Finite Group Databasebeta

← #13159583

Subgroups — #13159583

Order 1728.

1052 conjugacy classes of subgroups; only the 220 normal subgroups are shown.

normal subgroups only

This group has 1052 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
1C2×#88850C211yesmaximal, characteristic
2#11723452C211yesmaximal, characteristic
3C2×#88785C211yesmaximal, characteristic
4C3×#63642C211yesmaximal, characteristic, complemented
5C32×C8×Dic3C211yesmaximal, characteristic, complemented
6#11722019C211yesmaximal, characteristic, complemented
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10592388—31nomaximal
9#10590910—34nomaximal
10C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
11C2×C33×C8C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13#88850C414yescomplemented
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88806C412yes—
19#88804C412yes—
20#88785C414yescomplemented
21C2×#3743—31no—
22C2×#3743—34no—
23C2×#3743—64no—
24C2×C8×C3⋊D3—31no—
25#64081—31no—
26#63874—34no—
27C2×#3710—34no—
28C32×C4×C8—31no—
29#63642D311yescharacteristic, complemented
30C8×C3⋊Dic3—31no—
31#63630—31no—
32C3×#815—34no—
33C3×C8×Dic3D314yescomplemented
34#63554—34no—
35Dic3×C3⋊C8—34no—
36C22×C3⋊(C3⋊D3)C2×C411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
39C4×C3⋊(C3⋊D3)C2×C412yes—
40C4×C3⋊(C3⋊D3)C2×C412yes—
41C2×C3×C3⋊Dic3C2×C412yes—
42C2×C32×Dic3C2×C412yes—
43C33×C8C2×C412yes—
44C3⋊(C3⋊(C3⋊C8))C2×C412yes—
45C3×C3⋊(C3⋊C8)C2×C412yes—
46C32×C3⋊C8C2×C412yes—
47#4708—91no—
48#4689—94no—
49C2×C4×C3⋊D3—31no—
50C2×C4×C3⋊D3—34no—
51C2×C4×C3⋊D3—64no—
52C2×C32×C8—24no—
53C2×C32×C8—31no—
54C2×C32×C8C2×D314yes—
55C2×C32×C8C2×D311yescharacteristic
56C32×C42—31no—
57#3752—32no—
58#3750—32no—
59C4×C3⋊Dic3C2×D311yescharacteristic
60C2×C3⋊(C3⋊C8)—31no—
61C2×C3⋊(C3⋊C8)—34no—
62C2×C3⋊(C3⋊C8)—64no—
63C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
64#3743—34no—
65#3743—316no—
66#3743—616no—
67C8×C3⋊D3—34no—
68C3×C4×Dic3—34no—
69C3×C4×Dic3C2×D314yes—
70C2×C3×C3⋊C8—34no—
71C2×C3×C3⋊C8C2×D314yes—
72#3724—38no—
73#3722—38no—
74#3710—316no—
75C22×C33C22×C411yescharacteristic
76C2×C3⋊(C3⋊D3)—22no—
77C2×C3⋊(C3⋊D3)C4212yes—
78C33×C4C22×C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C3⋊(C3⋊Dic3)C4212yes—
81C3×C3⋊Dic3—22no—
82C32×Dic3—22no—
83C2×ΓL(1,𝔽25)—94no—
84C2×ΓL(1,𝔽25)—91no—
85C2×ΓL(1,𝔽25)—184no—
86C2×D3×C8—94no—
87C2×D3×C8—91no—
88#872—94no—
89#872—91no—
90C3×#97—91no—
91C3×C4×C8—34no—
92#815—34no—
