Finite Group Databasebeta

← #10767959

Subgroups — #10767959

Order 768.

1386 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×#3496C211yesmaximal, characteristic, complemented
2#83300C212yesmaximal, complemented
3#82347C211yesmaximal, characteristic, complemented
4#71845C214yesmaximal, complemented
5#71844C214yesmaximal, complemented
6#71812C212yesmaximal, complemented
7#71603C211yesmaximal, characteristic, complemented
8#18178—31nomaximal, Sylow
9C2×C3×H(5,𝔽2)—21no—
10C3×#547—22no—
11C3×#546—24no—
12C3×D42—22no—
13C3×#537—21no—
14C3×#534C2214yes—
15C3×#532C2212yes—
16C2×C3×#121—22no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C4×#67—22no—
19C2×C4×D12C2211yescharacteristic, complemented
20#5223—21no—
21#5084—22no—
22#5072—22no—
23#5036C2214yescomplemented
24#5034C2214yescomplemented
25#5014—24no—
26#4993—21no—
27#4960C2212yescomplemented
28#4949C2212yescomplemented
29#4932C2211yescharacteristic, complemented
30#4926C2212yescomplemented
31#4537C2218yescomplemented
32#4525C2214yescomplemented
33#3496D311yescharacteristic, complemented
34#3109—32no—
35#2397—31no—
36#1613—34no—
37#1595—34no—
38#1505—32no—
39#1466—31no—
40C3×H(5,𝔽2)—48no—
41C2×C3×#41—24no—
42C2×C3×#41—22no—
43C22×C3×D4—24no—
44C22×C3×D4—24no—
45C22×C3×D4—22no—
46C22×C3×D4D411yescharacteristic
47C23×C3×C4D412yes—
48C22×#67—22no—
49C22×D12C2311yescharacteristic
50C22×C4×D3—21no—
51C3×C4⋊Q8D412yescomplemented
52C3×#127D412yescomplemented
53C3×#124—22no—
54C3×#123—28no—
55C3×C22⋊Q8—24no—
56C3×#121—24no—
57C3×#121—24no—
58C3×#121—24no—
59C3×#121—22no—
60C3×#121—24no—
61C3×#121C2314yes—
62C3×#120—28no—
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4—22no—
66C3×C4×D4D414yescomplemented
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D411yescharacteristic
69C2×C3×C22⋊C4—24no—
70C2×C3×C22⋊C4C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#213—24no—
74C4×#67—48no—
75C2×(C2×D3)⋊C4—22no—
76C2×(C2×D3)⋊C4—21no—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—21no—
80C4×D12—22no—
81C4×D12—22no—
82#832—24no—
83C3⋊#95—21no—
84C3⋊#95C2312yes—
85#830C2314yes—
86#809—24no—
87#808—24no—
88C4⋊(C3⋊C8)—22no—
89#803—22no—
90C2×H(5,𝔽2)—21no—
91#547—22no—
92#546—24no—
93D42—22no—
94#537—21no—
95#534C2×D314yes—
96#532C2×D312yes—
97C2×#121—22no—
98C2×C4×D4—31no—
99C2×C4×D4—62no—
100C2×C4×D4C2×D311yescharacteristic
101#481—34no—
102#476—62no—
103#459—62no—
104#447—34no—
105#442—64no—
106#432—32no—
107#399—32no—
108#399—61no—
109#392—32no—
110#392—61no—
111#376—31no—
112#327—38no—
113#323—34no—
114C24×C3—24no—
115C24×C3C2×D412yes—
116C23×D3—21no—
117C3×#41—48no—
118C3×#41—416no—
119C2×C3×Q8—22no—
120C2×C3×D4—24no—
121C2×C3×D4—28no—
122C2×C3×D4—24no—
123C2×C3×D4—24no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—416no—
128C2×C3×D4—48no—
129C2×C3×D4—48no—
130C2×C3×D4—84no—
131C2×C3×D4C2×D414yes—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—24no—
135C22×C3×C4—22no—
136C22×C3×C4—44no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2411yescharacteristic
