Finite Group Databasebeta

← #11778723

Subgroups — #11778723

Order 1152.

624 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 117 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#10595986C211yesmaximal, characteristic
2C32×#400C211yesmaximal, characteristic, complemented
3#10592669C211yesmaximal, characteristic, complemented
4#10592667C211yesmaximal, characteristic
5#10589394C211yesmaximal, characteristic, complemented
6#10589318C211yesmaximal, characteristic, complemented
7#10589317C211yesmaximal, characteristic
8#69717—34nomaximal
9C2×C32×C4⋊C4C2211yescharacteristic
10C2×C32×C4⋊C4C2211yescharacteristic
11#64139C2211yescharacteristic
12#64138—21no—
13C4×#638—21no—
14#64133C2211yescharacteristic
15#64107—21no—
16#64105—21no—
17#64103—21no—
18#64098—21no—
19C32×#95—21no—
20C32×#95—21no—
21C32×#95C2211yescharacteristic
22#63657C2211yescharacteristic
23#63657C2211yescharacteristic
24#63637—21no—
25#63636—21no—
26#63635—21no—
27#63634—21no—
28#5515—34no—
29C3×#400D314yescomplemented
30#4970—34no—
31#4968—34no—
32#4528—34no—
33#4452—34no—
34#4451—34no—
35C22×C32×C4—21no—
36C22×C32×C4—21no—
37C22×C32×C4D411yescharacteristic
38C22×C32×C4D411yescharacteristic
39C22×C32×C4C2311yescharacteristic
40C2×#638—21no—
41C2×C4⋊(C3⋊D3)D411yescharacteristic
42C2×C4×C3⋊D3—21no—
43C2×#634D411yescharacteristic
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4—21no—
48#3757—21no—
49#3752—21no—
50#3752—21no—
51#3752—21no—
52#3752—21no—
53C4⋊(C3⋊Dic3)D411yescharacteristic
54C4⋊(C3⋊Dic3)D411yescharacteristic
55#3750—21no—
56#3750—21no—
57#3750—21no—
58#3750—21no—
59C4×C3⋊Dic3—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C2×C3⋊(C3⋊C8)—21no—
62#1615—91noSylow
63C2×C3×C4⋊C4C2×D314yes—
64C2×C3×C4⋊C4C2×D314yes—
65#930—34no—
66#929—64no—
67C4×#67—64no—
68#924—34no—
69#898—64no—
70#896—64no—
71#894—64no—
72#889—64no—
73C3×#95—24no—
74C3×#95—24no—
75C3×#95C2×D314yes—
76#830—34no—
77#830—34no—
78#810—64no—
79#809—64no—
80#808—64no—
81#807—64no—
82C23×C32C2×D411yescharacteristic
83C22×C3⋊D3—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4—41no—
91C2×C32×C4—41no—
92C2×C32×C4—42no—
93C2×C32×C4—41no—
94C2×C32×C4—41no—
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97#638—41no—
98#638—41no—
99C2×C3⋊Dic3—21no—
100C2×C3⋊Dic3—21no—
101C2×C3⋊Dic3—21no—
102C4⋊(C3⋊D3)—41no—
103C4×C3⋊D3—81no—
104#634—41no—
105C3⋊(C3⋊C8)—41no—
106C3⋊(C3⋊C8)—41no—
107#539—91no—
108#484—91no—
109#481—91no—
110#400C3⋊D311yescharacteristic, complemented
111#328—91no—
112#327—91no—
113#324—91no—
114C22×C3×C4—24no—
115C22×C3×C4—24no—
116C22×C3×C4#6714yes—
117C22×C3×C4#6714yes—
118C22×C3×C4C22×D314yes—
119C2×#67—64no—
120C2×D12—34no—
121C2×C4×D3—64no—
122C2×Dic6—34no—
123C3×C4⋊C4—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—24no—
126C3×C4⋊C4—24no—
127#217—64no—
128(C2×D3)⋊C4—64no—
129(C2×D3)⋊C4—64no—
130(C2×D3)⋊C4—64no—
131(C2×D3)⋊C4—64no—
132C4⋊Dic3—34no—
133C4⋊Dic3—34no—
134Dic3⋊C4—64no—
135Dic3⋊C4—64no—
136Dic3⋊C4—64no—
137Dic3⋊C4—64no—
138C4×Dic3—64no—
139C2×C3⋊C8—64no—
140C2×C3⋊C8—64no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32#12011yescharacteristic
144C22×C32#13511yescharacteristic
145C22×C32#13711yescharacteristic
146C2×C3⋊D3—41no—
147C2×C3⋊D3—41no—
148C2×C3⋊D3—41no—
149C32×C4—21no—
150C32×C4—21no—
151C32×C4—41no—
152C32×C4—41no—
153C32×C4—81no—
