Finite Group Databasebeta

← #11780096

Subgroups — #11780096

Order 1152.

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

normal subgroups only

This group has 720 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
2#10592718C211yesmaximal, characteristic, complemented
3#10592545C211yesmaximal, characteristic, complemented
4#10592381C211yesmaximal, characteristic, complemented
5C32×D4⋊C8C211yesmaximal, characteristic, complemented
6#10589352C211yesmaximal, characteristic, complemented
7#10589325C211yesmaximal, characteristic, complemented
8#68208—34nomaximal
9C2×#3830—21no—
10C32×C4×D4C2211yescharacteristic
11C2×#3754—21no—
12#64139—21no—
13C2×C4⋊(C3⋊Dic3)—21no—
14#64106—21no—
15#64095—22no—
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17#64078C2211yescharacteristic
18#64078C2211yescharacteristic
19C32×C4⋊C8C2211yescharacteristic
20C32×C4×C8C2211yescharacteristic
21#63648—21no—
22#63645—21no—
23#63645—22no—
24#63637—21no—
25C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
26#5555—34no—
27#5019—34no—
28#4846—34no—
29#4682—34no—
30C3×D4⋊C8D314yescomplemented
31#4486—34no—
32#4459—34no—
33C2×C32×D4D411yescharacteristic
34C22×C32×C4—21no—
35C2×#638—22no—
36C22×C3⋊Dic3—22no—
37#3830—42no—
38#3830—42no—
39C2×C4⋊(C3⋊D3)D411yescharacteristic
40C2×C4×C3⋊D3—21no—
41C2×#634—21no—
42C2×#634—21no—
43C2×C32×C8—21no—
44C2×C32×C8—21no—
45C32×C4⋊C4D411yescharacteristic
46C32×C22⋊C4—21no—
47C32×C42C2311yescharacteristic
48#3754—42no—
49#3752—22no—
50#3752—21no—
51C4⋊(C3⋊Dic3)—22no—
52C4⋊(C3⋊Dic3)—21no—
53C4⋊(C3⋊Dic3)—21no—
54C4⋊(C3⋊Dic3)—21no—
55C4⋊(C3⋊Dic3)D412yes—
56C4⋊(C3⋊Dic3)D411yescharacteristic
57C2×C3⋊(C3⋊C8)—21no—
58#1647—91noSylow
59C2×#237—64no—
60C3×C4×D4C2×D314yes—
61C2×#214—64no—
62#930—64no—
63C2×C4⋊Dic3—64no—
64#897—64no—
65#886—68no—
66C4×D12—34no—
67#869—34no—
68#869—34no—
69C3×C4⋊C8C2×D314yes—
70C3×C4×C8C2×D314yes—
71#821—64no—
72#818—68no—
73#818—64no—
74#810—64no—
75C4⋊(C3⋊C8)—34no—
76C23×C32—21no—
77C22×C3⋊D3—21no—
78C32×D4—21no—
79C32×D4D812yes—
80C2×C32×C4—21no—
81C2×C32×C4—41no—
82C2×C32×C4—41no—
83C2×C32×C4C2×D411yescharacteristic
84C2×C32×C4C2×D411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86#638—44no—
87C2×C3⋊Dic3—22no—
88C2×C3⋊Dic3—21no—
89C2×C3⋊Dic3—44no—
90C2×C3⋊Dic3—41no—
91C4⋊(C3⋊D3)—21no—
92C4⋊(C3⋊D3)—21no—
93C4×C3⋊D3—41no—
94C4×C3⋊D3—41no—
95#634—41no—
96#634—41no—
97#634—41no—
98#634—41no—
99C32×C8—22no—
100C32×C8—41no—
101C3⋊(C3⋊C8)—41no—
102#547—91no—
103#500—91no—
104#474—91no—
105#461—91no—
106#332—91no—
107D4⋊C8—91no—
108D4⋊C8C3⋊D311yescharacteristic, complemented
109C2×C3×D4D1214yes—
110C22×C3×C4—24no—
111C2×#67—68no—
112C22×Dic3—68no—
113#237—128no—
114#237—128no—
115C2×D12—34no—
116C2×C4×D3—64no—
117C2×Dic6—64no—
118C2×Dic6—64no—
119C2×C3×C8—24no—
120C2×C3×C8—24no—
121C3×C4⋊C4D1214yes—
122C3×C22⋊C4—24no—
123C3×C42C22×D314yes—
124#214—128no—
125(C2×D3)⋊C4—68no—
126(C2×D3)⋊C4—64no—
127C4⋊Dic3—38no—
128C4⋊Dic3—34no—
129C4⋊Dic3—68no—
130C4⋊Dic3—64no—
131C4⋊Dic3—64no—
132C4⋊Dic3—64no—
133C2×C3⋊C8—64no—
134C22×C32—22no—
135C22×C32—41no—
136C22×C32—41no—
137C22×C32#12011yescharacteristic
138C2×C3⋊D3—41no—
139C2×C3⋊D3—41no—
140C2×C3⋊D3—41no—
141C32×C4—21no—
142C32×C4—41no—
143C32×C4C2×D811yescharacteristic
144C32×C4#13511yescharacteristic
145C32×C4AGL(1,ℤ/8)11yescharacteristic
146C32×C4#13711yescharacteristic
147C3⋊Dic3—42no—
148C3⋊Dic3—41no—
149C3⋊Dic3—81no—
150C2×#41—181no—
151C2×SD16—181no—
