Finite Group Databasebeta

← #11779050

Subgroups — #11779050

Order 1152.

684 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 143 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
1Q8×#634C211yesmaximal, characteristic, complemented
2#10592726C211yesmaximal, characteristic
3#10592547C211yesmaximal, characteristic
4#10592392C211yesmaximal, characteristic
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589347C211yesmaximal, characteristic, complemented
7#10589322C211yesmaximal, characteristic, complemented
8#68264—34nomaximal
9C32×C4×Q8C2211yescharacteristic
10Q8×C3⋊Dic3—21no—
11C2×#3755—21no—
12C2×#3744—21no—
13C2×#3744—22no—
14#64097—22no—
15#64097—21no—
16#64083C2211yescharacteristic
17#64078—21no—
18#64078C2211yescharacteristic
19C4×#634—21no—
20C4×#634C2211yescharacteristic
21C32×C4⋊C8C2211yescharacteristic
22C32×C4×C8C2211yescharacteristic
23#63648—21no—
24#63636—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26Q8×Dic6—34no—
27#5027—34no—
28#4848—34no—
29#4693—34no—
30C3×Q8⋊C8D314yescomplemented
31#4481—34no—
32#4456—34no—
33C2×C32×Q8D411yescharacteristic
34C2×C4⋊(C3⋊D3)—21no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C4⋊(C3⋊D3)—21no—
37C2×#634—22no—
38C2×#634D411yescharacteristic
39C2×#634D412yes—
40C2×C32×C8—21no—
41C2×C32×C8—21no—
42C32×C4⋊C4—21no—
43C32×C4⋊C4D411yescharacteristic
44C32×C42—21no—
45C32×C42C2311yescharacteristic
46#3755—42no—
47C4⋊(C3⋊Dic3)—22no—
48C4⋊(C3⋊Dic3)—21no—
49C4⋊(C3⋊Dic3)—21no—
50C4⋊(C3⋊Dic3)D411yescharacteristic
51#3750—22no—
52#3750—21no—
53C4×C3⋊Dic3—22no—
54C4×C3⋊Dic3—21no—
55C2×C3⋊(C3⋊C8)—21no—
56#3744—44no—
57#3744—42no—
58#1623—91noSylow
59C3×C4×Q8C2×D314yes—
60Q8×Dic3—64no—
61C2×#215—64no—
62C2×#204—68no—
63C2×#204—64no—
64#888—68no—
65#888—64no—
66#874—34no—
67#869—34no—
68#869—64no—
69C4×Dic6—34no—
70C4×Dic6—64no—
71C3×C4⋊C8C2×D314yes—
72C3×C4×C8C2×D314yes—
73#821—64no—
74#809—64no—
75C4⋊(C3⋊C8)—34no—
76C22×C3⋊D3—41no—
77C32×Q8—21no—
78C32×Q8SD1612yescomplemented
79C2×C32×C4—21no—
80C2×C32×C4—21no—
81C2×C32×C4C2×D411yescharacteristic
82C2×C32×C4C2×D411yescharacteristic
83C2×C32×C4C2×D411yescharacteristic
84C2×C3⋊Dic3—21no—
85C2×C3⋊Dic3—22no—
86C2×C3⋊Dic3—21no—
87C4⋊(C3⋊D3)—42no—
88C4⋊(C3⋊D3)—41no—
89C4⋊(C3⋊D3)—41no—
90C4⋊(C3⋊D3)—41no—
91C4⋊(C3⋊D3)—41no—
92#634—21no—
93#634—42no—
94#634—82no—
95#634SD1612yescomplemented
96C32×C8—22no—
97C32×C8—41no—
98C3⋊(C3⋊C8)—41no—
99Q82—91no—
100#492—91no—
101#460—91no—
102#460—91no—
103#333—91no—
104Q8⋊C8—91no—
105Q8⋊C8C3⋊D311yescharacteristic, complemented
106C2×C3×Q8D1214yes—
107C2×D12—64no—
108C2×D12—64no—
109C2×D12—64no—
110C2×Dic6—34no—
111C2×Dic6—38no—
112C2×Dic6—68no—
113C2×C3×C8—24no—
114C2×C3×C8—24no—
115C3×C4⋊C4—24no—
116C3×C4⋊C4D1214yes—
117C3×C42—24no—
118C3×C42C22×D314yes—
119#215—128no—
120C4⋊Dic3—34no—
121C4⋊Dic3—68no—
122C4⋊Dic3—64no—
123C4⋊Dic3—64no—
124Dic3⋊C4—68no—
125Dic3⋊C4—64no—
126C4×Dic3—68no—
127C4×Dic3—64no—
128C2×C3⋊C8—64no—
129#204—1216no—
130#204—128no—
131C22×C32#12011yescharacteristic
132C2×C3⋊D3—81no—
133C2×C3⋊D3—81no—
134C2×C3⋊D3—81no—
135C32×C4—22no—
136C32×C4—21no—
137C32×C4—41no—
138C32×C4C2×SD1611yescharacteristic
139C32×C4C2×SD1611yescharacteristic
140C32×C4#13711yescharacteristic
141C32×C4#13711yescharacteristic
