Finite Group Databasebeta

← #11779514

Subgroups — #11779514

Order 1152.

702 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 141 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#10596026C211yesmaximal, characteristic
2C32×#476C211yesmaximal, characteristic, complemented
3#10592783C211yesmaximal, characteristic, complemented
4#10592778C211yesmaximal, characteristic, complemented
5D4⋊(C3⋊(C3⋊C8))C211yesmaximal, characteristic, complemented
6#10589349C211yesmaximal, characteristic, complemented
7#10589348C211yesmaximal, characteristic, complemented
8#71146—34nomaximal
9C2×#3830—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4×D4C2211yescharacteristic
12#64139—21no—
13C2×C4⋊(C3⋊Dic3)—21no—
14#64106—21no—
15#64095—22no—
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17#64078C2211yescharacteristic
18C32×#106—21no—
19C32×C4⋊C8C2211yescharacteristic
20C32×#102—21no—
21#63662—22no—
22#63636—21no—
23#63635—21no—
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4×C3⋊(C3⋊C8)C2211yescharacteristic
26#5555—34no—
27C3×#476D314yescomplemented
28#5084—34no—
29#5079—34no—
30D4⋊(C3⋊C8)—34no—
31#4483—34no—
32#4482—34no—
33C2×C32×Q8—21no—
34C2×C32×D4D411yescharacteristic
35C22×C32×C4—21no—
36C2×#638—22no—
37C22×C3⋊Dic3—22no—
38#3830—42no—
39#3830—42no—
40C2×C4⋊(C3⋊D3)D411yescharacteristic
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#3752—22no—
50#3752—21no—
51C4⋊(C3⋊Dic3)—22no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)D412yes—
54C4⋊(C3⋊Dic3)D411yescharacteristic
55C2×C3⋊(C3⋊C8)—21no—
56C2×C3⋊(C3⋊C8)—21no—
57#1635—91noSylow
58C2×#237—64no—
59C3×C4⋊Q8C2×D314yes—
60C3×C4×D4C2×D314yes—
61#930—64no—
62C2×C4⋊Dic3—64no—
63#897—64no—
64#886—68no—
65C4×D12—34no—
66#869—34no—
67C3×#106—24no—
68C3×C4⋊C8C2×D314yes—
69C3×#102—24no—
70#835—68no—
71#809—64no—
72#808—64no—
73C4⋊(C3⋊C8)—34no—
74C4×C3⋊C8—34no—
75C23×C32—21no—
76C22×C3⋊D3—21no—
77C32×Q8—42no—
78C32×D4—21no—
79C32×D4SD1612yes—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4—41no—
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D411yescharacteristic
87#638—44no—
88C2×C3⋊Dic3—22no—
89C2×C3⋊Dic3—21no—
90C2×C3⋊Dic3—44no—
91C4⋊(C3⋊D3)—21no—
92C4⋊(C3⋊D3)—21no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3—41no—
95#634—41no—
96#634—41no—
97C32×C8—41no—
98C3⋊(C3⋊C8)—41no—
99C3⋊(C3⋊C8)—41no—
100#547—91no—
101#487—91no—
102#476—91no—
103#476C3⋊D311yescharacteristic, complemented
104#332—91no—
105D4⋊C8—91no—
106D4⋊C8—91no—
107C2×C3×Q8—24no—
108C2×C3×D4#6714yes—
109C22×C3×C4—24no—
110C2×#67—68no—
111C22×Dic3—68no—
112#237—128no—
113#237—128no—
114C2×D12—34no—
115C2×C4×D3—64no—
116C2×Dic6—64no—
117C2×C3×C8—24no—
118C3×C4⋊C4—24no—
119C3×C4⋊C4—24no—
120C3×C4⋊C4#6714yes—
121C3×C22⋊C4—24no—
122C3×C42C22×D314yes—
123(C2×D3)⋊C4—68no—
124(C2×D3)⋊C4—64no—
125C4⋊Dic3—34no—
126C4⋊Dic3—38no—
127C4⋊Dic3—68no—
128C4⋊Dic3—64no—
129C2×C3⋊C8—64no—
130C2×C3⋊C8—64no—
131C22×C32—22no—
132C22×C32—41no—
133C22×C32—41no—
134C22×C32#12011yescharacteristic
135C2×C3⋊D3—41no—
136C2×C3⋊D3—41no—
137C2×C3⋊D3—41no—
138C32×C4—21no—
139C32×C4—41no—
140C32×C4—81no—
141C32×C4C2×SD1611yescharacteristic
142C32×C4#13511yescharacteristic
143C32×C4AGL(1,ℤ/8)11yescharacteristic
144C32×C4#13711yescharacteristic
145C3⋊Dic3—42no—
