Finite Group Databasebeta

← #13135174

Subgroups — #13135174

Order 1728.

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

normal subgroups only

This group has 383 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
1C2×#88538C211yesmaximal, characteristic
2#64135⋊C3C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#10589171—31nomaximal
5#10589169—31nomaximal
6C22×C4×H(3,𝔽3)C2211yescharacteristic
7#88538—21no—
8#88538—21no—
9C2×#6180C412yescomplemented
10#3757⋊C3C412yes—
11C4⋊((C3⋊Dic3)⋊C3)—21no—
12C4×(C3⋊Dic3)⋊C3—21no—
13#3748⋊C3—21no—
14#3748⋊C3C412yescomplemented
15C2×(C3⋊(C3⋊C8))⋊C3—21no—
16#64135D311yescharacteristic
17C2×#3748D311yescharacteristic
18C3×#926—31no—
19C2×C3×#208—31no—
20C2×C3×#208—31no—
21C2×C3×#208—31no—
22C2×C3×#208—61no—
23C23×H(3,𝔽3)C2×C411yescharacteristic
24C2×C4×H(3,𝔽3)C2×C411yescharacteristic
25C2×C4×H(3,𝔽3)C2×C411yescharacteristic
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)D411yescharacteristic
29C2×C4×H(3,𝔽3)Q811yescharacteristic
30C2×(C3⋊Dic3)⋊C3—22no—
31#6180—22no—
32(C3⋊(C3⋊C8))⋊C3—22no—
33#4559—91no—
34#4556—91no—
35#4533—91no—
36C22×C32×C4—21no—
37C22×C32×C4C2×D311yescharacteristic
38C22×C32×C4C2×D311yescharacteristic
39#3757Dic312yes—
40C4⋊(C3⋊Dic3)—21no—
41C4×C3⋊Dic3—21no—
42#3748—21no—
43#3748Dic312yescomplemented
44C2×C3⋊(C3⋊C8)—21no—
45C3×#217—32no—
46C3×C4⋊Dic3—61no—
47C3×C4×Dic3—61no—
48C3×#208—32no—
49C3×#208—62no—
50C3×#208—62no—
51C3×#208—61no—
52C3×#208—61no—
53C3×#208—61no—
54C3×#208—61no—
55C3×#208—61no—
56C2×C3×C3⋊C8—32no—
57C2×C3×C3⋊C8—32no—
58C2×C3×C3⋊C8—62no—
59C2×C3×C3⋊C8—61no—
60C22×H(3,𝔽3)—41no—
61C22×H(3,𝔽3)C4211yescharacteristic
62C22×H(3,𝔽3)C22⋊C411yescharacteristic
63C22×H(3,𝔽3)C4⋊C411yescharacteristic
64C4×H(3,𝔽3)C22⋊C411yescharacteristic
65C4×H(3,𝔽3)C22⋊C411yescharacteristic
66C4×H(3,𝔽3)C4⋊C411yescharacteristic
67C4×H(3,𝔽3)C4⋊C411yescharacteristic
68(C3⋊Dic3)⋊C3—42no—
69C2×C3×M16—91no—
70C2×C3×M16—91no—
71C3×#117—91no—
72#926—31no—
73#926—91no—
74C2×#208—31no—
75C2×#208—31no—
76C2×#208—31no—
77C2×#208—61no—
78C2×#208—91no—
79C23×C32—21no—
80C23×C32C4×D311yescharacteristic
81C23×C32C2×Dic311yescharacteristic
82C2×C32×C4—21no—
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—21no—
87C2×C32×C4—21no—
88C2×C32×C4Dic611yescharacteristic
89C2×C32×C4Dic611yescharacteristic
90C2×C32×C4C4×D311yescharacteristic
91C2×C32×C4C4×D311yescharacteristic
92C2×C32×C4C4×D311yescharacteristic
93C2×C32×C4D1211yescharacteristic
94C2×C32×C4D1211yescharacteristic
95C2×C32×C4C2×Dic311yescharacteristic
96C2×C32×C4#6711yescharacteristic
97C2×C32×C4#6711yescharacteristic
