Finite Group Databasebeta

← #10950857

Subgroups — #10950857

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×#3169C211yesmaximal, characteristic, complemented
2#78469C211yesmaximal, characteristic, complemented
3#71386C211yesmaximal, characteristic, complemented
4#71383C211yesmaximal, characteristic, complemented
5#71292C211yesmaximal, characteristic, complemented
6#71289C211yesmaximal, characteristic, complemented
7#71072C212yesmaximal, complemented
8#71071C212yesmaximal, complemented
9#70885C212yesmaximal, complemented
10#70884C212yesmaximal, complemented
11C2×#4461C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13C3×#532—21no—
14#5479—21no—
15C3×#500C2211yescharacteristic
16C3×#493C2211yescharacteristic
17C3×#486—21no—
18C3×#466—21no—
19C3×#466C2212yes—
20C3×C4×D8—21no—
21C3×C4×D8C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23C2×#827C2212yes—
24C2×#826C2212yes—
25#5116C2211yescharacteristic
26#5115C2211yescharacteristic
27#5114—21no—
28C22×C3⋊C16C2212yes—
29C4×Dic12C2211yescharacteristic
30#4697—21no—
31#4696C2211yescharacteristic
32C4×D24C2211yescharacteristic
33C4×#204—21no—
34#4683C2211yescharacteristic
35#4562C2212yes—
36#4562C2212yes—
37#4492C2212yes—
38#4491C2212yes—
39#4490C2212yes—
40#4489C2212yes—
41#4461C2211yescharacteristic
42#4461C2211yescharacteristic
43#4461C2212yes—
44#3169D311yescharacteristic, complemented
45#3168—31no—
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×C3×#41—41no—
56C2×C3×D8—21no—
57C2×C3×D8C2312yes—
58C22×C3×C8D411yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C3×C4⋊Q8—21no—
61C3×#127—21no—
62C3×#124—41no—
63C3×#121—22no—
64C3×#121—41no—
65C3×C4×D4—22no—
66C3×C4×D4—41no—
67C2×C3×C42D411yescharacteristic
68#930—21no—
69#929—41no—
70C4×#67—41no—
71#924—21no—
72C2×Dic12C2311yescharacteristic
73C2×D24C2311yescharacteristic
74C2×#204—21no—
75#876—41no—
76#875—21no—
77C4×D12—21no—
78#872—41no—
79#870—21no—
80C4×Dic6—21no—
81C3×#107—21no—
82C3×#107C2312yes—
83C3×#102—22no—
84C3×#102—22no—
85C3×C4×C8D411yescharacteristic
86C3×C4×C8D411yescharacteristic
87C3×C4×C8C2311yescharacteristic
88C3×C4×C8C2311yescharacteristic
89#827—24no—
90#826—24no—
91#821—21no—
92#821—21no—
93#818C2311yescharacteristic
94#818C2311yescharacteristic
95#817—21no—
96#816—21no—
97#816—21no—
98C2×C3⋊C16—24no—
99C2×C3⋊C16C2312yes—
100C2×C3⋊C16C2312yes—
101#532—21no—
102#529—61no—
103C2×SD32—32no—
104C2×D16—32no—
105C22×C16—32no—
106#500C2×D311yescharacteristic
107#499—31no—
108#495—31no—
109#493C2×D311yescharacteristic
110#488—61no—
111#486—21no—
112#467—31no—
113#466—21no—
114#466—31no—
115#466C2×D312yes—
116#465—61no—
117C4×Q16—31no—
118C4×SD16—61no—
119C4×D8—21no—
120C4×D8—31no—
121C4×D8C2×D312yes—
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—
132C3×#41—84no—
133C2×C3×Q8—41no—
134C2×C3×D4—22no—
135C2×C3×D4—42no—
136C2×C3×D4—41no—
137C2×C3×D4—41no—
138C22×C3×C4—42no—
139C22×C3×C4D811yescharacteristic
140C22×C3×C4D811yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C2×#67—41no—
143C2×D12—21no—
144C2×C4×D3—41no—
145C2×Dic6—21no—
146C3×D8—22no—
147C3×D8—22no—
148C3×D8—42no—
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—22no—
156C3×C4⋊C4—41no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—42no—
159C3×C42D811yescharacteristic
160C3×C42D811yescharacteristic
161C3×C42C2×D411yescharacteristic
162C3×C42C2×D411yescharacteristic
163#217—41no—
164(C2×D3)⋊C4—41no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167(C2×D3)⋊C4—41no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—41no—
173Dic3⋊C4—41no—
174C4×Dic3—41no—
175Dic12—22no—
176D24—22no—
177#204—42no—
178C3⋊C16—22no—
179C3⋊C16#4114yescomplemented
180C2×#41—41no—
181C2×Q16—31no—
182C2×SD16—61no—
183C2×D8—21no—
184C2×D8—31no—
185C2×D8C22×D312yes—
186C22×C8#6711yescharacteristic
187C22×C8C22×D311yescharacteristic
188C4⋊Q8—21no—
189#127—21no—
190#126—121no—
191#125—61no—
192#124—41no—
193#124—61no—
194#123—121no—
195C22⋊Q8—61no—
196#121—22no—
197#121—41no—
198#121—61no—
199C4×Q8—61no—
