Finite Group Databasebeta

← #10909607

Subgroups — #10909607

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#84038C212yesmaximal, complemented
2#78703C212yesmaximal, complemented
3C3×#1783C211yesmaximal, characteristic, complemented
4#70087C212yesmaximal, complemented
5#70078C212yesmaximal, complemented
6#70076C212yesmaximal, complemented
7#69993C211yesmaximal, characteristic, complemented
8#69992C212yesmaximal, complemented
9#69987C211yesmaximal, characteristic, complemented
10#30434—31nomaximal, Sylow
11C23×C3×D4C2211yescharacteristic
12C2×#932C2212yes—
13C2×#932C2212yes—
14#5791C2212yes—
15#5748C2211yescharacteristic, complemented
16#5745C2212yescomplemented
17C2×C3×#100C2212yes—
18C2×C3×#100C2212yes—
19C3×#407C2211yescharacteristic
20C3×#407C2211yescharacteristic
21#5218—24no—
22#5217—24no—
23#5215C2212yes—
24#5215C2212yes—
25#5207C2212yes—
26#5139C2212yescomplemented
27#5132C2212yescomplemented
28#5125C2211yescharacteristic, complemented
29#5124C2212yescomplemented
30#4749—24no—
31#4748—24no—
32C3⋊#338C2211yescharacteristic, complemented
33#4559C2212yescomplemented
34#4557C2211yescharacteristic, complemented
35#4556C2212yescomplemented
36#3068—32no—
37#3003—32no—
38#2091—31no—
39#2081—32no—
40#2048—32no—
41#2010—31no—
42#1903—32no—
43#1891—32no—
44#1783D311yescharacteristic, complemented
45C25×C3—21no—
46C22×C3×D4—24no—
47C22×C3×D4—21no—
48C22×C3×D4—41no—
49C22×C3×D4—41no—
50C22×C3×D4—42no—
51C22×C3×D4C2312yes—
52C23×C3×C4C2311yescharacteristic
53C2×#235C2312yes—
54C2×C3×M16C2311yescharacteristic
55C2×C3×M16C2311yescharacteristic
56C2×C3×M16C2312yes—
57#953—22no—
58C2×#214—22no—
59C2×#214D414yescomplemented
60#932—24no—
61#932—42no—
62#932—42no—
63#932—42no—
64C2×#213—22no—
65C2×#213D414yescomplemented
66#930—22no—
67#929—22no—
68C4×#67—42no—
69#926C2312yes—
70#924—22no—
71C2×#208C2312yes—
72C2×#208C2312yes—
73#909—22no—
74#908—22no—
75#907—24no—
76D3×M16—42no—
77#904—41no—
78#901—42no—
79C3×#100—24no—
80C3×#100—42no—
81C3×C22⋊C8—41no—
82C3×C22⋊C8—41no—
83#832—24no—
84#832—24no—
85#830—24no—
86#825—42no—
87#824—22no—
88#823—22no—
89#822—24no—
90#819—41no—
91C23×D4C2×D311yescharacteristic
92#575—31no—
93C2×AGL(1,ℤ/8)—32no—
94C2×AGL(1,ℤ/8)—32no—
95#568—32no—
96#525—32no—
97#472—64no—
98#471—64no—
99#450—64no—
100#447—64no—
101#430—32no—
102#430—31no—
103#421—32no—
104#418—32no—
105#418—32no—
106#413—32no—
107C2×#100C2×D312yes—
108C2×#100C2×D312yes—
109#407—32no—
110#407C2×D311yescharacteristic
111#407C2×D311yescharacteristic
112#344—32no—
113#344—32no—
114#338—31no—
115H(3,ℤ/4)—31no—
116C24×C3—22no—
117C24×C3—44no—
118C24×C3—42no—
119C24×C3—84no—
120C24×C3#4111yescharacteristic
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—22no—
126C2×C3×D4—44no—
127C2×C3×D4—42no—
128C2×C3×D4—44no—
129C2×C3×D4—48no—
130C2×C3×D4—44no—
131C2×C3×D4C2×D414yes—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—22no—
