Finite Group Databasebeta

← #13143658

Subgroups — #13143658

Order 1728.

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

normal subgroups only

This group has 726 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
2#64144⋊C3C213yesmaximal, complemented
3#63661⋊C3C213yesmaximal, complemented
4#10592942C311yesmaximal, characteristic, complemented
5C3×#5243—31nomaximal
6#41×H(3,𝔽3)—23no—
7C2×D4×H(3,𝔽3)—23no—
8C2×D4×H(3,𝔽3)C2213yes—
9C22×(C3⋊Dic3)⋊C3C2211yescharacteristic, complemented
10#3757⋊C3—23no—
11#3757⋊C3C2213yescomplemented
12#3750⋊C3—23no—
13C32×H(5,𝔽2)—21no—
14C32×H(5,𝔽2)D311yescharacteristic
15C32×H(5,𝔽2)C2×C311yescharacteristic
16#64144C2×C313yescomplemented
17C3×#935—33no—
18#63661C2×C313yescomplemented
19C3×#834—33no—
20C23×H(3,𝔽3)—21no—
21C23×H(3,𝔽3)D413yes—
22C23×H(3,𝔽3)C2311yescharacteristic
23Q8×H(3,𝔽3)—21no—
24D4×H(3,𝔽3)—23no—
25D4×H(3,𝔽3)—26no—
26C2×C4×H(3,𝔽3)—23no—
27C2×C4×H(3,𝔽3)D413yes—
28C2×(C3⋊Dic3)⋊C3—23no—
29C2×(C3⋊Dic3)⋊C3—23no—
30C3×#458—91no—
31#5243—31no—
32#5243—91no—
33C32×#41—26no—
34C32×#41—23no—
35C32×#41—23no—
36C2×C32×D4—26no—
37C2×C32×D4—23no—
38C2×C32×D4—23no—
39C2×C32×D4—23no—
40C2×C32×D4C2×D313yes—
41C2×C32×D4C22×C313yes—
42C22×C3⋊Dic3C22×C311yescharacteristic, complemented
43C22×C3×Dic3—31no—
44#3757—23no—
45#3757C22×C313yescomplemented
46#3750—23no—
47C3×#217—33no—
48C3×#217—63no—
49C3×Dic3⋊C4—63no—
50C22×H(3,𝔽3)—23no—
51C22×H(3,𝔽3)—41no—
52C22×H(3,𝔽3)—43no—
53C22×H(3,𝔽3)—81no—
54C22×H(3,𝔽3)C2×D413yes—
55C4×H(3,𝔽3)—23no—
56(C3⋊Dic3)⋊C3—43no—
57(C3⋊Dic3)⋊C3—41no—
58C3×H(5,𝔽2)—31no—
59C3×H(5,𝔽2)—31no—
60C3×H(5,𝔽2)—61no—
61C3×H(5,𝔽2)C3×D311yescharacteristic
62C3×#123—93no—
63#935—33no—
64#935—93no—
65C3×#99—93no—
66#834—33no—
67#834—93no—
68C23×C32—22no—
69C23×C32—21no—
70C23×C32—21no—
71C23×C32—23no—
72C23×C32—21no—
73C23×C32#6713yes—
74C23×C32C3×D413yes—
75C23×C32C22×D311yescharacteristic
76C23×C32C23×C311yescharacteristic
77C32×Q8—22no—
78C32×Q8—21no—
79C32×Q8—21no—
80C32×D4—26no—
81C32×D4—23no—
82C32×D4—23no—
83C32×D4—212no—
84C32×D4—26no—
85C32×D4—26no—
86C2×C32×C4—26no—
87C2×C32×C4—23no—
88C2×C32×C4—23no—
89C2×C32×C4—23no—
90C2×C32×C4#6713yes—
91C2×C32×C4C3×D413yes—
92C2×C3⋊Dic3—23no—
93C2×C3⋊Dic3—23no—
94C2×C3×Dic3—63no—
95C2×C3×Dic3—63no—
96#458—91noSylow
97C2×H(3,𝔽3)—23no—
98C2×H(3,𝔽3)—43no—
99C2×H(3,𝔽3)#12011yescharacteristic
100C3×#41—23no—
101C3×#41—63no—
102C3×#41—63no—
103C3×#41—66no—
104C2×C3×D4—23no—
105C2×C3×D4—33no—
106C2×C3×D4—33no—
107C2×C3×D4—63no—
108C2×C3×D4—63no—
109C2×C3×D4—66no—
110C2×C3×D4—63no—
111C2×C3×D4C2×C3×D313yes—
112C22×C3×C4—91no—
113C22×Dic3—31no—
114C22×Dic3—91no—
115C3×C4⋊C4—183no—
116C3×C22⋊C4—93no—
117C3×C22⋊C4—183no—
118#217—33no—
119#217—63no—
120#217—93no—
121#217—183no—
122Dic3⋊C4—63no—
123Dic3⋊C4—183no—
124C22×C32—26no—
125C22×C32—23no—
126C22×C32—23no—
