Finite Group Databasebeta

← #10950436

Subgroups — #10950436

Order 768.

526 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#84000C211yesmaximal, characteristic, complemented
2#80892C211yesmaximal, characteristic, complemented
3C3×#2191C211yesmaximal, characteristic, complemented
4#72112C211yesmaximal, characteristic
5#71521C212yesmaximal, complemented
6#71519C212yesmaximal, complemented
7#71090C211yesmaximal, characteristic, complemented
8#69249C212yesmaximal, complemented
9#69247C212yesmaximal, complemented
10#69116C211yesmaximal, characteristic, complemented
11#69114C211yesmaximal, characteristic, complemented
12#33690—31nomaximal, Sylow
13C3×#575C2211yescharacteristic
14#5819—21no—
15#5624—21no—
16C3×#510C2212yes—
17C3×#470C2211yescharacteristic
18C2×#837—21no—
19#5197—21no—
20#5197C2212yes—
21#5196—21no—
22#5196C2212yes—
23#5193C2212yes—
24#5191C2211yescharacteristic, complemented
25#5184C2211yescharacteristic, complemented
26#5135C2211yescharacteristic
27#5125C2211yescharacteristic
28Dic3×M16C2211yescharacteristic
29#5049—21no—
30#4869C2211yescharacteristic, complemented
31#4858C2211yescharacteristic, complemented
32#4817—21no—
33#4800—21no—
34C3×#360C2212yes—
35C3×#349C2211yescharacteristic
36#4566C2212yes—
37#4565C2212yescomplemented
38#4553C2211yescharacteristic
39#4516C2212yescomplemented
40#4513C2212yescomplemented
41#4512C2212yescomplemented
42#4511C2212yescomplemented
43#4487C2212yescomplemented
44#3183—31no—
45#2956—31no—
46#2239—31no—
47#2238—31no—
48#2231—32no—
49#2212—32no—
50#2191D311yescharacteristic, complemented
51#2190—31no—
52AGL(1,ℤ/16)—32no—
53#2180—32no—
54#2165—31no—
55C2×C3×#41—21no—
56C3×#137—22no—
57C3×AGL(1,ℤ/8)—22no—
58C3×#135—21no—
59C3×#135C2312yes—
60C2×C3×Q16C2311yescharacteristic
61C2×C3×M16C2311yescharacteristic
62C3×C22⋊Q8—21no—
63#948—22no—
64#948—22no—
65#948—42no—
66#947—21no—
67Q8×Dic3—21no—
68C2×#215—21no—
69#930—21no—
70C2×C4×Dic3D411yescharacteristic
71C2×#208—21no—
72C2×#208—42no—
73C2×#208D411yescharacteristic
74C2×D24C2311yescharacteristic
75#898—41no—
76#893—21no—
77#892—41no—
78#890—21no—
79#886—41no—
80#67⋊C4—41no—
81C3×Q32—22no—
82C3×SD32—22no—
83C3×M32C2312yes—
84C3×#106C2311yescharacteristic
85C3×#103—21no—
86#837—22no—
87#837—22no—
88#835—21no—
89#828—22no—
90#826—22no—
91#821—21no—
92#819—21no—
93#819C2312yes—
94#817C2311yescharacteristic
95#815D411yescharacteristic
96#815C2311yescharacteristic
97C8×Dic3D411yescharacteristic
98C8×Dic3C2311yescharacteristic
99#812C2312yes—
100#809—21no—
101#807—21no—
102#575C2×D311yescharacteristic
103#568—61no—
104#529—61no—
105#510C2×D312yes—
106#509—32no—
107#499—31no—
108#495—31no—
109#488—61no—
110#470C2×D311yescharacteristic
111#469—31no—
112#468—61no—
113#444—32no—
114#444—61no—
115#443—32no—
116#443—61no—
117#442—31no—
118#441—31no—
119#440—61no—
120#430—31no—
121C2×#104—61no—
122C4×M16—31no—
123#365—32no—
124#365—32no—
125#364—32no—
126#362—32no—
127#361—32no—
128#360—32no—
129#360C2×D312yes—
130#349—31no—
131#349C2×D311yescharacteristic
132#347—32no—
133C3×#41—22no—
134C3×#41—22no—
135C3×#41—41no—
136C3×#41—41no—
137C2×C3×Q8—21no—
138C2×C3×Q8—21no—
139C2×C3×D4—21no—
140C2×C3×D4—41no—
141C22×C3×C4—41no—
142C22×C3×C4C2×D411yescharacteristic
143C2×#67—41no—
144C22×Dic3—21no—
145C2×D12—21no—
146C2×C4×D3—41no—
147C3×Q16—21no—
148C3×Q16—22no—
149C3×Q16#4112yes—
150C3×SD16—22no—
151C3×SD16—22no—
152C3×D8—21no—
153C3×D8#4112yes—
154C3×M16—22no—
155C2×C3×C8C2×D411yescharacteristic
156C2×C3×C8C2411yescharacteristic
157C3×C4⋊C4—21no—
158C3×C4⋊C4—41no—
159C3×C22⋊C4—41no—
160#215—42no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165C4⋊Dic3—21no—
166C4⋊Dic3—41no—
167C4⋊Dic3—41no—
168Dic3⋊C4—41no—
169C4×Dic3—21no—
170C4×Dic3—41no—
171C4×Dic3C2×D411yescharacteristic
172C4×Dic3C2×D411yescharacteristic
173#208—22no—
174#208—22no—
175#208—42no—
176#208—41no—
177#208—42no—
178C2×C3⋊C8—42no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—42no—
181C2×C3⋊C8C2×D411yescharacteristic
182C2×C3⋊C8C2×D411yescharacteristic
183D24—22no—
184C3×C16—22no—
185C3⋊C16—22no—
186C2×#41—21no—
187#137—22no—
188AGL(1,ℤ/8)—22no—
189#135—21no—
190#135C22×D312yes—
191C2×Q16C22×D311yescharacteristic
192C2×SD16—61no—
