Finite Group Databasebeta

← #13139032

Subgroups — #13139032

Order 1728.

733 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 87 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×#88655C211yesmaximal, characteristic
2C2×#88527C211yesmaximal, characteristic, complemented
3#11721089C211yesmaximal, characteristic, complemented
4#63648⋊C3C212yesmaximal, complemented
5#11720318C212yesmaximal, complemented
6#10590939—32nomaximal
7C22×((C3⋊D3)⋊C3)⋊C2—21no—
8C2×C4×#272C2211yescharacteristic
9C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
10C2×#6198—21no—
11#88655—22no—
12#88655C414yescomplemented
13C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
14C2×#6180C2211yescharacteristic, complemented
15#88527—22no—
16C4⋊((C3⋊Dic3)⋊C3)C2212yescomplemented
17#88451—21no—
18#88447—21no—
19C2×D3×D12—32no—
20C2×C3×ΓL(1,𝔽25)—32no—
21C2×C3×ΓL(1,𝔽25)—61no—
22#63841—32no—
23#63648D312yescomplemented
24C3×#821—32no—
25#63564—32no—
26#63563—32no—
27C22×#272D411yescharacteristic
28C22×(C3⋊D3)⋊C3—22no—
29C2×((C3⋊D3)⋊C3)⋊C2—24no—
30C2×((C3⋊D3)⋊C3)⋊C2—44no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084D411yescharacteristic
33C4×#272C2×C412yes—
34C4×#272D412yes—
35C2×(C3⋊Dic3)⋊C3—22no—
36C4⋊((C3⋊D3)⋊C3)—22no—
37C4⋊((C3⋊D3)⋊C3)C2×C414yes—
38#6198—24no—
39C8×H(3,𝔽3)—21no—
40#6180—21no—
41#5131—91no—
42#4770—92no—
43C22×D32—62no—
44C2×C4⋊(C3⋊D3)C2×D312yes—
45C2×C3×D12—32no—
46C2×C3×C4×D3—32no—
47C2×C3×C4×D3—61no—
48C2×#626—62no—
49D3×D12—38no—
50D3×D12—64no—
51C2×C32×C8—21no—
52C2×C32×C8C2×D312yescomplemented
53C4⋊(C3⋊Dic3)C2×D312yescomplemented
54C3×C4⋊Dic3—32no—
55C2×C3×C3⋊C8—32no—
56C2×C3×C3⋊C8—61no—
57C3×ΓL(1,𝔽25)—64no—
58C3×ΓL(1,𝔽25)—64no—
59(C2×D3)⋊Dic3—62no—
60Dic32—62no—
61C22×H(3,𝔽3)C2×D411yescharacteristic
62C2×#272—22no—
63C2×#272C22⋊C412yes—
64C2×(C3⋊D3)⋊C3—24no—
65C2×(C3⋊D3)⋊C3—42no—
66C2×(C3⋊D3)⋊C3—42no—
67((C3⋊D3)⋊C3)⋊C2—48no—
68C4×H(3,𝔽3)C22×C411yescharacteristic
69C4×H(3,𝔽3)C2×D411yescharacteristic
70#1084C22⋊C412yes—
71(C3⋊Dic3)⋊C3—42no—
72C2×D3×D4—92no—
73C22×D12—91no—
74C2×C3×M16—91no—
75#926—91no—
76C2×ΓL(1,𝔽25)—32no—
77C2×ΓL(1,𝔽25)—61no—
78#892—92no—
79C3×#102—92no—
80#832—92no—
81#821—32no—
82#821—92no—
83#821—92no—
84#809—92no—
85C22×C3⋊D3—22no—
86C22×C3×D3—32no—
87C22×C3×D3—61no—
88C22×C3×D3—62no—
89C2×D32—68no—
90C2×D32—124no—
91C2×D32—124no—
92C2×C32×C4—21no—
93C2×C32×C4C22×D312yes—
94C2×C3⋊Dic3—22no—
95C4⋊(C3⋊D3)—22no—
96C4⋊(C3⋊D3)C4×D314yes—
97C2×C3×Dic3—32no—
98C2×C3×Dic3—61no—
99C2×C3×Dic3—62no—
100C3×D12—34no—
101C3×D12—62no—
102C3×C4×D3—34no—
103C3×C4×D3—34no—
104C3×C4×D3—62no—
105C3×C4×D3—62no—
106#626—68no—
107C32×C8—22no—
108C32×C8—22no—
109C3×C3⋊C8—62no—
110C3×C3⋊C8—62no—
111#418—271noSylow
112C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
113C2×H(3,𝔽3)AGL(1,ℤ/8)12yes—
114#272—22no—
115(C3⋊D3)⋊C3—44no—
116C23×D3—182no—
117C23×D3—181no—
118C2×C3×D4—92no—
119C22×C3×C4—91no—
120C2×#67—182no—
121D3×D4—98no—
122D3×D4—184no—
123C2×D12—32no—
124C2×D12—92no—
125C2×D12—94no—
126C2×D12—182no—
127C2×D12—181no—
128C2×C4×D3—32no—
129C2×C4×D3—61no—
130C3×M16—182no—
131C2×C3×C8—32no—
132C2×C3×C8—61no—
133C2×C3×C8—91no—
134C2×C3×C8D3211yescharacteristic
135C3×C4⋊C4—92no—
136#217—181no—
137(C2×D3)⋊C4—182no—
138C4⋊Dic3—32no—
139C4⋊Dic3—92no—
140C4×Dic3—182no—
141C4×Dic3—181no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—61no—
144ΓL(1,𝔽25)—64no—
145ΓL(1,𝔽25)—64no—
146C22×C32—21no—
147C22×C32D3×D412yes—
148C2×C3⋊D3—24no—
149C2×C3⋊D3—42no—
150C2×C3⋊D3—42no—
151C2×C3×D3—34no—
152C2×C3×D3—62no—
153C2×C3×D3—62no—
154C2×C3×D3—62no—
155C2×C3×D3—64no—
156C2×C3×D3—122no—
157C2×C3×D3—122no—
