Finite Group Databasebeta

← #11779796

Subgroups — #11779796

Order 1152.

750 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 750 conjugacy classes of subgroups — too many to draw. Only its 131 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#10596189C211yesmaximal, characteristic
2C32×#486C211yesmaximal, characteristic, complemented
3#10592789C211yesmaximal, characteristic, complemented
4#10592786C211yesmaximal, characteristic, complemented
5#10589410C211yesmaximal, characteristic, complemented
6#10589325C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#71207—34nomaximal
9C2×#3830—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×#127C2211yescharacteristic
12C2×#3756—21no—
13#64149—21no—
14#64148—22no—
15#64147—21no—
16D4×C3⋊Dic3—21no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18C4×#634C2211yescharacteristic
19C32×#102—22no—
20C32×C4×C8C2211yescharacteristic
21#63662—21no—
22#63636—21no—
23#63634—21no—
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5718—34no—
27C3×#486D314yescomplemented
28#5090—34no—
29#5087—34no—
30#4544—34no—
31#4459—34no—
32#4456—34no—
33C2×C32×Q8—21no—
34C2×C32×D4—21no—
35C2×C32×D4—21no—
36C2×C32×D4D412yes—
37C2×#638—22no—
38C2×#638—22no—
39C22×C3⋊Dic3—22no—
40#3830—42no—
41#3830—42no—
42C2×C4⋊(C3⋊D3)D411yescharacteristic
43C2×C4×C3⋊D3—21no—
44C2×#634D411yescharacteristic
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48C32×C42C2311yescharacteristic
49#3757—22no—
50#3756—42no—
51#3752—21no—
52C4⋊(C3⋊Dic3)D411yescharacteristic
53C4⋊(C3⋊Dic3)D411yescharacteristic
54#3750—21no—
55C4×C3⋊Dic3—21no—
56C2×C3⋊(C3⋊C8)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58#1641—91noSylow
59C2×#237—64no—
60C3×C4⋊Q8C2×D314yes—
61C3×#127C2×D314yes—
62C2×#216—64no—
63#940—64no—
64#939—68no—
65#938—64no—
66D4×Dic3—64no—
67C4×D12—34no—
68C4×Dic6—34no—
69C3×#102—28no—
70C3×C4×C8C2×D314yes—
71#835—64no—
72#809—64no—
73#807—64no—
74C4⋊(C3⋊C8)—34no—
75C4⋊(C3⋊C8)—34no—
76C23×C32—22no—
77C22×C3⋊D3—21no—
78C32×Q8—41no—
79C32×Q8—41no—
80C32×D4—42no—
81C32×D4—41no—
82C32×D4—41no—
83C32×D4—41no—
84C32×D4—41no—
85C2×C32×C4—41no—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D411yescharacteristic
88C2×C32×C4C2×D411yescharacteristic
89#638—44no—
90#638—82no—
91C2×C3⋊Dic3—21no—
92C2×C3⋊Dic3—21no—
93C2×C3⋊Dic3—21no—
94C2×C3⋊Dic3—44no—
95C4⋊(C3⋊D3)—21no—
96C4⋊(C3⋊D3)—21no—
97C4×C3⋊D3—41no—
98C4×C3⋊D3—41no—
99#634—21no—
100#634D812yes—
101C32×C8—41no—
102C3⋊(C3⋊C8)—41no—
103C3⋊(C3⋊C8)—41no—
104#550—91no—
105#486C3⋊D311yescharacteristic, complemented
106#474—91no—
107#462—91no—
108#331—91no—
109Q8⋊C8—91no—
110D4⋊C8—91no—
111C2×C3×Q8—24no—
112C2×C3×D4—24no—
113C2×C3×D4—24no—
114C2×C3×D4#6718yes—
115C2×#67—68no—
116C2×#67—68no—
117C22×Dic3—68no—
118#237—128no—
119#237—128no—
120C2×D12—34no—
121C2×C4×D3—64no—
122C2×Dic6—34no—
123C2×C3×C8—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—24no—
126C3×C42C22×D314yes—
127#217—68no—
128#216—128no—
129(C2×D3)⋊C4—64no—
130C4⋊Dic3—34no—
131C4⋊Dic3—34no—
132Dic3⋊C4—64no—
133C4×Dic3—64no—
134C2×C3⋊C8—64no—
135C2×C3⋊C8—64no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32#12011yescharacteristic
140C2×C3⋊D3—41no—
141C2×C3⋊D3—41no—
142C2×C3⋊D3—41no—
143C32×C4—21no—
144C32×C4—81no—
145C32×C4C2×D811yescharacteristic
146C32×C4#13511yescharacteristic
147C32×C4AGL(1,ℤ/8)11yescharacteristic
148C32×C4#13711yescharacteristic
149C3⋊Dic3—22no—
