Finite Group Databasebeta

← #13143659

Subgroups — #13143659

Order 1728.

859 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 102 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
1H(3,𝔽3)×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2#64146⋊C3C213yesmaximal, complemented
3#63661⋊C3C213yesmaximal, complemented
4#10592943C311yesmaximal, characteristic, complemented
5C3×#5244—31nomaximal
6C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
7#41×H(3,𝔽3)—23no—
8C2×D4×H(3,𝔽3)—23no—
9C2×D4×H(3,𝔽3)C2213yes—
10C2×#638⋊C3—23no—
11#3757⋊C3C2213yescomplemented
12#3752⋊C3—23no—
13C32×H(5,𝔽2)—21no—
14C32×H(5,𝔽2)D311yescharacteristic, complemented
15C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
16#64146C2×C313yescomplemented
17C3×#937—33no—
18#63661C2×C313yescomplemented
19C3×#834—33no—
20C23×H(3,𝔽3)—21no—
21C23×H(3,𝔽3)D413yescomplemented
22C23×H(3,𝔽3)C2311yescharacteristic
23C22×(C3⋊D3)⋊C3—23no—
24C22×(C3⋊D3)⋊C3—42no—
25Q8×H(3,𝔽3)—21no—
26D4×H(3,𝔽3)—23no—
27D4×H(3,𝔽3)—26no—
28C2×C4×H(3,𝔽3)—23no—
29C2×C4×H(3,𝔽3)D413yescomplemented
30#638⋊C3—46no—
31C2×(C3⋊Dic3)⋊C3—23no—
32C3×B(4,𝔽2)—91no—
33#5244—31no—
34#5244—91no—
35C23×C3⋊D3C22×C311yescharacteristic, complemented
36C23×C3×D3—31no—
37C32×#41—26no—
38C32×#41—23no—
39C32×#41—23no—
40C2×C32×D4—26no—
41C2×C32×D4—23no—
42C2×C32×D4—23no—
43C2×C32×D4—23no—
44C2×C32×D4C2×D313yes—
45C2×C32×D4C22×C313yes—
46C2×#638—23no—
47C2×C3×#67—63no—
48#3757C22×C313yescomplemented
49#3752—23no—
50C3×#217—33no—
51C3×(C2×D3)⋊C4—63no—
52C22×H(3,𝔽3)—23no—
53C22×H(3,𝔽3)—41no—
54C22×H(3,𝔽3)—43no—
55C22×H(3,𝔽3)—81no—
56C22×H(3,𝔽3)C2×D413yes—
57C2×(C3⋊D3)⋊C3—26no—
58C2×(C3⋊D3)⋊C3—43no—
59C2×(C3⋊D3)⋊C3—41no—
60C4×H(3,𝔽3)—23no—
61(C3⋊Dic3)⋊C3—43no—
62C3×H(5,𝔽2)—31no—
63C3×H(5,𝔽2)—31no—
64C3×H(5,𝔽2)—61no—
65C3×H(5,𝔽2)C3×D311yescharacteristic
66C3×#120—93no—
67#937—33no—
68#937—93no—
69C3×#99—93no—
70#834—33no—
71#834—93no—
72C23×C32—22no—
73C23×C32—21no—
74C23×C32—21no—
75C23×C32—23no—
76C23×C32—21no—
77C23×C32#6713yescomplemented
78C23×C32C3×D413yescomplemented
79C23×C32C22×D311yescharacteristic
80C23×C32C23×C311yescharacteristic
81C22×C3⋊D3—23no—
82C22×C3⋊D3—42no—
83C22×C3×D3—63no—
84C22×C3×D3—122no—
85C32×Q8—22no—
86C32×Q8—21no—
87C32×Q8—21no—
88C32×D4—26no—
89C32×D4—23no—
90C32×D4—23no—
91C32×D4—212no—
92C32×D4—26no—
93C32×D4—26no—
94C2×C32×C4—26no—
95C2×C32×C4—23no—
96C2×C32×C4—23no—
97C2×C32×C4—23no—
98C2×C32×C4#6713yescomplemented
