Finite Group Databasebeta

← #10876270

Subgroups — #10876270

Order 768.

794 conjugacy classes of subgroups.

The subgroup lattice has 408 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#69685C211yesmaximal, characteristic, complemented
7#69684C211yesmaximal, characteristic, complemented
8#69671C211yesmaximal, characteristic, complemented
9#69671C211yesmaximal, characteristic, complemented
10#69667C212yesmaximal, complemented
11#69666C212yesmaximal, complemented
12#27596—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C3×#391—22no—
17C3×#391C2211yescharacteristic
18C3×#391C2211yescharacteristic
19C3×#391C2211yescharacteristic
20C3×#391C2211yescharacteristic
21C3×#381—22no—
22C2×#835C2212yes—
23C2×#835C2212yes—
24C2×#835C2212yes—
25C2×#835C2212yes—
26#5212C2212yes—
27C2×#830C2212yes—
28#4980—22no—
29C2×#808C2212yes—
30C2×#807C2212yes—
31#4951—22no—
32#4936C2211yescharacteristic
33#4935C2211yescharacteristic
34C4×C4⋊Dic3C2212yes—
35#4529C2212yes—
36#4529C2212yes—
37#4524C2212yes—
38#4524C2211yescharacteristic
39#4524C2211yescharacteristic
40#2314—31no—
41#2310D311yescharacteristic
42#2096—31no—
43#2095—31no—
44#2074—31no—
45#2074—31no—
46#1928—32no—
47#1927—32no—
48#1854—32no—
49#1818—31no—
50#1787—32no—
51C22×C3×Q8—22no—
52C22×C3×Q8—22no—
53C22×C3×Q8—21no—
54C22×C3×Q8—24no—
55C22×C3×Q8—42no—
56C22×C3×Q8C2311yescharacteristic
57C22×C3×Q8C2311yescharacteristic
58C23×C3×C4—21no—
59C23×C3×C4C2311yescharacteristic
60C3×#117—24no—
61C3×#117—42no—
62C2×C3×C4⋊C4D412yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4D412yes—
67C2×C3×C42—21no—
68C2×C3×C42C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70C2×#217—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3—22no—
75C22×C3⋊C8C2312yes—
76C22×C3⋊C8C2312yes—
77C3×#95—22no—
78C3×#95—22no—
79C3×#95—22no—
80C3×#95—22no—
81#835—24no—
82#835—24no—
83#835—42no—
84#835—42no—
85C3⋊#95—22no—
86C3⋊#95—22no—
87C3⋊#95—22no—
88#830—24no—
89#808—42no—
90#807—42no—
91C23×Q8C2×D311yescharacteristic
92C2×#117C2×D311yescharacteristic
93C2×#117C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×#103—32no—
97C2×#103—32no—
98C2×#103—32no—
99C2×#103—32no—
100C2×C22⋊C8—32no—
101#391—22no—
102#391C2×D311yescharacteristic
103#391C2×D311yescharacteristic
104#391C2×D311yescharacteristic
105#391C2×D311yescharacteristic
106#388—62no—
107#387—62no—
108#384—31no—
109#382—31no—
110#381—22no—
111#380—32no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336—31no—
116#336—31no—
117#336—32no—
118C24×C3C2×D411yescharacteristic
119C2×C3×Q8—24no—
120C2×C3×Q8—21no—
121C2×C3×Q8—21no—
122C2×C3×Q8—22no—
123C2×C3×Q8—22no—
124C2×C3×Q8—22no—
125C2×C3×Q8—22no—
126C2×C3×Q8—48no—
127C2×C3×Q8—48no—
128C2×C3×Q8—42no—
129C2×C3×Q8—42no—
130C2×C3×Q8#4112yes—
131C2×C3×Q8#4112yes—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—22no—
135C22×C3×C4—24no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4#4114yes—
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—22no—
149C3×C4⋊C4—24no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—42no—
154C3×C42—22no—
155C3×C42—22no—
156C3×C42—42no—
157#217—82no—
158C4⋊Dic3—22no—
159C4⋊Dic3—22no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—82no—
163C4×Dic3—82no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—22no—
170C22×Q8—22no—
171C22×Q8—22no—
172C22×Q8—21no—
173C22×Q8—24no—
174C22×Q8—42no—
175C22×Q8C22×D311yescharacteristic
176C22×Q8C22×D311yescharacteristic
177C23×C4—21no—
178C23×C4C22×D311yescharacteristic
179C22×C8—32no—
180C22×C8—32no—
181#117—24no—
182#117—42no—
183C2×C4⋊C4—32no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4#6712yes—
187C2×C4⋊C4C22×D311yescharacteristic
