Finite Group Databasebeta

← #13135164

Subgroups — #13135164

Order 1728.

405 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 101 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#11721826C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#10589158—31nomaximal
5#10589157—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7#88552—21no—
8C4⋊#1084—21no—
9C4×#1084C412yescomplemented
10#3757⋊C3—21no—
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12C4×(C3⋊Dic3)⋊C3C412yescomplemented
13#3748⋊C3—21no—
14#3748⋊C3—21no—
15C2×(C3⋊(C3⋊C8))⋊C3C412yescomplemented
16#64135D311yescharacteristic
17C2×#3748D311yescharacteristic
18C3×#926—31no—
19C3×#926—31no—
20C3×#926—31no—
21C3×#926—61no—
22C2×C3×#208—31no—
23C23×H(3,𝔽3)D411yescharacteristic
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)C2×C411yescharacteristic
27C2×C4×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)Q811yescharacteristic
30C2×#1084—22no—
31C2×(C3⋊Dic3)⋊C3—22no—
32(C3⋊(C3⋊C8))⋊C3—22no—
33#4557—91no—
34#4531—91no—
35#4531—91no—
36C22×C32×C4—21no—
37C22×C32×C4C2×D311yescharacteristic
38C22×C32×C4C2×D311yescharacteristic
39#3757—21no—
40C4⋊(C3⋊Dic3)—21no—
41C4×C3⋊Dic3Dic312yescomplemented
42#3748—21no—
43#3748—21no—
44C2×C3⋊(C3⋊C8)Dic312yescomplemented
45C3×#217—62no—
46C3×#217—61no—
47C3×#217—61no—
48C3×#217—61no—
49C3×C4⋊Dic3—62no—
50C3×C4⋊Dic3—61no—
51C3×C4⋊Dic3—61no—
52C3×C4⋊Dic3—61no—
53C3×C4×Dic3—32no—
54C3×C4×Dic3—32no—
55C3×C4×Dic3—32no—
56C3×C4×Dic3—62no—
57C3×#208—61no—
58C3×#208—61no—
59C2×C3×C3⋊C8—32no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)C22⋊C411yescharacteristic
62C22×H(3,𝔽3)C22⋊C411yescharacteristic
63C22×H(3,𝔽3)C4⋊C411yescharacteristic
64C4×H(3,𝔽3)C4211yescharacteristic
65C4×H(3,𝔽3)C22⋊C411yescharacteristic
66C4×H(3,𝔽3)C4⋊C411yescharacteristic
67C4×H(3,𝔽3)C4⋊C411yescharacteristic
68#1084—42no—
69(C3⋊Dic3)⋊C3—42no—
70C2×C3×M16—91no—
71C3×#117—91no—
72C3×#117—91no—
73#926—31no—
74#926—31no—
75#926—31no—
76#926—61no—
77#926—91no—
78C2×#208—31no—
79C2×#208—91no—
80C23×C32—21no—
81C23×C32D1211yescharacteristic
82C23×C32#6711yescharacteristic
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4—21no—
89C2×C32×C4Dic611yescharacteristic
90C2×C32×C4Dic611yescharacteristic
91C2×C32×C4C4×D311yescharacteristic
92C2×C32×C4C4×D311yescharacteristic
93C2×C32×C4C4×D311yescharacteristic
94C2×C32×C4C4×D311yescharacteristic
95C2×C32×C4D1211yescharacteristic
96C2×C32×C4C2×Dic311yescharacteristic
97C2×C32×C4C2×Dic311yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4#6711yescharacteristic
100C2×C32×C4#6711yescharacteristic
101C2×C3⋊Dic3—22no—
102C2×C3×Dic3—64no—
103C2×C3×Dic3—62no—
104C2×C3×Dic3—62no—
105C2×C3×Dic3—62no—
106C3⋊(C3⋊C8)—22no—
107C3×C3⋊C8—62no—
108H(3,ℤ/4)—271noSylow
109C2×H(3,𝔽3)—21no—
110C2×H(3,𝔽3)—21no—
111C2×H(3,𝔽3)—21no—
112C2×H(3,𝔽3)#9511yescharacteristic
113C22×C3×C4—31no—
114C22×C3×C4—31no—
115C22×C3×C4—61no—
116C22×C3×C4D3211yescharacteristic
117C3×M16—181no—
118C3×M16—181no—
119C2×C3×C8—92no—
120C3×C4⋊C4—181no—
121C3×C4⋊C4—181no—
122C3×C22⋊C4—181no—
123C3×C22⋊C4—181no—
124C3×C42—92no—
125C3×C42—92no—
126#217—61no—
127#217—61no—
128#217—62no—
129#217—61no—
130#217—181no—
131C4⋊Dic3—61no—
132C4⋊Dic3—61no—
133C4⋊Dic3—62no—
134C4⋊Dic3—61no—
135C4⋊Dic3—181no—
136C4×Dic3—32no—
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3—62no—
140C4×Dic3—92no—
141#208—61no—
142#208—61no—
143#208—181no—
144#208—181no—
145C2×C3⋊C8—32no—
146C2×C3⋊C8—92no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32—41no—
151C22×C32—41no—
152C22×C32—42no—
153C22×C32Dic3⋊C411yescharacteristic
154C22×C32C4⋊Dic311yescharacteristic
155C22×C32(C2×D3)⋊C411yescharacteristic
156C22×C32(C2×D3)⋊C411yescharacteristic
157C22×C32(C2×D3)⋊C411yescharacteristic
158C22×C32#21711yescharacteristic
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4C4×Dic311yescharacteristic
164C32×C4C4×Dic311yescharacteristic
