Finite Group Databasebeta

← #11788198

Subgroups — #11788198

Order 1152.

1074 conjugacy classes of subgroups; only the 241 normal subgroups are shown.

normal subgroups only

This group has 1074 conjugacy classes of subgroups — too many to draw. Only its 241 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
1C2×C32×#117C211yesmaximal, characteristic, complemented
2#10596262C211yesmaximal, characteristic
3#10592906C211yesmaximal, characteristic, complemented
4C2×#63632C212yesmaximal, complemented
5#10589389C212yesmaximal, complemented
6#71591—34nomaximal
7C23×C32×C4C2211yescharacteristic
8C2×#3828C2211yescharacteristic
9C32×#117—24no—
10C2×C32×C4⋊C4—21no—
11C2×C32×C22⋊C4—21no—
12C2×C32×C42C2212yes—
13#64162—21no—
14#64139C412yescomplemented
15#64138—21no—
16C4×#638—22no—
17#64135C2211yescharacteristic
18#64133C412yescomplemented
19C2×#3748C2212yes—
20#63657—22no—
21#63632—24no—
22C2×C3×#117D314yescomplemented
23#5791—34no—
24#5207—34no—
25C2×#805—38no—
26#4523—38no—
27C24×C32D411yescharacteristic
28C22×C32×C4—22no—
29C22×C32×C4—22no—
30C22×C32×C4—22no—
31C22×C32×C4C2×C412yes—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D412yes—
34C22×C32×C4C2311yescharacteristic
35C2×#638—22no—
36C2×#638—21no—
37#3828—22no—
38#3828—22no—
39C2×C4⋊(C3⋊D3)C2×C411yescharacteristic
40C2×C4×C3⋊D3—21no—
41C2×#634C2×C411yescharacteristic
42C32×C4⋊C4—24no—
43C32×C22⋊C4—24no—
44C32×C42—22no—
45C32×C42D414yescomplemented
46#3757—22no—
47#3757—21no—
48#3752—22no—
49C4⋊(C3⋊Dic3)C2×C412yes—
50#3750—22no—
51C4×C3⋊Dic3—21no—
52#3748—22no—
53#3748—22no—
54C2×C3⋊(C3⋊C8)—22no—
55#1897—91noSylow
56C23×C3×C4C2×D314yes—
57C2×#235—34no—
58C3×#117—216no—
59C2×C3×C4⋊C4—24no—
60C2×C3×C22⋊C4—24no—
61C2×C3×C42C2×D318yes—
62#953—64no—
63#930—38no—
64#929—64no—
65C4×#67—68no—
66#926—34no—
67#924—38no—
68C2×#208—38no—
69#830—68no—
70#805—616no—
71C23×C32—21no—
72C23×C32—21no—
73C23×C32—42no—
74C23×C32C22⋊C412yes—
75C23×C32C2×D411yescharacteristic
76C22×C3⋊D3—21no—
77C2×C32×C4—21no—
78C2×C32×C4—21no—
79C2×C32×C4—24no—
80C2×C32×C4—24no—
81C2×C32×C4—24no—
82C2×C32×C4—44no—
83C2×C32×C4C22⋊C412yes—
84C2×C32×C4C22×C411yescharacteristic
85C2×C32×C4C2×D411yescharacteristic
86C2×C32×C4C2×D412yes—
87C2×C32×C4#4112yes—
88#638—44no—
89#638—41no—
90#638—41no—
91C2×C3⋊Dic3—22no—
92C2×C3⋊Dic3—21no—
93C4⋊(C3⋊D3)—22no—
94C4×C3⋊D3—42no—
95#634—22no—
96C3⋊(C3⋊C8)—42no—
97#525—91no—
98C2×#117C3⋊D311yescharacteristic, complemented
99C2×#104—92no—
100#407—91no—
101#339—92no—
102C24×C3#6714yes—
103C22×C3×C4—28no—
104C22×C3×C4—28no—
105C22×C3×C4—28no—
106C22×C3×C4C4×D318yes—
107C22×C3×C4D1218yes—
108C22×C3×C4#6714yes—
109C22×C3×C4C22×D314yes—
110C2×#67—68no—
111C2×#67—64no—
112#235—68no—
113#235—68no—
114C2×D12—34no—
115C2×C4×D3—64no—
116C2×Dic6—34no—
117C3×C4⋊C4—216no—
118C3×C22⋊C4—216no—
119C3×C42—28no—
120C3×C42#67116yescomplemented
121#217—68no—
122#217—64no—
123(C2×D3)⋊C4—68no—
124C4⋊Dic3—38no—
125Dic3⋊C4—68no—
126C4×Dic3—64no—
127#208—68no—
128#208—68no—
129C2×C3⋊C8—68no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—42no—
135C22×C32—42no—
136C22×C32C2×C22⋊C411yescharacteristic
137C22×C32#12011yescharacteristic
138C22×C32#12311yescharacteristic
139C2×C3⋊D3—41no—
140C2×C3⋊D3—42no—
141C32×C4—22no—
142C32×C4—44no—
143C32×C4C4×D412yes—
144C32×C4#12112yes—
145C3⋊Dic3—41no—
146C3⋊Dic3—42no—
147C2×#41—91no—
148C23×C4C2×C3⋊D311yescharacteristic
149C2×M16—92no—
