Finite Group Databasebeta

← #10950871

Subgroups — #10950871

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#81809C211yesmaximal, characteristic
2#81792C211yesmaximal, characteristic, complemented
3C3×#2251C211yesmaximal, characteristic, complemented
4#71504C212yesmaximal, complemented
5#71496C212yesmaximal, complemented
6#71487C212yesmaximal
7#71474C212yesmaximal, complemented
8#71319C211yesmaximal, characteristic, complemented
9#71311C211yesmaximal, characteristic, complemented
10D3×#346C211yesmaximal, characteristic, complemented
11#69069C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13#5728—21no—
14#5680—21no—
15C2×C3×Q32C2212yes—
16C2×C3×SD32C2212yes—
17C3×#495C2211yescharacteristic
18C3×#494C2211yescharacteristic
19#5188—21no—
20#5188C2212yes—
21#5188C2211yescharacteristic
22Dic3×Q16—21no—
23Dic3×Q16C2212yes—
24Dic3×Q16C2211yescharacteristic
25#5171—21no—
26Dic3×SD16—21no—
27#5157C2211yescharacteristic
28Dic3×D8C2211yescharacteristic
29#5079—21no—
30#5056—21no—
31C2×D3×C16C2212yes—
32C4×D3×C8C2211yescharacteristic
33#4683C2211yescharacteristic
34#4682C2211yescharacteristic
35C3×#346C2211yescharacteristic
36#4564C2212yes—
37#4564C2212yes—
38#4564C2212yes—
39#4562C2212yes—
40#4505C2212yes—
41#4504C2212yes—
42#4502C2212yes—
43#4461C2211yescharacteristic
44#3170—31no—
45#3168—31no—
46#2255—31no—
47#2251D311yescharacteristic, complemented
48#2245—31no—
49#2243—31no—
50#2217—32no—
51#2216—32no—
52#2176—32no—
53#2175—32no—
54C2×#346—31no—
55C2×C3×Q16—21no—
56C2×C3×Q16C2312yes—
57C2×C3×Q16C2311yescharacteristic
58C2×C3×SD16—21no—
59C2×C3×D8C2311yescharacteristic
60C3×C4⋊Q8—21no—
61C3×#124—21no—
62#946—21no—
63#946—22no—
64Q8×Dic3—21no—
65Q8×Dic3—22no—
66#938—21no—
67D4×Dic3—21no—
68C2×D3×C8D411yescharacteristic
69C2×D3×C8C2311yescharacteristic
70#897—41no—
71#892—41no—
72#890—41no—
73#887—41no—
74#883—41no—
75#67⋊C4—41no—
76#879—41no—
77#877—41no—
78C42×D3D411yescharacteristic
79#870—21no—
80#869—21no—
81C3×Q32—24no—
82C3×SD32—24no—
83C2×C3×C16C2312yes—
84C3×C4×C8C2311yescharacteristic
85#835—21no—
86#835—21no—
87#835—22no—
88#835—22no—
89#832—21no—
90#832—21no—
91#818—21no—
92#818C2311yescharacteristic
93#818C2311yescharacteristic
94#818C2312yes—
95#817—21no—
96C8×Dic3D411yescharacteristic
97C8×Dic3C2311yescharacteristic
98C2×C3⋊C16C2312yes—
99C4×C3⋊C8D411yescharacteristic
100D3×C16—24no—
101#533—61no—
102#529—61no—
103C2×Q32C2×D312yes—
104C2×SD32C2×D312yes—
105C22×C16—32no—
106#500—31no—
107#499—31no—
108#495C2×D311yescharacteristic
109#494C2×D311yescharacteristic
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#358—32no—
128#357—32no—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×Q8—21no—
133C2×C3×Q8—22no—
134C2×C3×D4—21no—
135C2×#67—41no—
136C22×Dic3—41no—
137C2×C4×D3D811yescharacteristic
138C2×C4×D3D811yescharacteristic
139C2×C4×D3C2×D411yescharacteristic
140C2×Dic6—41no—
141C3×Q16—22no—
142C3×Q16—22no—
143C3×Q16—22no—
144C3×Q16—42no—
145C3×SD16—42no—
146C3×D8—22no—
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2411yescharacteristic
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42C2×D411yescharacteristic
154#217—41no—
155#217—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—42no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—22no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163C4⋊Dic3—42no—