154C32×C4—81no—
155C3⋊Dic3—41no—
156C3⋊Dic3—41no—
157C3⋊Dic3—41no—
158#126—181no—
159#125—181no—
160#123—181no—
161#123—181no—
162C22⋊Q8—91no—
163C22⋊Q8—181no—
164#121—91no—
165C4×D4—181no—
166C2×C4⋊C4C2×C3⋊D311yescharacteristic
167C2×C4⋊C4C2×C3⋊D311yescharacteristic
168#107—181no—
169#106—181no—
170#103—181no—
171#102—181no—
172C22⋊C8—91no—
173C22⋊C8—91no—
174#95—21no—
175#95—21no—
176#95C2×C3⋊D311yescharacteristic
177C23×C3C2×#6714yes—
178C22×D3—64no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—24no—
182C2×C3×C4—24no—
183C2×C3×C4—44no—
184C2×C3×C4—44no—
185C2×C3×C4—44no—
186C2×C3×C4—44no—
187C2×C3×C4—48no—
188C2×C3×C4—44no—
189C2×C3×C4—44no—
190C2×C3×C4D3×D414yes—
191C2×C3×C4D3×D414yes—
192#67—124no—
193#67—124no—
194C2×Dic3—64no—
195C2×Dic3—64no—
196C2×Dic3—64no—
197D12—124no—
198C4×D3—244no—
199Dic6—124no—
200C3⋊C8—124no—
201C3⋊C8—124no—
202C2×C32—21no—
203C2×C32—21no—
204C2×C32#45211yescharacteristic
205C2×C32#45411yescharacteristic
206C2×C32#45811yescharacteristic
207C3⋊D3—81no—
208C2×Q8—91no—
209C2×D4—91no—
210C2×D4—181no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—181no—
214C22×C4#63811yescharacteristic
215C22×C4#63811yescharacteristic
216C22×C4C22×C3⋊D311yescharacteristic
217C2×C8—181no—
218C2×C8—181no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—91no—
224C4⋊C4—91no—
225C4⋊C4—181no—
226C4⋊C4—181no—
227C4⋊C4—181no—
228C4⋊C4—181no—
229C22⋊C4—181no—
230C22⋊C4—181no—
231C22⋊C4—181no—
232C22⋊C4—181no—
233C22⋊C4—181no—
234C42—181no—
235C22×C3—24no—
236C22×C3—24no—
237C22×C3#93714yes—
238C22×C3#94214yes—
239C22×C3#95014yes—
240C2×D3—124no—
241C2×D3—124no—
242C2×D3—124no—
243C3×C4—24no—
244C3×C4—24no—
245C3×C4—44no—
246C3×C4—44no—
247C3×C4—84no—
248C3×C4—84no—
249Dic3—124no—
250Dic3—124no—
251Dic3—124no—
252C32#161511yescharacteristic, complemented, Sylow
253C23—181no—
254C23C2×#63811yescharacteristic
255Q8—361no—
256D4—361no—
257D4—361no—
258D4—361no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—181no—
271C2×C4—181no—
272C2×C4—181no—
273C2×C4—721no—
274C2×C4D4×C3⋊D311yescharacteristic
275C2×C4D4×C3⋊D311yescharacteristic
276C8—361no—
277C8—361no—
278C2×C3—24no—
279C2×C3—24no—
280C2×C3#504714yes—
281C2×C3#505314yes—
282C2×C3#506214yes—
283D3—244no—
284C22—21no—
285C22—21no—
286C22—361no—
287C22—361no—
288C22—361no—
289C22#6414611yescharacteristic
290C22#6415111yescharacteristic
291C22#6415911yescharacteristic
292C4—21no—
293C4—21no—
294C4—41no—
295C4—41no—
296C4—81no—
297C4—81no—
298C4—361no—
299C4—361no—
300C4—361no—
301C3#6971714yescomplemented
302C2—21no—
303C2—21no—
304C2—721no—
305C2#1059274611yescharacteristic
306C2#1059275211yescharacteristic
307C2#1059276111yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C22×C3⋊D3
FittingC32×#400C2
socleC22×C32#120
prime radical#64139C22
hypercenter#400C3⋊D3
hypocenterC32#1615
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #400

Complements

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

NH ≅ G/Ntype
2. C32×#400C2semidirect
3. #10592669C2semidirect
5. #10589394C2semidirect
6. #10589318C2semidirect
29. C3×#400D3semidirect
110. #400C3⋊D3semidirect
252. C32#1615semidirect
301. C3#69717semidirect

Maximal subgroups

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

subgroupindex
#105959862
C32×#4002
#105926692
#105926672
#105893942
#105893182
#105893172
#697173