99C2×C32×C4C3×D413yescomplemented
100#638—46no—
101C2×C3⋊Dic3—23no—
102C3×#67—126no—
103C2×C3×Dic3—63no—
104B(4,𝔽2)—91noSylow
105C2×H(3,𝔽3)—23no—
106C2×H(3,𝔽3)—43no—
107C2×H(3,𝔽3)#12011yescharacteristic
108(C3⋊D3)⋊C3—42no—
109C24×C3—91no—
110C23×D3—31no—
111C23×D3—91no—
112C3×#41—23no—
113C3×#41—63no—
114C3×#41—63no—
115C3×#41—66no—
116C2×C3×D4—23no—
117C2×C3×D4—33no—
118C2×C3×D4—33no—
119C2×C3×D4—63no—
120C2×C3×D4—63no—
121C2×C3×D4—66no—
122C2×C3×D4—63no—
123C2×C3×D4—183no—
124C2×C3×D4C2×C3×D313yes—
125C2×#67—63no—
126C2×#67—183no—
127C3×C22⋊C4—93no—
128C3×C22⋊C4—183no—
129#217—33no—
130#217—93no—
131(C2×D3)⋊C4—63no—
132(C2×D3)⋊C4—183no—
133C22×C32—26no—
134C22×C32—23no—
135C22×C32—23no—
136C22×C32—23no—
137C22×C32—41no—
138C22×C32—41no—
139C22×C32—43no—
140C22×C32—43no—
141C22×C32—81no—
142C22×C32—83no—
143C22×C32—82no—
144C22×C32—81no—
145C22×C32—81no—
146C22×C32C2×#6713yes—
147C22×C32C2×C3×D413yes—
148C2×C3⋊D3—26no—
149C2×C3⋊D3—43no—
150C2×C3⋊D3—41no—
151C2×C3×D3—66no—
152C2×C3×D3—123no—
153C2×C3×D3—121no—
154C32×C4—26no—
155C32×C4—23no—
156C32×C4—23no—
157C3⋊Dic3—43no—
158C3×Dic3—123no—
159H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic, complemented
160#120—93no—
161#99—93no—
162H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
163C23×C3—21no—
164C23×C3—33no—
165C23×C3—33no—
166C23×C3—31no—
167C23×C3—31no—
168C23×C3—61no—
169C23×C3—61no—
170C23×C3—62no—
171C23×C3—63no—
172C23×C3—61no—
173C23×C3—183no—
174C23×C3—362no—
175C23×C3C3×#6713yes—
176C23×C3C22×C3×D311yescharacteristic
177C22×D3—63no—
178C22×D3—122no—
179C22×D3—183no—
180C22×D3—362no—
181C3×Q8—21no—
182C3×Q8—61no—
183C3×Q8—61no—
184C3×Q8—62no—
185C3×D4—23no—
186C3×D4—26no—
187C3×D4—66no—
188C3×D4—66no—
189C3×D4—63no—
190C3×D4—63no—
191C3×D4—66no—
192C3×D4—612no—
193C3×D4—366no—
194C2×C3×C4—23no—
195C2×C3×C4—33no—
196C2×C3×C4—33no—
197C2×C3×C4—63no—
198C2×C3×C4—63no—
199C2×C3×C4—66no—
200C2×C3×C4—63no—
201C2×C3×C4—183no—
202C2×C3×C4C3×#6713yes—
203#67—126no—
204#67—366no—
205C2×Dic3—63no—
206C2×Dic3—183no—
207C2×C32—23no—
208C2×C32—23no—
209C2×C32—21no—
210C2×C32—43no—
211C2×C32—46no—
212C2×C32—43no—
213C2×C32—43no—
214C2×C32#95311yescharacteristic
215C2×C32C3×#12011yescharacteristic
216C3⋊D3—42no—
217C3×D3—122no—
218C24—91no—
219#41—23no—
220C2×D4—23no—
221C2×D4—183no—
222C2×D4C2×(C3⋊D3)⋊C313yes—
223C22⋊C4—93no—
224C22⋊C4—183no—
225C22×C3—23no—
226C22×C3—33no—
227C22×C3—33no—
228C22×C3—41no—
229C22×C3—43no—
