Finite Group Databasebeta

← #11780413

Subgroups — #11780413

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#10596197C211yesmaximal, characteristic
2C32×#498C211yesmaximal, characteristic, complemented
3#10592786C211yesmaximal, characteristic, complemented
4#10592782C211yesmaximal, characteristic, complemented
5#10589413C211yesmaximal, characteristic, complemented
6#10589325C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#71343—34nomaximal
9C2×#3832—21no—
10C32×C4⋊Q8C2211yescharacteristic
11C32×C4⋊Q8C2211yescharacteristic
12#64155—21no—
13C2×#3755—21no—
14#64104—21no—
15#64103—22no—
16C4⋊C4×C3⋊D3—21no—
17C4×C4⋊(C3⋊D3)C2211yescharacteristic
18C4×C4⋊(C3⋊D3)C2211yescharacteristic
19C32×#106—22no—
20C32×C4×C8C2211yescharacteristic
21#63662—21no—
22#63636—21no—
23#63634—21no—
24C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5726—34no—
27C3×#498D314yescomplemented
28#5087—34no—
29#5083—34no—
30#4547—34no—
31#4459—34no—
32#4459—34no—
33C2×C32×Q8—21no—
34C2×C32×Q8—21no—
35#3832—42no—
36#3832—42no—
37C2×C4⋊(C3⋊D3)—22no—
38C2×C4⋊(C3⋊D3)D411yescharacteristic
39C2×C4⋊(C3⋊D3)D411yescharacteristic
40C2×C4×C3⋊D3—22no—
41C2×C4×C3⋊D3—21no—
42C2×C4×C3⋊D3—21no—
43C2×C32×C8—21no—
44C32×C4⋊C4—21no—
45C32×C4⋊C4—21no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4D412yes—
48C32×C42C2311yescharacteristic
49#3755—42no—
50#3752—22no—
51#3752—21no—
52#3752—21no—
53C4⋊(C3⋊Dic3)D411yescharacteristic
54C4⋊(C3⋊Dic3)D411yescharacteristic
55#3750—22no—
56C2×C3⋊(C3⋊C8)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58#1654—91noSylow
59C2×#239—64no—
60C3×C4⋊Q8C2×D314yes—
61C3×C4⋊Q8C2×D314yes—
62#946—64no—
63C2×#215—64no—
64#895—64no—
65#894—68no—
66D3×C4⋊C4—64no—
67C4×D12—34no—
68C4×D12—34no—
69C3×#106—28no—
70C3×C4×C8C2×D314yes—
71#835—64no—
72#809—64no—
73#807—64no—
74C4⋊(C3⋊C8)—34no—
75C4⋊(C3⋊C8)—34no—
76C22×C3⋊D3—21no—
77C22×C3⋊D3—21no—
78C32×Q8—41no—
79C32×Q8—41no—
80C32×Q8—41no—
81C32×Q8—41no—
82C2×C32×C4—22no—
83C2×C32×C4—41no—
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D411yescharacteristic
87C2×C3⋊Dic3—21no—
88C2×C3⋊Dic3—21no—
89C4⋊(C3⋊D3)—21no—
90C4⋊(C3⋊D3)—21no—
91C4⋊(C3⋊D3)—21no—
92C4⋊(C3⋊D3)—44no—
93C4⋊(C3⋊D3)SD1612yescomplemented
94C4×C3⋊D3—44no—
95C4×C3⋊D3—41no—
96C4×C3⋊D3—41no—
97C4×C3⋊D3—41no—
98C4×C3⋊D3—41no—
99C32×C8—22no—
100C3⋊(C3⋊C8)—41no—
101C3⋊(C3⋊C8)—41no—
102#547—91no—
103#498C3⋊D311yescharacteristic, complemented
104#474—91no—
105#461—91no—
106#332—91no—
107D4⋊C8—91no—
108D4⋊C8—91no—
109C2×C3×Q8—24no—
110C2×C3×Q8—24no—
111#239—128no—
112#239—128no—
113C2×D12—34no—
114C2×D12—34no—
115C2×D12—68no—
116C2×C4×D3—68no—
117C2×C4×D3—64no—
118C2×C4×D3—64no—
119C2×C3×C8—24no—
120C3×C4⋊C4—24no—
121C3×C4⋊C4—24no—
122C3×C4⋊C4—24no—
123C3×C4⋊C4#6718yes—
124C3×C42C22×D314yes—
125#215—128no—
126(C2×D3)⋊C4—68no—
127(C2×D3)⋊C4—64no—
128(C2×D3)⋊C4—64no—
129C4⋊Dic3—34no—
130C4⋊Dic3—34no—
131Dic3⋊C4—68no—
132C2×C3⋊C8—64no—
133C2×C3⋊C8—64no—
134C22×C32#12011yescharacteristic
135C2×C3⋊D3—22no—
136C2×C3⋊D3—41no—
137C2×C3⋊D3—41no—
138C2×C3⋊D3—41no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3—41no—
141C32×C4—21no—
142C32×C4—42no—
143C32×C4—81no—
144C32×C4C2×SD1611yescharacteristic
145C32×C4#13511yescharacteristic
146C32×C4AGL(1,ℤ/8)11yescharacteristic
147C32×C4AGL(1,ℤ/8)11yescharacteristic
148C3⋊Dic3—41no—
149C3⋊Dic3—41no—
150C2×#41—181no—
151C2×SD16—181no—