141C2×#67—44no—
142C2×#67—44no—
143C2×#67—44no—
144C22×Dic3—22no—
145C22×Dic3—21no—
146C2×D12—22no—
147C2×D12—42no—
148C2×C4×D3—44no—
149C2×C4×D3—41no—
150C2×C4×D3—41no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4C2×D414yes—
155C3×C22⋊C4—28no—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C42C2×D412yes—
160C3×C42C2×D412yes—
161#217—44no—
162#213—48no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—42no—
165C4⋊Dic3—22no—
166Dic3⋊C4—44no—
167C4×Dic3—24no—
168C2×C3⋊C8—24no—
169H(5,𝔽2)—48no—
170C2×#41—24no—
171C2×#41—22no—
172C22×D4—24no—
173C22×D4—24no—
174C22×D4—22no—
175C22×D4—31no—
176C22×D4—62no—
177C22×D4#6711yescharacteristic
178C23×C4—61no—
179C23×C4#6712yes—
180C2×D8—64no—
181C4⋊Q8#6712yescomplemented
182#127#6712yescomplemented
183#124—22no—
184#123—28no—
185C22⋊Q8—24no—
186#121—24no—
187#121—24no—
188#121—24no—
189#121—22no—
190#121—24no—
191#121C22×D314yes—
192#120—28no—
193C4×D4—22no—
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—62no—
197C4×D4—62no—
198C4×D4—128no—
199C4×D4#6714yescomplemented
200C2×C4⋊C4—22no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—62no—
203C2×C4⋊C4#6711yescharacteristic
204C2×C22⋊C4—24no—
205C2×C22⋊C4—62no—
206C2×C22⋊C4—61no—
207C2×C22⋊C4—62no—
208C2×C22⋊C4C22×D312yes—
209C2×C42—61no—
210C2×C42C22×D311yescharacteristic
211#106—64no—
212C4⋊C8—62no—
213#102—64no—
214#102—64no—
215C22⋊C8—34no—
216#97—62no—
217#95—32no—
218#95—61no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—24no—
222C23×C3—42no—
223C23×C3—42no—
224C23×C3—48no—
225C23×C3—84no—
226C23×C3—84no—
227C23×C3AGL(1,ℤ/8)14yes—
228C23×C3C22×D411yescharacteristic
229C22×D3—22no—
230C22×D3—42no—
231C22×D3—41no—
232C22×D3—41no—
233C22×D3—42no—
234C3×Q8—44no—
235C3×D4—28no—
236C3×D4—44no—
237C3×D4—44no—
238C3×D4—416no—
239C3×D4—48no—
240C3×D4—44no—
241C3×D4—84no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—24no—
246C2×C3×C4—41no—
247C2×C3×C4—48no—
248C2×C3×C4—48no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—82no—
252C2×C3×C4#12012yes—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4C22×D412yes—
256#67—416no—
257C2×Dic3—24no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—41no—
262D12—24no—
263D12—42no—
264C4×D3—82no—
265C4×D3—82no—
266C3⋊C8—44no—
267C24—24no—
268C24—61no—
269C24C2×#6712yes—
270#41—48no—
271#41—416no—
272C2×Q8—22no—
273C2×D4—24no—
274C2×D4—28no—
275C2×D4—24no—
276C2×D4—24no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—416no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—62no—
284C2×D4—84no—
285C2×D4—122no—
286C2×D4—124no—
287C2×D4—124no—
288C2×D4—124no—
289C2×D4C2×#6714yes—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—44no—
295C22×C4—62no—
296C22×C4—61no—
297C22×C4—124no—
298C22×C4—121no—
299C22×C4—121no—
300C22×C4C2×#6711yescharacteristic
301C22×C4C2×#6711yescharacteristic
302C22×C4C2×#6712yes—
303C22×C4C23×D311yescharacteristic
304D8—128no—
305C2×C8—64no—
306C4⋊C4—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—62no—