165C32×C4Dic3⋊C411yescharacteristic
166C32×C4Dic3⋊C411yescharacteristic
167C32×C4Dic3⋊C411yescharacteristic
168C32×C4C4⋊Dic311yescharacteristic
169C32×C4(C2×D3)⋊C411yescharacteristic
170C32×C4#21711yescharacteristic
171C3⋊Dic3—42no—
172C3×Dic3—122no—
173C3×Dic3—124no—
174C3×Dic3—122no—
175C3×Dic3—122no—
176C2×M16—91no—
177#117—91no—
178#117—91no—
179H(3,𝔽3)H(3,ℤ/4)11yescharacteristic, complemented, Sylow
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—61no—
183C23×C3#62611yescharacteristic
184C2×C3×C4—31no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—31no—
193C2×C3×C4—31no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—61no—
199C2×C3×C4—61no—
200C2×C3×C4—61no—
201C2×C3×C4—61no—
202C2×C3×C4—182no—
203C2×C3×C4—182no—
204C2×C3×C4D3×Dic311yescharacteristic
205C2×C3×C4D3×Dic311yescharacteristic
206C2×C3×C4#62411yescharacteristic
207C2×C3×C4#62511yescharacteristic
208C2×C3×C4#62611yescharacteristic
209C2×C3×C4#62711yescharacteristic
210C2×Dic3—62no—
211C2×Dic3—62no—
212C2×Dic3—64no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C3×C8—182no—
216C3⋊C8—62no—
217C3⋊C8—182no—
218C2×C32—22no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32—21no—
225C2×C32—21no—
226C2×C32—41no—
227C2×C32—41no—
228C2×C32C3⋊#9511yescharacteristic
229C2×C32C3⋊#9511yescharacteristic
230C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
231M16—181no—
232M16—181no—
233C2×C8—92no—
234C4⋊C4—181no—
235C4⋊C4—181no—
236C22⋊C4—181no—
237C22⋊C4—181no—
238C42—92no—
239C42—92no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—31no—
243C22×C3—31no—
244C22×C3—31no—
245C22×C3—31no—
246C22×C3—41no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—121no—
251C22×C3—121no—
252C22×C3—122no—
253C22×C3(C2×D3)⋊Dic311yescharacteristic
254C22×C3#372211yescharacteristic
255C22×C3Dic3⋊Dic311yescharacteristic
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—31no—
261C3×C4—31no—
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4—61no—
267C3×C4—61no—
268C3×C4—362no—
269C3×C4—362no—
270C3×C4Dic3211yescharacteristic
271C3×C4(C2×D3)⋊Dic311yescharacteristic
272C3×C4Dic3⋊Dic311yescharacteristic
273C3×C4#372411yescharacteristic
274Dic3—122no—
275Dic3—122no—
276Dic3—122no—
277Dic3—124no—
278Dic3—362no—
279C32—21no—
280C32#453111yescharacteristic, complemented
281C32#455711yescharacteristic, complemented
282C23#620011yescharacteristic
283C2×C4—182no—
284C2×C4—182no—
285C2×C4#619611yescharacteristic
286C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
287C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
288C2×C4#619811yescharacteristic
289C2×C4#619911yescharacteristic
290C2×C4#620011yescharacteristic
291C8—182no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3—31no—
296C2×C3—31no—
297C2×C3—61no—
298C2×C3—61no—
299C2×C3—61no—
300C2×C3—61no—
301C2×C3—61no—
302C2×C3—61no—
303C2×C3—62no—
304C2×C3—61no—
305C2×C3—121no—
306C2×C3—121no—
307C2×C3#6357911yescharacteristic
308C22—41no—
309C22#8844811yescharacteristic
310C22#8845011yescharacteristic
311C22#8845111yescharacteristic
312C4—362no—
313C4—362no—
314C4#8844711yescharacteristic
315C4#8844811yescharacteristic
316C4#8844911yescharacteristic
317C4#8845011yescharacteristic
318C3—31no—
319C3—31no—
320C3—61no—
321C3#1058915811yescharacteristic
322C2—21no—
323C2—21no—
324C2—21no—
325C2#1172033411yescharacteristic
subgroupG/H
centerC4#88447
commutatorC4×H(3,𝔽3)C42
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC2×C3#63579
prime radicalC23×H(3,𝔽3)D4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)H(3,ℤ/4)
LCSG ▷ C4×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
9. C4×#1084C4semidirect
12. C4×(C3⋊Dic3)⋊C3C4semidirect
15. C2×(C3⋊(C3⋊C8))⋊C3C4semidirect
41. C4×C3⋊Dic3Dic3semidirect
44. C2×C3⋊(C3⋊C8)Dic3semidirect
179. H(3,𝔽3)H(3,ℤ/4)semidirect
280. C32#4531semidirect
281. C32#4557semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64135⋊C32
C2×#3748⋊C32
#105891583
#105891573