Finite Group Databasebeta

← #10950875

Subgroups — #10950875

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
1C3×#3168C211yesmaximal, characteristic
2#78475C211yesmaximal, characteristic, complemented
3#71355C211yesmaximal, characteristic, complemented
4#71353C211yesmaximal, characteristic, complemented
5#71322C211yesmaximal, characteristic, complemented
6#71319C211yesmaximal, characteristic, complemented
7#71081C212yesmaximal, complemented
8#71072C212yesmaximal
9#70909C212yesmaximal, complemented
10#70884C212yesmaximal, complemented
11C2×#4461C211yesmaximal, characteristic, complemented
12#33747—31nomaximal, Sylow
13C3×#529—21no—
14#5470—21no—
15C3×#499C2211yescharacteristic
16C3×#495C2211yescharacteristic
17C3×#488—21no—
18C3×#467C2211yescharacteristic
19C3×#466C2211yescharacteristic
20C3×#465—21no—
21C3×C4×Q16C2211yescharacteristic
22C3×C4×SD16—21no—
23C3×C4×D8C2211yescharacteristic
24C2×C3×C4×C8C2211yescharacteristic
25C2×#829C2212yes—
26C2×#827C2212yes—
27#5116—21no—
28#5116C2212yes—
29C22×C3⋊C16C2212yes—
30#4699C2211yescharacteristic
31C4×Dic12—21no—
32C4×Dic12C2212yes—
33#4682C2211yescharacteristic
34#4681—21no—
35#4564C2212yes—
36#4562C2212yes—
37#4492C2212yes—
38#4492C2212yes—
39#4490C2212yes—
40#4489C2212yes—
41#4461C2211yescharacteristic
42#4461C2211yescharacteristic
43#4461C2212yes—
44#3170—31no—
45#3168D311yescharacteristic
46#2255—31no—
47#2251—31no—
48#2245—31no—
49#2243—31no—
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
58C22×C3×C8D411yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×#126—41no—
61C3×#125—21no—
62C3×#124—21no—
63C3×#123—41no—
64C3×C22⋊Q8—21no—
65C3×#121—21no—
66C3×C4×Q8—21no—
67C3×C4×D4—21no—
68C3×C4×D4—41no—
69C3×#117—41no—
70C2×C3×C42D411yescharacteristic
71#926—41no—
72#924—22no—
73#924—41no—
74C2×Dic12—21no—
75C2×Dic12C2312yes—
76#870—41no—
77#869—21no—
78#869—21no—
79C4×Dic6—22no—
80C4×Dic6—41no—
81C3×#107C2311yescharacteristic
82C3×#107C2311yescharacteristic
83C3×#106—21no—
84C3×#103—21no—
85C3×#103—21no—
86C3×#102—21no—
87C3×#102—21no—
88C3×C4×C8D411yescharacteristic
89C3×C4×C8D411yescharacteristic
90C3×C4×C8C2311yescharacteristic
91C3×C4×C8C2311yescharacteristic
92#829—24no—
93#827—24no—
94#818—21no—
95#818C2312yes—
96#816—22no—
97#816—22no—
98C2×C3⋊C16—24no—
99C2×C3⋊C16C2312yes—
100C2×C3⋊C16C2312yes—
101#533—61no—
102#529—21no—
103C2×Q32—32no—
104C2×SD32—32no—
105C22×C16—32no—
106#500—31no—
107#499C2×D311yescharacteristic
108#495C2×D311yescharacteristic
109#494—31no—
110#488—21no—
111#487—61no—
112#467—32no—
113#467—61no—
114#467C2×D311yescharacteristic
115#466C2×D311yescharacteristic
116#465—21no—
117C4×Q16—32no—
118C4×Q16—61no—
119C4×Q16C2×D311yescharacteristic
120C4×SD16—21no—
121C4×D8C2×D311yescharacteristic
122C2×C4×C8C2×D311yescharacteristic
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#346—31no—
132C2×C3×Q8—21no—
133C2×C3×D4—21no—
134C2×C3×D4—41no—
135C22×C3×C4—41no—
136C22×C3×C4D811yescharacteristic
137C22×C3×C4D811yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C2×Dic6—22no—
140C2×Dic6—41no—
141C3×Q16—22no—
142C3×SD16—42no—
143C3×D8—22no—
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2×D411yescharacteristic
148C2×C3×C8C2×D411yescharacteristic
149C2×C3×C8C2411yescharacteristic
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4—41no—
155C3×C4⋊C4—41no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C22⋊C4—41no—
161C3×C42—41no—
162C3×C42D811yescharacteristic
163C3×C42D811yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166#217—42no—
167C4⋊Dic3—22no—
168C4⋊Dic3—41no—
169C4⋊Dic3—41no—
170C4⋊Dic3—41no—
171Dic3⋊C4—42no—
172Dic3⋊C4—42no—
173C4×Dic3—42no—
174Dic12—22no—
175Dic12—22no—
176Dic12—42no—
177C3⋊C16—22no—
178C3⋊C16#4114yes—
179C2×Q16—32no—
180C2×Q16—61no—
181C2×Q16C22×D311yescharacteristic
182C2×SD16—21no—
183C2×D8C22×D311yescharacteristic
184C22×C8#6711yescharacteristic
185C22×C8C22×D311yescharacteristic
186C4⋊Q8—61no—
187C4⋊Q8—61no—
188#126—41no—
189#125—21no—
190#125—121no—
191#124—21no—
192#123—41no—
193C22⋊Q8—21no—