150#123—181no—
151C22⋊Q8—92no—
152#121—92no—
153#120—181no—
154C4×D4—182no—
155#117—24no—
156#117—91no—
157C2×C4⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C42C2×C3⋊D312yes—
160#104—184no—
161C22⋊C8—182no—
162C23×C3—24no—
163C23×C3—24no—
164C23×C3—48no—
165C23×C3(C2×D3)⋊C418yes—
166C23×C3C2×#6714yes—
167C22×D3—64no—
168C2×C3×C4—24no—
169C2×C3×C4—24no—
170C2×C3×C4—216no—
171C2×C3×C4—216no—
172C2×C3×C4—216no—
173C2×C3×C4—416no—
174C2×C3×C4(C2×D3)⋊C418yes—
175C2×C3×C4C2×C4×D314yes—
176C2×C3×C4C2×D1214yes—
177C2×C3×C4#23518yes—
178C2×C3×C4C2×#6718yes—
179#67—1216no—
180#67—124no—
181#67—124no—
182C2×Dic3—68no—
183C2×Dic3—64no—
184D12—68no—
185C4×D3—128no—
186Dic6—68no—
187C3⋊C8—128no—
188C2×C32—21no—
189C2×C32—21no—
190C2×C32—42no—
191C2×C32#38611yescharacteristic
192C2×C32#42112yes—
193C3⋊D3—81no—
194C24#63811yescharacteristic
195#41—182no—
196#41—182no—
197C2×Q8—91no—
198C2×D4—91no—
199C2×D4—182no—
200C2×D4—181no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—181no—
205C22×C4C4×C3⋊D312yes—
206C22×C4C4⋊(C3⋊D3)12yes—
207C22×C4#63811yescharacteristic
208C22×C4C22×C3⋊D311yescharacteristic
209M16—182no—
210M16—182no—
211C2×C8—182no—
212C4⋊C4—24no—
213C4⋊C4—92no—
214C4⋊C4—182no—
215C22⋊C4—24no—
216C22⋊C4—182no—
217C22⋊C4—181no—
218C22⋊C4—182no—
219C42—22no—
220C42—181no—
221C42#63814yescomplemented
222C22×C3—28no—
223C22×C3—28no—
224C22×C3—28no—
225C22×C3—28no—
226C22×C3—48no—
227C22×C3—48no—
228C22×C3C2×(C2×D3)⋊C414yes—
229C22×C3#92914yes—
230C22×C3#95314yes—
231C2×D3—124no—
232C2×D3—128no—
233C3×C4—28no—
234C3×C4—416no—
235C3×C4C4×#6718yes—
236C3×C4#93018yes—
237Dic3—124no—
238Dic3—128no—
239C32#189711yescharacteristic, complemented, Sylow
240C23—21no—
241C23—21no—
242C23—42no—
243C23—181no—
244C23#375212yes—
245C23C2×#63811yescharacteristic
246Q8—182no—
247D4—182no—
248D4—364no—
249D4—361no—
250D4—361no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—24no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—44no—
257C2×C4—182no—
258C2×C4—181no—
259C2×C4—362no—
260C2×C4#375212yes—
261C2×C4C2×C4×C3⋊D311yescharacteristic
262C2×C4C2×C4⋊(C3⋊D3)11yescharacteristic
263C2×C4#382812yes—
264C2×C4C2×#63812yes—
265C8—362no—
266C2×C3—24no—
267C2×C3—24no—
268C2×C3—48no—
269C2×C3#500518yes—
270C2×C3#521314yes—
271D3—244no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—361no—
279C22—362no—
280C22C2×#375211yescharacteristic
281C22#6413811yescharacteristic
282C22#6416211yescharacteristic
283C4—22no—
284C4—44no—
285C4—361no—
286C4—362no—
287C4C4×#63812yes—
288C4#6413912yes—
289C3#7159114yescomplemented
290C2—21no—
291C2—21no—
292C2—42no—
293C2—721no—
294C2#1059270412yes—
295C2#1059291211yescharacteristic
subgroupG/H
centerC2×C4C2×C4⋊(C3⋊D3)
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C22×C3⋊D3
FittingC2×C32×#117C2
socleC22×C32C2×C22⋊C4
prime radical#10596262C2
cosocleC22×C32×C4C23
hypercenterC2×#117C3⋊D3
hypocenterC32#1897
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
1. C2×C32×#117C2semidirect
3. #10592906C2semidirect
4. C2×#63632C2semidirect
5. #10589389C2semidirect
14. #64139C4semidirect
18. #64133C4semidirect
22. C2×C3×#117D3semidirect
45. C32×C42D4semidirect
98. C2×#117C3⋊D3semidirect
120. C3×C42#67semidirect
221. C42#638semidirect
239. C32#1897semidirect
289. C3#71591semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1172
#105962622
#105929062
C2×#636322
#105893892
#715913