152C4⋊Q8C2×C3⋊D311yescharacteristic
153C4⋊Q8C2×C3⋊D311yescharacteristic
154#123—182no—
155C22⋊Q8—181no—
156#121—181no—
157C4×D4—91no—
158C4×D4—91no—
159C2×C4⋊C4—181no—
160#107—181no—
161#106—22no—
162C4⋊C8—91no—
163C4⋊C8—91no—
164#103—181no—
165#102—181no—
166C4×C8C2×C3⋊D311yescharacteristic
167C22×D3—64no—
168C22×D3—64no—
169C3×Q8—44no—
170C3×Q8—44no—
171C3×Q8—44no—
172C3×Q8—44no—
173C2×C3×C4—28no—
174C2×C3×C4—44no—
175C2×C3×C4D3×D414yes—
176C2×C3×C4D3×D414yes—
177C2×C3×C4C2×#6714yes—
178C2×Dic3—64no—
179C2×Dic3—64no—
180D12—38no—
181D12—64no—
182D12—64no—
183D12—64no—
184D12—1216no—
185C4×D3—1216no—
186C4×D3—124no—
187C4×D3—124no—
188C4×D3—124no—
189C4×D3—124no—
190C3×C8—28no—
191C3⋊C8—124no—
192C3⋊C8—124no—
193C2×C32#44911yescharacteristic
194C2×C32#45011yescharacteristic
195C2×C32#45611yescharacteristic
196C3⋊D3—42no—
197C3⋊D3—81no—
198#41—362no—
199#41—362no—
200C2×Q8—21no—
201C2×Q8—21no—
202C2×D4—91no—
203C2×D4—91no—
204C2×D4—182no—
205C22×C4—182no—
206C22×C4—181no—
207C22×C4—181no—
208SD16—362no—
209C2×C8—21no—
210C2×C8—181no—
211C2×C8—181no—
212C4⋊C4—21no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—91no—
216C4⋊C4—91no—
217C4⋊C4—182no—
218C4⋊C4#63812yes—
219C22⋊C4—182no—
220C22⋊C4—181no—
221C22⋊C4—181no—
222C42C22×C3⋊D311yescharacteristic
223C22×C3#93714yes—
224C2×D3—68no—
225C2×D3—124no—
226C2×D3—124no—
227C2×D3—124no—
228C2×D3—124no—
229C2×D3—124no—
230C3×C4—24no—
231C3×C4—48no—
232C3×C4—84no—
233C3×C4D3×SD1614yes—
234C3×C4#91414yes—
235C3×C4#91414yes—
236C3×C4#91614yes—
237Dic3—124no—
238Dic3—124no—
239C32#165411yescharacteristic, complemented, Sylow
240C23—181no—
241C23—181no—
242Q8—41no—
243Q8—41no—
244Q8—41no—
245Q8—41no—
246D4—92no—
247D4—181no—
248D4—181no—
249D4—181no—
250D4—364no—
251C2×C4—22no—
252C2×C4—41no—
253C2×C4—181no—
254C2×C4—181no—
255C2×C4—364no—
256C2×C4—361no—
257C2×C4—361no—
258C2×C4—361no—
259C2×C4—361no—
260C2×C4D4×C3⋊D311yescharacteristic
261C2×C4D4×C3⋊D311yescharacteristic
262C2×C4C2×#63811yescharacteristic
263C8—22no—
264C8—361no—
265C8—361no—
266C2×C3#509514yes—
267C2×C3#516914yes—
268C2×C3#517214yes—
269D3—128no—
270D3—244no—
271C22—182no—
272C22—361no—
273C22—361no—
274C22—361no—
275C22—361no—
276C22—361no—
277C22#6414611yescharacteristic
278C4—21no—
279C4—42no—
280C4—81no—
281C4—361no—
282C4—361no—
283C4SD16×C3⋊D311yescharacteristic
284C4#6412311yescharacteristic
285C4#6412311yescharacteristic
286C4#6412511yescharacteristic
287C3#7134314yescomplemented
288C2—362no—
289C2—721no—
290C2#1059279411yescharacteristic
291C2#1059286811yescharacteristic
292C2#1059287111yescharacteristic
subgroupG/H
centerC22#64146
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×#498C2
socleC22×C32#120
prime radical#10596197C2
hypercenter#498C3⋊D3
hypocenterC32#1654
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #498

Complements

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

NH ≅ G/Ntype
2. C32×#498C2semidirect
3. #10592786C2semidirect
4. #10592782C2semidirect
5. #10589413C2semidirect
6. #10589325C2semidirect
7. #10589325C2semidirect
27. C3×#498D3semidirect
93. C4⋊(C3⋊D3)SD16semidirect
103. #498C3⋊D3semidirect
239. C32#1654semidirect
287. C3#71343semidirect

Maximal subgroups

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

subgroupindex
#105961972
C32×#4982
#105927862
#105927822
#105894132
#105893252
#713433