Finite Group Databasebeta

← #11886917

Subgroups — #11886917

Order 1152.

1005 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1005 conjugacy classes of subgroups — too many to draw. Only its 178 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×#5769C212yesmaximal, complemented
2#10595444C212yesmaximal, complemented
3#10592895C211yesmaximal, characteristic, complemented
4#10591678C212yesmaximal, complemented
5#10591665C214yesmaximal, complemented
6#10590926C212yesmaximal, complemented
7#10590921C211yesmaximal, characteristic, complemented
8#10590886C211yesmaximal, characteristic, complemented
9#84010—32nomaximal
10C3×#1009C2214yescomplemented
11C32×#135C2211yescharacteristic
12#64157C2212yescomplemented
13C3×#949C2214yescomplemented
14C3×#919—22no—
15C3×#914—24no—
16C3×D3×D8—22no—
17C2×C3×D24C2212yescomplemented
18C3×#901C2212yescomplemented
19#63914C2212yescomplemented
20#63823—24no—
21#63823C2214yescomplemented
22#63822—22no—
23#63822C2211yescharacteristic
24C2×#3713—22no—
25C2×#3713C2212yescomplemented
26#63664C2212yescomplemented
27#63646C2211yescharacteristic, complemented
28C8×C3⋊Dic3C2211yescharacteristic, complemented
29#63578C2211yescharacteristic, complemented
30#63568C2212yescomplemented
31#63558C2214yescomplemented
32C3×#576—32no—
33#5835—34no—
34#5769D312yescomplemented
35#5241—34no—
36#5199—34no—
37#5198—34no—
38#5196—32no—
39#5196—61no—
40#5160—32no—
41#5151—32no—
42#5144—32no—
43#5142—32no—
44C32×#41C2312yes—
45C3×#239—24no—
46C3×D3×D4—24no—
47C3×D3×D4—44no—
48C3×#235—24no—
49C3×#235C2312yes—
50C2×C3×D12—24no—
51C2×C3×D12C2314yes—
52C2×#625—42no—
53C32×Q16D411yescharacteristic, complemented
54C32×SD16D412yescomplemented
55C32×D8D411yescharacteristic, complemented
56C2×C32×C8C2311yescharacteristic
57C4×C3⋊Dic3C2311yescharacteristic
58#3748—22no—
59#3748C2312yes—
60C2×C3⋊(C3⋊C8)—22no—
61C2×C3⋊(C3⋊C8)C2311yescharacteristic
62C3×#215—24no—
63C3×#213—24no—
64C3×#208C2312yes—
65C3×D24—22no—
66C3×D24—22no—
67C3×ΓL(1,𝔽25)—22no—
68C3×D3×C8—22no—
69#3718D411yescharacteristic, complemented
70#3715—24no—
71#3715D412yescomplemented
72#3713—24no—
73#3713—22no—
74#3713—22no—
75#3713D411yescharacteristic, complemented
76#3292—91noSylow
77C3×H(5,𝔽2)—34no—
78#1009C2×D314yescomplemented
79C3×AGL(1,ℤ/8)—34no—
80C3×AGL(1,ℤ/8)—64no—
81C3×#135—21no—
82C3×#135—62no—
83C3×#135C2×D312yes—
84C2×C3×D8—32no—
85C2×C3×D8—62no—
86C3×#131—32no—
87#950—32no—
88#950—64no—
89#949—34no—
90#949—64no—
91#949C2×D314yescomplemented
92#948—34no—
93#948—62no—
94#940—34no—
95#932—34no—
96#932—64no—
97C2×#213—34no—
98C2×#213—64no—
99#919—22no—
100#914—24no—
101D3×D8—22no—
102C2×D24C2×D312yescomplemented
103#901C2×D312yescomplemented
104#837—34no—
105#837—32no—
106#837—62no—
107#833—34no—
108#825—32no—
109#823—34no—
110#822—32no—
111#819—32no—
112#819—61no—
113C8×Dic3—32no—
114C8×Dic3—61no—
115C22×C3×D3—24no—
116C22×C3×D3—44no—
117C32×Q8C2×D412yes—
118C32×D4—22no—
119C32×D4C2×D412yes—
120C2×C32×C4—22no—
121C2×C32×C4C2411yescharacteristic
122C2×C3⋊Dic3—21no—
123C3×#67—22no—
124C3×#67—44no—
125C3×D12—24no—
126C3×D12—24no—
127C3×D12—44no—
128C3×D12C2×D412yes—
129C3×C4×D3—22no—
130C3×C4×D3—44no—
131C3×Dic6C2×D412yes—
132#625—84no—
133C32×C8C2×D411yescharacteristic
134C32×C8C2×D411yescharacteristic
135C3⋊(C3⋊C8)—22no—
136C3⋊(C3⋊C8)C2×D411yescharacteristic
137C3⋊(C3⋊C8)C2×D411yescharacteristic
138C3×C3⋊C8—22no—
139#576—32no—
140#576—92no—
141#493—91no—
142#472—92no—
143#472—92no—
144#453—94no—
145#443—91no—
146#443—91no—
147C3×#41—22no—
148C3×#41—32no—
149C3×#41—64no—
150C3×#41—64no—
151C3×#41C22×D314yes—
152C2×C3×D4—34no—
153C2×C3×D4—64no—
154C2×C3×D4—64no—
155C2×C3×D4—124no—
156C2×#67—124no—
157#239—24no—
158D3×D4—24no—
159D3×D4—44no—
160#235—24no—
161#235C22×D312yes—
162C2×D12—24no—
163C2×D12C22×D314yes—
164C3×Q16—21no—
165C3×Q16#6712yescomplemented
166C3×SD16—22no—
167C3×SD16—64no—
168C3×SD16#6714yescomplemented
169C3×D8—21no—
170C3×D8—64no—
171C3×D8—62no—
172C3×D8—62no—
173C3×D8#6712yescomplemented
174C3×M16—32no—
175C3×M16—62no—