98C2×C32×C4#6711yescharacteristic
99C2×C32×C4#6711yescharacteristic
100C2×C3⋊Dic3—22no—
101C2×C3×Dic3—62no—
102C3⋊(C3⋊C8)—22no—
103C3×C3⋊C8—64no—
104C3×C3⋊C8—62no—
105C3×C3⋊C8—62no—
106C3×C3⋊C8—62no—
107#344—271noSylow
108C2×H(3,𝔽3)—21no—
109C2×H(3,𝔽3)—21no—
110C2×H(3,𝔽3)—21no—
111C2×H(3,𝔽3)#9511yescharacteristic
112C22×C3×C4—31no—
113C22×C3×C4—31no—
114C22×C3×C4—61no—
115C22×C3×C4D3211yescharacteristic
116C3×M16—92no—
117C3×M16—181no—
118C3×M16—181no—
119C3×M16—181no—
120C2×C3×C8—92no—
121C2×C3×C8—181no—
122C3×C4⋊C4—181no—
123C3×C22⋊C4—92no—
124C3×C42—181no—
125#217—32no—
126#217—92no—
127C4⋊Dic3—61no—
128C4⋊Dic3—181no—
129C4×Dic3—61no—
130C4×Dic3—181no—
131#208—32no—
132#208—61no—
133#208—61no—
134#208—61no—
135#208—61no—
136#208—62no—
137#208—62no—
138#208—61no—
139#208—92no—
140#208—181no—
141C2×C3⋊C8—32no—
142C2×C3⋊C8—32no—
143C2×C3⋊C8—62no—
144C2×C3⋊C8—61no—
145C2×C3⋊C8—181no—
146C22×C32—21no—
147C22×C32—21no—
148C22×C32—21no—
149C22×C32—41no—
150C22×C32—41no—
151C22×C32—42no—
152C22×C32C4×Dic311yescharacteristic
153C22×C32C4×Dic311yescharacteristic
154C22×C32Dic3⋊C411yescharacteristic
155C22×C32C4⋊Dic311yescharacteristic
156C22×C32(C2×D3)⋊C411yescharacteristic
157C22×C32#21711yescharacteristic
158C32×C4—21no—
159C32×C4—21no—
160C32×C4—21no—
161C32×C4—21no—
162C32×C4Dic3⋊C411yescharacteristic
163C32×C4Dic3⋊C411yescharacteristic
164C32×C4Dic3⋊C411yescharacteristic
165C32×C4C4⋊Dic311yescharacteristic
166C32×C4(C2×D3)⋊C411yescharacteristic
167C32×C4(C2×D3)⋊C411yescharacteristic
168C32×C4(C2×D3)⋊C411yescharacteristic
169C32×C4#21711yescharacteristic
170C3⋊Dic3—42no—
171C3×Dic3—122no—
172C2×M16—91no—
173C2×M16—91no—
174#117—91no—
175H(3,𝔽3)#34411yescharacteristic, complemented, Sylow
176C23×C3—31no—
177C23×C3—31no—
178C23×C3—61no—
179C23×C3D3×Dic311yescharacteristic
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—31no—
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—61no—
193C2×C3×C4—61no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—61no—
197C2×C3×C4—61no—
198C2×C3×C4—182no—
199C2×C3×C4D3×Dic311yescharacteristic
200C2×C3×C4#62411yescharacteristic
201C2×C3×C4#62511yescharacteristic
202C2×C3×C4#62611yescharacteristic
203C2×C3×C4#62611yescharacteristic
204C2×C3×C4#62711yescharacteristic
205C2×Dic3—62no—
206C2×Dic3—182no—
207C3×C8—182no—
208C3×C8—182no—
209C3⋊C8—62no—
210C3⋊C8—62no—
211C3⋊C8—64no—
212C3⋊C8—62no—
213C3⋊C8—182no—
214C2×C32—22no—
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—21no—
219C2×C32—21no—
220C2×C32—21no—
221C2×C32—21no—
222C2×C32—41no—
223C2×C32—41no—
