Finite Group Databasebeta

← #10950877

Subgroups — #10950877

Order 768.

512 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81934C211yesmaximal, characteristic, complemented
2#81809C211yesmaximal, characteristic
3C3×#2243C211yesmaximal, characteristic, complemented
4#71386C211yesmaximal, characteristic, complemented
5#71322C211yesmaximal, characteristic, complemented
6#69227C212yesmaximal, complemented
7#69209C212yesmaximal, complemented
8#69196C212yesmaximal
9#69176C212yesmaximal, complemented
10#69083C211yesmaximal, characteristic, complemented
11D3×#346C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13#5728—21no—
14#5680—21no—
15C3×#500C2211yescharacteristic
16C3×#495C2211yescharacteristic
17#5188C2211yescharacteristic
18Dic3×Q16C2211yescharacteristic
19C2×#829C2212yes—
20#5171—21no—
21Dic3×SD16—21no—
22#5157C2211yescharacteristic
23Dic3×D8C2211yescharacteristic
24C2×#827C2212yes—
25#5093—21no—
26#5056—21no—
27C2×D3×C16C2212yes—
28#4887—21no—
29#4887C2212yes—
30#4873—21no—
31#4873C2212yes—
32#4699C2211yescharacteristic
33C4×D3×C8C2211yescharacteristic
34#4683C2211yescharacteristic
35C3×#358C2212yes—
36C3×#357C2212yes—
37C3×#346C2211yescharacteristic
38#4503C2212yes—
39#4503C2212yes—
40#4502C2212yes—
41#4490C2212yes—
42#4489C2212yes—
43#4461C2211yescharacteristic
44#3170—31no—
45#3168—31no—
46#2255—31no—
47#2251—31no—
48#2245—31no—
49#2243D311yescharacteristic, complemented
50#2217—32no—
51#2216—32no—
52#2176—32no—
53#2175—32no—
54C2×#346—31no—
55C2×C3×Q16C2311yescharacteristic
56C2×C3×SD16—21no—
57C2×C3×D8C2311yescharacteristic
58C3×C4⋊Q8—21no—
59C3×#124—21no—
60#946—21no—
61#946—41no—
62Q8×Dic3—21no—
63Q8×Dic3—41no—
64#938—21no—
65D4×Dic3—21no—
66C2×Dic12—21no—
67C2×Dic12C2312yes—
68C2×D3×C8D411yescharacteristic
69C2×D3×C8C2311yescharacteristic
70#897—22no—
71#892—41no—
72#890—41no—
73#887—22no—
74#883—41no—
75#67⋊C4—41no—
76#879—41no—
77#877—41no—
78C42×D3D411yescharacteristic
79#870—21no—
80#869—21no—
81C2×C3×C16C2312yes—
82C3×#107—21no—
83C3×#107C2312yes—
84C3×C4×C8C2311yescharacteristic
85#835—21no—
86#835—21no—
87#832—21no—
88#832—21no—
89#829—24no—
90#827—24no—
91#818C2311yescharacteristic
92#818C2311yescharacteristic
93#817—21no—
94C8×Dic3D411yescharacteristic
95C8×Dic3C2311yescharacteristic
96C2×C3⋊C16C2312yes—
97#810—22no—
98#810—22no—
99C4×C3⋊C8D411yescharacteristic
100D3×C16—24no—
101#533—61no—
102#529—61no—
103C2×Q32—32no—
104C2×SD32—32no—
105C22×C16—32no—
106#500C2×D311yescharacteristic
107#499—31no—
108#495C2×D311yescharacteristic
109#494—31no—
110#488—61no—
111#487—61no—
112#467—31no—
113#467—32no—
114#467—61no—
115#466—31no—
116#465—61no—
117C4×Q16—31no—
118C4×Q16—32no—
119C4×Q16—61no—
120C4×SD16—61no—
121C4×D8—31no—
122C2×C4×C8—31no—
123#367—32no—
124#366—32no—
125#358—32no—
126#358—32no—
127#358C2×D312yes—
128#357C2×D312yes—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×Q8—21no—
133C2×C3×Q8—41no—
134C2×C3×D4—21no—
135C2×#67—41no—
136C22×Dic3—41no—
137C2×C4×D3D811yescharacteristic
138C2×C4×D3D811yescharacteristic
139C2×C4×D3C2×D411yescharacteristic
140C2×Dic6—22no—
141C3×Q16—22no—
142C3×SD16—42no—
143C3×D8—22no—
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8C2411yescharacteristic
146C3×C4⋊C4—22no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C42C2×D411yescharacteristic
151#217—41no—
152#217—41no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—42no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—41no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3—42no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164Dic3⋊C4—42no—
165C4×Dic3—41no—
166C4×Dic3—42no—
167C4×Dic3D811yescharacteristic
168C4×Dic3D811yescharacteristic
169C4×Dic3C2×D411yescharacteristic
170C2×C3⋊C8C2×D411yescharacteristic
171C2×C3⋊C8C2×D411yescharacteristic
172Dic12—22no—
173Dic12—22no—
174Dic12—42no—
175D3×C8—22no—
176D3×C8—22no—
177C3×C16—22no—
178C3⋊C16#4114yes—
179C2×Q16—32no—
180C2×Q16—61no—
181C2×Q16C22×D311yescharacteristic