176C2×C3×C8—21no—
177C2×C3×C8—62no—
178C2×C3×C8C22×D312yes—
179#216—32no—
180#215—24no—
181#215—32no—
182#215—64no—
183#214—32no—
184#214—64no—
185#213—24no—
186#213—32no—
187#213—64no—
188#213—64no—
189#213—64no—
190#213—64no—
191C4×Dic3—32no—
192C4×Dic3—61no—
193#208—34no—
194#208—62no—
195#208—64no—
196#208—122no—
197#208C22×D312yes—
198C2×C3⋊C8—32no—
199C2×C3⋊C8—61no—
200C2×C3⋊C8—64no—
201C2×C3⋊C8—122no—
202D24—22no—
203D24—22no—
204ΓL(1,𝔽25)—22no—
205D3×C8—22no—
206C22×C32—22no—
207C22×C32H(5,𝔽2)11yescharacteristic
208C2×C3×D3—22no—
209C2×C3×D3—44no—
210C2×C3×D3—84no—
211C2×C3×D3—164no—
212C32×C4—22no—
213C32×C4C22×D411yescharacteristic
214C32×C4C22×D411yescharacteristic
215C3⋊Dic3—22no—
216C3×Dic3—22no—
217H(5,𝔽2)—34no—
218AGL(1,ℤ/8)—34no—
219AGL(1,ℤ/8)—64no—
220AGL(1,ℤ/8)—94no—
221AGL(1,ℤ/8)—184no—
222#135—62no—
223#135—91no—
224#135—182no—
225#135D3211yescharacteristic
226C2×D8—32no—
227C2×D8—62no—
228C2×D8—92no—
229C2×D8—182no—
230#131—32no—
231#131—92no—
232#127—92no—
233#108—91no—
234#108—91no—
235#104—92no—
236#104—92no—
237#100—94no—
238C4×C8—91no—
239C23×C3—64no—
240C23×C3—124no—
241C22×D3—24no—
242C22×D3—44no—
243C3×Q8—22no—
244C3×Q8—32no—
245C3×Q8C2×#6714yes—
246C3×D4—24no—
247C3×D4—22no—
248C3×D4—32no—
249C3×D4—42no—
250C3×D4—62no—
251C3×D4—64no—
252C3×D4—64no—
253C3×D4—124no—
254C3×D4—124no—
255C3×D4C2×#6714yes—
256C2×C3×C4—24no—
257C2×C3×C4—21no—
258C2×C3×C4—42no—
259C2×C3×C4—62no—
260C2×C3×C4—124no—
261C2×C3×C4C23×D312yes—
262#67—22no—
263#67—44no—
264#67—248no—
265C2×Dic3—62no—
266C2×Dic3—121no—
267D12—24no—
268D12—24no—
269D12—44no—
270D12D3×D412yes—
271C4×D3—22no—
272C4×D3—44no—
273Dic6D3×D412yes—
274C3×C8—21no—
275C3×C8—21no—
276C3×C8—62no—
277C3×C8C2×#6712yes—
278C3×C8C2×#6712yes—
279C3⋊C8—22no—
280C3⋊C8—32no—
281C3⋊C8—32no—
282C3⋊C8—61no—
283C3⋊C8—61no—
284C3⋊C8—64no—
285C3⋊C8—122no—
286C2×C32—21no—
287C2×C32—42no—
288C2×C32D4211yescharacteristic
289C3×D3—42no—
290C3×D3—84no—
291#41—32no—
292#41—64no—
293#41—64no—
294#41C2×D3212yes—
295C2×D4—34no—
296C2×D4—64no—
297C2×D4—64no—
298C2×D4—124no—
299C2×D4—362no—
300Q16—91no—
301Q16#62511yescharacteristic, complemented
302SD16—64no—
303SD16—92no—
304SD16—184no—
305SD16#62512yescomplemented
306D8—64no—
307D8—62no—
308D8—62no—
309D8—91no—
310D8—182no—
311D8—182no—
312D8—184no—
313D8#62511yescharacteristic, complemented
314M16—32no—
315M16—62no—
316M16—92no—
317M16—182no—
318C2×C8—62no—
319C2×C8—91no—
320C2×C8—182no—
321C2×C8C2×D3211yescharacteristic
322C42—91no—
323C22×C3—24no—
324C22×C3—21no—
325C22×C3—42no—
326C22×C3—62no—
327C22×C3—124no—
328C22×C3—244no—
329C22×C3—484no—
330C22×C3#100412yes—
331C2×D3—22no—
332C2×D3—44no—
333C2×D3—84no—
334C2×D3—164no—
335C3×C4—24no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—42no—
339C3×C4—62no—
340C3×C4C2×D3×D412yes—
341C3×C4C22×#6712yes—
342Dic3—22no—
343Dic3—64no—
344Dic3—122no—
345C32#329211yescharacteristic, complemented, Sylow
346C23—64no—
347C23—124no—
348Q8—32no—
349Q8C2×#62512yes—
350D4—22no—
351D4—32no—
352D4—62no—
353D4—64no—
354D4—64no—
355D4—124no—
356D4—124no—
357D4—724no—
358D4C2×#62512yes—
359C2×C4—22no—
360C2×C4—62no—
361C2×C4—124no—
362C2×C4—181no—
363C2×C4C22×D3211yescharacteristic
364C8—62no—
365C8—91no—
366C8—91no—
367C8—182no—
368C8C2×#62511yescharacteristic
369C8C2×#62511yescharacteristic
370C2×C3—22no—
371C2×C3—21no—
372C2×C3—41no—
373C2×C3—44no—
374C2×C3—82no—
375C2×C3—122no—
376C2×C3—244no—
377C2×C3D4×#6712yes—
378D3—42no—
379D3—84no—
380C22—22no—
381C22—62no—
382C22—124no—
383C22—244no—
384C22—484no—
385C22#6431311yescharacteristic
386C4—22no—
387C4—62no—
388C4—182no—
389C4D32×D411yescharacteristic
390C4C22×#62511yescharacteristic
391C3—21no—
392C3#8401012yescomplemented
393C2—21no—
394C2—42no—
395C2—122no—
396C2—244no—
397C2D4×#62511yescharacteristic
subgroupG/H
centerC2D4×#625
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×#135C22
socleC2×C32D42
hypercenter#135D32
hypocenterC32#3292
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C2×C4 ◁ #135

