Finite Group Databasebeta

← #11775072

Subgroups — #11775072

Order 1152.

720 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 139 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#10596093C211yesmaximal, characteristic
2#10592417C211yesmaximal, characteristic
3#10592348C211yesmaximal, characteristic
4C32×#328C211yesmaximal, characteristic
5#10589404C211yesmaximal, characteristic
6#10589333C211yesmaximal, characteristic
7#10589318C211yesmaximal, characteristic
8#68309—34nomaximal
9C32×C22⋊Q8C2211yescharacteristic
10Q8×C3⋊Dic3C411yescharacteristic, complemented
11#64153C411yescharacteristic, complemented
12#64133C2211yescharacteristic
13C2×#3750—21no—
14C2×C4×C3⋊Dic3—21no—
15C2×C4×C3⋊Dic3C2211yescharacteristic
16#64098—21no—
17#64097C411yescharacteristic, complemented
18#64096—21no—
19#64096C411yescharacteristic, complemented
20#64087—21no—
21#64086—21no—
22C32×C22⋊C8C2211yescharacteristic
23C32×#95C2211yescharacteristic
24#63658C2211yescharacteristic
25#63657C2211yescharacteristic
26#63641—21no—
27C8×C3⋊Dic3—21no—
28#5622—34no—
29#4718—34no—
30#4649—34no—
31C3×#328D314yes—
32#4538—34no—
33#4467—34no—
34#4452—34no—
35C2×C32×Q8C2×C411yescharacteristic
36C22×C32×C4—21no—
37C22×C32×C4C2311yescharacteristic
38C22×C3⋊Dic3—21no—
39C22×C3⋊Dic3D411yescharacteristic
40C22×C3⋊Dic3D411yescharacteristic
41C2×#634—21no—
42C2×#634C2×C411yescharacteristic
43C2×C32×C8—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4C2×C411yescharacteristic
46C32×C22⋊C4—21no—
47#3757—21no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)C2×C411yescharacteristic
50#3750—24no—
51#3750—21no—
52#3750—21no—
53#3750—21no—
54#3750—21no—
55#3750—21no—
56C4×C3⋊Dic3—21no—
57C4×C3⋊Dic3—21no—
58C4×C3⋊Dic3—42no—
59C4×C3⋊Dic3C2×C411yescharacteristic
60C4×C3⋊Dic3C2×C411yescharacteristic
61C4×C3⋊Dic3D411yescharacteristic
62C4×C3⋊Dic3D411yescharacteristic
63C2×C3⋊(C3⋊C8)—21no—
64#1503—91noSylow
65C3×C22⋊Q8C2×D314yes—
66Q8×Dic3—34no—
67#944—34no—
68#924—34no—
69C2×Dic3⋊C4—64no—
70C2×C4×Dic3—34no—
71C2×C4×Dic3—64no—
72#889—64no—
73#888—34no—
74#887—34no—
75#887—64no—
76#878—64no—
77#877—64no—
78C3×C22⋊C8C2×D314yes—
79C3×#95C2×D314yes—
80C3⋊#95—34no—
81#830—34no—
82Dic3⋊C8—64no—
83C8×Dic3—64no—
84C23×C32C2×D411yescharacteristic
85C32×Q8—41no—
86C2×C32×C4—22no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4C22×C411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C3⋊Dic3—22no—
95C2×C3⋊Dic3—22no—
96C2×C3⋊Dic3—21no—
97C2×C3⋊Dic3—21no—
98C2×C3⋊Dic3—41no—
99C2×C3⋊Dic3—41no—
100C2×C3⋊Dic3C22⋊C411yescharacteristic
101C2×C3⋊Dic3C22⋊C411yescharacteristic
102C2×C3⋊Dic3C22⋊C411yescharacteristic
103C2×C3⋊Dic3C22⋊C411yescharacteristic
104C2×C3⋊Dic3SD1612yes—
105C2×C3⋊Dic3Q1612yes—
106#634—42no—
107#634—41no—
108C32×C8—41no—
109C3⋊(C3⋊C8)—41no—
110#533—91no—
111#431—91no—
112C4×C4⋊C4—91no—
113#328—91no—
114#328—91no—
115#328—91no—
116#328C3⋊D311yescharacteristic
117C2×C3×Q8C4×D314yes—
118C22×C3×C4—24no—
119C22×C3×C4C22×D314yes—
120C22×Dic3—34no—
121C22×Dic3—34no—
122C22×Dic3—64no—
123C2×Dic6—34no—
124C2×Dic6—64no—
125C2×C3×C8—24no—
126C3×C4⋊C4—24no—
127C3×C4⋊C4C4×D314yes—
128C3×C22⋊C4—24no—
129#217—64no—
130C4⋊Dic3—34no—
131C4⋊Dic3—64no—
132Dic3⋊C4—64no—
133Dic3⋊C4—64no—
134Dic3⋊C4—64no—
135Dic3⋊C4—64no—
136Dic3⋊C4—616no—
137Dic3⋊C4—64no—
138C4×Dic3—34no—
139C4×Dic3—34no—
140C4×Dic3—34no—
141C4×Dic3—34no—
142C4×Dic3—64no—
143C4×Dic3—64no—
144C4×Dic3—128no—
145C2×C3⋊C8—64no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32C2×C22⋊C411yescharacteristic
149C22×C32C2×SD1611yescharacteristic
150C22×C32C2×Q1611yescharacteristic
