Finite Group Databasebeta

← #13139934

Subgroups — #13139934

Order 1728.

475 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 475 conjugacy classes of subgroups — too many to draw. Only its 131 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
3C22×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic
4#11720307C212yesmaximal, complemented
5#11720306C212yesmaximal, complemented
6#10591138—31nomaximal
7#10591122—31nomaximal
8C22×C4×H(3,𝔽3)C2211yescharacteristic
9#88552—21no—
10C4⋊#1084C2212yes—
11C4×#1084—21no—
12#3757⋊C3—21no—
13C4⋊((C3⋊Dic3)⋊C3)C2212yes—
14C4×(C3⋊Dic3)⋊C3—21no—
15C2×(C3⋊(C3⋊C8))⋊C3C414yescomplemented
16C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
17C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
18#64135D311yescharacteristic
19C22×C3⋊(C3⋊C8)D311yescharacteristic
20C3×#926—31no—
21C3×#926—31no—
22C3×#926—31no—
23C3×#926—61no—
24C22×C3×C3⋊C8—31no—
25#63576—32no—
26#63575—32no—
27#63574—32no—
28#63573—32no—
29C23×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)C2×C412yes—
31C2×C4×H(3,𝔽3)D411yescharacteristic
32C2×C4×H(3,𝔽3)Q812yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×#1084—22no—
35C2×(C3⋊Dic3)⋊C3—22no—
36(C3⋊(C3⋊C8))⋊C3C2×C412yes—
37(C3⋊(C3⋊C8))⋊C3C2×C412yes—
38#5106—91no—
39#4998—91no—
40#4998—91no—
41C22×C32×C4—21no—
42C22×C32×C4C2×D311yescharacteristic
43C22×C32×C4C2×D311yescharacteristic
44#3757—21no—
45C4⋊(C3⋊Dic3)C2×D312yes—
46C4×C3⋊Dic3—21no—
47C2×C3⋊(C3⋊C8)Dic314yescomplemented
48C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
50C3×#217—62no—
51C3×#217—61no—
52C3×#217—61no—
53C3×#217—61no—
54C3×C4⋊Dic3—32no—
55C3×C4⋊Dic3—32no—
56C3×C4⋊Dic3—32no—
57C3×C4⋊Dic3—62no—
58C3×C4×Dic3—62no—
59C3×C4×Dic3—61no—
60C3×C4×Dic3—61no—
61C3×C4×Dic3—61no—
62C2×C3×C3⋊C8—34no—
63C2×C3×C3⋊C8—31no—
64C2×C3×C3⋊C8—31no—
65C22×H(3,𝔽3)—22no—
66C22×H(3,𝔽3)C4⋊C412yes—
67C22×H(3,𝔽3)C2×D411yescharacteristic
68C4×H(3,𝔽3)C4⋊C412yes—
69C4×H(3,𝔽3)C22×C411yescharacteristic
70C4×H(3,𝔽3)C2×Q811yescharacteristic
71#1084—42no—
72(C3⋊Dic3)⋊C3—42no—
73C22×C3×C8—91no—
74C3×#117—91no—
75C3×#117—91no—
76#926—31no—
77#926—31no—
78#926—31no—
79#926—61no—
80#926—91no—
81C22×C3⋊C8—31no—
82C22×C3⋊C8—91no—
83#818—92no—
84#817—92no—
85#808—92no—
86#808—92no—
87#807—92no—
88#807—92no—
89C23×C32—21no—
90C23×C32D1211yescharacteristic
91C23×C32#6711yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—21no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4Dic612yes—
97C2×C32×C4Dic612yes—
98C2×C32×C4C4×D312yes—
99C2×C32×C4D1211yescharacteristic
100C2×C32×C4C2×Dic312yes—
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—22no—
105C2×C3×Dic3—64no—
106C2×C3×Dic3—62no—
107C2×C3×Dic3—62no—
108C2×C3×Dic3—62no—
109C3⋊(C3⋊C8)C2×Dic312yes—
110C3⋊(C3⋊C8)C2×Dic312yes—
111C3×C3⋊C8—32no—
112C3×C3⋊C8—32no—
113#427—271noSylow
114C2×H(3,𝔽3)—22no—
115C2×H(3,𝔽3)C2×C4⋊C411yescharacteristic
116C2×H(3,𝔽3)#13512yes—
117C22×C3×C4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—61no—
120C22×C3×C4D3211yescharacteristic
121C2×C3×C8—91no—
122C2×C3×C8—91no—
123C2×C3×C8—94no—
124C3×C4⋊C4—92no—
125C3×C4⋊C4—92no—
126C3×C22⋊C4—181no—
127C3×C22⋊C4—181no—
128C3×C42—181no—
129C3×C42—181no—
130#217—61no—
131#217—61no—
132#217—62no—
133#217—61no—
134#217—181no—
135C4⋊Dic3—32no—
136C4⋊Dic3—32no—
137C4⋊Dic3—32no—
138C4⋊Dic3—62no—
139C4⋊Dic3—92no—
140C4×Dic3—61no—
141C4×Dic3—61no—
142C4×Dic3—62no—
143C4×Dic3—61no—
144C4×Dic3—181no—
145C2×C3⋊C8—34no—
146C2×C3⋊C8—31no—
147C2×C3⋊C8—31no—
148C2×C3⋊C8—91no—
149C2×C3⋊C8—91no—
150C2×C3⋊C8—94no—
151C22×C32—22no—
152C22×C32—22no—
153C22×C32—22no—
154C22×C32—21no—
155C22×C32—42no—
156C22×C32Dic3⋊C412yes—
157C22×C32C4⋊Dic312yes—
158C22×C32C2×D1211yescharacteristic
