Finite Group Databasebeta

← #11780087

Subgroups — #11780087

Order 1152.

720 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 131 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#10596022C211yesmaximal, characteristic
2C32×#474C211yesmaximal, characteristic, complemented
3#10592782C211yesmaximal, characteristic, complemented
4#10592777C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10589352C211yesmaximal, characteristic, complemented
7#10589349C211yesmaximal, characteristic, complemented
8#71110—34nomaximal
9C2×#3828—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4×D4C2211yescharacteristic
12C2×#3755—21no—
13#64147—21no—
14#64133—21no—
15C4⋊C4×C3⋊D3—21no—
16#64092—22no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18#64078C2211yescharacteristic
19C32×#107—21no—
20C32×C4⋊C8C2211yescharacteristic
21C32×#102—21no—
22#63662—21no—
23#63634—22no—
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4×C3⋊(C3⋊C8)C2211yescharacteristic
26#5551—34no—
27C3×#474D314yescomplemented
28#5083—34no—
29#5078—34no—
30D4⋊(C3⋊C8)—34no—
31#4486—34no—
32#4483—34no—
33C2×C32×Q8—21no—
34C2×C32×D4D411yescharacteristic
35C22×C32×C4—21no—
36C2×#638—22no—
37#3828—42no—
38#3828—42no—
39C2×C4⋊(C3⋊D3)D411yescharacteristic
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—21no—
42C2×#634—21no—
43C2×C32×C8—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4D411yescharacteristic
47C32×C22⋊C4—21no—
48C32×C42C2311yescharacteristic
49#3757—22no—
50#3755—42no—
51#3752—21no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)D412yes—
54C4⋊(C3⋊Dic3)D411yescharacteristic
55#3750—22no—
56C2×C3⋊(C3⋊C8)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58#1647—91noSylow
59C2×#235—64no—
60C3×C4⋊Q8C2×D314yes—
61C3×C4×D4C2×D314yes—
62C2×#215—64no—
63#938—64no—
64#924—64no—
65D3×C4⋊C4—64no—
66#883—68no—
67C4×D12—34no—
68#869—34no—
69C3×#107—24no—
70C3×C4⋊C8C2×D314yes—
71C3×#102—24no—
72#835—64no—
73#807—68no—
74C4⋊(C3⋊C8)—34no—
75C4×C3⋊C8—34no—
76C23×C32—21no—
77C22×C3⋊D3—21no—
78C32×Q8—41no—
79C32×Q8—41no—
80C32×D4—21no—
81C32×D4—21no—
82C2×C32×C4—21no—
83C2×C32×C4—41no—
84C2×C32×C4—41no—
85C2×C32×C4—41no—
86C2×C32×C4C2×D411yescharacteristic
87C2×C32×C4C2×D411yescharacteristic
88C2×C32×C4C2×D411yescharacteristic
89#638—44no—
90C2×C3⋊Dic3—21no—
91C2×C3⋊Dic3—22no—
92C4⋊(C3⋊D3)—21no—
93C4⋊(C3⋊D3)D812yes—
94C4×C3⋊D3—44no—
95C4×C3⋊D3—41no—
96C4×C3⋊D3—41no—
97#634—41no—
98#634—41no—
99C32×C8—41no—
100C3⋊(C3⋊C8)—22no—
101C3⋊(C3⋊C8)—41no—
102#547—91no—
103#500—91no—
104#474C3⋊D311yescharacteristic, complemented
105#461—91no—
106#332—91no—
107D4⋊C8—91no—
108D4⋊C8—91no—
109C2×C3×Q8—24no—
110C2×C3×D4#6714yes—
111C22×C3×C4—24no—
112C2×#67—68no—
113#235—128no—
114#235—128no—
115C2×D12—34no—
116C2×C4×D3—68no—
117C2×C4×D3—64no—
118C2×Dic6—64no—
119C2×C3×C8—24no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4#6714yes—
123C3×C22⋊C4—24no—
124C3×C42C22×D314yes—
125#217—68no—
126#215—128no—
127(C2×D3)⋊C4—64no—
128C4⋊Dic3—38no—
129C4⋊Dic3—34no—
130C4⋊Dic3—64no—
131Dic3⋊C4—68no—
132C2×C3⋊C8—64no—
133C2×C3⋊C8—64no—
134C22×C32—41no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32#12011yescharacteristic
138C2×C3⋊D3—22no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3—41no—
141C32×C4—21no—
142C32×C4—41no—
143C32×C4—81no—
144C32×C4C2×D811yescharacteristic
145C32×C4#13511yescharacteristic
146C32×C4AGL(1,ℤ/8)11yescharacteristic