93#815—91no—
94C8×Dic3—94no—
95C8×Dic3C3⋊D311yescharacteristic, complemented
96#803—94no—
97C4×C3⋊C8—91no—
98C22×C3⋊D3—31no—
99C22×C3⋊D3—34no—
100C22×C3⋊D3—64no—
101C2×C32×C4—24no—
102C2×C32×C4—32no—
103C2×C32×C4C22×D314yes—
104C2×C32×C4C22×D311yescharacteristic
105C2×C3⋊Dic3—31no—
106C2×C3⋊Dic3—34no—
107C2×C3⋊Dic3—64no—
108C2×C3⋊Dic3C4×D312yes—
109C4×C3⋊D3—32no—
110C4×C3⋊D3—32no—
111C4×C3⋊D3—38no—
112C4×C3⋊D3—38no—
113C4×C3⋊D3—68no—
114C4×C3⋊D3—68no—
115C2×C3×Dic3—38no—
116C2×C3×Dic3C4×D318yes—
117C32×C8—28no—
118C32×C8—32no—
119C32×C8C4×D318yes—
120C32×C8C4×D312yes—
121C3⋊(C3⋊C8)—32no—
122C3⋊(C3⋊C8)—38no—
123C3⋊(C3⋊C8)—68no—
124C3⋊(C3⋊C8)C4×D312yes—
125C3×C3⋊C8—38no—
126C3×C3⋊C8C4×D318yes—
127#405—271noSylow
128C2×C33C2×C4211yescharacteristic
129C2×C33#13112yes—
130C3⋊(C3⋊D3)—22no—
131C2×C4×D3—94no—
132C2×C4×D3—91no—
133C2×C4×D3—184no—
134C2×C3×C8—24no—
135C2×C3×C8—34no—
136C2×C3×C8—91no—
137C2×C3×C8D3214yes—
138C2×C3×C8C2×C3⋊D311yescharacteristic
139C3×C42—34no—
140C3×C42—91no—
141(C2×D3)⋊C4—98no—
142(C2×D3)⋊C4—92no—
143Dic3⋊C4—98no—
144Dic3⋊C4—92no—
145C4×Dic3—34no—
146C4×Dic3C2×C3⋊D311yescharacteristic
147C2×C3⋊C8—34no—
148C2×C3⋊C8—94no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—184no—
151C2×C3⋊C8C2×C3⋊D311yescharacteristic
152ΓL(1,𝔽25)—916no—
153ΓL(1,𝔽25)—94no—
154ΓL(1,𝔽25)—1816no—
155D3×C8—916no—
156D3×C8—94no—
157C22×C32—24no—
158C22×C32C2×C4×D314yes—
159C22×C32C2×C4×D311yescharacteristic
160C2×C3⋊D3—32no—
161C2×C3⋊D3—38no—
162C2×C3⋊D3—68no—
163C2×C3⋊D3—62no—
164C2×C3⋊D3—68no—
165C2×C3⋊D3—128no—
166C32×C4—24no—
167C32×C4—24no—
168C32×C4—62no—
169C32×C4C2×C4×D314yes—
170C32×C4C2×C4×D314yes—
171C32×C4C2×C4×D311yescharacteristic
172C32×C4C2×C4×D311yescharacteristic
173C3⋊Dic3—22no—
174C3⋊Dic3—32no—
175C3⋊Dic3—38no—
176C3⋊Dic3—68no—
177C3×Dic3—28no—
178C3×Dic3—68no—
179C2×M16—271no—
180C22×C8—271no—
181#117—271no—
182#97—91no—
183#97—271no—
184C4×C8—31no—
185C4×C8—271no—
186C33#40511yescharacteristic, complemented, Sylow
187C22×D3—94no—
188C22×D3—91no—
189C22×D3—184no—
190C2×C3×C4—24no—
191C2×C3×C4—38no—
192C2×C3×C4—92no—
193C2×C3×C4C2×D3214yes—
194C2×C3×C4C22×C3⋊D311yescharacteristic
195C2×Dic3—38no—
196C2×Dic3—94no—
197C2×Dic3—91no—
198C2×Dic3—184no—
199C2×Dic3C4×C3⋊D312yes—
200C4×D3—98no—
201C4×D3—98no—
202C4×D3—92no—
203C4×D3—92no—
204C4×D3—188no—
205C4×D3—188no—
206C3×C8—28no—
207C3×C8—38no—
208C3×C8—92no—
209C3×C8#62418yes—
210C3×C8C4×C3⋊D312yes—
211C3⋊C8—38no—
212C3⋊C8—98no—