Complements

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

NH ≅ G/Ntype
1. C3×#5769C2semidirect
2. #10595444C2semidirect
3. #10592895C2semidirect
4. #10591678C2semidirect
5. #10591665C2semidirect
6. #10590926C2semidirect
7. #10590921C2semidirect
8. #10590886C2semidirect
10. C3×#1009C22semidirect
12. #64157C22semidirect
13. C3×#949C22semidirect
17. C2×C3×D24C22semidirect
18. C3×#901C22semidirect
19. #63914C22semidirect
21. #63823C22semidirect
25. C2×#3713C22semidirect
26. #63664C22semidirect
27. #63646C22semidirect
28. C8×C3⋊Dic3C22semidirect
29. #63578C22semidirect
30. #63568C22semidirect
31. #63558C22semidirect
34. #5769D3semidirect
53. C32×Q16D4semidirect
54. C32×SD16D4semidirect
55. C32×D8D4semidirect
69. #3718D4semidirect
71. #3715D4semidirect
75. #3713D4semidirect
78. #1009C2×D3semidirect
91. #949C2×D3semidirect
102. C2×D24C2×D3semidirect
103. #901C2×D3semidirect
165. C3×Q16#67semidirect
168. C3×SD16#67semidirect
173. C3×D8#67semidirect
301. Q16#625semidirect
305. SD16#625semidirect
313. D8#625semidirect
345. C32#3292semidirect
392. C3#84010semidirect

Maximal subgroups

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

subgroupindex
C3×#57692
#105954442
#105928952
#105916782
#105916652
#105909262
#105909212
#105908862
#840103