224C2×C32C3⋊#9511yescharacteristic
225C2×C32C3⋊#9511yescharacteristic
226C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
227M16—92no—
228M16—181no—
229M16—181no—
230M16—181no—
231C2×C8—92no—
232C2×C8—181no—
233C4⋊C4—181no—
234C22⋊C4—92no—
235C42—181no—
236C22×C3—31no—
237C22×C3—31no—
238C22×C3—31no—
239C22×C3—31no—
240C22×C3—31no—
241C22×C3—31no—
242C22×C3—41no—
243C22×C3—61no—
244C22×C3—61no—
245C22×C3—61no—
246C22×C3—121no—
247C22×C3—121no—
248C22×C3—122no—
249C22×C3Dic3211yescharacteristic
250C22×C3(C2×D3)⋊Dic311yescharacteristic
251C22×C3Dic3⋊Dic311yescharacteristic
252C3×C4—31no—
253C3×C4—31no—
254C3×C4—31no—
255C3×C4—31no—
256C3×C4—31no—
257C3×C4—31no—
258C3×C4—31no—
259C3×C4—31no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4—61no—
264C3×C4—362no—
265C3×C4(C2×D3)⋊Dic311yescharacteristic
266C3×C4#372211yescharacteristic
267C3×C4Dic3⋊Dic311yescharacteristic
268C3×C4#372411yescharacteristic
269Dic3—122no—
270Dic3—362no—
271C32—21no—
272C32#453311yescharacteristic, complemented
273C32#455911yescharacteristic, complemented
274C23((C3⋊D3)⋊C3)⋊C411yescharacteristic
275C2×C4—182no—
276C2×C4#619611yescharacteristic
277C2×C4((C3⋊D3)⋊C3)⋊C411yescharacteristic
278C2×C4#619811yescharacteristic
279C2×C4#619911yescharacteristic
280C2×C4#620011yescharacteristic
281C2×C4#620011yescharacteristic
282C8—182no—
283C8—182no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—31no—
288C2×C3—31no—
289C2×C3—61no—
290C2×C3—61no—
291C2×C3—61no—
292C2×C3—61no—
293C2×C3—61no—
294C2×C3—61no—
295C2×C3—62no—
296C2×C3—61no—
297C2×C3—121no—
298C2×C3—121no—
299C2×C3#6357911yescharacteristic
300C22—41no—
301C22#8844711yescharacteristic
302C22#8844811yescharacteristic
303C22#8845011yescharacteristic
304C4—362no—
305C4#8844811yescharacteristic
306C4#8844911yescharacteristic
307C4#8845011yescharacteristic
308C4#8845111yescharacteristic
309C3—31no—
310C3—31no—
311C3—61no—
312C3#1058917011yescharacteristic
313C2—21no—
314C2—21no—
315C2—21no—
316C2#1172033411yescharacteristic
subgroupG/H
centerC4#88451
commutatorC22×H(3,𝔽3)C42
FrattiniC22×C3×C4D32
Fitting, cosocleC22×C4×H(3,𝔽3)C22
socleC2×C3#63579
prime radicalC23×H(3,𝔽3)C2×C4
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#344
LCSG ▷ C22×H(3,𝔽3) ▷ C2×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. C2×#6180C4semidirect
14. #3748⋊C3C4semidirect
43. #3748Dic3semidirect
175. H(3,𝔽3)#344semidirect
272. C32#4533semidirect
273. C32#4559semidirect

Maximal subgroups

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

subgroupindex
C2×#885382
#64135⋊C32
C2×#3748⋊C32
#105891713
#105891693