Finite Group Databasebeta

← #10950865

Subgroups — #10950865

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
1#81933C211yesmaximal, characteristic, complemented
2#81907C211yesmaximal, characteristic, complemented
3C3×#2255C211yesmaximal, characteristic, complemented
4#71383C211yesmaximal, characteristic, complemented
5#71352C211yesmaximal, characteristic, complemented
6#69347C212yesmaximal, complemented
7#69339C212yesmaximal, complemented
8#69327C212yesmaximal, complemented
9#69319C212yesmaximal, complemented
10#69081C211yesmaximal, characteristic, complemented
11D3×#346C211yesmaximal, characteristic, complemented
12#33746—31nomaximal, Sylow
13#5725—21no—
14#5691—21no—
15C3×#500C2211yescharacteristic
16C3×#499C2211yescharacteristic
17#5088—21no—
18#5068—21no—
19C2×#800C2212yes—
20C2×D48C2212yes—
21C2×D3×C16C2212yes—
22#4887C2211yescharacteristic
23#4883—21no—
24#4883C2212yes—
25#4883C2211yescharacteristic
26#4873C2211yescharacteristic
27#4872—21no—
28#4872C2212yes—
29#4872C2211yescharacteristic
30#4864—21no—
31#4852—21no—
32#4696C2211yescharacteristic
33#4695C2211yescharacteristic
34C4×D3×C8C2211yescharacteristic
35C3×#367C2212yes—
36C3×#366C2212yes—
37C3×#346C2211yescharacteristic
38#4502C2212yes—
39#4492C2212yes—
40#4491C2212yes—
41#4491C2212yes—
42#4491C2212yes—
43#4461C2211yescharacteristic
44#3169—31no—
45#3168—31no—
46#2255D311yescharacteristic, complemented
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×#239—41no—
56C3×C4⋊Q8—21no—
57C3×#125—21no—
58#947—41no—
59C2×Dic12C2311yescharacteristic
60C2×D24—21no—
61C2×D24C2311yescharacteristic
62C2×D24C2312yes—
63C2×#204—21no—
64C2×D3×C8D411yescharacteristic
65C2×D3×C8C2311yescharacteristic
66#898—41no—
67#897—21no—
68#895—21no—
69#895—22no—
70#895—41no—
71#894—41no—
72#893—22no—
73#893—21no—
74#893—41no—
75#893—41no—
76#892—41no—
77#887—21no—
78#875—21no—
79#874—21no—
80C42×D3D411yescharacteristic
81C2×C3×C16C2312yes—
82C3×#107—21no—
83C3×#107C2311yescharacteristic
84C3×#107C2312yes—
85C3×#107C2311yescharacteristic
86C3×#106—21no—
87C3×C4×C8C2311yescharacteristic
88C8×Dic3D411yescharacteristic
89C8×Dic3C2311yescharacteristic
90C2×C3⋊C16C2312yes—
91#810—21no—
92#810—21no—
93#809—21no—
94#809—22no—
95#809—22no—
96#809—21no—
97C4×C3⋊C8D411yescharacteristic
98#800—24no—
99D48—24no—
100D3×C16—24no—
101#532—61no—
102#529—61no—
103C2×SD32—32no—
104C2×D16—32no—
105C22×C16—32no—
106#500C2×D311yescharacteristic
107#499C2×D311yescharacteristic
108#495—31no—
109#493—31no—
110#488—61no—
111#486—61no—
112#467—31no—
113#466—31no—
114#466—32no—
115#466—61no—
116#465—61no—
117C4×Q16—31no—
118C4×SD16—61no—
119C4×D8—31no—
120C4×D8—32no—
121C4×D8—61no—
122C2×C4×C8—31no—
123#367C2×D312yes—
124#366C2×D312yes—
125#358—32no—
126#357—32no—
127#357—32no—
128#357—32no—
129#346—32no—
130#346—31no—
131#346C2×D311yescharacteristic
132C2×C3×Q8—41no—
133#239—84no—
134C2×D12—22no—
135C2×D12—21no—
136C2×D12—41no—
137C2×D12—42no—
138C2×D12—41no—
139C2×D12—41no—
140C2×C4×D3—41no—
141C2×C4×D3—42no—
142C2×C4×D3D811yescharacteristic
143C2×C4×D3D811yescharacteristic
144C2×C4×D3C2×D411yescharacteristic
145C2×Dic6—21no—
146C2×C3×C8C2×D411yescharacteristic
147C2×C3×C8C2411yescharacteristic
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—41no—
152C3×C4⋊C4—41no—
153C3×C4⋊C4—41no—
154C3×C42C2×D411yescharacteristic
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—42no—
161(C2×D3)⋊C4—42no—
162C4⋊Dic3—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165C4×Dic3D811yescharacteristic
166C4×Dic3D811yescharacteristic
167C4×Dic3C2×D411yescharacteristic
168C2×C3⋊C8C2×D411yescharacteristic
169C2×C3⋊C8C2×D411yescharacteristic
170Dic12—22no—
171D24—22no—
172D24—22no—
173D24—22no—
174D24—42no—
175#204—42no—
176D3×C8—22no—
177D3×C8—22no—
178C3×C16#4114yescomplemented
179C3⋊C16—22no—
180C2×#41—121no—
181C2×Q16—31no—
182C2×SD16—61no—
183C2×D8—32no—
184C2×D8—31no—
185C2×D8—61no—
186C22×C8—31no—
187C22×C8—31no—
188C4⋊Q8—21no—
189#127—61no—
190#126—121no—
191#125—21no—
192#124—61no—
193#124—121no—
194#123—121no—
195C22⋊Q8—61no—
196#121—61no—
197#121—62no—
198#121—121no—
199C4×Q8—61no—
200C4×D4—61no—
201C4×D4—62no—