200C4×D4—22no—
201C4×D4—41no—
202C4×D4—61no—
203C4×D4—121no—
204#117—121no—
205C2×C42#6711yescharacteristic
206SD32—64no—
207D16—64no—
208C2×C16—32no—
209C2×C16—32no—
210C2×C16—64no—
211#107—21no—
212#107—31no—
213#107—31no—
214#107C22×D312yes—
215#106—61no—
216#103—61no—
217#103—61no—
218#102—22no—
219#102—22no—
220#102—61no—
221#102—61no—
222C4×C8#6711yescharacteristic
223C4×C8#6711yescharacteristic
224C4×C8C22×D311yescharacteristic
225C4×C8C22×D311yescharacteristic
226C23×C3—42no—
227C23×C3C2×D811yescharacteristic
228C22×D3—41no—
229C3×Q8—82no—
230C3×D4—42no—
231C3×D4—42no—
232C3×D4—82no—
233C3×D4—82no—
234C3×D4—84no—
235C2×C3×C4—22no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—41no—
239C2×C3×C4—42no—
240C2×C3×C4—84no—
241C2×C3×C4C2×D811yescharacteristic
242C2×C3×C4C2×D811yescharacteristic
243C2×C3×C4C2×D811yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×D811yescharacteristic
246C2×C3×C4C22×D411yescharacteristic
247#67—82no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251D12—41no—
252D12—41no—
253C4×D3—82no—
254Dic6—41no—
255Dic6—41no—
256C3×C8—21no—
257C3×C8—21no—
258C3×C8#13512yes—
259C3×C8C2×#4112yes—
260#41—84no—
261C2×Q8—41no—
262C2×Q8—61no—
263C2×D4—22no—
264C2×D4—42no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—61no—
268C2×D4—121no—
269C22×C4—42no—
270C22×C4—121no—
271C22×C4#21311yescharacteristic
272C22×C4#21311yescharacteristic
273C22×C4C2×#6711yescharacteristic
274Q16—62no—
275SD16—122no—
276D8—22no—
277D8—22no—
278D8—42no—
279D8—62no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8C2×#6711yescharacteristic
283C2×C8C2×#6711yescharacteristic
284C2×C8C2×#6711yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—41no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—42no—
295C22⋊C4—42no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C42—121no—
302C42#21311yescharacteristic
303C42#21311yescharacteristic
304C42C2×#6711yescharacteristic
305C42C2×#6711yescharacteristic
306C16—34no—
307C16—62no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—82no—
311C22×C3—82no—
312C22×C3—82no—
313C22×C3#50912yes—
314C22×C3C22×D811yescharacteristic
315C2×D3—81no—
316C2×D3—81no—
317C2×D3—81no—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—82no—
324C3×C4#53211yescharacteristic
325C3×C4C2×#13511yescharacteristic
326Dic3—81no—
327Dic3—81no—
328Dic3—81no—
329C23—42no—
330C23—121no—
331C23C2×#21311yescharacteristic
332Q8—82no—
333Q8—121no—
334Q8—121no—
335D4—42no—
336D4—42no—
337D4—82no—
338D4—82no—
339D4—84no—
340D4—121no—
341D4—121no—
342D4—242no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—84no—
349C2×C4—121no—
350C2×C4—121no—
351C2×C4—121no—
352C2×C4—242no—
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—162no—
365C2×C3#316911yescharacteristic
366C2×C3C2×#50911yescharacteristic
367C2×C3#341411yescharacteristic
368D3—161no—
369C22—21no—
370C22—21no—
371C22—82no—
372C22—82no—
373C22—82no—
374C22—241no—
375C22—241no—
376C22—241no—
377C22#514712yes—
378C22C22×#21311yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—241no—
386C4—241no—
387C4—241no—
388C4C2×#95011yescharacteristic
389C4#583211yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—22no—
392C2—162no—
393C2—481no—
394C2#8329111yescharacteristic
395C2C2×#514711yescharacteristic
396C2#8389411yescharacteristic
subgroupG/H
centerC22C22×#213
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#3169C2
socleC22×C3C22×D8
hypercenter#3169D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ #3169

Complements

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

NH ≅ G/Ntype
1. C3×#3169C2semidirect
2. #78469C2semidirect
3. #71386C2semidirect
4. #71383C2semidirect
5. #71292C2semidirect
6. #71289C2semidirect
7. #71072C2semidirect
8. #71071C2semidirect
9. #70885C2semidirect
10. #70884C2semidirect
11. C2×#4461C2semidirect
44. #3169D3semidirect
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×#31692
#784692
#713862
#713832
#712922
#712892
#710722
#710712
#708852
#708842
C2×#44612
#337463