194C22⋊Q8—62no—
195C22⋊Q8—121no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—62no—
199C4×Q8—121no—
200C4×D4—21no—
201C4×D4—41no—
202#117—41no—
203#117—121no—
204C2×C42#6711yescharacteristic
205Q32—64no—
206SD32—64no—
207C2×C16—32no—
208C2×C16—32no—
209C2×C16—64no—
210#107—32no—
211#107—61no—
212#107C22×D311yescharacteristic
213#107C22×D311yescharacteristic
214#106—21no—
215#103—21no—
216#103—21no—
217#103—62no—
218#103—62no—
219#102—21no—
220#102—21no—
221C4×C8#6711yescharacteristic
222C4×C8#6711yescharacteristic
223C4×C8C22×D311yescharacteristic
224C4×C8C22×D311yescharacteristic
225C23×C3—41no—
226C23×C3C2×D811yescharacteristic
227C3×Q8—41no—
228C3×Q8—41no—
229C3×D4—41no—
230C3×D4—41no—
231C3×D4—82no—
232C2×C3×C4—22no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—82no—
240C2×C3×C4C2×D811yescharacteristic
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4C2×D811yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C22×D411yescharacteristic
246C2×Dic3—42no—
247C2×Dic3—42no—
248Dic6—42no—
249Dic6—42no—
250Dic6—82no—
251C3×C8—21no—
252C3×C8—21no—
253C3×C8#13512yes—
254C3×C8C2×#4112yes—
255C2×Q8—21no—
256C2×Q8—62no—
257C2×Q8—121no—
258C2×D4—21no—
259C2×D4—41no—
260C22×C4—41no—
261C22×C4#21311yescharacteristic
262C22×C4#21311yescharacteristic
263C22×C4C2×#6711yescharacteristic
264Q16—22no—
265Q16—62no—
266Q16—62no—
267Q16—122no—
268SD16—42no—
269D8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8C2×#6711yescharacteristic
273C2×C8C2×#6711yescharacteristic
274C2×C8C2×#6711yescharacteristic
275C2×C8C23×D311yescharacteristic
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—62no—
283C4⋊C4—121no—
284C4⋊C4—121no—
285C4⋊C4—122no—
286C4⋊C4—122no—
287C4⋊C4—121no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—122no—
294C42—41no—
295C42—122no—
296C42#21311yescharacteristic
297C42#21311yescharacteristic
298C42C2×#6711yescharacteristic
299C42C2×#6711yescharacteristic
300C16—34no—
301C16—62no—
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—81no—
305C22×C3—81no—
306C22×C3—81no—
307C22×C3#51012yes—
308C22×C3C22×D811yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—81no—
315C3×C4—81no—
316C3×C4—81no—
317C3×C4#53211yescharacteristic
318C3×C4C2×#13511yescharacteristic
319Dic3—82no—
320Dic3—82no—
321C23—41no—
322C23C2×#21311yescharacteristic
323Q8—41no—
324Q8—41no—
325Q8—122no—
326Q8—122no—
327Q8—242no—
328D4—41no—
329D4—41no—
330D4—82no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—82no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4C2×#21311yescharacteristic
342C2×C4C2×#21311yescharacteristic
343C2×C4C2×#21311yescharacteristic
344C2×C4C2×#21311yescharacteristic
345C2×C4C2×#21311yescharacteristic
346C2×C4C22×#6711yescharacteristic
347C8—21no—
348C8—21no—
349C8#95012yes—
350C8D3×#4112yes—
351C2×C3—22no—
352C2×C3—161no—
353C2×C3#316911yescharacteristic
354C2×C3C2×#51011yescharacteristic
355C2×C3#341411yescharacteristic
356C22—21no—
357C22—21no—
358C22—81no—
359C22—81no—
360C22—81no—
361C22#519512yes—
362C22C22×#21311yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4—242no—
372C4—242no—
373C4C2×#95011yescharacteristic
374C4#583211yescharacteristic
375C3#3374711yescharacteristic, complemented, Sylow
376C2—22no—
377C2—161no—
378C2#8329111yescharacteristic
379C2#8389411yescharacteristic
380C2C2×#519511yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3168C2
socleC22×C3C22×D8
prime radicalC3×#466C22
hypercenter#3168D3
hypocenterC3#33747
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3168

Complements

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

NH ≅ G/Ntype
2. #78475C2semidirect
3. #71355C2semidirect
4. #71353C2semidirect
5. #71322C2semidirect
6. #71319C2semidirect
7. #71081C2semidirect
9. #70909C2semidirect
10. #70884C2semidirect
11. C2×#4461C2semidirect
375. C3#33747semidirect

Maximal subgroups

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

subgroupindex
C3×#31682
#784752
#713552
#713532
#713222
#713192
#710812
#710722
#709092
#708842
C2×#44612
#337473