202C4×D4—121no—
203C4×D4—121no—
204#117—121no—
205C2×C42—31no—
206SD32—64no—
207D16—64no—
208C2×C16—32no—
209C2×C16—64no—
210C2×C16C22×D312yes—
211#107—21no—
212#107C22×D311yescharacteristic
213#107C22×D311yescharacteristic
214#107C22×D312yes—
215#106—21no—
216#103—61no—
217#103—61no—
218#102—61no—
219#102—62no—
220#102—62no—
221#102—61no—
222C4×C8—31no—
223C4×C8—31no—
224C4×C8—31no—
225C4×C8C22×D311yescharacteristic
226C22×D3—41no—
227C22×D3—42no—
228C22×D3C2×D811yescharacteristic
229C3×Q8—82no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4C2×D811yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4C22×D411yescharacteristic
236C2×Dic3—41no—
237C2×Dic3C2×D811yescharacteristic
238C2×Dic3C2×D811yescharacteristic
239C2×Dic3C2×D811yescharacteristic
240D12—41no—
241D12—42no—
242D12—41no—
243D12—42no—
244D12—82no—
245D12—82no—
246D12—82no—
247D12—84no—
248C4×D3—22no—
249C4×D3—21no—
250C4×D3—21no—
251C4×D3—41no—
252C4×D3—82no—
253C4×D3—84no—
254Dic6—41no—
255Dic6—41no—
256C3×C8#13512yes—
257C3×C8C2×#4112yes—
258C3⋊C8—21no—
259C3⋊C8—21no—
260#41—244no—
261C2×Q8—41no—
262C2×Q8—61no—
263C2×D4—61no—
264C2×D4—62no—
265C2×D4—121no—
266C2×D4—122no—
267C2×D4—121no—
268C2×D4—121no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—121no—
273C22×C4—122no—
274Q16—62no—
275SD16—122no—
276D8—62no—
277D8—62no—
278D8—62no—
279D8—122no—
280C2×C8—31no—
281C2×C8—31no—
282C2×C8—62no—
283C2×C8—62no—
284C2×C8D3×D411yescharacteristic
285C2×C8C23×D311yescharacteristic
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—121no—
293C4⋊C4—121no—
294C22⋊C4—121no—
295C22⋊C4—122no—
296C22⋊C4—121no—
297C22⋊C4—122no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C42—31no—
302C42—31no—
303C42—31no—
304C42—121no—
305C42D3×D411yescharacteristic
306C16—62no—
307C16#23914yescomplemented
308C22×C3C22×D811yescharacteristic
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—82no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—82no—
315C2×D3—81no—
316C2×D3—82no—
317C2×D3#50912yes—
318C3×C4—21no—
319C3×C4—21no—
320C3×C4—82no—
321C3×C4—81no—
322C3×C4—81no—
323C3×C4#53211yescharacteristic
324C3×C4C2×#13511yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—21no—
328Dic3—81no—
329C23—31no—
330C23—121no—
331C23—122no—
332Q8—82no—
333Q8—121no—
334Q8—121no—
335D4—121no—
336D4—122no—
337D4—122no—
338D4—121no—
339D4—242no—
340D4—242no—
341D4—244no—
342D4—242no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—31no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—42no—
349C2×C4—62no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—121no—
353C2×C4—121no—
354C2×C4—244no—
355C2×C4—242no—
356C2×C4D3×D811yescharacteristic
357C2×C4D3×D811yescharacteristic
358C2×C4C2×D3×D411yescharacteristic
359C8—61no—
360C8—61no—
361C8#91912yes—
362C8C2×#23912yes—
363C2×C3#316911yescharacteristic
364C2×C3C2×#50911yescharacteristic
365C2×C3#341411yescharacteristic
366D3—22no—
367D3—162no—
368D3—161no—
369C22—32no—
370C22—61no—
371C22—61no—
372C22—242no—
373C22—241no—
374C22—242no—
375C22—241no—
376C22—241no—
377C22—242no—
378C22C2×D3×D811yescharacteristic
379C4—21no—
380C4—21no—
381C4—61no—
382C4—61no—
383C4—61no—
384C4—82no—
385C4—81no—
386C4—81no—
387C4—241no—
388C4#572511yescharacteristic
389C4C2×#91911yescharacteristic
390C3#3374611yescharacteristic, complemented, Sylow
391C2—62no—
392C2—482no—
393C2—481no—
394C2#8193311yescharacteristic
395C2D3×#50911yescharacteristic
396C2#8203911yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2255C2
socleC22×C3C22×D8
hypercenter#2255D3
hypocenterC3#33746
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
1. #81933C2semidirect
2. #81907C2semidirect
3. C3×#2255C2semidirect
4. #71383C2semidirect
5. #71352C2semidirect
6. #69347C2semidirect
7. #69339C2semidirect
8. #69327C2semidirect
9. #69319C2semidirect
10. #69081C2semidirect
11. D3×#346C2semidirect
46. #2255D3semidirect
178. C3×C16#41semidirect
307. C16#239semidirect
390. C3#33746semidirect

Maximal subgroups

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

subgroupindex
#819332
#819072
C3×#22552
#713832
#713522
#693472
#693392
#693272
#693192
#690812
D3×#3462
#337463