164Dic3⋊C4—41no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167Dic3⋊C4—42no—
168C4×Dic3—41no—
169C4×Dic3—42no—
170C4×Dic3D811yescharacteristic
171C4×Dic3D811yescharacteristic
172C4×Dic3C2×D411yescharacteristic
173C2×C3⋊C8C2×D411yescharacteristic
174C2×C3⋊C8C2×D411yescharacteristic
175D3×C8—22no—
176D3×C8—22no—
177C3×C16#4114yes—
178C3⋊C16—22no—
179C2×Q16—21no—
180C2×Q16C22×D312yes—
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—61no—
194C22⋊Q8—62no—
195C22⋊Q8—121no—
196#121—61no—
197C4×Q8—61no—
198C4×Q8—62no—
199C4×Q8—121no—
200C4×D4—61no—
201C4×D4—121no—
202#117—121no—
203#117—121no—
204C2×C42—31no—
205Q32—24no—
206SD32—24no—
207C2×C16—32no—
208C2×C16—64no—
209C2×C16C22×D312yes—
210#107—32no—
211#107—31no—
212#107—31no—
213#107—61no—
214#106—61no—
215#103—61no—
216#103—61no—
217#103—62no—
218#103—62no—
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—42no—
228C3×Q8—41no—
229C3×Q8—41no—
230C3×Q8—42no—
231C3×D4—41no—
232C3×D4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4C2×D811yescharacteristic
236C2×C3×C4C2×D811yescharacteristic
237C2×C3×C4C22×D411yescharacteristic
238#67—82no—
239C2×Dic3—41no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—82no—
243C2×Dic3C2×D811yescharacteristic
244C2×Dic3C2×D811yescharacteristic
245C2×Dic3C2×D811yescharacteristic
246C4×D3—22no—
247C4×D3—21no—
248C4×D3—21no—
249C4×D3—41no—
250Dic6—82no—
251C3×C8#13512yes—
252C3×C8C2×#4112yes—
253C3⋊C8—21no—
254C3⋊C8—21no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—121no—
258C2×D4—21no—
259C2×D4—121no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—121no—
264Q16—22no—
265Q16—22no—
266Q16—22no—
267Q16—42no—
268SD16—42no—
269D8—22no—
270C2×C8—31no—
271C2×C8—31no—
272C2×C8—62no—
273C2×C8—62no—
274C2×C8D3×D411yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—62no—
281C4⋊C4—121no—
282C4⋊C4—122no—
283C4⋊C4—121no—
284C4⋊C4—122no—
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—62no—
301C16#23714yes—
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—81no—
312C3×C4—82no—
313C3×C4#53211yescharacteristic
314C3×C4C2×#13511yescharacteristic
315Dic3—21no—
316Dic3—21no—
317Dic3—21no—
318Dic3—81no—
319Dic3—82no—
320Dic3—81no—
321C23—31no—
322C23—41no—
323Q8—42no—
324Q8—41no—
325Q8—41no—
326Q8—42no—
327Q8—242no—
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#91212yes—
350C8C2×#23712yes—
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—81no—
369C4—82no—
370C4—241no—
371C4—241no—
372C4—242no—
373C4#571611yescharacteristic
374C4C2×#91211yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—62no—
377C2—161no—
378C2#8177511yescharacteristic
379C2D3×#51011yescharacteristic
380C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2251C2
socleC22×C3C22×D8
prime radical#5157C22
hypercenter#2251D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2251

Complements

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

NH ≅ G/Ntype
2. #81792C2semidirect
3. C3×#2251C2semidirect
4. #71504C2semidirect
5. #71496C2semidirect
7. #71474C2semidirect
8. #71319C2semidirect
9. #71311C2semidirect
10. D3×#346C2semidirect
11. #69069C2semidirect
47. #2251D3semidirect
375. C3#33747semidirect

Maximal subgroups

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

subgroupindex
#818092
#817922
C3×#22512
#715042
#714962
#714872
#714742
#713192
#713112
D3×#3462
#690692
#337473