Finite Group Databasebeta

← #11865140

Subgroups — #11865140

Order 1152.

2208 conjugacy classes of subgroups.

The subgroup lattice has 405 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#585C211yesmaximal, characteristic, complemented
2#10596301C211yesmaximal, characteristic, complemented
3C2×#64158C212yesmaximal, complemented
4#10592944C214yesmaximal, complemented
5#10592940C214yesmaximal, complemented
6#10592936C211yesmaximal, characteristic, complemented
7#10592935C212yesmaximal, complemented
8#84122—34nomaximal
9C32×2−1+4—22no—
10C32×2−1+4C2212yescomplemented
11C32×H(5,𝔽2)—22no—
12C32×H(5,𝔽2)—22no—
13C32×H(5,𝔽2)C2212yescomplemented
14C2×C32×#41—24no—
15C2×C32×#41—22no—
16C2×C32×#41C2212yes—
17C2×C32×#41C2211yescharacteristic
18C22×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
19#64158—24no—
20#64158C2218yescomplemented
21#64156C2212yescomplemented
22C2×#3755—22no—
23#64149C2212yescomplemented
24#64146—22no—
25C2×#3753—22no—
26#64139—22no—
27#64135C2211yescharacteristic, complemented
28C2×#3748C2212yescomplemented
29#63664C2218yescomplemented
30#63663C2212yescomplemented
31#63660C2212yescomplemented
32C3×#585D314yescomplemented
33#5830—34no—
34C2×#949—38no—
35#5245—316no—
36#5241—316no—
37#5237—34no—
38#5236—38no—
39C23×C3⋊D3—21no—
40C32×#41—24no—
41C32×#41—28no—
42C32×#41—24no—
43C32×#41—24no—
44C32×#41—24no—
45C32×#41—22no—
46C32×#41—22no—
47C32×#41—24no—
48C32×#41—22no—
49C32×#41—22no—
50C32×#41D414yescomplemented
51C32×#41C2314yes—
52C2×C32×Q8—22no—
53C2×C32×Q8—24no—
54C2×C32×Q8D412yescomplemented
55C2×C32×Q8C2311yescharacteristic
56C2×C32×D4—28no—
57C2×C32×D4—22no—
58C2×C32×D4—24no—
59C2×C32×D4—22no—
60C2×C32×D4—24no—
61C2×C32×D4D412yes—
62C2×C32×D4D412yescomplemented
63C2×C32×D4C2311yescharacteristic
64C22×C32×C4—24no—
65C22×C32×C4—22no—
66C22×C32×C4D412yes—
67C22×C32×C4C2311yescharacteristic
68C2×#638—24no—
69C2×C4⋊(C3⋊D3)—22no—
70C2×C4⋊(C3⋊D3)—22no—
71C2×C4⋊(C3⋊D3)—21no—
72C2×C4⋊(C3⋊D3)C2312yes—
73#3757—21no—
74#3755—28no—
75#3753—28no—
76#3752—24no—
77C4⋊(C3⋊Dic3)—21no—
78C4×C3⋊Dic3C2312yes—
79#3748—22no—
80#3748C2314yes—
81C2×C3⋊(C3⋊C8)—22no—
82#3021—91noSylow
83C3×2−1+4—28no—
84C3×2−1+4C2×D318yescomplemented
85C3×H(5,𝔽2)—28no—
86C3×H(5,𝔽2)—28no—
87C3×H(5,𝔽2)C2×D318yescomplemented
88C2×C3×#41—216no—
89C2×C3×#41—28no—
90C2×C3×#41C2×D318yes—
91C2×C3×#41C2×D314yes—
92C22×D12—34no—
93#949—332no—
94#949—616no—
95#947—38no—
96C2×#215—68no—
97#940—38no—
98#937—68no—
99C2×#213—68no—
100#930—68no—
101#926—34no—
102C2×#208—38no—
103#837—332no—
104#836—38no—
105#833—38no—
106C23×C32—24no—
107C23×C32—22no—
108C23×C32C2×D412yes—
109C23×C32C2×D411yescharacteristic
110C22×C3⋊D3—22no—
111C22×C3⋊D3—42no—
112C22×C3⋊D3—82no—
113C32×Q8—24no—
114C32×Q8—22no—
115C32×Q8—22no—
116C32×Q8C2×D414yes—
117C32×D4—28no—
118C32×D4—24no—
119C32×D4—24no—
120C32×D4—24no—
121C32×D4—24no—
122C32×D4—22no—
123C32×D4—22no—
124C32×D4C2×D414yes—
125C2×C32×C4—24no—
126C2×C32×C4—28no—
127C2×C32×C4—22no—
128C2×C32×C4—24no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—24no—
132C2×C32×C4C2×D412yes—
