Finite Group Databasebeta

← #13147868

Subgroups — #13147868

Order 1728.

762 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 129 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#11721721C211yesmaximal, characteristic
2#64079⋊C3C212yesmaximal, complemented
3#11721151C212yesmaximal
4#11721150C214yesmaximal, complemented
5#11721088C214yesmaximal, complemented
6#11721080C212yesmaximal, complemented
7#10593952—32nomaximal
8C2×C4⋊#272C2211yescharacteristic
9C2×#6229C2211yescharacteristic
10C2×#6199C2212yes—
11C2×#6196C2212yes—
12C42×H(3,𝔽3)C2211yescharacteristic
13#88542C2214yes—
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15C4⋊((C3⋊Dic3)⋊C3)C2212yes—
16#3750⋊C3C2214yes—
17#3750⋊C3C2214yes—
18#88451C2212yes—
19#88451C2214yes—
20#88448C2214yes—
21#88447C2212yes—
22#64079D312yescomplemented
23C3×#875—32no—
24C3×#875—61no—
25C3×#870—32no—
26#63899—34no—
27#63893—34no—
28#63884—34no—
29#63861—34no—
30#63858—34no—
31#63839—34no—
32C22×#272C2312yes—
33C2×C4×H(3,𝔽3)C2311yescharacteristic
34C2×C4×H(3,𝔽3)C2312yes—
35C2×#1084C2312yes—
36C4⋊#272—41no—
37#6229—41no—
38C2×(C3⋊Dic3)⋊C3C2314yes—
39C2×(C3⋊Dic3)⋊C3C2314yes—
40#6199—42no—
41#6196—42no—
42#5699—92no—
43#5672—91no—
44C2×C3×D12—32no—
45C2×C3×D12—61no—
46C2×C3×Dic6—32no—
47C2×C3×Dic6—61no—
48C2×#627—34no—
49C2×D3×Dic3—34no—
50C32×C42—21no—
51C32×C42C2×D312yes—
52C4⋊(C3⋊Dic3)C2×D312yes—
53C4⋊(C3⋊Dic3)C2×D312yes—
54#3750C2×D314yes—
55#3750C2×D314yes—
56C3×(C2×D3)⋊C4—34no—
57C3×(C2×D3)⋊C4—34no—
58C3×(C2×D3)⋊C4—64no—
59C3×C4⋊Dic3—32no—
60C3×C4⋊Dic3—32no—
61C3×Dic3⋊C4—34no—
62C3×Dic3⋊C4—34no—
63#3724—34no—
64Dic3⋊Dic3—34no—
65(C2×D3)⋊Dic3—34no—
66(C2×D3)⋊Dic3—34no—
67(C2×D3)⋊Dic3—34no—
68Dic32—34no—
69C22×H(3,𝔽3)C2411yescharacteristic
70C2×#272—22no—
71C2×#272—24no—
72C4×H(3,𝔽3)—21no—
73C4×H(3,𝔽3)—22no—
74#1084—22no—
75(C3⋊Dic3)⋊C3—24no—
76(C3⋊Dic3)⋊C3—24no—
77C3×#125—92no—
78C3×#124—91no—
79#944—92no—
80#935—92no—
81#898—94no—
82#898—94no—
83#897—94no—
84#896—94no—
85#894—94no—
86#888—94no—
87#879—94no—
88#878—94no—
89#875—32no—
90#875—61no—
91#870—32no—
92#870—92no—
93C22×C3×D3—34no—
94C22×C3×D3—62no—
95C2×C32×C4—21no—
96C2×C32×C4—22no—
97C2×C32×C4C22×D312yes—
98C2×C32×C4C22×D312yes—
99C2×C32×C4C22×D312yes—
100C2×C3⋊Dic3C22×D314yes—
101C2×C3⋊Dic3C22×D314yes—
102C2×C3×Dic3—34no—
103C2×C3×Dic3—34no—
104C2×C3×Dic3—34no—
105C2×C3×Dic3—62no—
106C3×D12—122no—
107C3×D12—122no—
108C3×Dic6—122no—
109C3×Dic6—122no—
110#627—124no—
111D3×Dic3—124no—
112#563—271noSylow
113C2×H(3,𝔽3)H(5,𝔽2)11yescharacteristic
114C2×H(3,𝔽3)2−1+412yes—
115#272—42no—
116C2×C3×Q8—91no—
117C2×C3×D4—91no—
118C22×Dic3—92no—
119C2×D12—32no—
120C2×D12—61no—
121C2×C4×D3—94no—
122C2×Dic6—32no—
123C2×Dic6—61no—
124C2×Dic6—92no—
125C2×Dic6—94no—
126C3×C4⋊C4—94no—
127C3×C4⋊C4—92no—
128C3×C4⋊C4—94no—
129C3×C4⋊C4—92no—
130C3×C22⋊C4—94no—
131C3×C42—32no—
132C3×C42—61no—
133C3×C42D3211yescharacteristic
134#217—94no—
135#217—92no—
136(C2×D3)⋊C4—34no—
137(C2×D3)⋊C4—34no—
138(C2×D3)⋊C4—64no—
139(C2×D3)⋊C4—94no—
140(C2×D3)⋊C4—94no—
141(C2×D3)⋊C4—94no—
142C4⋊Dic3—32no—
143C4⋊Dic3—32no—
144C4⋊Dic3—92no—
145C4⋊Dic3—92no—
146C4⋊Dic3—94no—
147Dic3⋊C4—34no—
148Dic3⋊C4—34no—
149Dic3⋊C4—94no—
150Dic3⋊C4—94no—
