Finite Group Databasebeta

← #13158916

Subgroups — #13158916

Order 1728.

1084 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1084 conjugacy classes of subgroups — too many to draw. Only its 220 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#11723913C211yesmaximal, characteristic
2C2×#88788C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×C4×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C32×#803C211yesmaximal, characteristic, complemented
6#11722021C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10592407—31nomaximal
9#10590321—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C33×C42C2211yescharacteristic
12#88859C412yes—
13#88857C412yes—
14C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88788C414yescomplemented
20#88785C414yescomplemented
21C2×#3743—31no—
22C2×C8×C3⋊D3—31no—
23#64081—31no—
24#64081—34no—
25#64081—64no—
26C2×#3712—34no—
27C2×#3710—34no—
28C32×#97—31no—
29#63642—31no—
30C8×C3⋊Dic3—31no—
31C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
32C3×#803D314yescomplemented
33C3×C4×C3⋊C8—34no—
34#63555—34no—
35#63553—34no—
36C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C33×C4C2×C412yes—
38C2×C33×C4C2311yescharacteristic
39C2×C3⋊(C3⋊Dic3)C2×C412yes—
40C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
41C4×C3⋊(C3⋊D3)C2×C412yes—
42C4×C3⋊(C3⋊D3)C2×C412yes—
43C3×C3⋊(C3⋊C8)C2×C412yes—
44C3×C3⋊(C3⋊C8)C2×C412yes—
45C32×C3⋊C8C2×C412yes—
46C32×C3⋊C8C2×C412yes—
47#4708—94no—
48#4689—91no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—34no—
51C2×C4×C3⋊D3—64no—
52C2×C32×C8—31no—
53C2×C32×C8—31no—
54C32×C42—24no—
55C32×C42C2×D314yes—
56C32×C42C2×D311yescharacteristic
57#3752—32no—
58#3752—38no—
59#3752—68no—
60#3750—32no—
61#3750—38no—
62#3750—68no—
63C4×C3⋊Dic3—31no—
64C4×C3⋊Dic3—34no—
65C4×C3⋊Dic3—64no—
66C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
67C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
68#3743—34no—
69C8×C3⋊D3—34no—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8—34no—
72C2×C3×C3⋊C8C2×D314yes—
73C2×C3×C3⋊C8C2×D314yes—
74#3712—316no—
75#3710—316no—
76C22×C33C22×C411yescharacteristic
77C2×C3⋊(C3⋊D3)—22no—
78C2×C3⋊(C3⋊D3)C4212yes—
79C33×C4—22no—
80C33×C4C22×C411yescharacteristic
81C33×C4C22×C411yescharacteristic
82C3⋊(C3⋊Dic3)—22no—
83C3⋊(C3⋊Dic3)C4212yes—
84C2×ΓL(1,𝔽25)—94no—
85C2×ΓL(1,𝔽25)—91no—
86C2×D3×C8—94no—
87C2×D3×C8—91no—
88#872—94no—
89#872—91no—
90#872—184no—
91C3×#97—34no—
92C3×C4×C8—91no—
93#815—94no—
94#815—91no—
95C8×Dic3—94no—
96C8×Dic3—91no—
97#803C3⋊D311yescharacteristic, complemented
98C4×C3⋊C8—34no—
99C22×C3⋊D3—31no—
100C22×C3⋊D3—34no—
101C22×C3⋊D3—64no—
102C2×C32×C4—28no—
103C2×C32×C4—24no—
104C2×C32×C4C4×D318yes—
105C2×C32×C4C4×D312yes—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108C2×C3⋊Dic3—31no—
109C2×C3⋊Dic3—32no—
110C2×C3⋊Dic3—34no—
111C2×C3⋊Dic3—38no—
112C2×C3⋊Dic3—64no—
113C2×C3⋊Dic3—68no—
114C4×C3⋊D3—32no—
115C4×C3⋊D3—32no—
116C4×C3⋊D3—38no—
117C4×C3⋊D3—38no—
118C4×C3⋊D3—68no—
119C4×C3⋊D3—68no—
120C32×C8—32no—
121C32×C8—32no—
122C3⋊(C3⋊C8)C4×D312yes—
123C3⋊(C3⋊C8)C4×D312yes—
124C3×C3⋊C8—38no—
125C3×C3⋊C8—38no—
126C3×C3⋊C8C4×D318yes—
127C3×C3⋊C8C4×D318yes—
128#405—271noSylow
129C2×C33C2×C4211yescharacteristic
130C2×C33#13112yes—
131C3⋊(C3⋊D3)—22no—
132C2×C4×D3—94no—
133C2×C4×D3—91no—
134C2×C4×D3—184no—
135C2×C3×C8—34no—
136C2×C3×C8—34no—
137C2×C3×C8—91no—
138C2×C3×C8—91no—
139C3×C42—24no—
140C3×C42D3214yes—
141C3×C42C2×C3⋊D311yescharacteristic
142(C2×D3)⋊C4—98no—
143(C2×D3)⋊C4—92no—
144(C2×D3)⋊C4—188no—
145Dic3⋊C4—98no—
146Dic3⋊C4—92no—
147Dic3⋊C4—188no—
148C4×Dic3—94no—
149C4×Dic3—91no—
150C4×Dic3—184no—
151C2×C3⋊C8—34no—
152C2×C3⋊C8—34no—
