Finite Group Databasebeta

← #11780740

Subgroups — #11780740

Order 1152.

552 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 125 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
1C32×#563C211yesmaximal, characteristic
2#10592768C211yesmaximal, characteristic, complemented
3#10592763C211yesmaximal, characteristic, complemented
4#10592754C211yesmaximal, characteristic
5#10589409C211yesmaximal, characteristic, complemented
6#10589409C211yesmaximal, characteristic, complemented
7#10589406C211yesmaximal, characteristic
8#71957—34nomaximal
9C32×C4⋊Q8—21no—
10C32×#126—21no—
11C32×#126—21no—
12C32×#125C2211yescharacteristic
13C32×#125C2211yescharacteristic
14C32×#124C2211yescharacteristic
15C32×#123—21no—
16C32×C22⋊Q8—21no—
17C32×C22⋊Q8—21no—
18#64078C2211yescharacteristic
19#63662—21no—
20#63659—21no—
21#63637—21no—
22#63637—21no—
23#63635—21no—
24#63634—21no—
25#63630C2211yescharacteristic
26#63630C2211yescharacteristic
27#63630C2211yescharacteristic
28C3×#563D314yes—
29#5069—34no—
30#5064—34no—
31#5055—34no—
32#4543—34no—
33#4543—34no—
34#4540—34no—
35C2×C32×Q8—21no—
36C2×C32×Q8D411yescharacteristic
37C2×C32×D4D411yescharacteristic
38C22×C32×C4—21no—
39C2×#634—21no—
40C32×C4⋊C4—21no—
41C32×C4⋊C4—21no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C4⋊C4D411yescharacteristic
48C32×C4⋊C4D411yescharacteristic
49C32×C4⋊C4D411yescharacteristic
50C32×C22⋊C4—21no—
51C32×C22⋊C4—21no—
52C32×C22⋊C4—21no—
53C32×C22⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42—21no—
56C32×C42C2311yescharacteristic
57C4⋊(C3⋊Dic3)—21no—
58C4⋊(C3⋊Dic3)—21no—
59C2×C3⋊(C3⋊C8)—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C2×C3⋊(C3⋊C8)—21no—
62#1664—91noSylow
63C3×C4⋊Q8—24no—
64C3×#126—24no—
65C3×#126—24no—
66C3×#125C2×D314yes—
67C3×#125C2×D314yes—
68C3×#124C2×D314yes—
69C3×#123—24no—
70C3×C22⋊Q8—24no—
71C3×C22⋊Q8—24no—
72#869—34no—
73#835—64no—
74#832—64no—
75#810—64no—
76#810—64no—
77#808—64no—
78#807—64no—
79#803—34no—
80#803—34no—
81#803—34no—
82C23×C32—21no—
83C32×Q8—41no—
84C32×Q8—81no—
85C32×D4—41no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4—21no—
90C2×C32×C4—21no—
91C2×C32×C4—81no—
92C2×C32×C4C2×D411yescharacteristic
93C2×C32×C4C2×D411yescharacteristic
94C2×C32×C4C2×D411yescharacteristic
95C2×C3⋊Dic3—41no—
96#634—42no—
97C3⋊(C3⋊C8)—22no—
98C3⋊(C3⋊C8)—41no—
99C3⋊(C3⋊C8)—41no—
100#563C3⋊D311yescharacteristic
101#501—91no—
102#491—91no—
103#489—91no—
104#330—91no—
105#330—91no—
106#329—91no—
107C2×C3×Q8—24no—
108C2×C3×Q8#6714yes—
109C2×C3×D4#6714yes—
110C22×C3×C4—24no—
111C2×Dic6—64no—
112C3×C4⋊C4—24no—
113C3×C4⋊C4—24no—
114C3×C4⋊C4—24no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—24no—
118C3×C4⋊C4#6714yes—
119C3×C4⋊C4#6714yes—
120C3×C4⋊C4#6714yes—
121C3×C4⋊C4#6714yes—
122C3×C22⋊C4—24no—
123C3×C22⋊C4—24no—
124C3×C22⋊C4—24no—
125C3×C22⋊C4—24no—
126C3×C22⋊C4—24no—
127C3×C42—24no—
128C3×C42C22×D314yes—
129C4⋊Dic3—64no—
130C4⋊Dic3—64no—
131C2×C3⋊C8—64no—
132C2×C3⋊C8—64no—
133C2×C3⋊C8—64no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32#12011yescharacteristic
