Finite Group Databasebeta

← #10950864

Subgroups — #10950864

Order 768.

556 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3168C211yesmaximal, characteristic, complemented
2#78470C211yesmaximal, characteristic, complemented
3#71355C211yesmaximal, characteristic, complemented
4#71352C211yesmaximal, characteristic, complemented
5#71321C211yesmaximal, characteristic, complemented
6#71319C211yesmaximal, characteristic, complemented
7#71082C212yesmaximal, complemented
8#71071C212yesmaximal, complemented
9#70913C212yesmaximal, complemented
10#70885C212yesmaximal, complemented
11C2×#4461C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13C3×#529—21no—
14#5477—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×#828C2212yes—
26C2×#826C2212yes—
27#5115—21no—
28#5115C2212yes—
29C22×C3⋊C16C2212yes—
30#4695C2211yescharacteristic
31#4694—21no—
32C4×D24—21no—
33C4×D24C2212yes—
34#4682C2211yescharacteristic
35#4564C2212yes—
36#4562C2212yes—
37#4491C2212yes—
38#4491C2212yes—
39#4490C2212yes—
40#4489C2212yes—
41#4461C2212yes—
42#4461C2211yescharacteristic
43#4461C2211yescharacteristic
44#3169—31no—
45#3168D311yescharacteristic, complemented
46#2255—31no—
47#2250—31no—
48#2244—31no—
49#2243—31no—
50#2217—32no—
51#2215—32no—
52#2177—32no—
53#2175—32no—
54C2×#346—31no—
55C2×#235—41no—
56C2×C3×Q16C2311yescharacteristic
57C2×C3×SD16—21no—
58C2×C3×D8C2311yescharacteristic
59C22×C3×C8D411yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#126—41no—
62C3×#125—21no—
63C3×#124—21no—
64C3×#123—41no—
65C3×C22⋊Q8—21no—
66C3×#121—21no—
67C3×C4×Q8—21no—
68C3×C4×D4—21no—
69C3×C4×D4—41no—
70C3×#117—41no—
71C2×C3×C42D411yescharacteristic
72#930—22no—
73#930—41no—
74C2×D24—21no—
75C2×D24C2312yes—
76#875—41no—
77#874—21no—
78C4×D12—22no—
79C4×D12—41no—
80#869—21no—
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#828—24no—
93#826—24no—
94#821—22no—
95#821—22no—
96#818—21no—
97#818C2312yes—
98C2×C3⋊C16—24no—
99C2×C3⋊C16C2312yes—
100C2×C3⋊C16C2312yes—
101#532—61no—
102#529—21no—
103C2×SD32—32no—
104C2×D16—32no—
105C22×C16—32no—
106#500—31no—
107#499C2×D311yescharacteristic
108#495C2×D311yescharacteristic
109#493—31no—
110#488—21no—
111#486—61no—
112#467C2×D311yescharacteristic
113#466—32no—
114#466—61no—
115#466C2×D311yescharacteristic
116#465—21no—
117C4×Q16C2×D311yescharacteristic
118C4×SD16—21no—
119C4×D8—32no—
120C4×D8—61no—
121C4×D8C2×D311yescharacteristic
122C2×C4×C8C2×D311yescharacteristic
123#367—32no—
124#366—32no—
125#358—32no—
126#357—32no—
127#357—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×#67—42no—
140#235—84no—
141C2×D12—22no—
142C2×D12—41no—
143C2×D12—41no—
144C2×C4×D3—42no—
145C2×Dic6—41no—
146C3×Q16—22no—
147C3×SD16—42no—
148C3×D8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—22no—
151C2×C3×C8C2×D411yescharacteristic
152C2×C3×C8C2×D411yescharacteristic
153C2×C3×C8C2×D411yescharacteristic
154C2×C3×C8C2411yescharacteristic
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—41no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C22⋊C4—41no—
164C3×C22⋊C4—41no—
165C3×C22⋊C4—41no—
166C3×C42—41no—
167C3×C42D811yescharacteristic
168C3×C42D811yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171(C2×D3)⋊C4—42no—
172(C2×D3)⋊C4—42no—
173C4⋊Dic3—22no—
174C4⋊Dic3—41no—
175D24—22no—
176D24—22no—
177D24—42no—
178C3⋊C16—22no—
179C3⋊C16#4114yescomplemented
180C2×#41—121no—
181C2×Q16C22×D311yescharacteristic
182C2×SD16—21no—
183C2×D8—32no—
184C2×D8—61no—
185C2×D8C22×D311yescharacteristic
186C22×C8#6711yescharacteristic
187C22×C8C22×D311yescharacteristic
188C4⋊Q8—61no—
189#127—61no—
190#126—41no—
191#125—21no—
192#124—21no—
193#124—121no—
194#123—41no—
195C22⋊Q8—21no—
196#121—21no—
197#121—62no—
198#121—121no—
199C4×Q8—21no—
200C4×D4—21no—
201C4×D4—41no—
202C4×D4—62no—
203C4×D4—121no—
204#117—41no—
205C2×C42#6711yescharacteristic
206SD32—64no—
207D16—64no—
