Finite Group Databasebeta

← #11788472

Subgroups — #11788472

Order 1152.

1038 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×AGL(1,ℤ/8)C211yesmaximal, characteristic
2#10596269C211yesmaximal, characteristic, complemented
3#10592939C211yesmaximal, characteristic, complemented
4#10592935C211yesmaximal, characteristic, complemented
5#10592914C211yesmaximal, characteristic, complemented
6#10589423C211yesmaximal, characteristic, complemented
7#10589422C211yesmaximal, characteristic, complemented
8#72089—34nomaximal
9C2×C32×#41C2211yescharacteristic
10C22×C32×D4C2211yescharacteristic
11C32×AGL(1,ℤ/8)—22no—
12C32×AGL(1,ℤ/8)—22no—
13C2×C32×SD16C412yescomplemented
14C2×C32×D8C412yescomplemented
15C2×C32×M16C2211yescharacteristic
16C2×#3757—22no—
17D4×C3⋊Dic3—22no—
18D4×C3⋊Dic3—22no—
19#64135C2211yescharacteristic
20#64135C2211yescharacteristic
21#64135C2211yescharacteristic
22C2×#3748C2211yescharacteristic
23#63664—22no—
24#63661—22no—
25#63659—22no—
26C2×C3×AGL(1,ℤ/8)D314yes—
27#5798—34no—
28#5240—34no—
29#5236—34no—
30#5215—34no—
31#4557—34no—
32#4556—34no—
33C24×C32—21no—
34C32×#41—21no—
35C32×#41—21no—
36C32×#41—21no—
37C32×#41—21no—
38C2×C32×Q8C2×C411yescharacteristic, complemented
39C2×C32×D4—21no—
40C2×C32×D4—21no—
41C2×C32×D4—21no—
42C2×C32×D4—21no—
43C2×C32×D4C2×C412yes—
44C2×C32×D4C2×C411yescharacteristic, complemented
45C2×C32×D4D412yes—
46C2×C32×D4D412yes—
47C22×C32×C4—21no—
48C22×C32×C4C2311yescharacteristic
49C22×C3⋊Dic3—24no—
50C32×SD16—24no—
51C32×D8—24no—
52C32×M16—21no—
53C32×M16—21no—
54C2×C32×C8C2×C412yescomplemented
55#3757—24no—
56#3757—21no—
57#3757—21no—
58#3757D412yes—
59C4⋊(C3⋊Dic3)—21no—
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C4×C3⋊Dic3—21no—
63C4×C3⋊Dic3—21no—
64C4×C3⋊Dic3D412yes—
65#3748—21no—
66#3748—21no—
67C2×C3⋊(C3⋊C8)—21no—
68#1904—91noSylow
69C2×C3×#41C2×D314yes—
70C22×C3×D4C2×D314yes—
71C3×AGL(1,ℤ/8)—28no—
72C3×AGL(1,ℤ/8)—28no—
73C2×C3×SD16Dic318yescomplemented
74C2×C3×D8Dic318yescomplemented
75C2×C3×M16C2×D314yes—
76C2×#217—68no—
77D4×Dic3—68no—
78D4×Dic3—68no—
79#926—34no—
80#926—34no—
81#926—34no—
82C2×#208—34no—
83#837—68no—
84#834—68no—
85#832—68no—
86C23×C32—22no—
87C23×C32—22no—
88C23×C32—22no—
89C23×C32—21no—
90C23×C32—82no—
91C23×C32C2×D411yescharacteristic
92C32×Q8—21no—
93C32×Q8—21no—
94C32×D4—21no—
95C32×D4—22no—
96C32×D4—21no—
97C32×D4—22no—
98C32×D4—22no—
99C32×D4—41no—
100C32×D4—41no—
101C32×D4C22⋊C414yes—
102C2×C32×C4—21no—
103C2×C32×C4—41no—
104C2×C32×C4—41no—
105C2×C32×C4C22×C411yescharacteristic
106C2×C32×C4C2×D411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4#4111yescharacteristic
110C2×C32×C4#4111yescharacteristic
111C2×C3⋊Dic3—22no—
112C2×C3⋊Dic3—22no—
113C2×C3⋊Dic3—41no—
114C2×C3⋊Dic3—84no—
115C32×C8—22no—
116C3⋊(C3⋊C8)—41no—