147C32×C4#13711yescharacteristic
148C3⋊Dic3—42no—
149C3⋊Dic3—41no—
150C2×#41—181no—
151C2×SD16—181no—
152C4⋊Q8—91no—
153C4⋊Q8C2×C3⋊D311yescharacteristic
154#123—182no—
155C22⋊Q8—181no—
156#121—181no—
157C4×D4—91no—
158C4×D4C2×C3⋊D311yescharacteristic
159C2×C4⋊C4—181no—
160#107—21no—
161#107—182no—
162C4⋊C8—91no—
163C4⋊C8C2×C3⋊D311yescharacteristic
164#103—181no—
165#102—21no—
166C4×C8—91no—
167C23×C3—24no—
168C22×D3—64no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×D4—24no—
172C3×D4—24no—
173C2×C3×C4—24no—
174C2×C3×C4—44no—
175C2×C3×C4—44no—
176C2×C3×C4—44no—
177C2×C3×C4D3×D414yes—
178C2×C3×C4D3×D414yes—
179C2×C3×C4C2×#6714yes—
180#67—1216no—
181C2×Dic3—64no—
182C2×Dic3—68no—
183D12—38no—
184D12—64no—
185C4×D3—1216no—
186C4×D3—124no—
187C4×D3—124no—
188Dic6—124no—
189Dic6—124no—
190C3×C8—44no—
191C3⋊C8—68no—
192C3⋊C8—124no—
193C2×C32—81no—
194C2×C32#44711yescharacteristic
195C2×C32#45211yescharacteristic
196C2×C32#45611yescharacteristic
197C3⋊D3—42no—
198#41—362no—
199#41—362no—
200C2×Q8—21no—
201C2×Q8—181no—
202C2×D4—91no—
203C2×D4—182no—
204C2×D4#63811yescharacteristic
205C22×C4—21no—
206C22×C4—182no—
207C22×C4—181no—
208SD16—362no—
209C2×C8—21no—
210C2×C8—181no—
211C2×C8—181no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—91no—
215C4⋊C4—92no—
216C4⋊C4—182no—
217C4⋊C4—181no—
218C4⋊C4#63811yescharacteristic
219C22⋊C4—21no—
220C22⋊C4—182no—
221C22⋊C4—181no—
222C42C22×C3⋊D311yescharacteristic
223C22×C3—44no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3#93714yes—
227C2×D3—68no—
228C2×D3—124no—
229C2×D3—124no—
230C3×C4—24no—
231C3×C4—44no—
232C3×C4—84no—
233C3×C4D3×D814yes—
234C3×C4#91114yes—
235C3×C4#94214yes—
236C3×C4#95014yes—
237Dic3—128no—
238Dic3—124no—
239C32#164711yescharacteristic, complemented, Sylow
240C23—21no—
241C23—181no—
242Q8—41no—
243Q8—41no—
244Q8—361no—
245Q8—361no—
246D4—21no—
247D4—21no—
248D4—92no—
249D4—181no—
250D4—364no—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—181no—
256C2×C4—182no—
257C2×C4—364no—
258C2×C4—361no—
259C2×C4—361no—
260C2×C4D4×C3⋊D311yescharacteristic
261C2×C4D4×C3⋊D311yescharacteristic
262C2×C4C2×#63811yescharacteristic
263C8—41no—
264C8—182no—
265C8—361no—
266C2×C3—84no—
267C2×C3#504714yes—
268C2×C3#515614yes—
269C2×C3#520514yes—
270D3—128no—
271C22—41no—
272C22—41no—
273C22—41no—
274C22—182no—
275C22—361no—
276C22—361no—
277C22#6414611yescharacteristic
278C4—21no—
279C4—41no—
280C4—81no—
281C4—362no—
282C4—361no—
283C4D8×C3⋊D311yescharacteristic
284C4#6412011yescharacteristic
285C4#6415111yescharacteristic
286C4#6415911yescharacteristic
287C3#7111014yescomplemented
288C2—81no—
289C2—362no—
290C2#1059274611yescharacteristic
291C2#1059285511yescharacteristic
292C2#1059290411yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#474C2
socleC22×C32#120
prime radical#10596022C2
hypercenter#474C3⋊D3
hypocenterC32#1647
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. C32×#474C2semidirect
3. #10592782C2semidirect
4. #10592777C2semidirect
5. D4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10589352C2semidirect
7. #10589349C2semidirect
27. C3×#474D3semidirect
104. #474C3⋊D3semidirect
239. C32#1647semidirect
287. C3#71110semidirect

Maximal subgroups

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

subgroupindex
#105960222
C32×#4742
#105927822
#105927772
D4⋊(C3⋊(C3⋊C8))2
#105893522
#105893492
#711103