151Dic3⋊C4—94no—
152Dic3⋊C4—94no—
153C4×Dic3—92no—
154C4×Dic3—94no—
155C22×C32—21no—
156C22×C32C23×D312yes—
157C2×C3×D3—64no—
158C2×C3×D3—64no—
159C2×C3×D3—64no—
160C2×C3×D3—122no—
161C2×C3×D3—124no—
162C32×C4—22no—
163C32×C4—22no—
164C32×C4—22no—
165C32×C4—22no—
166C32×C4—42no—
167C3⋊Dic3—24no—
168C3⋊Dic3—24no—
169C3×Dic3—64no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—122no—
173C4⋊Q8—272no—
174#126—274no—
175#125—92no—
176#124—91no—
177#123—272no—
178C22⋊Q8—274no—
179H(3,𝔽3)#56311yescharacteristic, complemented, Sylow
180C23×C3—92no—
181C22×D3—34no—
182C22×D3—62no—
183C3×Q8—361no—
184C3×D4—361no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—32no—
188C2×C3×C4—61no—
189C2×C3×C4—62no—
190C2×C3×C4—94no—
191C2×C3×C4—94no—
192C2×C3×C4—92no—
193C2×C3×C4C2×D3211yescharacteristic
194C2×C3×C4C2×D3212yes—
195C2×Dic3—34no—
196C2×Dic3—34no—
197C2×Dic3—34no—
198C2×Dic3—62no—
199C2×Dic3—94no—
200C2×Dic3—94no—
201C2×Dic3—362no—
202D12—122no—
203D12—122no—
204C4×D3—364no—
205Dic6—122no—
206Dic6—122no—
207Dic6—362no—
208Dic6—364no—
209C2×C32—22no—
210C2×C32—21no—
211C2×C32#100712yes—
212C2×C32#100912yes—
213C2×C32#101012yes—
214C3×D3—124no—
215C3×D3—242no—
216C2×Q8—91no—
217C2×Q8—272no—
218C2×D4—91no—
219C22×C4—272no—
220C4⋊C4—92no—
221C4⋊C4—94no—
222C4⋊C4—94no—
223C4⋊C4—92no—
224C4⋊C4—274no—
225C22⋊C4—94no—
226C22⋊C4—274no—
227C22⋊C4—272no—
228C42—272no—
229C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
230C22×C3—32no—
231C22×C3—61no—
232C22×C3—184no—
233C22×C3—182no—
234C22×C3C22×D3211yescharacteristic
235C2×D3—64no—
236C2×D3—64no—
237C2×D3—64no—
238C2×D3—122no—
239C2×D3—124no—
240C3×C4—22no—
241C3×C4—21no—
242C3×C4—62no—
243C3×C4—62no—
244C3×C4—62no—
245C3×C4—62no—
246C3×C4—122no—
247C3×C4—184no—
248C3×C4—184no—
249C3×C4—182no—
250Dic3—64no—
251Dic3—64no—
252Dic3—64no—
253Dic3—122no—
254Dic3—184no—
255Dic3—184no—
256C32—21no—
257C32#569912yescomplemented
258C23—92no—
259Q8—361no—
260Q8—1082no—
261D4—361no—
262C2×C4—92no—
263C2×C4—94no—
264C2×C4—94no—
265C2×C4—1082no—
266C2×C4C2×((C3⋊D3)⋊C3)⋊C212yes—
267C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
268C2×C3—32no—
269C2×C3—32no—
270C2×C3—32no—
271C2×C3—62no—
272C2×C3—61no—
273C2×C3—362no—
274C2×C3#6429712yes—
275C2×C3#6430811yescharacteristic
276D3—124no—
277D3—242no—
278C22—184no—
279C22—182no—
280C22C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
281C4—22no—
282C4—21no—
283C4—184no—
284C4—184no—
285C4—182no—
286C3—32no—
287C3—61no—
288C3#1059385711yescharacteristic
289C2—362no—
290C2#1172248212yes—
291C2#1172249311yescharacteristic
subgroupG/H
centerC22C22×((C3⋊D3)⋊C3)⋊C2
commutator, cosocleC22×H(3,𝔽3)C24
Frattini, socleC22×C3C22×D32
FittingC42×H(3,𝔽3)C22
prime radicalC2×C4⋊#272C22
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#563
LCSG ▷ C22×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
2. #64079⋊C3C2semidirect
4. #11721150C2semidirect
5. #11721088C2semidirect
6. #11721080C2semidirect
22. #64079D3semidirect
179. H(3,𝔽3)#563semidirect
257. C32#5699semidirect

Maximal subgroups

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

subgroupindex
#117217212
#64079⋊C32
#117211512
#117211502
#117210882
#117210802
#105939523