159C22×C32C2×#6711yescharacteristic
160C32×C4—22no—
161C32×C4—21no—
162C32×C4—21no—
163C32×C4Dic3⋊C412yes—
164C32×C4C4⋊Dic312yes—
165C32×C4C2×Dic611yescharacteristic
166C32×C4C2×Dic611yescharacteristic
167C32×C4C2×C4×D311yescharacteristic
168C32×C4C22×Dic311yescharacteristic
169C3⋊Dic3—42no—
170C3×Dic3—124no—
171C3×Dic3—122no—
172C3×Dic3—122no—
173C3×Dic3—122no—
174C22×C8—91no—
175#117—91no—
176#117—91no—
177#107—272no—
178#106—272no—
179H(3,𝔽3)#42711yescharacteristic, complemented, Sylow
180C23×C3—31no—
181C23×C3—31no—
182C23×C3—61no—
183C23×C3#62611yescharacteristic
184C2×C3×C4—32no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—31no—
189C2×C3×C4—31no—
190C2×C3×C4—31no—
191C2×C3×C4—31no—
192C2×C3×C4—62no—
193C2×C3×C4—62no—
194C2×C3×C4—61no—
195C2×C3×C4—61no—
196C2×C3×C4—182no—
197C2×C3×C4—182no—
198C2×C3×C4D3×Dic312yes—
199C2×C3×C4#62611yescharacteristic
200C2×C3×C4#62712yes—
201C2×C3×C4C2×D3211yescharacteristic
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—64no—
205C2×Dic3—62no—
206C2×Dic3—182no—
207C3×C8—92no—
208C3×C8—92no—
209C3⋊C8—32no—
210C3⋊C8—32no—
211C3⋊C8—92no—
212C3⋊C8—92no—
213C2×C32—22no—
214C2×C32—22no—
215C2×C32—22no—
216C2×C32—21no—
217C2×C32—42no—
218C2×C32#90412yes—
219C2×C32C2×Dic3⋊C411yescharacteristic
220C2×C32C2×C4⋊Dic311yescharacteristic
221C2×C32#95012yes—
222C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
223C2×C8—94no—
224C2×C8—91no—
225C2×C8—91no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C22⋊C4—181no—
229C22⋊C4—181no—
230C42—181no—
231C42—181no—
232C22×C3—22no—
233C22×C3—31no—
234C22×C3—31no—
235C22×C3—32no—
236C22×C3—32no—
237C22×C3—62no—
238C22×C3—62no—
239C22×C3—61no—
240C22×C3—62no—
241C22×C3—122no—
242C22×C3Dic3⋊Dic312yes—
243C22×C3C2×#62611yescharacteristic
244C3×C4—31no—
245C3×C4—31no—
246C3×C4—31no—
247C3×C4—31no—
248C3×C4—32no—
249C3×C4—32no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4—62no—
253C3×C4—362no—
254C3×C4—362no—
255C3×C4Dic3⋊Dic312yes—
256C3×C4C2×D3×Dic311yescharacteristic
257C3×C4C2×#62711yescharacteristic
258Dic3—122no—
259Dic3—122no—
260Dic3—124no—
261Dic3—122no—
262Dic3—362no—
263C32—21no—
264C32#499811yescharacteristic, complemented
265C32#510611yescharacteristic, complemented
266C23#620011yescharacteristic
267C2×C4—182no—
268C2×C4—182no—
269C2×C4#619612yes—
270C2×C4((C3⋊D3)⋊C3)⋊C412yes—
271C2×C4#620011yescharacteristic
272C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
273C8—92no—
274C8—92no—
275C2×C3—22no—
276C2×C3—31no—
277C2×C3—31no—
278C2×C3—32no—
279C2×C3—32no—
280C2×C3—62no—
281C2×C3—62no—
282C2×C3—61no—
283C2×C3—62no—
284C2×C3—122no—
285C2×C3#6382912yes—
286C2×C3C2×Dic3⋊Dic311yescharacteristic
287C22—22no—
288C22#8844812yes—
289C22C2×#620011yescharacteristic
290C4—362no—
291C4—362no—
292C4#8844812yes—
293C4C2×#619611yescharacteristic
294C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
295C3—31no—
296C3—31no—
297C3—61no—
298C3#1059112211yescharacteristic
299C2—22no—
300C2#1172107312yes—
301C2C2×#8844811yescharacteristic
subgroupG/H
centerC2×C4#6200
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC22×C4×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radicalC23×H(3,𝔽3)D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC22×C4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#427
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720307C2semidirect
5. #11720306C2semidirect
15. C2×(C3⋊(C3⋊C8))⋊C3C4semidirect
47. C2×C3⋊(C3⋊C8)Dic3semidirect
179. H(3,𝔽3)#427semidirect
264. C32#4998semidirect
265. C32#5106semidirect

Maximal subgroups

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

subgroupindex
#117218262
#64135⋊C32
C22×(C3⋊(C3⋊C8))⋊C32
#117203072
#117203062
#105911383
#105911223