Finite Group Databasebeta

← #10909601

Subgroups — #10909601

Order 768.

1000 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 1000 conjugacy classes of subgroups — too many to draw. Only its 143 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#84002C212yesmaximal, complemented
2#82927C212yesmaximal, complemented
3C3×#2010C211yesmaximal, characteristic, complemented
4#71714C212yesmaximal, complemented
5#70402C212yesmaximal, complemented
6#70400C212yesmaximal, complemented
7#69988C211yesmaximal, characteristic, complemented
8#69987C211yesmaximal, characteristic, complemented
9#68706C212yesmaximal, complemented
10#30434—31nomaximal, Sylow
11C23×D12C2211yescharacteristic
12C3×#575C2211yescharacteristic, complemented
13C3×#525C2212yes—
14C2×#949C2212yes—
15C2×#949C2212yes—
16#5819C2212yescomplemented
17C3×#421C2212yescomplemented
18C3×#407C2211yescharacteristic
19#5237C2212yescomplemented
20#5207C2212yes—
21#5207C2211yescharacteristic
22#5203—24no—
23#5199—24no—
24C2×#823C2212yes—
25C2×#823C2212yes—
26#5125C2211yescharacteristic, complemented
27#5124C2212yescomplemented
28#5046—24no—
29#5036—24no—
30#5001C2212yes—
31#5001C2212yes—
32C3×H(3,ℤ/4)C2211yescharacteristic, complemented
33C3⋊#338C2211yescharacteristic, complemented
34#4533C2212yescomplemented
35#4533C2212yescomplemented
36#3068—32no—
37#3003—32no—
38#2091—31no—
39#2081—32no—
40#2048—32no—
41#2010D311yescharacteristic, complemented
42#1903—32no—
43#1891—32no—
44#1783—31no—
45C24×D3—21no—
46C2×C3×#41C2312yes—
47C23×C3×C4C2311yescharacteristic
48C22×D12—24no—
49C22×D12—21no—
50C22×D12—41no—
51C22×D12—41no—
52C22×D12—42no—
53C22×D12C2312yes—
54C3×#137—22no—
55C3×AGL(1,ℤ/8)—22no—
56C3×#135—41no—
57C2×C3×M16C2311yescharacteristic
58C2×C3×M16C2311yescharacteristic
59C3×#123—22no—
60C3×C22⋊Q8—22no—
61C3×#121—22no—
62C3×#120—22no—
63C3×C4×D4—42no—
64C3×#117C2312yes—
65#949—24no—
66#949—42no—
67#949—42no—
68#949—42no—
69#948—24no—
70#948—42no—
71C2×#215—22no—
72C2×#215D414yescomplemented
73C2×#213—22no—
74C2×#213D414yescomplemented
75C2×#208—42no—
76C2×#208C2312yes—
77C2×#208C2312yes—
78C2×#208C2312yes—
79C3×#104—42no—
80C3×C22⋊C8—41no—
81#836—22no—
82#833—22no—
83#830—24no—
84#830—41no—
85#823—24no—
86#823—42no—
87#822—24no—
88#819—41no—
89#809—24no—
90#809—24no—
91C23×D4—31no—
92#575C2×D311yescharacteristic, complemented
93C2×AGL(1,ℤ/8)—32no—
94C2×AGL(1,ℤ/8)—32no—
95#568—32no—
96#525C2×D312yes—
97#472—64no—
98#471—64no—
99#450—64no—
100#447—64no—
101#430—32no—
102#430—31no—
103#421C2×D312yescomplemented
104#418—32no—
105#418—32no—
106#413—32no—
107C2×#100—32no—
108C2×#100—32no—
109#407—32no—
110#407—31no—
111#407C2×D311yescharacteristic
112#344—32no—
113#344—32no—
114#338—31no—
115H(3,ℤ/4)C2×D311yescharacteristic, complemented
116C24×C3#4111yescharacteristic
117C23×D3—22no—
118C23×D3—44no—
119C23×D3—42no—
120C23×D3—84no—
121C3×#41—22no—
122C3×#41—22no—
123C3×#41—22no—
124C3×#41—22no—
125C2×C3×Q8C2×D412yes—