136C22×C3×C4#4111yescharacteristic
137C22×C3×C4C2411yescharacteristic
138C2×#67—22no—
139C2×#67—42no—
140#235—22no—
141#235—22no—
142#235—22no—
143#235—22no—
144C2×D12C2×D412yes—
145C2×C4×D3—22no—
146C2×Dic6C2×D412yes—
147C3×M16—22no—
148C3×M16—22no—
149C3×M16—22no—
150C3×M16—22no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C2×C3×C8—21no—
154#217—22no—
155#217—42no—
156#214—44no—
157#214—44no—
158#213—44no—
159#213—44no—
160(C2×D3)⋊C4—42no—
161C4⋊Dic3—22no—
162Dic3⋊C4—42no—
163C4×Dic3—22no—
164#208—22no—
165#208—22no—
166#208—22no—
167#208—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8C2×D414yes—
170Dic12—41no—
171D24—41no—
172#204—42no—
173ΓL(1,𝔽25)—44no—
174D3×C8—44no—
175C25—21no—
176C2×#41—32no—
177C22×D4—24no—
178C22×D4—21no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4—42no—
182C22×D4C22×D312yes—
183C23×C4C22×D311yescharacteristic
184#137—62no—
185AGL(1,ℤ/8)—62no—
186AGL(1,ℤ/8)—64no—
187AGL(1,ℤ/8)—122no—
188AGL(1,ℤ/8)—122no—
189AGL(1,ℤ/8)—122no—
190#135—121no—
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—122no—
200C2×M16C22×D312yes—
201C2×M16C22×D311yescharacteristic
202C2×M16C22×D311yescharacteristic
203#123—62no—
204C22⋊Q8—62no—
205#121—62no—
206#120—62no—
207C4×D4—122no—
208#117—32no—
209#108—64no—
210#108—121no—
211#104—122no—
212#102—64no—
213#102—64no—
214#101—62no—
215#100—24no—
216#100—42no—
217#100—62no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—64no—
221C23×C3—21no—
222C23×C3—22no—
223C23×C3—24no—
224C23×C3—42no—
225C23×C3—48no—
226C23×C3—42no—
227C23×C3—42no—
228C23×C3—84no—
229C23×C3—88no—
230C23×C3—84no—
231C23×C3—88no—
232C23×C3—84no—
233C23×C3C2×#4111yescharacteristic
234C22×D3—22no—
235C3×D4—24no—
236C3×D4—24no—
237C3×D4—48no—
238C3×D4—42no—
239C3×D4—42no—
240C2×C3×C4—21no—
241C2×C3×C4—22no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4C22×D412yes—
246C2×C3×C4H(5,𝔽2)11yescharacteristic
247C2×C3×C4H(5,𝔽2)12yes—
248C2×C3×C4H(5,𝔽2)11yescharacteristic
249#67—42no—
250#67—42no—
251#67—84no—
252C2×Dic3—22no—
253C2×Dic3—42no—
254D12—22no—
255D12—22no—
256C4×D3—42no—
257C4×D3—42no—
258Dic6—22no—
259Dic6—22no—
260C3×C8—42no—
261C3×C8—41no—
262C3×C8—41no—
263C3⋊C8—24no—
264C3⋊C8—42no—
265C24—22no—
266C24—44no—
267C24—42no—
268C24—84no—
269C24#23511yescharacteristic
270#41—62no—
271#41—62no—
272#41—62no—
273#41—62no—
274C2×Q8—32no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—32no—
281C2×D4—44no—
282C2×D4—42no—
283C2×D4—44no—
284C2×D4—48no—
285C2×D4—44no—
286C2×D4—62no—
287C2×D4—122no—
288C2×D4D3×D414yes—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—62no—
294C22×C4#23511yescharacteristic
295C22×C4C23×D311yescharacteristic
296Q16—121no—
297SD16—122no—
298SD16—124no—
299SD16—124no—
300D8—121no—
301D8—124no—
302D8—124no—
303M16—22no—
304M16—22no—
305M16—22no—
306M16—22no—