138C32×C4—21no—
139C32×C4—21no—
140C32×C4—21no—
141C32×C4—41no—
142C32×C4—41no—
143C32×C4—41no—
144C32×C4—41no—
145C32×C4—41no—
146C3⋊Dic3—81no—
147C4⋊Q8—21no—
148C4⋊Q8—91no—
149#126—21no—
150#126—21no—
151#125C2×C3⋊D311yescharacteristic
152#125C2×C3⋊D311yescharacteristic
153#124C2×C3⋊D311yescharacteristic
154#123—21no—
155C22⋊Q8—21no—
156C22⋊Q8—21no—
157#107—181no—
158#106—181no—
159#103—181no—
160#103—181no—
161#103—181no—
162#102—181no—
163#97—91no—
164#97—91no—
165#97—91no—
166C23×C3—24no—
167C3×Q8—44no—
168C3×Q8—84no—
169C3×D4—44no—
170C2×C3×C4—24no—
171C2×C3×C4—24no—
172C2×C3×C4—24no—
173C2×C3×C4—24no—
174C2×C3×C4—24no—
175C2×C3×C4—84no—
176C2×C3×C4C2×#6714yes—
177C2×C3×C4C2×#6714yes—
178C2×C3×C4C2×#6714yes—
179C2×Dic3—124no—
180Dic6—128no—
181C3⋊C8—68no—
182C3⋊C8—124no—
183C3⋊C8—124no—
184C2×C32—81no—
185C2×C32#45411yescharacteristic
186C2×C32#45511yescharacteristic
187C2×C32#45611yescharacteristic
188C2×Q8—21no—
189C2×Q8—181no—
190C2×Q8#63811yescharacteristic
191C2×D4#63811yescharacteristic
192C22×C4—21no—
193C2×C8—181no—
194C2×C8—181no—
195C2×C8—181no—
196C4⋊C4—21no—
197C4⋊C4—21no—
198C4⋊C4—21no—
199C4⋊C4—21no—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—181no—
203C4⋊C4—181no—
204C4⋊C4#63811yescharacteristic
205C4⋊C4#63811yescharacteristic
206C4⋊C4#63811yescharacteristic
207C4⋊C4#63811yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—21no—
210C22⋊C4—21no—
211C22⋊C4—21no—
212C22⋊C4—21no—
213C42—21no—
214C42C22×C3⋊D311yescharacteristic
215C22×C3—44no—
216C22×C3—44no—
217C22×C3—44no—
218C22×C3#95314yes—
219C3×C4—24no—
220C3×C4—24no—
221C3×C4—24no—
222C3×C4—44no—
223C3×C4—44no—
224C3×C4—44no—
225C3×C4—44no—
226C3×C4—44no—
227Dic3—244no—
228C32#166411yescharacteristic, complemented, Sylow
229C23—21no—
230Q8—41no—
231Q8—81no—
232Q8—362no—
233D4—41no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—81no—
240C2×C4—361no—
241C2×C4C2×#63811yescharacteristic
242C2×C4C2×#63811yescharacteristic
243C2×C4C2×#63811yescharacteristic
244C8—182no—
245C8—361no—
246C8—361no—
247C2×C3—84no—
248C2×C3#524214yes—
249C2×C3#524514yes—
250C2×C3#524614yes—
251C22—41no—
252C22—41no—
253C22—41no—
254C22#6416211yescharacteristic
255C4—21no—
256C4—21no—
257C4—21no—
258C4—41no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—41no—
263C4—721no—
264C3#7195714yescomplemented
265C2—81no—
266C2#1059294111yescharacteristic
267C2#1059294411yescharacteristic
268C2#1059294511yescharacteristic
subgroupG/H
centerC22#64162
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#563C2
socleC22×C32#120
prime radicalC2×C32×D4D4
hypercenter#563C3⋊D3
hypocenterC32#1664
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #563

Complements

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

NH ≅ G/Ntype
2. #10592768C2semidirect
3. #10592763C2semidirect
5. #10589409C2semidirect
6. #10589409C2semidirect
228. C32#1664semidirect
264. C3#71957semidirect

Maximal subgroups

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

subgroupindex
C32×#5632
#105927682
#105927632
#105927542
#105894092
#105894062
#719573