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—42no—
154C32×C4—41no—
155C32×C4—41no—
156C3⋊Dic3—21no—
157C3⋊Dic3—42no—
158C3⋊Dic3—41no—
159C3⋊Dic3—41no—
160C3⋊Dic3#10312yes—
161C3⋊Dic3#10312yes—
162C3⋊Dic3#10412yes—
163C4⋊Q8—91no—
164C4⋊Q8—91no—
165#125—181no—
166C22⋊Q8—91no—
167C22⋊Q8—181no—
168C22⋊Q8C2×C3⋊D311yescharacteristic
169C4×Q8—91no—
170C4×Q8—91no—
171C4×Q8—181no—
172#117—181no—
173C2×C4⋊C4—181no—
174C2×C42—91no—
175C2×C42—181no—
176C4⋊C8—181no—
177C22⋊C8—91no—
178C22⋊C8C2×C3⋊D311yescharacteristic
179C4×C8—181no—
180#95—91no—
181#95C2×C3⋊D311yescharacteristic
182C23×C3D3×D414yes—
183C3×Q8—44no—
184C2×C3×C4—28no—
185C2×C3×C4—24no—
186C2×C3×C4—24no—
187C2×C3×C4—44no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4C2×C4×D314yes—
191C2×C3×C4D3×D414yes—
192C2×Dic3—34no—
193C2×Dic3—34no—
194C2×Dic3—34no—
195C2×Dic3—38no—
196C2×Dic3—38no—
197C2×Dic3—34no—
198C2×Dic3—68no—
199C2×Dic3—68no—
200C2×Dic3—64no—
201C2×Dic3—64no—
202C2×Dic3—124no—
203C2×Dic3—124no—
204Dic6—128no—
205Dic6—124no—
206C3×C8—44no—
207C3⋊C8—124no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32#41011yescharacteristic
211C2×C32C2×#10311yescharacteristic
212C2×C32C2×#10411yescharacteristic
213C2×Q8—91no—
214C2×Q8—181no—
215C2×Q8C4×C3⋊D311yescharacteristic
216C22×C4—21no—
217C22×C4—91no—
218C22×C4—91no—
219C22×C4—181no—
220C22×C4C22×C3⋊D311yescharacteristic
221C2×C8—21no—
222C2×C8—181no—
223C4⋊C4—21no—
224C4⋊C4—91no—
225C4⋊C4—181no—
226C4⋊C4—181no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C4⋊C4—181no—
230C4⋊C4—184no—
231C4⋊C4—181no—
232C4⋊C4C4×C3⋊D311yescharacteristic
233C22⋊C4—21no—
234C22⋊C4—181no—
235C42—91no—
236C42—91no—
237C42—91no—
238C42—91no—
239C42—181no—
240C42—181no—
241C42—362no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3D3×C22⋊C414yes—
245C22×C3D3×SD1614yes—
246C22×C3D3×Q1614yes—
247C3×C4—24no—
248C3×C4—24no—
249C3×C4—48no—
250C3×C4—44no—
251C3×C4—44no—
252Dic3—38no—
253Dic3—38no—
254Dic3—38no—
255Dic3—64no—
256Dic3—128no—
257Dic3—124no—
258Dic3—124no—
259C32#150311yescharacteristic, complemented, Sylow
260C23D4×C3⋊D311yescharacteristic
261Q8—41no—
262Q8—361no—
263Q8—362no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—91no—
271C2×C4—91no—
272C2×C4—91no—
273C2×C4—92no—
274C2×C4—92no—
275C2×C4—91no—
276C2×C4—182no—
277C2×C4—182no—
278C2×C4—181no—
279C2×C4—181no—
280C2×C4—361no—
281C2×C4—361no—
282C2×C4C2×C4×C3⋊D311yescharacteristic
283C2×C4D4×C3⋊D311yescharacteristic
284C8—41no—
285C8—361no—
286C2×C3—24no—
287C2×C3—24no—
288C2×C3#474014yes—
289C2×C3D3×#10314yes—
290C2×C3D3×#10414yes—
291C22—21no—
292C22—21no—
293C22C22⋊C4×C3⋊D311yescharacteristic
294C22SD16×C3⋊D311yescharacteristic
295C22Q16×C3⋊D311yescharacteristic
296C4—21no—
297C4—21no—
298C4—42no—
299C4—41no—
300C4—41no—
301C4—92no—
302C4—92no—
303C4—92no—
304C4—181no—
305C4—362no—
306C4—361no—
307C4—361no—
308C3#6830914yescomplemented
309C2—21no—
310C2—21no—
311C2#1059243911yescharacteristic
312C2C3⋊D3×#10311yescharacteristic
313C2C3⋊D3×#10411yescharacteristic
subgroupG/H
centerC22C22⋊C4×C3⋊D3
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC32×#328C2
socleC22×C32C2×C22⋊C4
prime radicalC23×C32C2×D4
cosocleC22×C32×C4C23
hypercenter#328C3⋊D3
hypocenterC32#1503
LCSG ▷ C2×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
10. Q8×C3⋊Dic3C4semidirect
11. #64153C4semidirect
17. #64097C4semidirect
19. #64096C4semidirect
259. C32#1503semidirect
308. C3#68309semidirect

Maximal subgroups

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

subgroupindex
#105960932
#105924172
#105923482
C32×#3282
#105894042
#105893332
#105893182
#683093