146C3⋊Dic3—41no—
147C2×#41—181no—
148C4⋊Q8—91no—
149C4⋊Q8C2×C3⋊D311yescharacteristic
150#123—182no—
151C22⋊Q8—181no—
152#121—181no—
153C4×D4—91no—
154C4×D4C2×C3⋊D311yescharacteristic
155C2×C4⋊C4—181no—
156#106—21no—
157#106—181no—
158C4⋊C8—91no—
159C4⋊C8C2×C3⋊D311yescharacteristic
160#103—182no—
161#102—21no—
162#102—181no—
163C4×C8—91no—
164C23×C3—24no—
165C22×D3—64no—
166C3×Q8—48no—
167C3×D4—24no—
168C3×D4#21518yes—
169C2×C3×C4—24no—
170C2×C3×C4—44no—
171C2×C3×C4—44no—
172C2×C3×C4—44no—
173C2×C3×C4D3×D414yes—
174C2×C3×C4D3×D414yes—
175C2×C3×C4C2×#6714yes—
176#67—1216no—
177C2×Dic3—68no—
178C2×Dic3—64no—
179C2×Dic3—1216no—
180D12—64no—
181D12—64no—
182C4×D3—124no—
183C4×D3—124no—
184Dic6—124no—
185Dic6—124no—
186C3×C8—44no—
187C3⋊C8—124no—
188C3⋊C8—124no—
189C2×C32—42no—
190C2×C32#45011yescharacteristic
191C2×C32#45211yescharacteristic
192C2×C32#45411yescharacteristic
193C3⋊D3—81no—
194#41—362no—
195#41—362no—
196C2×Q8—21no—
197C2×Q8—181no—
198C2×D4—91no—
199C2×D4—182no—
200C2×D4#63811yescharacteristic
201C22×C4—21no—
202C22×C4—182no—
203C22×C4—181no—
204C2×C8—21no—
205C2×C8—181no—
206C2×C8—181no—
207C4⋊C4—21no—
208C4⋊C4—21no—
209C4⋊C4—92no—
210C4⋊C4—91no—
211C4⋊C4—182no—
212C4⋊C4—181no—
213C4⋊C4#63811yescharacteristic
214C22⋊C4—21no—
215C22⋊C4—182no—
216C22⋊C4—181no—
217C42C22×C3⋊D311yescharacteristic
218C22×C3—28no—
219C22×C3—44no—
220C22×C3—44no—
221C22×C3#93714yes—
222C2×D3—124no—
223C2×D3—124no—
224C2×D3—124no—
225C3×C4—24no—
226C3×C4—44no—
227C3×C4—84no—
228C3×C4#91514yes—
229C3×C4#91614yes—
230C3×C4C2×#21514yes—
231C3×C4#94914yes—
232Dic3—128no—
233Dic3—124no—
234C32#163511yescharacteristic, complemented, Sylow
235C23—21no—
236C23—181no—
237Q8—42no—
238Q8—361no—
239Q8—361no—
240D4—21no—
241D4—181no—
242D4—181no—
243D4—364no—
244D4#375512yes—
245C2×C4—21no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—181no—
250C2×C4—182no—
251C2×C4—364no—
252C2×C4—361no—
253C2×C4—361no—
254C2×C4D4×C3⋊D311yescharacteristic
255C2×C4D4×C3⋊D311yescharacteristic
256C2×C4C2×#63811yescharacteristic
257C8—41no—
258C8—361no—
259C8—361no—
260C2×C3—48no—
261C2×C3#504614yes—
262C2×C3#517314yes—
263C2×C3#520114yes—
264D3—244no—
265C22—22no—
266C22—41no—
267C22—41no—
268C22—361no—
269C22—361no—
270C22—361no—
271C22#6414611yescharacteristic
272C4—21no—
273C4—41no—
274C4—81no—
275C4—361no—
276C4—362no—
277C4#6412411yescharacteristic
278C4#6412511yescharacteristic
279C4C2×#375511yescharacteristic
280C4#6415811yescharacteristic
281C3#7114614yescomplemented
282C2—42no—
283C2—721no—
284C2#1059274511yescharacteristic
285C2#1059287211yescharacteristic
286C2#1059290011yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#476C2
socleC22×C32#120
prime radical#10596026C2
hypercenter#476C3⋊D3
hypocenterC32#1635
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. C32×#476C2semidirect
3. #10592783C2semidirect
4. #10592778C2semidirect
5. D4⋊(C3⋊(C3⋊C8))C2semidirect
6. #10589349C2semidirect
7. #10589348C2semidirect
27. C3×#476D3semidirect
103. #476C3⋊D3semidirect
234. C32#1635semidirect
281. C3#71146semidirect

Maximal subgroups

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

subgroupindex
#105960262
C32×#4762
#105927832
#105927782
D4⋊(C3⋊(C3⋊C8))2
#105893492
#105893482
#711463