153C2×C3⋊C8C2×C3⋊D311yescharacteristic
154C2×C3⋊C8C2×C3⋊D311yescharacteristic
155ΓL(1,𝔽25)—916no—
156ΓL(1,𝔽25)—94no—
157D3×C8—916no—
158D3×C8—94no—
159C22×C32—24no—
160C22×C32C2×C4×D314yes—
161C22×C32C2×C4×D311yescharacteristic
162C2×C3⋊D3—32no—
163C2×C3⋊D3—38no—
164C2×C3⋊D3—68no—
165C2×C3⋊D3—62no—
166C2×C3⋊D3—68no—
167C2×C3⋊D3—128no—
168C32×C4—22no—
169C32×C4—24no—
170C32×C4—24no—
171C32×C4—28no—
172C32×C4—48no—
173C32×C4C2×C4×D314yes—
174C32×C4C2×C4×D314yes—
175C32×C4C2×C4×D311yescharacteristic
176C32×C4C2×C4×D311yescharacteristic
177C3⋊Dic3—32no—
178C3⋊Dic3—38no—
179C3⋊Dic3—68no—
180C3⋊Dic3—62no—
181C3⋊Dic3—68no—
182C3⋊Dic3—128no—
183C2×M16—271no—
184C22×C8—271no—
185#117—271no—
186#97—31no—
187#97—271no—
188C4×C8—91no—
189C4×C8—271no—
190C33#40511yescharacteristic, complemented, Sylow
191C22×D3—94no—
192C22×D3—91no—
193C22×D3—184no—
194C2×C3×C4—28no—
195C2×C3×C4—24no—
196C2×C3×C4#62418yes—
197C2×C3×C4C4×C3⋊D312yes—
198C2×C3×C4C2×D3214yes—
199C2×C3×C4C22×C3⋊D311yescharacteristic
200C2×Dic3—94no—
201C2×Dic3—91no—
202C2×Dic3—98no—
203C2×Dic3—92no—
204C2×Dic3—184no—
205C2×Dic3—188no—
206C4×D3—98no—
207C4×D3—98no—
208C4×D3—92no—
209C4×D3—92no—
210C4×D3—188no—
211C4×D3—188no—
212C3×C8—38no—
213C3×C8—38no—
214C3×C8—92no—
215C3×C8—92no—
216C3⋊C8—38no—
217C3⋊C8—38no—
218C3⋊C8C4×C3⋊D312yes—
219C3⋊C8C4×C3⋊D312yes—
220C2×C32—28no—
221C2×C32—24no—
222C2×C32C42×D314yes—
223C2×C32C42×D311yescharacteristic
224C2×C32#90112yes—
225C2×C32#90718yes—
226C3⋊D3—62no—
227C3⋊D3—68no—
228C3⋊D3—128no—
229C22×C4—271no—
230M16—274no—
231C2×C8—31no—
232C2×C8—31no—
233C2×C8—91no—
234C2×C8—91no—
235C2×C8—274no—
236C4⋊C4—272no—
237C22⋊C4—272no—
238C42—271no—
239C42D3×C3⋊D311yescharacteristic
240C22×C3—24no—
241C22×C3C2×#62414yes—
242C22×C3C2×C4×C3⋊D311yescharacteristic
243C2×D3—92no—
244C2×D3—98no—
245C2×D3—188no—
246C2×D3—188no—
247C2×D3—182no—
248C2×D3—368no—
249C3×C4—28no—
250C3×C4—22no—
251C3×C4—24no—
252C3×C4—24no—
253C3×C4—48no—
254C3×C4C4×D3214yes—
255C3×C4C4×D3214yes—
256C3×C4C2×C4×C3⋊D311yescharacteristic
257C3×C4C2×C4×C3⋊D311yescharacteristic
258Dic3—92no—
259Dic3—98no—
260Dic3—188no—
261Dic3—188no—
262Dic3—182no—
263Dic3—368no—
264C32—24no—
265C32#468911yescharacteristic, complemented
266C32#470814yescomplemented
267C23—271no—
268C2×C4—271no—
269C2×C4—272no—
270C2×C4—272no—
271C2×C4—272no—
272C2×C4#628912yes—
273C2×C4C2×D3×C3⋊D311yescharacteristic
274C8—32no—
275C8—32no—
276C8—92no—
277C8—92no—
278C2×C3—28no—
279C2×C3—24no—
280C2×C3#6381718yes—
281C2×C3C4×#62414yes—
282C2×C3C42×C3⋊D311yescharacteristic
283C2×C3#6411612yes—
284D3—188no—
285D3—182no—
286D3—368no—
287C22—272no—
288C22—542no—
289C22C2×#628911yescharacteristic
290C4—22no—
291C4—272no—
292C4—542no—
293C4C4×D3×C3⋊D311yescharacteristic
294C4C4×D3×C3⋊D311yescharacteristic
295C3—24no—
296C3#1059032114yescomplemented
297C3#1059240711yescharacteristic, complemented
298C2—542no—
299C2#1172335712yes—
300C2C4×#628911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC33×C42C22
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC42D3×C3⋊D3
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
4. C3×C4×C3⋊(C3⋊C8)C2semidirect
5. C32×#803C2semidirect
6. #11722021C2semidirect
7. #11722018C2semidirect
19. #88788C4semidirect
20. #88785C4semidirect
31. C4×C3⋊(C3⋊C8)D3semidirect
32. C3×#803D3semidirect
97. #803C3⋊D3semidirect
190. C33#405semidirect
265. C32#4689semidirect
266. C32#4708semidirect
296. C3#10590321semidirect
297. C3#10592407semidirect

Maximal subgroups

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

subgroupindex
#117239132
C2×#887882
C2×#887852
C3×C4×C3⋊(C3⋊C8)2
C32×#8032
#117220212
#117220182
#105924073
#105903213