142C3⋊Dic3—22no—
143C3⋊Dic3—42no—
144C3⋊Dic3—41no—
145C2×SD16—181no—
146C2×SD16—182no—
147C2×SD16—181no—
148C4⋊Q8—91no—
149C4⋊Q8—181no—
150C4⋊Q8—182no—
151C4⋊Q8—181no—
152#127—91no—
153C4×Q8—91no—
154C4×Q8—181no—
155C4×Q8—181no—
156C4×Q8C2×C3⋊D311yescharacteristic
157C4⋊C8—91no—
158C4⋊C8C2×C3⋊D311yescharacteristic
159#102—181no—
160#102—181no—
161C4×C8C2×C3⋊D311yescharacteristic
162C22×D3—124no—
163C3×Q8—24no—
164C3×Q8#20418yescomplemented
165C2×C3×C4—24no—
166C2×C3×C4—24no—
167C2×C3×C4C2×D1214yes—
168C2×C3×C4D3×D414yes—
169C2×C3×C4D3×D414yes—
170C2×Dic3—64no—
171C2×Dic3—64no—
172C2×Dic3—68no—
173D12—128no—
174D12—124no—
175D12—124no—
176D12—124no—
177D12—124no—
178Dic6—38no—
179Dic6—64no—
180Dic6—128no—
181Dic6—248no—
182C3×C8—28no—
183C3×C8—44no—
184C3⋊C8—124no—
185C2×C32#44811yescharacteristic
186C2×C32#44811yescharacteristic
187C2×C32#45311yescharacteristic
188C3⋊D3—161no—
189C2×Q8—91no—
190C2×Q8—92no—
191C2×Q8—182no—
192C2×Q8C4⋊(C3⋊D3)11yescharacteristic
193C2×D4—181no—
194C2×D4—181no—
195C2×D4—181no—
196SD16—364no—
197SD16—362no—
198SD16—362no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—181no—
202C4⋊C4—21no—
203C4⋊C4—91no—
204C4⋊C4—182no—
205C4⋊C4—182no—
206C4⋊C4—181no—
207C4⋊C4—181no—
208C4⋊C4—181no—
209C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
210C42—21no—
211C42—182no—
212C42—181no—
213C42C22×C3⋊D311yescharacteristic
214C22×C3#88214yes—
215C2×D3—244no—
216C2×D3—244no—
217C2×D3—244no—
218C3×C4—28no—
219C3×C4—24no—
220C3×C4—44no—
221C3×C4C2×#20414yes—
222C3×C4#90914yes—
223C3×C4D3×SD1614yes—
224C3×C4#91814yes—
225Dic3—68no—
226Dic3—128no—
227Dic3—124no—
228C32#162311yescharacteristic, complemented, Sylow
229C23—361no—
230Q8—21no—
231Q8—92no—
232Q8—181no—
233Q8—362no—
234Q8—722no—
235Q8#374412yescomplemented
236D4—362no—
237D4—361no—
238D4—361no—
239D4—361no—
240D4—361no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—181no—
244C2×C4—181no—
245C2×C4—182no—
246C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
247C2×C4D4×C3⋊D311yescharacteristic
248C2×C4D4×C3⋊D311yescharacteristic
249C8—22no—
250C8—41no—
251C8—361no—
252C2×C3#473814yes—
253C2×C3#480614yes—
254C2×C3#482214yes—
255D3—484no—
256C22—721no—
257C22—721no—
258C22—721no—
259C22#6409111yescharacteristic
260C4—21no—
261C4—22no—
262C4—41no—
263C4—182no—
264C4—362no—
265C4—361no—
266C4C2×#374411yescharacteristic
267C4#6411811yescharacteristic
268C4SD16×C3⋊D311yescharacteristic
269C4#6412711yescharacteristic
270C3#6826414yescomplemented
271C2—1441no—
272C2#1059243711yescharacteristic
273C2#1059250511yescharacteristic
274C2#1059252111yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radical#64083C22
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1623
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
1. Q8×#634C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589347C2semidirect
7. #10589322C2semidirect
30. C3×Q8⋊C8D3semidirect
78. C32×Q8SD16semidirect
95. #634SD16semidirect
105. Q8⋊C8C3⋊D3semidirect
164. C3×Q8#204semidirect
228. C32#1623semidirect
235. Q8#3744semidirect
270. C3#68264semidirect

Maximal subgroups

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

subgroupindex
Q8×#6342
#105927262
#105925472
#105923922
C32×Q8⋊C82
#105893472
#105893222
#682643