Finite Group Databasebeta

← #10876223

Subgroups — #10876223

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×#2310C211yesmaximal, characteristic
2#82231C211yesmaximal, characteristic
3#72132C212yesmaximal
4#71584C211yesmaximal, characteristic
5#71576C212yesmaximal
6#69686C212yesmaximal, complemented
7#69670C212yesmaximal, complemented
8#69667C212yesmaximal, complemented
9#69666C212yesmaximal, complemented
10#27592—31nomaximal, Sylow
11C23×C3×Q8C2211yescharacteristic
12C2×C3×#117C2211yescharacteristic
13C2×C3×#117C2211yescharacteristic
14C3×#391—22no—
15C3×#391C2212yes—
16C3×#391C2212yes—
17C3×#381—22no—
18C2×#835C2212yes—
19C2×#835C2212yes—
20C2×#835C2212yes—
21C2×#835C2212yes—
22#5212C2212yes—
23C2×#830C2212yes—
24#4980—22no—
25C2×#808C2212yes—
26C2×#807C2212yes—
27#4951—22no—
28#4936C2211yescharacteristic
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31#4529C2212yes—
32#4529C2212yes—
33#4524C2211yescharacteristic
34#4524C2211yescharacteristic
35#4524C2212yes—
36#2314—31no—
37#2310D311yescharacteristic
38#2097—32no—
39#2072—32no—
40#1928—32no—
41#1927—32no—
42#1854—32no—
43#1818—31no—
44#1787—32no—
45C22×C3×Q8—24no—
46C22×C3×Q8—21no—
47C22×C3×Q8—24no—
48C22×C3×Q8—41no—
49C22×C3×Q8—41no—
50C22×C3×Q8C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C3×#117—24no—
54C3×#117—41no—
55C3×#117—41no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4D412yes—
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×#217—22no—
64C2×C4⋊Dic3—22no—
65C2×C4⋊Dic3C2312yes—
66C2×C4⋊Dic3C2312yes—
67C2×C4×Dic3—22no—
68C22×C3⋊C8C2312yes—
69C22×C3⋊C8C2312yes—
70C3×#95—24no—
71C3×#95—24no—
72#835—24no—
73#835—24no—
74#835—42no—
75#835—42no—
76C3⋊#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79#830—24no—
80#808—42no—
81#807—42no—
82C23×Q8C2×D311yescharacteristic
83C2×#117C2×D311yescharacteristic
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—22no—
93#391C2×D312yes—
94#391C2×D312yes—
95#388—62no—
96#387—62no—
97#384—31no—
98#382—31no—
99#381—22no—
100#380—32no—
101C4×C4⋊C4—32no—
102#340—32no—
103#340—32no—
104#336—32no—
105#336—31no—
106#336—31no—
107C24×C3C2×D411yescharacteristic
108C2×C3×Q8—24no—
109C2×C3×Q8—22no—
110C2×C3×Q8—22no—
111C2×C3×Q8—22no—
112C2×C3×Q8—22no—
113C2×C3×Q8—22no—
114C2×C3×Q8—48no—
115C2×C3×Q8—48no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8#4114yes—
119C22×C3×C4—22no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—42no—
126C22×C3×C4—42no—
127C22×C3×C4C2×D412yes—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4#4114yes—
131C22×C3×C4C2411yescharacteristic
132C22×Dic3—22no—
133C22×Dic3—22no—
134C3×C4⋊C4—24no—
135C3×C4⋊C4—22no—
136C3×C4⋊C4—22no—
137C3×C22⋊C4—24no—
138C3×C22⋊C4—42no—
139C3×C42—22no—
140C3×C42—22no—
141C3×C42—42no—
142#217—82no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3—82no—
148C4×Dic3—82no—
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—24no—
156C22×Q8—21no—
157C22×Q8—24no—
158C22×Q8—41no—
159C22×Q8—41no—
160C22×Q8C22×D312yes—
161C23×C4—21no—
162C23×C4C22×D311yescharacteristic
163C22×C8—32no—
164C22×C8—32no—
165#117—24no—
166#117—41no—
167#117—41no—
168C2×C4⋊C4—32no—
169C2×C4⋊C4—32no—
170C2×C4⋊C4—62no—
171C2×C4⋊C4#6712yes—
172C2×C4⋊C4C22×D312yes—
173C2×C22⋊C4—21no—
174C2×C22⋊C4—62no—
175C2×C22⋊C4#6712yes—
176C2×C42—21no—
177C2×C42—62no—
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—24no—
188#95—24no—
189#95—62no—
190#95—62no—
191#95—62no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—22no—
195C23×C3—22no—
196C23×C3#12312yes—
197C23×C3C22×D411yescharacteristic
198C3×Q8—24no—
199C3×Q8—24no—
200C3×Q8—48no—
201C3×Q8—42no—
202C3×Q8—42no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—24no—
206C2×C3×C4—24no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—24no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—48no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4#12312yes—
221C2×C3×C4C2×#4112yes—
222C2×C3×C4C2×#4112yes—
223C2×C3×C4H(5,𝔽2)11yescharacteristic
224C2×C3×C4H(5,𝔽2)11yescharacteristic
225C2×Dic3—42no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—42no—
231C3⋊C8—42no—
232C3⋊C8—42no—
233C24C2×#6711yescharacteristic
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—48no—
241C2×Q8—48no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8#23514yes—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—62no—
254C22×C4—62no—
255C22×C4#23714yes—
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—62no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—242no—
274C22⋊C4—24no—
275C22⋊C4—42no—
276C22⋊C4—242no—
277C42—22no—
278C42—22no—
279C42—42no—
280C42—242no—
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—44no—
295C3×C4—82no—
296C3×C4#55112yes—
297C3×C4#55112yes—
298Dic3—82no—
299Dic3—82no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—22no—
304C23#93512yes—
305C23C22×#6711yescharacteristic
306Q8—24no—
307Q8—24no—
308Q8—48no—
309Q8—42no—
310Q8—42no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—122no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4#93512yes—
335C2×C4C2×#23512yes—
336C2×C4C2×#23712yes—
337C2×C4#100411yescharacteristic
338C2×C4#100411yescharacteristic
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—41no—
350C22—41no—
351C22—42no—
352C22#579111yescharacteristic
353C22C2×#93511yescharacteristic
354C22#580511yescharacteristic
355C22#583712yes—
356C22#583812yes—
357C4—22no—
358C4—44no—
359C4—42no—
360C4—42no—
361C4—82no—
362C4—242no—
363C4—242no—
364C4#549512yes—
365C4#549512yes—
366C3#2759211yescharacteristic, complemented, Sylow
367C2—42no—
368C2#8211311yescharacteristic
369C2#8255311yescharacteristic
370C2#8255511yescharacteristic
371C2#8339712yes—
372C2#8341012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2310D3
hypocenterC3#27592
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2310

Complements

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

NH ≅ G/Ntype
6. #69686C2semidirect
7. #69670C2semidirect
8. #69667C2semidirect
9. #69666C2semidirect
366. C3#27592semidirect

Maximal subgroups

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

subgroupindex
C3×#23102
#822312
#721322
#715842
#715762
#696862
#696702
#696672
#696662
#275923