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D412yes—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2411yescharacteristic
137#638—84no—
138C2×C3⋊Dic3—22no—
139C4⋊(C3⋊D3)—24no—
140C4⋊(C3⋊D3)—41no—
141C4⋊(C3⋊D3)—41no—
142C3⋊(C3⋊C8)—24no—
143#585C3⋊D311yescharacteristic, complemented
144C2×AGL(1,ℤ/8)—92no—
145#535—91no—
146#454—94no—
147#453—94no—
148#421—92no—
149#413—91no—
150C23×D3—64no—
151C3×#41—216no—
152C3×#41—232no—
153C3×#41—216no—
154C3×#41—216no—
155C3×#41—216no—
156C3×#41—28no—
157C3×#41—28no—
158C3×#41—216no—
159C3×#41—28no—
160C3×#41—28no—
161C3×#41#67116yescomplemented
162C3×#41C22×D3116yes—
163C2×C3×Q8—28no—
164C2×C3×Q8—216no—
165C2×C3×Q8#6718yescomplemented
166C2×C3×Q8C22×D314yes—
167C2×C3×D4—232no—
168C2×C3×D4—28no—
169C2×C3×D4—216no—
170C2×C3×D4—28no—
171C2×C3×D4—216no—
172C2×C3×D4#6718yes—
173C2×C3×D4#6718yescomplemented
174C2×C3×D4C22×D314yes—
175C22×C3×C4—216no—
176C22×C3×C4—28no—
177C22×C3×C4#6718yes—
178C22×C3×C4C22×D314yes—
179C2×#67—616no—
180C2×D12—38no—
181C2×D12—68no—
182C2×D12—68no—
183C2×D12—64no—
184#217—64no—
185#215—632no—
186#213—632no—
187(C2×D3)⋊C4—616no—
188C4⋊Dic3—64no—
189C4×Dic3—38no—
190#208—316no—
191#208—68no—
192C2×C3⋊C8—68no—
193C22×C32—22no—
194C22×C32—24no—
195C22×C32—41no—
196C22×C32—42no—
197C22×C32—82no—
198C22×C32—84no—
199C22×C32#12012yes—
200C22×C32C22×D411yescharacteristic
201C2×C3⋊D3—44no—
202C2×C3⋊D3—42no—
203C2×C3⋊D3—84no—
204C32×C4—24no—
205C32×C4—22no—
206C32×C4#12011yescharacteristic
207C32×C4#12011yescharacteristic
208C32×C4C22×D412yes—
209C3⋊Dic3—42no—
2102−1+4—22no—
2112−1+4C2×C3⋊D312yescomplemented
212H(5,𝔽2)—22no—
213H(5,𝔽2)—22no—
214H(5,𝔽2)C2×C3⋊D312yescomplemented
215C2×#41—24no—
216C2×#41—22no—
217C2×#41C2×C3⋊D312yes—
218C2×#41C2×C3⋊D311yescharacteristic
219C22×D4—91no—
220AGL(1,ℤ/8)—98no—
221AGL(1,ℤ/8)—184no—
222C2×SD16—182no—
223C2×D8—182no—
224C2×M16—92no—
225#127—92no—
226#124—92no—
227#121—182no—
228#120—182no—
229#117—91no—
230#104—98no—
231#101—92no—
232#100—92no—
233C23×C3—216no—
234C23×C3—28no—
235C23×C3C2×#6718yes—
236C23×C3C2×#6714yes—
237C22×D3—68no—
238C22×D3—128no—
239C22×D3—248no—
240C3×Q8—216no—
241C3×Q8—28no—
242C3×Q8—28no—
243C3×Q8C2×#67116yes—
244C3×D4—232no—
245C3×D4—216no—
246C3×D4—216no—
247C3×D4—216no—
248C3×D4—216no—
249C3×D4—28no—
250C3×D4—28no—
251C3×D4C2×#67116yes—
252C2×C3×C4—216no—
253C2×C3×C4—232no—
254C2×C3×C4—28no—
255C2×C3×C4—216no—
256C2×C3×C4—28no—
257C2×C3×C4—28no—
258C2×C3×C4—216no—
259C2×C3×C4C2×#6718yes—
260C2×C3×C4C2×#6714yes—
261C2×C3×C4C2×#6718yes—
262C2×C3×C4C2×#6718yes—
263C2×C3×C4C23×D314yes—
264#67—2416no—
265C2×Dic3—68no—
266D12—616no—
267D12—124no—
268D12—124no—
269C3⋊C8—616no—
270C2×C32—21no—
271C2×C32—22no—
272C2×C32—42no—
273C2×C32—44no—
274C2×C32C2×#12011yescharacteristic
275C3⋊D3—82no—
276C24—181no—
277#41—24no—
278#41—28no—
279#41—24no—
280#41—24no—
281#41—24no—
282#41—22no—
283#41—22no—
284#41—24no—
285#41—22no—
286#41—22no—
287#41#63814yescomplemented
288#41C22×C3⋊D314yes—
289C2×Q8—22no—
290C2×Q8—24no—
291C2×Q8#63812yescomplemented
292C2×Q8C22×C3⋊D311yescharacteristic
293C2×D4—28no—
294C2×D4—22no—
295C2×D4—24no—
296C2×D4—22no—