126C2×C3×D4—22no—
127C2×C3×D4—42no—
128C2×C3×D4C2×D412yes—
129C22×C3×C4—22no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—22no—
133C22×C3×C4—22no—
134C22×C3×C4#4111yescharacteristic
135C22×C3×C4C2411yescharacteristic
136C2×D12—22no—
137C2×D12—22no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—44no—
142C2×D12—48no—
143C2×D12—44no—
144C2×D12—42no—
145C2×D12—44no—
146C2×D12C2×D414yes—
147C3×Q16—41no—
148C3×SD16—42no—
149C3×D8—41no—
150C3×M16—22no—
151C3×M16—22no—
152C2×C3×C8—21no—
153C2×C3×C8—21no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—42no—
156C3×C22⋊C4—22no—
157C3×C22⋊C4—42no—
158C3×C22⋊C4—42no—
159C3×C42—22no—
160#215—44no—
161#215—44no—
162#213—44no—
163#213—44no—
164#208—22no—
165#208—22no—
166#208—22no—
167#208—22no—
168#208—22no—
169#208—22no—
170#208—44no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—44no—
174C2×C3⋊C8C2×D414yes—
175C25—61no—
176C2×#41C22×D312yes—
177C22×D4—32no—
178C22×D4—64no—
179C22×D4—61no—
180C22×D4—121no—
181C22×D4—122no—
182C22×D4—121no—
183C23×C4C22×D311yescharacteristic
184#137—22no—
185AGL(1,ℤ/8)—22no—
186AGL(1,ℤ/8)—64no—
187AGL(1,ℤ/8)—122no—
188AGL(1,ℤ/8)—122no—
189AGL(1,ℤ/8)—122no—
190#135—41no—
191C2×SD16—34no—
192C2×SD16—62no—
193C2×D8—34no—
194C2×D8—62no—
195#131—64no—
196#131—122no—
197C2×M16—32no—
198C2×M16—32no—
199C2×M16—32no—
200C2×M16—122no—
201C2×M16C22×D311yescharacteristic
202C2×M16C22×D311yescharacteristic
203#123—22no—
204C22⋊Q8—22no—
205#121—22no—
206#120—22no—
207C4×D4—42no—
208#117C22×D312yes—
209#108—64no—
210#108—121no—
211#104—42no—
212#102—64no—
213#102—64no—
214#101—62no—
215#100—64no—
216#100—62no—
217#100—122no—
218C22⋊C8—41no—
219C22⋊C8—64no—
220C22⋊C8—121no—
221C23×C3—21no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—42no—
225C23×C3C2×#4111yescharacteristic
226C22×D3—24no—
227C22×D3—48no—
228C22×D3—42no—
229C22×D3—42no—
230C22×D3—84no—
231C22×D3—88no—
232C22×D3—84no—
233C22×D3—88no—
234C22×D3—84no—
235C3×Q8—22no—
236C3×Q8—22no—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—42no—
240C3×D4—42no—
241C3×D4—84no—
242C2×C3×C4—21no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—42no—
251C2×C3×C4C22×D412yes—
252C2×C3×C4H(5,𝔽2)12yes—
253C2×C3×C4H(5,𝔽2)11yescharacteristic
254C2×C3×C4H(5,𝔽2)11yescharacteristic
255D12—24no—
256D12—24no—
257D12—48no—
258D12—42no—
259D12—42no—
260C3×C8—41no—
261C3×C8—41no—
262C3⋊C8—24no—
263C3⋊C8—42no—
264C3⋊C8—42no—
265C24—62no—
266C24—124no—
267C24—122no—
268C24—244no—
269C24#23711yescharacteristic
270#41—22no—
271#41—22no—
272#41—22no—
273#41—22no—
274C2×Q8D3×D412yes—
275C2×D4—22no—
276C2×D4—34no—
277C2×D4—42no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—62no—
283C2×D4—124no—
284C2×D4—122no—
285C2×D4—124no—
286C2×D4—128no—
287C2×D4—124no—
288C2×D4D3×D412yes—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—21no—
294C22×C4#23711yescharacteristic
295C22×C4C23×D311yescharacteristic
296Q16—41no—
297SD16—42no—
298SD16—124no—