307M16—62no—
308M16—62no—
309M16—62no—
310M16—62no—
311M16—124no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—34no—
316C2×C8—62no—
317C2×C8—124no—
318C4⋊C4—62no—
319C4⋊C4—122no—
320C22⋊C4—62no—
321C22⋊C4—122no—
322C22⋊C4—122no—
323C42—62no—
324C22×C3—24no—
325C22×C3—22no—
326C22×C3—41no—
327C22×C3—42no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—44no—
331C22×C3—84no—
332C22×C3—84no—
333C22×C3—84no—
334C22×C3—816no—
335C22×C3#53811yescharacteristic
336C22×C3#54612yes—
337C2×D3—42no—
338C2×D3—82no—
339C3×C4—41no—
340C3×C4#53412yes—
341C3×C4D4211yescharacteristic
342C3×C4#56011yescharacteristic
343Dic3—42no—
344Dic3—82no—
345C23—22no—
346C23—21no—
347C23—24no—
348C23—42no—
349C23—48no—
350C23—42no—
351C23—42no—
352C23—62no—
353C23—88no—
354C23—84no—
355C23—84no—
356C23—88no—
357C23—84no—
358C23C2×#23511yescharacteristic
359Q8—62no—
360Q8—62no—
361D4—24no—
362D4—24no—
363D4—48no—
364D4—42no—
365D4—42no—
366D4—62no—
367D4—62no—
368D4—122no—
369D4—122no—
370D4—244no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—62no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4C2×D3×D412yes—
381C2×C4#100411yescharacteristic
382C2×C4#100412yes—
383C2×C4#100411yescharacteristic
384C8—42no—
385C8—41no—
386C8—41no—
387C8—64no—
388C8—122no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—42no—
392C2×C3—84no—
393C2×C3#266811yescharacteristic
394D3—82no—
395C22—24no—
396C22—22no—
397C22—41no—
398C22—42no—
399C22—41no—
400C22—41no—
401C22—44no—
402C22—84no—
403C22—84no—
404C22—84no—
405C22—816no—
406C22—122no—
407C22—242no—
408C22#549012yes—
409C22#549411yescharacteristic
410C4—41no—
411C4—122no—
412C4—242no—
413C4#558912yes—
414C4#559011yescharacteristic
415C4#559511yescharacteristic
416C3#3043411yescharacteristic, complemented, Sylow
417C2—22no—
418C2—21no—
419C2—42no—
420C2—84no—
421C2—242no—
422C2#7385711yescharacteristic
subgroupG/H
centerC2#73857
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1783C2
socleC2×C3#2668
hypercenter#1783D3
hypocenterC3#30434
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1783

Complements

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

NH ≅ G/Ntype
1. #84038C2semidirect
2. #78703C2semidirect
3. C3×#1783C2semidirect
4. #70087C2semidirect
5. #70078C2semidirect
6. #70076C2semidirect
7. #69993C2semidirect
8. #69992C2semidirect
9. #69987C2semidirect
15. #5748C22semidirect
16. #5745C22semidirect
26. #5139C22semidirect
27. #5132C22semidirect
28. #5125C22semidirect
29. #5124C22semidirect
32. C3⋊#338C22semidirect
33. #4559C22semidirect
34. #4557C22semidirect
35. #4556C22semidirect
44. #1783D3semidirect
59. C2×#214D4semidirect
65. C2×#213D4semidirect
416. C3#30434semidirect

Maximal subgroups

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

subgroupindex
#840382
#787032
C3×#17832
#700872
#700782
#700762
#699932
#699922
#699872
#304343