152C4⋊Q8—91no—
153C4⋊Q8—91no—
154#123—182no—
155C22⋊Q8—181no—
156#121—181no—
157C4×D4—91no—
158C4×D4C2×C3⋊D311yescharacteristic
159C2×C4⋊C4—181no—
160#107—181no—
161#107—182no—
162C4⋊C8—91no—
163C4⋊C8C2×C3⋊D311yescharacteristic
164#103—181no—
165#102—181no—
166C4×C8C2×C3⋊D311yescharacteristic
167C23×C3—24no—
168C22×D3—64no—
169C3×D4—24no—
170C3×D4D2418yes—
171C2×C3×C4—24no—
172C2×C3×C4—44no—
173C2×C3×C4—44no—
174C2×C3×C4C2×D1214yes—
175C2×C3×C4D3×D414yes—
176C2×C3×C4D3×D414yes—
177#67—1216no—
178C2×Dic3—68no—
179C2×Dic3—64no—
180C2×Dic3—1216no—
181C2×Dic3—124no—
182D12—64no—
183D12—64no—
184C4×D3—124no—
185C4×D3—124no—
186Dic6—124no—
187Dic6—124no—
188Dic6—124no—
189Dic6—124no—
190C3×C8—28no—
191C3×C8—44no—
192C3⋊C8—124no—
193C2×C32—42no—
194C2×C32#44711yescharacteristic
195C2×C32#45211yescharacteristic
196C2×C32#45611yescharacteristic
197C3⋊D3—81no—
198#41—362no—
199#41—362no—
200C2×Q8—181no—
201C2×Q8—181no—
202C2×D4—91no—
203C2×D4—182no—
204C2×D4C4⋊(C3⋊D3)11yescharacteristic
205C22×C4—21no—
206C22×C4—182no—
207C22×C4—181no—
208SD16—362no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—181no—
212C4⋊C4—91no—
213C4⋊C4—92no—
214C4⋊C4—182no—
215C4⋊C4—181no—
216C4⋊C4—181no—
217C4⋊C4—181no—
218C4⋊C4C4⋊(C3⋊D3)11yescharacteristic
219C22⋊C4—21no—
220C22⋊C4—182no—
221C22⋊C4—181no—
222C42C22×C3⋊D311yescharacteristic
223C22×C3—28no—
224C22×C3—44no—
225C22×C3—44no—
226C22×C3#88214yes—
227C2×D3—124no—
228C2×D3—124no—
229C2×D3—124no—
230C3×C4—24no—
231C3×C4—44no—
232C3×C4C2×D2414yes—
233C3×C4#90814yes—
234C3×C4#91214yes—
235C3×C4#91514yes—
236Dic3—128no—
237Dic3—124no—
238Dic3—244no—
239C32#164711yescharacteristic, complemented, Sylow
240C23—21no—
241C23—181no—
242Q8—361no—
243Q8—361no—
244Q8—361no—
245Q8—361no—
246D4—21no—
247D4—181no—
248D4—181no—
249D4—364no—
250D4#374512yes—
251C2×C4—21no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—181no—
255C2×C4—182no—
256C2×C4—364no—
257C2×C4—361no—
258C2×C4—361no—
259C2×C4—361no—
260C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
261C2×C4D4×C3⋊D311yescharacteristic
262C2×C4D4×C3⋊D311yescharacteristic
263C8—22no—
264C8—41no—
265C8—361no—
266C2×C3—48no—
267C2×C3#473214yes—
268C2×C3#478314yes—
269C2×C3#482614yes—
270D3—244no—
271C22—22no—
272C22—41no—
273C22—41no—
274C22—361no—
275C22—361no—
276C22—361no—
277C22#6409111yescharacteristic
278C4—21no—
279C4—41no—
280C4—361no—
281C4—362no—
282C4—721no—
283C4C2×#374511yescharacteristic
284C4#6411711yescharacteristic
285C4#6412111yescharacteristic
286C4#6412411yescharacteristic
287C3#6820814yescomplemented
288C2—42no—
289C2—721no—
290C2#1059243111yescharacteristic
291C2#1059248211yescharacteristic
292C2#1059252511yescharacteristic
subgroupG/H
centerC22#64091
commutator, cosocleC32×C42C23
FrattiniC42C22×C3⋊D3
FittingC32×D4⋊C8C2
socleC22×C32#120
prime radical#10596026C2
hypercenterD4⋊C8C3⋊D3
hypocenterC32#1647
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ D4⋊C8

Complements

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

NH ≅ G/Ntype
2. #10592718C2semidirect
3. #10592545C2semidirect
4. #10592381C2semidirect
5. C32×D4⋊C8C2semidirect
6. #10589352C2semidirect
7. #10589325C2semidirect
30. C3×D4⋊C8D3semidirect
108. D4⋊C8C3⋊D3semidirect
239. C32#1647semidirect
287. C3#68208semidirect

Maximal subgroups

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

subgroupindex
#105960262
#105927182
#105925452
#105923812
C32×D4⋊C82
#105893522
#105893252
#682083