Finite Group Databasebeta

← #11779211

Subgroups — #11779211

Order 1152.

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

normal subgroups only

This group has 750 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×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2#10592726C211yesmaximal, characteristic, complemented
3#10592542C211yesmaximal, characteristic
4#10592393C211yesmaximal, characteristic, complemented
5C32×Q8⋊C8C211yesmaximal, characteristic, complemented
6#10589352C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#68266—34nomaximal
9C2×Q8×C3⋊D3—21no—
10C32×C4×Q8C2211yescharacteristic
11C2×#3755—21no—
12C2×#3745—21no—
13#64106—21no—
14#64106—22no—
15#64083C2211yescharacteristic
16C4×C4⋊(C3⋊D3)—21no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078—21no—
19#64078C2211yescharacteristic
20C32×C4⋊C8C2211yescharacteristic
21C32×C4×C8C2211yescharacteristic
22#63648—21no—
23#63648—22no—
24#63636—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26Q8×D12—34no—
27#5027—34no—
28#4843—34no—
29#4694—34no—
30C3×Q8⋊C8D314yescomplemented
31#4486—34no—
32#4459—34no—
33C2×C32×Q8D411yescharacteristic
34Q8×C3⋊D3—42no—
35Q8×C3⋊D3—42no—
36C2×C4⋊(C3⋊D3)—21no—
37C2×C4⋊(C3⋊D3)—21no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)D411yescharacteristic
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—21no—
42C2×#634—22no—
43C2×#634—21no—
44C2×C32×C8—21no—
45C2×C32×C8—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4D411yescharacteristic
48C32×C42—21no—
49C32×C42C2311yescharacteristic
50#3755—42no—
51#3752—22no—
52#3752—21no—
53C4⋊(C3⋊Dic3)—22no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)D411yescharacteristic
56C4⋊(C3⋊Dic3)D412yes—
57C2×C3⋊(C3⋊C8)—21no—
58#3745—42no—
59#1627—91noSylow
60C2×D3×Q8—64no—
61C3×C4×Q8C2×D314yes—
62C2×#215—64no—
63C2×D24—64no—
64#897—68no—
65#897—64no—
66#874—34no—
67C4×D12—34no—
68C4×D12—64no—
69#869—34no—
70#869—64no—
71C3×C4⋊C8C2×D314yes—
72C3×C4×C8C2×D314yes—
73#821—68no—
74#821—64no—
75#809—64no—
76C4⋊(C3⋊C8)—34no—
77C22×C3⋊D3—21no—
78C22×C3⋊D3—41no—
79C32×Q8—21no—
80C32×Q8D812yescomplemented
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C3⋊Dic3—21no—
87C2×C3⋊Dic3—22no—
88C4⋊(C3⋊D3)—21no—
89C4⋊(C3⋊D3)—42no—
90C4⋊(C3⋊D3)—41no—
91C4⋊(C3⋊D3)—41no—
92C4⋊(C3⋊D3)—41no—
93C4⋊(C3⋊D3)—41no—
94C4⋊(C3⋊D3)SD1612yescomplemented
95C4×C3⋊D3—44no—
96C4×C3⋊D3—41no—
97C4×C3⋊D3—41no—
98#634—44no—
99#634—41no—
100#634—41no—
101C32×C8—41no—
102C32×C8—41no—
103C3⋊(C3⋊C8)—41no—
104D4×Q8—91no—
105#486—91no—
106#460—91no—
107#459—91no—
108#332—91no—
109Q8⋊C8C3⋊D311yescharacteristic, complemented
110D4⋊C8—91no—
111C2×C3×Q8D1214yes—
112D3×Q8—128no—
113D3×Q8—128no—
114C2×D12—34no—
115C2×D12—64no—
116C2×D12—64no—
117C2×D12—64no—
118C2×C4×D3—68no—
119C2×C4×D3—64no—
120C2×Dic6—68no—
121C2×Dic6—64no—
122C2×C3×C8—24no—
123C2×C3×C8—24no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4D1214yes—
126C3×C42—24no—
127C3×C42C22×D314yes—
128#215—128no—
129(C2×D3)⋊C4—68no—
130(C2×D3)⋊C4—64no—
131C4⋊Dic3—38no—
132C4⋊Dic3—34no—
133C4⋊Dic3—68no—
134C4⋊Dic3—64no—
135C2×C3⋊C8—64no—
136D24—128no—
137C22×C32#12011yescharacteristic
138C2×C3⋊D3—22no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3—41no—
141C2×C3⋊D3—81no—
142C2×C3⋊D3—81no—
143C2×C3⋊D3—81no—
144C32×C4—22no—
145C32×C4—21no—
146C32×C4—41no—
147C32×C4C2×D811yescharacteristic
148C32×C4C2×SD1611yescharacteristic
149C32×C4AGL(1,ℤ/8)11yescharacteristic
150C32×C4#13711yescharacteristic