213C3⋊C8—92no—
214C3⋊C8—188no—
215C3⋊C8C4×C3⋊D312yes—
216C2×C32—28no—
217C2×C32—24no—
218C2×C32C42×D314yes—
219C2×C32C42×D311yescharacteristic
220C2×C32#90118yes—
221C2×C32#90712yes—
222C3⋊D3—62no—
223C3⋊D3—68no—
224C3⋊D3—128no—
225C22×C4—271no—
226M16—274no—
227C2×C8—31no—
228C2×C8—91no—
229C2×C8—274no—
230C2×C8—271no—
231C2×C8D3×C3⋊D311yescharacteristic
232C4⋊C4—272no—
233C22⋊C4—272no—
234C42—31no—
235C42—91no—
236C22×C3—24no—
237C22×C3C4×D3214yes—
238C22×C3C2×C4×C3⋊D311yescharacteristic
239C2×D3—92no—
240C2×D3—98no—
241C2×D3—188no—
242C2×D3—188no—
243C2×D3—182no—
244C2×D3—368no—
245C3×C4—24no—
246C3×C4—24no—
247C3×C4—68no—
248C3×C4—182no—
249C3×C4C4×D3214yes—
250C3×C4C2×#62414yes—
251C3×C4C2×C4×C3⋊D311yescharacteristic
252C3×C4C2×C4×C3⋊D311yescharacteristic
253Dic3—22no—
254Dic3—68no—
255Dic3—92no—
256Dic3—98no—
257Dic3—188no—
258C32—24no—
259C32#468914yescomplemented
260C32#470811yescharacteristic, complemented
261C23—271no—
262C2×C4—32no—
263C2×C4—92no—
264C2×C4—271no—
265C2×C4—272no—
266C2×C4—272no—
267C2×C4C2×D3×C3⋊D311yescharacteristic
268C8—32no—
269C8—92no—
270C8—272no—
271C8#628912yes—
272C2×C3—28no—
273C2×C3—24no—
274C2×C3#6380418yes—
275C2×C3C4×#62414yes—
276C2×C3C42×C3⋊D311yescharacteristic
277C2×C3#6411012yes—
278D3—182no—
279D3—188no—
280D3—368no—
281C22—272no—
282C22—542no—
283C22C4×D3×C3⋊D311yescharacteristic
284C4—62no—
285C4—182no—
286C4—272no—
287C4C4×D3×C3⋊D311yescharacteristic
288C4C2×#628911yescharacteristic
289C3—24no—
290C3#1059091014yescomplemented
291C3#1059238811yescharacteristic, complemented
292C2—542no—
293C2#1172333212yes—
294C2C4×#628911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C2×D3×C3⋊D3
commutatorC2×C33C2×C42
FittingC2×C33×C8C22
socleC22×C33C22×C4
prime radicalC22×C3⋊(C3⋊D3)C2×C4
cosocleC2×C33×C4C23
hypercenterC2×C8D3×C3⋊D3
hypocenterC33#405
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
4. C3×#63642C2semidirect
5. C32×C8×Dic3C2semidirect
6. #11722019C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
13. #88850C4semidirect
20. #88785C4semidirect
29. #63642D3semidirect
33. C3×C8×Dic3D3semidirect
95. C8×Dic3C3⋊D3semidirect
186. C33#405semidirect
259. C32#4689semidirect
260. C32#4708semidirect
290. C3#10590910semidirect
291. C3#10592388semidirect

Maximal subgroups

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

subgroupindex
C2×#888502
#117234522
C2×#887852
C3×#636422
C32×C8×Dic32
#117220192
C3⋊C8×C3⋊Dic32
#105923883
#105909103