188C2×C4⋊C4C22×D311yescharacteristic
189C2×C22⋊C4—21no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4#6712yes—
192C2×C42—21no—
193C2×C42—62no—
194C2×C42C22×D311yescharacteristic
195C2×C42C22×D311yescharacteristic
196#107—122no—
197#106—122no—
198#103—64no—
199#103—64no—
200#103—122no—
201#103—122no—
202C22⋊C8—64no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—62no—
208#95—62no—
209#95—62no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3#12312yes—
215C23×C3C22×D411yescharacteristic
216C3×Q8—24no—
217C3×Q8—24no—
218C3×Q8—48no—
219C3×Q8—42no—
220C3×Q8—42no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—24no—
224C2×C3×C4—24no—
225C2×C3×C4—22no—
226C2×C3×C4—22no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—22no—
232C2×C3×C4—22no—
233C2×C3×C4—48no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4#12312yes—
243C2×C3×C4C2×#4111yescharacteristic
244C2×C3×C4C2×#4111yescharacteristic
245C2×C3×C4C2×#4112yes—
246C2×C3×C4H(5,𝔽2)12yes—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C3⋊C8—42no—
254C3⋊C8—42no—
255C24C2×#6711yescharacteristic
256C2×Q8—24no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8#23512yes—
268C2×Q8#23512yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4#23714yes—
282C22×C4C2×#6712yes—
283C22×C4C2×#6712yes—
284C22×C4C2×#6711yescharacteristic
285C22×C4C23×D311yescharacteristic
286C2×C8—62no—
287C2×C8—62no—
288C2×C8—62no—
289C2×C8—62no—
290C2×C8—62no—
291C2×C8—62no—
292C4⋊C4—24no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—62no—
296C4⋊C4—62no—
297C4⋊C4—62no—
298C4⋊C4—62no—
299C4⋊C4—242no—
300C22⋊C4—24no—
301C22⋊C4—42no—
302C22⋊C4—242no—
303C42—22no—
304C42—22no—
305C42—42no—
306C42—242no—
307C22×C3—24no—
308C22×C3—22no—
309C22×C3—42no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3C2×#12311yescharacteristic
313C22×C3#52511yescharacteristic
314C22×C3#53511yescharacteristic
315C22×C3#57712yes—
316C22×C3#57812yes—
317C3×C4—22no—
318C3×C4—42no—
319C3×C4—42no—
320C3×C4—42no—
321C3×C4—42no—
322C3×C4—81no—
323C3×C4—81no—
324C3×C4#55112yes—
325C3×C4#55112yes—
326Dic3—82no—
327Dic3—82no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23#93512yes—
333C23C22×#6711yescharacteristic
334Q8—24no—
335Q8—24no—
336Q8—48no—
337Q8—42no—
338Q8—42no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—48no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—122no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4#93512yes—
367C2×C4C2×#23511yescharacteristic
368C2×C4C2×#23511yescharacteristic
369C2×C4C2×#23712yes—
370C2×C4#100412yes—
371C8—122no—
372C8—122no—
373C2×C3—42no—
374C2×C3#248211yescharacteristic
375C2×C3#309712yes—
376C2×C3#312112yes—
377C2×C3#322111yescharacteristic
378C2×C3#322311yescharacteristic
379C22—22no—
380C22—24no—
381C22—42no—
382C22—41no—
383C22—41no—
384C22#579111yescharacteristic
385C22C2×#93511yescharacteristic
386C22#580511yescharacteristic
387C22#583712yes—
388C22#583812yes—
389C4—22no—
390C4—42no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—242no—
397C4—242no—
398C4#549512yes—
399C4#549512yes—
400C3#2759611yescharacteristic, complemented, Sylow
401C2—42no—
402C2#8211311yescharacteristic
403C2#8255212yes—
404C2#8339712yes—
405C2#8340911yescharacteristic
406C2#8341411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2310C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2310D3
hypocenterC3#27596
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. #69685C2semidirect
7. #69684C2semidirect
8. #69671C2semidirect
9. #69671C2semidirect
10. #69667C2semidirect
11. #69666C2semidirect
400. C3#27596semidirect

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
#696852
#696842
#696712
#696672
#696662
#275963