150C3⋊Dic3—41no—
151C3⋊Dic3—41no—
152C2×#41—181no—
153C2×Q16—181no—
154C4⋊Q8C2×C3⋊D311yescharacteristic
155#127—181no—
156#127C2×C3⋊D311yescharacteristic
157#121—181no—
158#121—182no—
159C4×Q8—91no—
160C4×D4—91no—
161C4×D4—181no—
162#107—181no—
163C4⋊C8—91no—
164C4⋊C8—91no—
165#103—181no—
166#102—22no—
167#102—181no—
168C4×C8C2×C3⋊D311yescharacteristic
169C23×C3—28no—
170C22×D3—64no—
171C3×Q8—44no—
172C3×Q8—44no—
173C3×D4—48no—
174C3×D4—44no—
175C3×D4—44no—
176C3×D4—44no—
177C3×D4—44no—
178C2×C3×C4—44no—
179C2×C3×C4D3×D414yes—
180C2×C3×C4D3×D414yes—
181C2×C3×C4C2×#6714yes—
182#67—1216no—
183#67—248no—
184C2×Dic3—64no—
185C2×Dic3—64no—
186C2×Dic3—64no—
187C2×Dic3—1216no—
188D12—64no—
189D12—64no—
190C4×D3—124no—
191C4×D3—124no—
192Dic6—38no—
193Dic6—64no—
194C3×C8—44no—
195C3⋊C8—124no—
196C3⋊C8—124no—
197C2×C32—82no—
198C2×C32#44711yescharacteristic
199C2×C32#45211yescharacteristic
200C2×C32#45511yescharacteristic
201C3⋊D3—81no—
202#41—362no—
203#41—362no—
204C2×Q8—21no—
205C2×Q8—91no—
206C2×D4—21no—
207C2×D4—21no—
208C2×D4—91no—
209C2×D4—182no—
210C2×D4—182no—
211C2×D4#63812yes—
212C22×C4—182no—
213C22×C4—181no—
214Q16—362no—
215C2×C8—21no—
216C2×C8—181no—
217C2×C8—181no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—91no—
221C4⋊C4—91no—
222C4⋊C4—181no—
223C22⋊C4—182no—
224C22⋊C4—181no—
225C42—181no—
226C42C22×C3⋊D311yescharacteristic
227C22×C3—48no—
228C22×C3—48no—
229C22×C3—48no—
230C22×C3#93714yes—
231C2×D3—124no—
232C2×D3—124no—
233C2×D3—124no—
234C3×C4—24no—
235C3×C4—84no—
236C3×C4D3×D814yes—
237C3×C4#91114yes—
238C3×C4#91514yes—
239C3×C4#91614yes—
240Dic3—68no—
241Dic3—124no—
242Dic3—124no—
243C32#164111yescharacteristic, complemented, Sylow
244C23—22no—
245C23—181no—
246Q8—41no—
247Q8—41no—
248Q8—92no—
249Q8—181no—
250D4—42no—
251D4—41no—
252D4—41no—
253D4—41no—
254D4—41no—
255D4—181no—
256D4—181no—
257D4—364no—
258D4—722no—
259C2×C4—41no—
260C2×C4—181no—
261C2×C4—181no—
262C2×C4—181no—
263C2×C4—364no—
264C2×C4—361no—
265C2×C4—361no—
266C2×C4D4×C3⋊D311yescharacteristic
267C2×C4D4×C3⋊D311yescharacteristic
268C2×C4C2×#63811yescharacteristic
269C8—41no—
270C8—361no—
271C8—361no—
272C2×C3—88no—
273C2×C3#506114yes—
274C2×C3#515614yes—
275C2×C3#517314yes—
276D3—244no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—361no—
281C22—361no—
282C22—361no—
283C22#6414611yescharacteristic
284C4—21no—
285C4—81no—
286C4—182no—
287C4—361no—
288C4—361no—
289C4D8×C3⋊D311yescharacteristic
290C4#6412011yescharacteristic
291C4#6412411yescharacteristic
292C4#6412511yescharacteristic
293C3#7120714yescomplemented
294C2—82no—
295C2—721no—
296C2#1059276011yescharacteristic
297C2#1059285511yescharacteristic
298C2#1059287211yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#486C2
socleC22×C32#120
prime radical#10596189C2
hypercenter#486C3⋊D3
hypocenterC32#1641
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #486

Complements

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

NH ≅ G/Ntype
2. C32×#486C2semidirect
3. #10592789C2semidirect
4. #10592786C2semidirect
5. #10589410C2semidirect
6. #10589325C2semidirect
7. #10589322C2semidirect
27. C3×#486D3semidirect
105. #486C3⋊D3semidirect
243. C32#1641semidirect
293. C3#71207semidirect

Maximal subgroups

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

subgroupindex
#105961892
C32×#4862
#105927892
#105927862
#105894102
#105893252
#105893222
#712073