230C22×C3—63no—
231C22×C3—63no—
232C22×C3—66no—
233C22×C3—63no—
234C22×C3—81no—
235C22×C3—121no—
236C22×C3—121no—
237C22×C3—123no—
238C22×C3—123no—
239C22×C3—186no—
240C22×C3—241no—
241C22×C3—243no—
242C22×C3—241no—
243C22×C3—241no—
244C22×C3—242no—
245C22×C3—361no—
246C22×C3—363no—
247C22×C3C2×C3×#6713yes—
248C2×D3—66no—
249C2×D3—123no—
250C2×D3—121no—
251C2×D3—186no—
252C2×D3—361no—
253C2×D3—363no—
254C3×C4—23no—
255C3×C4—63no—
256C3×C4—63no—
257C3×C4—66no—
258C3×C4—363no—
259Dic3—123no—
260Dic3—363no—
261C32—21no—
262C32#524411yescharacteristic, complemented
263C32C3×B(4,𝔽2)11yescharacteristic, complemented
264C23—21no—
265C23—183no—
266C23—362no—
267C23#638⋊C313yescomplemented
268C23C22×(C3⋊D3)⋊C311yescharacteristic
269Q8—21no—
270D4—23no—
271D4—26no—
272D4—366no—
273C2×C4—23no—
274C2×C4—183no—
275C2×C4#638⋊C313yescomplemented
276C2×C3—23no—
277C2×C3—31no—
278C2×C3—31no—
279C2×C3—43no—
280C2×C3—63no—
281C2×C3—63no—
282C2×C3—61no—
283C2×C3—123no—
284C2×C3—123no—
285C2×C3—123no—
286C2×C3—126no—
287C2×C3—362no—
288C2×C3C3×#95311yescharacteristic
289D3—122no—
290D3—362no—
291C22—23no—
292C22—41no—
293C22—43no—
294C22—81no—
295C22—186no—
296C22—363no—
297C22—361no—
298C22C2×#638⋊C313yes—
299C4—23no—
300C4—363no—
301C3—31no—
302C3—31no—
303C3—61no—
304C3C3×#524411yescharacteristic
305C2—23no—
306C2—43no—
307C2—362no—
308C2#1172170211yescharacteristic
subgroupG/H
centerC2#11721702
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
FittingH(3,𝔽3)×H(5,𝔽2)C2
socleC2×C3C3×#953
hypercenterH(5,𝔽2)(C3⋊D3)⋊C3
hypocenterC32C3×B(4,𝔽2)
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C23 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽3)×H(5,𝔽2)C2semidirect
2. #64146⋊C3C2semidirect
3. #63661⋊C3C2semidirect
4. #10592943C3semidirect
6. C23×(C3⋊D3)⋊C3C22semidirect
11. #3757⋊C3C22semidirect
14. C32×H(5,𝔽2)D3semidirect
15. C32×H(5,𝔽2)C2×C3semidirect
16. #64146C2×C3semidirect
18. #63661C2×C3semidirect
21. C23×H(3,𝔽3)D4semidirect
29. C2×C4×H(3,𝔽3)D4semidirect
35. C23×C3⋊D3C22×C3semidirect
48. #3757C22×C3semidirect
77. C23×C32#67semidirect
78. C23×C32C3×D4semidirect
98. C2×C32×C4#67semidirect
99. C2×C32×C4C3×D4semidirect
159. H(5,𝔽2)(C3⋊D3)⋊C3semidirect
162. H(3,𝔽3)B(4,𝔽2)semidirect
262. C32#5244semidirect
263. C32C3×B(4,𝔽2)semidirect
267. C23#638⋊C3semidirect
275. C2×C4#638⋊C3semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽3)×H(5,𝔽2)2
#64146⋊C32
#63661⋊C32
#105929433
C3×#52443