151C3⋊Dic3—41no—
152C3⋊Dic3—42no—
153C22×Q8—181no—
154C2×SD16—181no—
155C2×D8—181no—
156C4⋊Q8—91no—
157C4⋊Q8—181no—
158#127—91no—
159C22⋊Q8—182no—
160C22⋊Q8—181no—
161C4×Q8C2×C3⋊D311yescharacteristic
162C4×D4—91no—
163C4×D4—181no—
164C4⋊C8—91no—
165C4⋊C8C2×C3⋊D311yescharacteristic
166#102—181no—
167#102—182no—
168#102—181no—
169C4×C8C2×C3⋊D311yescharacteristic
170C22×D3—64no—
171C22×D3—124no—
172C3×Q8—24no—
173C3×Q8D2418yescomplemented
174C2×C3×C4—24no—
175C2×C3×C4—24no—
176C2×C3×C4C2×D1214yes—
177C2×C3×C4D3×D414yes—
178C2×C3×C4D3×D414yes—
179C2×Dic3—68no—
180C2×Dic3—64no—
181D12—38no—
182D12—64no—
183D12—128no—
184D12—124no—
185D12—124no—
186D12—124no—
187D12—124no—
188C4×D3—1216no—
189C4×D3—124no—
190C4×D3—124no—
191Dic6—1216no—
192Dic6—124no—
193Dic6—124no—
194C3×C8—44no—
195C3×C8—44no—
196C3⋊C8—124no—
197C2×C32#44711yescharacteristic
198C2×C32#44811yescharacteristic
199C2×C32#45411yescharacteristic
200C3⋊D3—42no—
201C3⋊D3—161no—
202C2×Q8—182no—
203C2×Q8—181no—
204C2×Q8—362no—
205C2×Q8—362no—
206C2×Q8C4⋊(C3⋊D3)11yescharacteristic
207C2×D4—91no—
208C2×D4—181no—
209C2×D4—181no—
210C2×D4—181no—
211C22×C4—182no—
212C22×C4—181no—
213SD16—362no—
214D8—362no—
215C2×C8—21no—
216C2×C8—21no—
217C2×C8—181no—
218C4⋊C4—21no—
219C4⋊C4—91no—
220C4⋊C4—92no—
221C4⋊C4—182no—
222C4⋊C4—181no—
223C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
224C22⋊C4—182no—
225C22⋊C4—181no—
226C42—21no—
227C42C22×C3⋊D311yescharacteristic
228C22×C3#88214yes—
229C2×D3—68no—
230C2×D3—124no—
231C2×D3—124no—
232C2×D3—244no—
233C2×D3—244no—
234C2×D3—244no—
235C3×C4—28no—
236C3×C4—24no—
237C3×C4—44no—
238C3×C4C2×D2414yes—
239C3×C4#90814yes—
240C3×C4D3×SD1614yes—
241C3×C4#91814yes—
242Dic3—124no—
243Dic3—128no—
244C32#162711yescharacteristic, complemented, Sylow
245C23—181no—
246C23—361no—
247Q8—21no—
248Q8—364no—
249Q8—361no—
250Q8—361no—
251Q8#374512yescomplemented
252D4—92no—
253D4—181no—
254D4—362no—
255D4—361no—
256D4—361no—
257D4—361no—
258D4—361no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—182no—
262C2×C4—181no—
263C2×C4—364no—
264C2×C4—361no—
265C2×C4—361no—
266C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
267C2×C4D4×C3⋊D311yescharacteristic
268C2×C4D4×C3⋊D311yescharacteristic
269C8—41no—
270C8—41no—
271C8—361no—
272C2×C3#473214yes—
273C2×C3#480614yes—
274C2×C3#482714yes—
275D3—128no—
276D3—484no—
277C22—182no—
278C22—361no—
279C22—361no—
280C22—721no—
281C22—721no—
282C22—721no—
283C22#6409111yescharacteristic
284C4—21no—
285C4—22no—
286C4—41no—
287C4—362no—
288C4—361no—
289C4C2×#374511yescharacteristic
290C4#6411711yescharacteristic
291C4SD16×C3⋊D311yescharacteristic
292C4#6412711yescharacteristic
293C3#6826614yescomplemented
294C2—362no—
295C2—1441no—
296C2#1059243111yescharacteristic
297C2#1059250511yescharacteristic
298C2#1059252611yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×Q8⋊C8C2
socleC22×C32#120
prime radical#10592542C2
hypercenterQ8⋊C8C3⋊D3
hypocenterC32#1627
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×C4⋊(C3⋊D3)C2semidirect
2. #10592726C2semidirect
4. #10592393C2semidirect
5. C32×Q8⋊C8C2semidirect
6. #10589352C2semidirect
7. #10589325C2semidirect
30. C3×Q8⋊C8D3semidirect
80. C32×Q8D8semidirect
94. C4⋊(C3⋊D3)SD16semidirect
109. Q8⋊C8C3⋊D3semidirect
173. C3×Q8D24semidirect
244. C32#1627semidirect
251. Q8#3745semidirect
293. C3#68266semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊(C3⋊D3)2
#105927262
#105925422
#105923932
C32×Q8⋊C82
#105893522
#105893252
#682663