182C2×SD16—21no—
183C2×D8C22×D311yescharacteristic
184C22×C8—31no—
185C22×C8—31no—
186C4⋊Q8—21no—
187C4⋊Q8—61no—
188#126—121no—
189#125—61no—
190#125—121no—
191#124—21no—
192#123—121no—
193C22⋊Q8—62no—
194C22⋊Q8—61no—
195C22⋊Q8—121no—
196#121—61no—
197C4×Q8—62no—
198C4×Q8—61no—
199C4×Q8—121no—
200C4×D4—61no—
201C4×D4—121no—
202#117—121no—
203#117—121no—
204C2×C42—31no—
205Q32—64no—
206SD32—64no—
207C2×C16—32no—
208C2×C16—64no—
209C2×C16C22×D312yes—
210#107—21no—
211#107—31no—
212#107—31no—
213#107C22×D312yes—
214#106—61no—
215#103—62no—
216#103—62no—
217#103—61no—
218#103—61no—
219#102—61no—
220#102—61no—
221C4×C8—31no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8C22×D311yescharacteristic
225C23×C3—41no—
226C22×D3C2×D811yescharacteristic
227C3×Q8—41no—
228C3×Q8—41no—
229C3×Q8—82no—
230C3×D4—41no—
231C3×D4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4C22×D411yescharacteristic
237#67—82no—
238C2×Dic3—41no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—82no—
242C2×Dic3C2×D811yescharacteristic
243C2×Dic3C2×D811yescharacteristic
244C2×Dic3C2×D811yescharacteristic
245C4×D3—22no—
246C4×D3—21no—
247C4×D3—21no—
248C4×D3—41no—
249Dic6—42no—
250Dic6—42no—
251C3×C8#13512yes—
252C3×C8C2×#4112yes—
253C3⋊C8—21no—
254C3⋊C8—21no—
255C2×Q8—21no—
256C2×Q8—41no—
257C2×Q8—62no—
258C2×D4—21no—
259C2×D4—121no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—121no—
264Q16—22no—
265Q16—62no—
266Q16—62no—
267Q16—122no—
268SD16—42no—
269D8—22no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8D3×D411yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—121no—
281C4⋊C4—122no—
282C4⋊C4—121no—
283C4⋊C4—122no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—122no—
293C22⋊C4—121no—
294C42—31no—
295C42—31no—
296C42—31no—
297C42—121no—
298C42—122no—
299C42D3×D411yescharacteristic
300C16—22no—
301C16—34no—
302C22×C3—81no—
303C22×C3—81no—
304C22×C3—81no—
305C22×C3C22×D811yescharacteristic
306C2×D3—21no—
307C2×D3—21no—
308C2×D3#51012yes—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—82no—
312C3×C4—81no—
313C3×C4#53211yescharacteristic
314C3×C4C2×#13511yescharacteristic
315Dic3—21no—
316Dic3—21no—
317Dic3—21no—
318Dic3—81no—
319Dic3—81no—
320Dic3—82no—
321C23—31no—
322C23—41no—
323Q8—41no—
324Q8—41no—
325Q8—82no—
326Q8—122no—
327Q8—122no—
328D4—41no—
329D4—41no—
330D4—242no—
331C2×C4—31no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—62no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—121no—
340C2×C4—121no—
341C2×C4—122no—
342C2×C4—121no—
343C2×C4—242no—
344C2×C4D3×D811yescharacteristic
345C2×C4D3×D811yescharacteristic
346C2×C4C2×D3×D411yescharacteristic
347C8—61no—
348C8—61no—
349C8#91612yes—
350C8D3×#4112yes—
351C2×C3—161no—
352C2×C3#316911yescharacteristic
353C2×C3C2×#51011yescharacteristic
354C2×C3#341411yescharacteristic
355D3—22no—
356C22—32no—
357C22—61no—
358C22—61no—
359C22—81no—
360C22—81no—
361C22—81no—
362C22C2×D3×D811yescharacteristic
363C4—21no—
364C4—21no—
365C4—61no—
366C4—61no—
367C4—61no—
368C4—82no—
369C4—81no—
370C4—241no—
371C4—241no—
372C4—242no—
373C4#566811yescharacteristic
374C4C2×#91611yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—62no—
377C2—161no—
378C2#8155711yescharacteristic
379C2#8203511yescharacteristic
380C2D3×#51011yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2243C2
socleC22×C3C22×D8
prime radical#5157C22
hypercenter#2243D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2243

Complements

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

NH ≅ G/Ntype
1. #81934C2semidirect
3. C3×#2243C2semidirect
4. #71386C2semidirect
5. #71322C2semidirect
6. #69227C2semidirect
7. #69209C2semidirect
9. #69176C2semidirect
10. #69083C2semidirect
11. D3×#346C2semidirect
49. #2243D3semidirect
375. C3#33747semidirect

Maximal subgroups

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

subgroupindex
#819342
#818092
C3×#22432
#713862
#713222
#692272
#692092
#691962
#691762
#690832
D3×#3462
#337473