299SD16—124no—
300D8—41no—
301D8—124no—
302D8—124no—
303M16—22no—
304M16—22no—
305M16—62no—
306M16—62no—
307M16—62no—
308M16—62no—
309M16—62no—
310M16—62no—
311M16—124no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—34no—
315C2×C8—62no—
316C2×C8—62no—
317C2×C8—124no—
318C4⋊C4—22no—
319C4⋊C4—42no—
320C22⋊C4—22no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—22no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—82no—
332C22×C3#53811yescharacteristic
333C22×C3#54612yes—
334C2×D3—44no—
335C2×D3—84no—
336C2×D3—84no—
337C2×D3—816no—
338C2×D3—84no—
339C3×C4—41no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4#53412yes—
343C3×C4D4211yescharacteristic
344C3×C4#56011yescharacteristic
345C23—22no—
346C23—21no—
347C23—22no—
348C23—42no—
349C23—64no—
350C23—128no—
351C23—122no—
352C23—122no—
353C23—248no—
354C23—244no—
355C23—244no—
356C23—248no—
357C23—244no—
358C23C2×#23711yescharacteristic
359Q8—22no—
360Q8—22no—
361D4—22no—
362D4—22no—
363D4—42no—
364D4—42no—
365D4—64no—
366D4—64no—
367D4—84no—
368D4—128no—
369D4—122no—
370D4—122no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4C2×D3×D412yes—
381C2×C4#100411yescharacteristic
382C2×C4#100412yes—
383C2×C4#100911yescharacteristic
384C8—41no—
385C8—41no—
386C8—64no—
387C8—122no—
388C8—122no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—82no—
393C2×C3#266811yescharacteristic
394D3—84no—
395C22—24no—
396C22—22no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—42no—
402C22—82no—
403C22—124no—
404C22—244no—
405C22—244no—
406C22—244no—
407C22—2416no—
408C22#559112yes—
409C22#561711yescharacteristic
410C4—41no—
411C4—42no—
412C4—82no—
413C4#559011yescharacteristic
414C4#560512yes—
415C4#572011yescharacteristic
416C3#3043411yescharacteristic, complemented, Sylow
417C2—22no—
418C2—21no—
419C2—42no—
420C2—82no—
421C2—244no—
422C2#7676311yescharacteristic
subgroupG/H
centerC2#76763
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2010C2
socleC2×C3#2668
hypercenter#2010D3
hypocenterC3#30434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #2010

Complements

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

NH ≅ G/Ntype
1. #84002C2semidirect
2. #82927C2semidirect
3. C3×#2010C2semidirect
4. #71714C2semidirect
5. #70402C2semidirect
6. #70400C2semidirect
7. #69988C2semidirect
8. #69987C2semidirect
9. #68706C2semidirect
12. C3×#575C22semidirect
16. #5819C22semidirect
17. C3×#421C22semidirect
19. #5237C22semidirect
26. #5125C22semidirect
27. #5124C22semidirect
32. C3×H(3,ℤ/4)C22semidirect
33. C3⋊#338C22semidirect
34. #4533C22semidirect
35. #4533C22semidirect
41. #2010D3semidirect
72. C2×#215D4semidirect
74. C2×#213D4semidirect
92. #575C2×D3semidirect
103. #421C2×D3semidirect
115. H(3,ℤ/4)C2×D3semidirect
416. C3#30434semidirect

Maximal subgroups

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

subgroupindex
#840022
#829272
C3×#20102
#717142
#704022
#704002
#699882
#699872
#687062
#304343