117C2×AGL(1,ℤ/8)C3⋊D311yescharacteristic
118#518—91no—
119#421—91no—
120#418—91no—
121#410—91no—
122#344—91no—
123H(3,ℤ/4)—91no—
124C24×C3—24no—
125C3×#41—24no—
126C3×#41—24no—
127C3×#41—24no—
128C3×#41—24no—
129C2×C3×Q8C2×Dic314yescomplemented
130C2×C3×D4—24no—
131C2×C3×D4—24no—
132C2×C3×D4—24no—
133C2×C3×D4—24no—
134C2×C3×D4C2×Dic318yes—
135C2×C3×D4C2×Dic314yescomplemented
136C2×C3×D4#6718yes—
137C2×C3×D4#6718yes—
138C22×C3×C4—24no—
139C22×C3×C4C22×D314yes—
140C22×Dic3—616no—
141C3×SD16—216no—
142C3×D8—216no—
143C3×M16—24no—
144C3×M16—24no—
145C2×C3×C8C2×Dic318yescomplemented
146#217—38no—
147#217—616no—
148#217—64no—
149#217—64no—
150C4⋊Dic3—64no—
151C4⋊Dic3—64no—
152C4⋊Dic3—64no—
153C4×Dic3—38no—
154C4×Dic3—64no—
155C4×Dic3—64no—
156#208—64no—
157#208—64no—
158C2×C3⋊C8—64no—
159C22×C32—24no—
160C22×C32—41no—
161C22×C32—44no—
162C22×C32—41no—
163C22×C32—82no—
164C22×C32—82no—
165C22×C32—81no—
166C22×C32C4×D411yescharacteristic
167C22×C32#12011yescharacteristic
168C22×C32#12111yescharacteristic
169C32×C4—41no—
170C32×C4C2×C22⋊C411yescharacteristic
171C32×C4C4×D411yescharacteristic
172C32×C4#12111yescharacteristic
173C32×C4#12311yescharacteristic
174C3⋊Dic3—42no—
175C3⋊Dic3—42no—
176C3⋊Dic3—81no—
177C2×#41C2×C3⋊D311yescharacteristic
178C22×D4C2×C3⋊D311yescharacteristic
179AGL(1,ℤ/8)—22no—
180AGL(1,ℤ/8)—22no—
181C2×SD16C3⋊Dic312yescomplemented
182C2×D8C3⋊Dic312yescomplemented
183C2×M16—91no—
184C2×M16C2×C3⋊D311yescharacteristic
185C4×D4—182no—
186C4×D4—182no—
187#117—91no—
188#117—91no—
189#117—91no—
190C2×C22⋊C4—182no—
191#104—182no—
192#102—182no—
193#99—182no—
194C23×C3—28no—
195C23×C3—28no—
196C23×C3—28no—
197C23×C3—24no—
198C23×C3—88no—
199C23×C3D3×D414yes—
200C3×Q8—24no—
201C3×Q8—24no—
202C3×D4—28no—
203C3×D4—24no—
204C3×D4—28no—
205C3×D4—28no—
206C3×D4—24no—
207C3×D4—44no—
208C3×D4—44no—
209C3×D4#217116yes—
210C2×C3×C4—24no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4D3×D414yes—
214C2×C3×C4#23714yes—
215C2×C3×C4#23714yes—
216C2×C3×C4C22×Dic314yes—
217C2×C3×C4C2×#6714yes—
218C2×C3×C4C2×#6714yes—
219C2×Dic3—68no—
220C2×Dic3—68no—
221C2×Dic3—124no—
222C2×Dic3—2416no—
223C3×C8—28no—
224C3⋊C8—124no—
225C2×C32—21no—
226C2×C32—21no—
227C2×C32—21no—
228C2×C32—44no—
229C2×C32—81no—
230C2×C32#38611yescharacteristic
231C24—21no—
232#41—21no—
233#41—21no—
234#41—21no—
235#41—21no—
236C2×Q8C2×C3⋊Dic311yescharacteristic, complemented
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4C2×C3⋊Dic312yes—
242C2×D4C2×C3⋊Dic311yescharacteristic, complemented
243C2×D4#63812yes—
244C2×D4#63812yes—
245C22×C4—21no—
246C22×C4—184no—
247C22×C4C22×C3⋊D311yescharacteristic
248SD16—24no—
249D8—24no—
250M16—21no—
251M16—21no—
252M16—181no—
253M16—181no—
254C2×C8—181no—
255C2×C8C2×C3⋊Dic312yescomplemented
256C4⋊C4—181no—