193C2×D8—31no—
194#131—62no—
195#131—62no—
196#131—122no—
197C2×M16—31no—
198C2×M16—61no—
199C2×M16—122no—
200C2×M16C22×D311yescharacteristic
201#126—121no—
202#125—61no—
203#124—61no—
204#123—121no—
205C22⋊Q8—21no—
206#121—61no—
207C4×Q8—61no—
208C4×D4—61no—
209C4×D4—121no—
210#117—121no—
211C2×C42—31no—
212Q32—22no—
213SD32—22no—
214SD32—62no—
215D16—62no—
216M32—32no—
217M32C22×D312yes—
218#108—32no—
219#108—61no—
220#107—61no—
221#106—31no—
222#106C22×D311yescharacteristic
223#104—62no—
224#104—62no—
225#103—21no—
226#103—61no—
227#102—61no—
228#102—61no—
229#97—31no—
230#97—31no—
231C4×C8—31no—
232C4×C8—31no—
233C23×C3—41no—
234C23×C3AGL(1,ℤ/8)11yescharacteristic
235C22×D3—41no—
236C3×Q8—22no—
237C3×Q8—41no—
238C3×Q8—41no—
239C3×Q8—41no—
240C3×D4—22no—
241C3×D4—41no—
242C3×D4—42no—
243C3×D4—42no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
252C2×C3×C4C22×D411yescharacteristic
253#67—82no—
254C2×Dic3—21no—
255C2×Dic3—21no—
256C2×Dic3—41no—
257C2×Dic3—81no—
258D12—41no—
259D12—41no—
260C4×D3—82no—
261C3×C8—21no—
262C3×C8C2×#4112yes—
263C3⋊C8—21no—
264C3⋊C8—42no—
265C3⋊C8AGL(1,ℤ/8)12yes—
266#41—22no—
267#41—22no—
268#41—41no—
269#41—41no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—21no—
273C2×D4—41no—
274C2×D4—61no—
275C2×D4—121no—
276C22×C4—41no—
277C22×C4—61no—
278C22×C4—121no—
279C22×C4D3×D411yescharacteristic
280Q16—21no—
281Q16—22no—
282Q16#23712yes—
283SD16—22no—
284SD16—22no—
285SD16—122no—
286D8—21no—
287D8—62no—
288D8#23712yes—
289M16—22no—
290M16—62no—
291M16—62no—
292M16—122no—
293M16—122no—
294M16—121no—
295C2×C8—31no—
296C2×C8—31no—
297C2×C8—122no—
298C2×C8—122no—
299C2×C8—121no—
300C2×C8D3×D411yescharacteristic
301C2×C8C23×D311yescharacteristic
302C4⋊C4—21no—
303C4⋊C4—41no—
304C4⋊C4—61no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C22⋊C4—41no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C42—31no—
314C42—31no—
315C42—61no—
316C42—121no—
317C16—22no—
318C16—62no—
319C22×C3—21no—
320C22×C3—41no—
321C22×C3—42no—
322C22×C3—161no—
323C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—81no—
330C3×C4—81no—
331C3×C4#52511yescharacteristic
332C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
333Dic3—41no—
334Dic3—41no—
335Dic3—81no—
336C23—41no—
337C23—121no—
338C23#91111yescharacteristic
339Q8—22no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343D4—22no—
344D4—41no—
345D4—42no—
346D4—42no—
347D4—121no—
348D4—121no—
349D4—242no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—121no—
360C2×C4—242no—
361C2×C4—241no—
362C2×C4#91111yescharacteristic
363C2×C4C2×D3×D411yescharacteristic
364C8—21no—
365C8—32no—
366C8—61no—
367C8—122no—
368C8C2×#23712yes—
369C2×C3—21no—
370C2×C3—41no—
371C2×C3—82no—
372C2×C3#310911yescharacteristic
373D3—161no—
374C22—21no—
375C22—41no—
376C22—42no—
377C22—161no—
378C22—241no—
379C22—241no—
380C22—241no—
381C22C2×#91111yescharacteristic
382C4—21no—
383C4—42no—
384C4—81no—
385C4—81no—
386C4—121no—
387C4—121no—
388C4—241no—
389C4#558611yescharacteristic
390C4D3×AGL(1,ℤ/8)11yescharacteristic
391C3#3369011yescharacteristic, complemented, Sylow
392C2—21no—
393C2—41no—
394C2—82no—
395C2—481no—
396C2#7999011yescharacteristic
subgroupG/H
centerC2#79990
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC3×#2191C2
socleC2×C3#3109
prime radical#72112C2
hypercenter#2191D3
hypocenterC3#33690
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×M16 ◁ #2191

Complements

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

NH ≅ G/Ntype
1. #84000C2semidirect
2. #80892C2semidirect
3. C3×#2191C2semidirect
5. #71521C2semidirect
6. #71519C2semidirect
7. #71090C2semidirect
8. #69249C2semidirect
9. #69247C2semidirect
10. #69116C2semidirect
11. #69114C2semidirect
24. #5191C22semidirect
25. #5184C22semidirect
30. #4869C22semidirect
31. #4858C22semidirect
37. #4565C22semidirect
39. #4516C22semidirect
40. #4513C22semidirect
41. #4512C22semidirect
42. #4511C22semidirect
43. #4487C22semidirect
50. #2191D3semidirect
391. C3#33690semidirect

Maximal subgroups

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

subgroupindex
#840002
#808922
C3×#21912
#721122
#715212
#715192
#710902
#692492
#692472
#691162
#691142
#336903