297C2×D4—24no—
298C2×D4—92no—
299C2×D4—184no—
300C2×D4—182no—
301C2×D4—182no—
302C2×D4—181no—
303C2×D4#63812yes—
304C2×D4#63812yescomplemented
305C2×D4C22×C3⋊D311yescharacteristic
306C22×C4—24no—
307C22×C4—22no—
308C22×C4#63812yes—
309C22×C4C22×C3⋊D311yescharacteristic
310SD16—188no—
311D8—188no—
312M16—94no—
313M16—182no—
314C2×C8—182no—
315C4⋊C4—181no—
316C22⋊C4—181no—
317C22⋊C4—184no—
318C42—92no—
319C22×C3—28no—
320C22×C3—216no—
321C22×C3—44no—
322C22×C3—48no—
323C22×C3—88no—
324C22×C3—816no—
325C22×C3#95318yes—
326C22×C3C22×#6714yes—
327C2×D3—1216no—
328C2×D3—128no—
329C2×D3—2416no—
330C3×C4—216no—
331C3×C4—28no—
332C3×C4#95314yes—
333C3×C4#95314yes—
334C3×C4C22×#6718yes—
335Dic3—128no—
336C32#302111yescharacteristic, complemented, Sylow
337C23—24no—
338C23—22no—
339C23—182no—
340C23—362no—
341C23—722no—
342C23C2×#63812yes—
343C23C2×#63811yescharacteristic
344Q8—24no—
345Q8—22no—
346Q8—22no—
347Q8C2×#63814yes—
348D4—28no—
349D4—24no—
350D4—24no—
351D4—24no—
352D4—24no—
353D4—22no—
354D4—22no—
355D4—184no—
356D4—361no—
357D4—361no—
358D4—724no—
359D4C2×#63814yes—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—182no—
368C2×C4C2×#63812yes—
369C2×C4C2×#63812yes—
370C2×C4C2×#63812yes—
371C2×C4C2×#63811yescharacteristic
372C2×C4C23×C3⋊D311yescharacteristic
373C8—184no—
374C2×C3—24no—
375C2×C3—28no—
376C2×C3—48no—
377C2×C3—416no—
378C2×C3C2×#95314yes—
379D3—248no—
380C22—22no—
381C22—24no—
382C22—41no—
383C22—42no—
384C22—82no—
385C22—84no—
386C22—364no—
387C22—362no—
388C22—724no—
389C22#6416212yes—
390C22C22×#63811yescharacteristic
391C4—24no—
392C4—22no—
393C4—362no—
394C4#6416211yescharacteristic
395C4#6416211yescharacteristic
396C4C22×#63812yes—
397C3#8412214yescomplemented
398C2—21no—
399C2—22no—
400C2—42no—
401C2—44no—
402C2—722no—
403C2C2×#6416211yescharacteristic
subgroupG/H
centerC2C2×#64162
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC32×#585C2
socleC2×C32C2×#120
hypercenter#585C3⋊D3
hypocenterC32#3021
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C32×#585C2semidirect
2. #10596301C2semidirect
3. C2×#64158C2semidirect
4. #10592944C2semidirect
5. #10592940C2semidirect
6. #10592936C2semidirect
7. #10592935C2semidirect
10. C32×2−1+4C22semidirect
13. C32×H(5,𝔽2)C22semidirect
18. C22×C4⋊(C3⋊D3)C22semidirect
20. #64158C22semidirect
21. #64156C22semidirect
23. #64149C22semidirect
27. #64135C22semidirect
28. C2×#3748C22semidirect
29. #63664C22semidirect
30. #63663C22semidirect
31. #63660C22semidirect
32. C3×#585D3semidirect
50. C32×#41D4semidirect
54. C2×C32×Q8D4semidirect
62. C2×C32×D4D4semidirect
84. C3×2−1+4C2×D3semidirect
87. C3×H(5,𝔽2)C2×D3semidirect
143. #585C3⋊D3semidirect
161. C3×#41#67semidirect
165. C2×C3×Q8#67semidirect
173. C2×C3×D4#67semidirect
211. 2−1+4C2×C3⋊D3semidirect
214. H(5,𝔽2)C2×C3⋊D3semidirect
287. #41#638semidirect
291. C2×Q8#638semidirect
304. C2×D4#638semidirect
336. C32#3021semidirect
397. C3#84122semidirect

Maximal subgroups

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

subgroupindex
C32×#5852
#105963012
C2×#641582
#105929442
#105929402
#105929362
#105929352
#841223