208C2×C16—32no—
209C2×C16—32no—
210C2×C16—64no—
211#107—32no—
212#107—61no—
213#107C22×D311yescharacteristic
214#107C22×D311yescharacteristic
215#106—21no—
216#103—21no—
217#103—21no—
218#102—21no—
219#102—21no—
220#102—62no—
221#102—62no—
222C4×C8#6711yescharacteristic
223C4×C8#6711yescharacteristic
224C4×C8C22×D311yescharacteristic
225C4×C8C22×D311yescharacteristic
226C23×C3—41no—
227C23×C3C2×D811yescharacteristic
228C22×D3—42no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—41no—
232C3×D4—41no—
233C3×D4—82no—
234C2×C3×C4—22no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—82no—
242C2×C3×C4C2×D811yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4C2×D811yescharacteristic
247C2×C3×C4C22×D411yescharacteristic
248#67—84no—
249C2×Dic3—42no—
250D12—42no—
251D12—42no—
252D12—82no—
253D12—82no—
254C4×D3—84no—
255Dic6—82no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8#13512yes—
259C3×C8C2×#4112yes—
260#41—244no—
261C2×Q8—21no—
262C2×Q8—121no—
263C2×D4—21no—
264C2×D4—41no—
265C2×D4—62no—
266C2×D4—122no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—41no—
270C22×C4—122no—
271C22×C4#21311yescharacteristic
272C22×C4#21311yescharacteristic
273C22×C4C2×#6711yescharacteristic
274Q16—22no—
275SD16—42no—
276D8—22no—
277D8—62no—
278D8—62no—
279D8—122no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8C2×#6711yescharacteristic
283C2×C8C2×#6711yescharacteristic
284C2×C8C2×#6711yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—62no—
293C4⋊C4—121no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—122no—
300C22⋊C4—122no—
301C42—41no—
302C42#21311yescharacteristic
303C42#21311yescharacteristic
304C42C2×#6711yescharacteristic
305C42C2×#6711yescharacteristic
306C16—34no—
307C16—62no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3#50912yes—
314C22×C3C22×D811yescharacteristic
315C2×D3—82no—
316C2×D3—82no—
317C2×D3—82no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—81no—
324C3×C4—81no—
325C3×C4—81no—
326C3×C4#53211yescharacteristic
327C3×C4C2×#13511yescharacteristic
328Dic3—82no—
329C23—41no—
330C23—122no—
331C23C2×#21311yescharacteristic
332Q8—41no—
333Q8—41no—
334Q8—242no—
335D4—41no—
336D4—41no—
337D4—82no—
338D4—122no—
339D4—122no—
340D4—242no—
341D4—242no—
342D4—244no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—82no—
351C2×C4—122no—
352C2×C4—244no—
353C2×C4C2×#21311yescharacteristic
354C2×C4C2×#21311yescharacteristic
355C2×C4C2×#21311yescharacteristic
356C2×C4C2×#21311yescharacteristic
357C2×C4C2×#21311yescharacteristic
358C2×C4C22×#6711yescharacteristic
359C8—21no—
360C8—21no—
361C8#95012yes—
362C8D3×#4112yes—
363C2×C3—22no—
364C2×C3—161no—
365C2×C3#316911yescharacteristic
366C2×C3C2×#50911yescharacteristic
367C2×C3#341411yescharacteristic
368D3—162no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—242no—
375C22—242no—
376C22—242no—
377C22#519312yes—
378C22C22×#21311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—81no—
385C4—81no—
386C4—81no—
387C4—242no—
388C4C2×#95011yescharacteristic
389C4#583211yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—22no—
392C2—161no—
393C2—482no—
394C2#8329111yescharacteristic
395C2C2×#519311yescharacteristic
396C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3168C2
socleC22×C3C22×D8
hypercenter#3168D3
hypocenterC3#33746
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
1. C3×#3168C2semidirect
2. #78470C2semidirect
3. #71355C2semidirect
4. #71352C2semidirect
5. #71321C2semidirect
6. #71319C2semidirect
7. #71082C2semidirect
8. #71071C2semidirect
9. #70913C2semidirect
10. #70885C2semidirect
11. C2×#4461C2semidirect
45. #3168D3semidirect
179. C3⋊C16#41semidirect
390. C3#33746semidirect

Maximal subgroups

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

subgroupindex
C3×#31682
#784702
#713552
#713522
#713212
#713192
#710822
#710712
#709132
#708852
C2×#44612
#337463