q8—128no—
309Q8—122no—
310Q8—122no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—64no—
328C2×C4—128no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4#92912yes—
335C2×C4C2×#23512yes—
336C2×C4D3×#4112yes—
337C2×C4#100911yescharacteristic
338C2×C4#100911yescharacteristic
339C8—122no—
340C8—122no—
341C2×C3—42no—
342C2×C3#248211yescharacteristic
343C2×C3#309712yes—
344C2×C3#312611yescharacteristic
345C2×C3#312711yescharacteristic
346C2×C3#322512yes—
347C22—22no—
348C22—24no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22C2×#92911yescharacteristic
353C22#582811yescharacteristic
354C22#583011yescharacteristic
355C22#583612yes—
356C22#583812yes—
357C4—22no—
358C4—42no—
359C4—42no—
360C4—82no—
361C4—82no—
362C4—124no—
363C4—242no—
364C4#552712yes—
365C4#552712yes—
366C3#2759211yescharacteristic, complemented, Sylow
367C2—42no—
368C2#8228211yescharacteristic
369C2#8257312yes—
370C2#8343911yescharacteristic
371C2#8345111yescharacteristic
372C2#8346412yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2314C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2314D3
hypocenterC3#27592
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2314

Complements

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

NH ≅ G/Ntype
4. #69589C2semidirect
5. #69561C2semidirect
8. #69465C2semidirect
9. #69464C2semidirect
366. C3#27592semidirect

Maximal subgroups

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

subgroupindex
C3×#23142
#822302
#715842
#695892
#695612
#695252
#695062
#694652
#694642
#275923