310C4⋊C4—124no—
311C4⋊C4C2×#6714yes—
312C22⋊C4—28no—
313C22⋊C4—24no—
314C22⋊C4—24no—
315C22⋊C4—24no—
316C22⋊C4—124no—
317C22⋊C4—124no—
318C22⋊C4—122no—
319C42—64no—
320C42C2×#6712yes—
321C42C2×#6712yes—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—24no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—42no—
328C22×C3—44no—
329C22×C3—44no—
330C22×C3—44no—
331C22×C3—82no—
332C22×C3—168no—
333C22×C3C2×#12011yescharacteristic
334C22×C3C2×AGL(1,ℤ/8)12yes—
335C2×D3—44no—
336C2×D3—42no—
337C2×D3—82no—
338C2×D3—81no—
339C2×D3—81no—
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—22no—
343C3×C4—44no—
344C3×C4—42no—
345C3×C4#45312yes—
346Dic3—44no—
347Dic3—81no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—42no—
352C23—42no—
353C23—48no—
354C23—62no—
355C23—84no—
356C23—84no—
357C23—122no—
358C23—121no—
359C23—122no—
360C23—121no—
361C23#93214yes—
362C23C22×#6711yescharacteristic
363Q8—44no—
364D4—28no—
365D4—44no—
366D4—44no—
367D4—416no—
368D4—48no—
369D4—44no—
370D4—64no—
371D4—84no—
372D4—122no—
373D4—1216no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—64no—
384C2×C4—82no—
385C2×C4—122no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—121no—
389C2×C4—242no—
390C2×C4—242no—
391C2×C4#95311yescharacteristic
392C2×C4#95312yes—
393C2×C4#95311yescharacteristic
394C2×C4C22×#6712yes—
395C8—124no—
396C2×C3—22no—
397C2×C3—44no—
398C2×C3—84no—
399C2×C3#300311yescharacteristic
400C2×C3C2×#45311yescharacteristic
401C2×C3#302511yescharacteristic
402D3—82no—
403C22—21no—
404C22—21no—
405C22—24no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22—82no—
413C22—124no—
414C22—122no—
415C22—168no—
416C22—242no—
417C22—241no—
418C22—241no—
419C22C2×#93212yes—
420C22C2×#95311yescharacteristic
421C4—22no—
422C4—21no—
423C4—22no—
424C4—44no—
425C4—42no—
426C4—124no—
427C4—241no—
428C4#524112yes—
429C3#1817811yescharacteristic, complemented, Sylow
430C2—22no—
431C2—44no—
432C2—84no—
433C2—242no—
434C2#8403811yescharacteristic
435C2C2×#524111yescharacteristic
436C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3496C2
socleC22×C3C2×#120
hypercenter#3496D3
hypocenterC3#18178
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3496

Complements

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

NH ≅ G/Ntype
1. C3×#3496C2semidirect
2. #83300C2semidirect
3. #82347C2semidirect
4. #71845C2semidirect
5. #71844C2semidirect
6. #71812C2semidirect
7. #71603C2semidirect
19. C2×C4×D12C22semidirect
23. #5036C22semidirect
24. #5034C22semidirect
27. #4960C22semidirect
28. #4949C22semidirect
29. #4932C22semidirect
30. #4926C22semidirect
31. #4537C22semidirect
32. #4525C22semidirect
33. #3496D3semidirect
51. C3×C4⋊Q8D4semidirect
52. C3×#127D4semidirect
66. C3×C4×D4D4semidirect
181. C4⋊Q8#67semidirect
182. #127#67semidirect
199. C4×D4#67semidirect
429. C3#18178semidirect

Maximal subgroups

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

subgroupindex
C3×#34962
#833002
#823472
#718452
#718442
#718122
#716032
#181783