158C2×C3×D3—122no—
159D32—1216no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4C2×C4×D312yes—
163C32×C4D3×D412yes—
164C3⋊Dic3—42no—
165C3×Dic3—34no—
166C3×Dic3—62no—
167C3×Dic3—122no—
168C22×D4—271no—
169C2×M16—91no—
170#117—271no—
171#102—92no—
172#102—272no—
173H(3,𝔽3)#41811yescharacteristic, complemented, Sylow
174C23×C3—91no—
175C23×C3—182no—
176C22×D3—32no—
177C22×D3—61no—
178C22×D3—62no—
179C22×D3—182no—
180C22×D3—188no—
181C22×D3—184no—
182C22×D3—364no—
183C22×D3—364no—
184C22×D3—364no—
185C3×D4—94no—
186C3×D4—182no—
187C2×C3×C4—32no—
188C2×C3×C4—61no—
189C2×C3×C4—91no—
190C2×C3×C4—92no—
191C2×C3×C4—92no—
192C2×C3×C4—182no—
193C2×C3×C4C2×D3211yescharacteristic
194#67—188no—
195C2×Dic3—32no—
196C2×Dic3—61no—
197C2×Dic3—62no—
198C2×Dic3—182no—
199D12—34no—
200D12—62no—
201D12—94no—
202D12—182no—
203D12—184no—
204C4×D3—34no—
205C4×D3—34no—
206C4×D3—62no—
207C4×D3—62no—
208C3×C8—21no—
209C3×C8—62no—
210C3×C8—62no—
211C3×C8—181no—
212C3⋊C8—62no—
213C3⋊C8—62no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32D3×C22⋊C412yes—
217C2×C32#91112yes—
218C2×C32#91412yes—
219C3⋊D3—44no—
220C3×D3—64no—
221C3×D3—122no—
222C3×D3—124no—
223C24—541no—
224C2×D4—92no—
225C2×D4—274no—
226C2×D4—542no—
227C2×D4—541no—
228C22×C4—91no—
229M16—182no—
230C2×C8—91no—
231C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
232C4⋊C4—92no—
233C22⋊C4—541no—
234C42—541no—
235C22×C3—32no—
236C22×C3—61no—
237C22×C3—92no—
238C22×C3—184no—
239C22×C3—182no—
240C22×C3—362no—
241C22×C3—362no—
242C22×C3#380211yescharacteristic
243C2×D3—34no—
244C2×D3—62no—
245C2×D3—62no—
246C2×D3—62no—
247C2×D3—64no—
248C2×D3—122no—
249C2×D3—122no—
250C2×D3—122no—
251C2×D3—184no—
252C2×D3—362no—
253C2×D3—362no—
254C2×D3—3616no—
255C2×D3—368no—
256C3×C4—32no—
257C3×C4—32no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—92no—
261C3×C4—362no—
262C3×C4C4×D3211yescharacteristic
263C3×C4#381111yescharacteristic
264Dic3—34no—
265Dic3—62no—
266Dic3—122no—
267Dic3—362no—
268C32—21no—
269C32#477012yescomplemented
270C23—91no—
271C23—182no—
272C23—544no—
273C23—1084no—
274D4—94no—
275D4—182no—
276D4—544no—
277C2×C4—91no—
278C2×C4—92no—
279C2×C4—92no—
280C2×C4—182no—
281C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
282C8—21no—
283C8—181no—
284C2×C3—32no—
285C2×C3—32no—
286C2×C3—32no—
287C2×C3—62no—
288C2×C3—61no—
289C2×C3—182no—
290C2×C3—364no—
291C2×C3#6379812yes—
292C2×C3#6392111yescharacteristic
293D3—64no—
294D3—122no—
295D3—124no—
296D3—364no—
297C22—92no—
298C22—182no—
299C22—184no—
300C22—362no—
301C22—362no—
302C22—1088no—
303C22#8865511yescharacteristic
304C4—92no—
305C4—362no—
306C4C4×((C3⋊D3)⋊C3)⋊C211yescharacteristic
307C4#8867711yescharacteristic
308C3—32no—
309C3—61no—
310C3#1059094111yescharacteristic
311C2—182no—
312C2—364no—
313C2#1172083912yes—
314C2#1172116111yescharacteristic
subgroupG/H
centerC22#88655
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3#3802
prime radicalC2×#88655C2
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#418
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
2. C2×#88527C2semidirect
3. #11721089C2semidirect
4. #63648⋊C3C2semidirect
5. #11720318C2semidirect
12. #88655C4semidirect
13. C2×C8×H(3,𝔽3)C22semidirect
14. C2×#6180C22semidirect
16. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
23. #63648D3semidirect
52. C2×C32×C8C2×D3semidirect
53. C4⋊(C3⋊Dic3)C2×D3semidirect
173. H(3,𝔽3)#418semidirect
231. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
269. C32#4770semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#885272
#117210892
#63648⋊C32
#117203182
#105909393