257C4⋊C4—181no—
258C4⋊C4—181no—
259C22⋊C4—92no—
260C22⋊C4—184no—
261C22⋊C4—181no—
262C22⋊C4—181no—
263C42—92no—
264C42—181no—
265C42—181no—
266C22×C3—216no—
267C22×C3—44no—
268C22×C3—416no—
269C22×C3—44no—
270C22×C3—88no—
271C22×C3—88no—
272C22×C3—84no—
273C22×C3D4×Dic314yes—
274C22×C3#93714yes—
275C22×C3#93914yes—
276C3×C4—44no—
277C3×C4D4×Dic314yes—
278C3×C4#93514yes—
279C3×C4#93914yes—
280C3×C4C2×#21714yes—
281Dic3—128no—
282Dic3—128no—
283Dic3—244no—
284C32#190411yescharacteristic, complemented, Sylow
285C23—22no—
286C23—22no—
287C23—22no—
288C23—21no—
289C23—82no—
290C23D4×C3⋊D311yescharacteristic
291Q8—21no—
292Q8—21no—
293D4—22no—
294D4—21no—
295D4—22no—
296D4—22no—
297D4—21no—
298D4—41no—
299D4—41no—
300D4#375714yes—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—182no—
305C2×C4—182no—
306C2×C4—361no—
307C2×C4—724no—
308C2×C4D4×C3⋊D311yescharacteristic
309C2×C4#383011yescharacteristic
310C2×C4#383011yescharacteristic
311C2×C4C22×C3⋊Dic311yescharacteristic
312C2×C4C2×#63811yescharacteristic
313C2×C4C2×#63811yescharacteristic
314C8—22no—
315C8—361no—
316C2×C3—24no—
317C2×C3—24no—
318C2×C3—24no—
319C2×C3—416no—
320C2×C3—84no—
321C2×C3#522014yes—
322C22—24no—
323C22—41no—
324C22—44no—
325C22—41no—
326C22—82no—
327C22—82no—
328C22—81no—
329C22D4×C3⋊Dic311yescharacteristic
330C22#6414611yescharacteristic
331C22#6414811yescharacteristic
332C4—41no—
333C4—362no—
334C4—362no—
335C4—721no—
336C4D4×C3⋊Dic311yescharacteristic
337C4#6414411yescharacteristic
338C4#6414811yescharacteristic
339C4C2×#375711yescharacteristic
340C3#7208914yescomplemented
341C2—21no—
342C2—21no—
343C2—21no—
344C2—44no—
345C2—81no—
346C2#1059291911yescharacteristic
subgroupG/H
centerC2#10592919
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
Fitting, prime radicalC2×C32×AGL(1,ℤ/8)C2
socleC2×C32#386
cosocleC22×C32×C4C23
hypercenterC2×AGL(1,ℤ/8)C3⋊D3
hypocenterC32#1904
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
2. #10596269C2semidirect
3. #10592939C2semidirect
4. #10592935C2semidirect
5. #10592914C2semidirect
6. #10589423C2semidirect
7. #10589422C2semidirect
13. C2×C32×SD16C4semidirect
14. C2×C32×D8C4semidirect
38. C2×C32×Q8C2×C4semidirect
44. C2×C32×D4C2×C4semidirect
54. C2×C32×C8C2×C4semidirect
73. C2×C3×SD16Dic3semidirect
74. C2×C3×D8Dic3semidirect
129. C2×C3×Q8C2×Dic3semidirect
135. C2×C3×D4C2×Dic3semidirect
145. C2×C3×C8C2×Dic3semidirect
181. C2×SD16C3⋊Dic3semidirect
182. C2×D8C3⋊Dic3semidirect
236. C2×Q8C2×C3⋊Dic3semidirect
242. C2×D4C2×C3⋊Dic3semidirect
255. C2×C8C2×C3⋊Dic3semidirect
284. C32#1904semidirect
340. C3#72089semidirect

Maximal subgroups

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

subgroupindex
C2×C32×AGL(1,ℤ/8)2
#105962692
#105929392
#105929352
#105929142
#105894232
#105894222
#720893