127C22×C32—23no—
128C22×C32—41no—
129C22×C32—41no—
130C22×C32—43no—
131C22×C32—43no—
132C22×C32—81no—
133C22×C32—83no—
134C22×C32—82no—
135C22×C32—81no—
136C22×C32—81no—
137C22×C32C2×#6713yes—
138C22×C32C2×C3×D413yes—
139C32×C4—26no—
140C32×C4—23no—
141C32×C4—23no—
142C3⋊Dic3—43no—
143C3⋊Dic3—41no—
144C3×Dic3—123no—
145C3×Dic3—121no—
146H(5,𝔽2)(C3⋊D3)⋊C311yescharacteristic
147#123—93no—
148#99—93no—
149H(3,𝔽3)#45811yescharacteristic, complemented, Sylow
150C23×C3—21no—
151C23×C3—33no—
152C23×C3—33no—
153C23×C3—31no—
154C23×C3—31no—
155C23×C3—61no—
156C23×C3—61no—
157C23×C3—62no—
158C23×C3—63no—
159C23×C3—61no—
160C23×C3C3×#6713yes—
161C23×C3C22×C3×D311yescharacteristic
162C3×Q8—21no—
163C3×Q8—61no—
164C3×Q8—61no—
165C3×Q8—62no—
166C3×D4—23no—
167C3×D4—26no—
168C3×D4—66no—
169C3×D4—66no—
170C3×D4—63no—
171C3×D4—63no—
172C3×D4—66no—
173C3×D4—612no—
174C2×C3×C4—23no—
175C2×C3×C4—33no—
176C2×C3×C4—33no—
177C2×C3×C4—63no—
178C2×C3×C4—63no—
179C2×C3×C4—66no—
180C2×C3×C4—63no—
181C2×C3×C4—183no—
182C2×C3×C4—183no—
183C2×C3×C4C3×#6713yes—
184C2×Dic3—63no—
185C2×Dic3—63no—
186C2×Dic3—183no—
187C2×Dic3—183no—
188C2×C32—23no—
189C2×C32—23no—
190C2×C32—21no—
191C2×C32—43no—
192C2×C32—46no—
193C2×C32—43no—
194C2×C32—43no—
195C2×C32#95311yescharacteristic
196C2×C32C3×#12011yescharacteristic
197#41—23no—
198C2×D4—23no—
199C2×D4C2×(C3⋊D3)⋊C313yes—
200C22×C4—91no—
201C4⋊C4—183no—
202C22⋊C4—93no—
203C22⋊C4—183no—
204C22×C3—23no—
205C22×C3—33no—
206C22×C3—33no—
207C22×C3—41no—
208C22×C3—43no—
209C22×C3—63no—
210C22×C3—63no—
211C22×C3—66no—
212C22×C3—63no—
213C22×C3—81no—
214C22×C3—121no—
215C22×C3—121no—
216C22×C3—123no—
217C22×C3—123no—
218C22×C3—241no—
219C22×C3—243no—
220C22×C3—241no—
221C22×C3—241no—
222C22×C3—242no—
223C22×C3C2×C3×#6713yes—
224C3×C4—23no—
225C3×C4—63no—
226C3×C4—63no—
227C3×C4—66no—
228C3×C4—361no—
229C3×C4—363no—
230Dic3—123no—
231Dic3—121no—
232Dic3—361no—
233Dic3—363no—
234C32—21no—
235C32#524311yescharacteristic, complemented
236C32C3×#45811yescharacteristic, complemented
237C23—21no—
238C23#638⋊C313yes—
239C23C22×(C3⋊D3)⋊C311yescharacteristic
240Q8—21no—
241D4—23no—
242D4—26no—
243C2×C4—23no—
244C2×C4—183no—
245C2×C4—183no—
246C2×C4#638⋊C313yes—
247C2×C3—23no—
248C2×C3—31no—
249C2×C3—31no—
250C2×C3—43no—
251C2×C3—63no—
252C2×C3—63no—
253C2×C3—61no—
254C2×C3—123no—
255C2×C3—123no—
256C2×C3—123no—
257C2×C3—126no—
258C2×C3C3×#95311yescharacteristic
259C22—23no—
260C22—41no—
261C22—43no—
262C22—81no—
263C22C2×#638⋊C313yes—
264C4—23no—
265C4—363no—
266C4—361no—
267C3—31no—
268C3—31no—
269C3—61no—
270C3C3×#524311yescharacteristic
271C2—23no—
272C2—43no—
273C2#1172170211yescharacteristic
subgroupG/H
centerC2#11721702
commutator, cosocleC23×C32C23×C3
FrattiniC23×C3C22×C3×D3
Fitting, prime radicalH(3,𝔽3)×H(5,𝔽2)C2
socleC2×C3C3×#953
hypercenterH(5,𝔽2)(C3⋊D3)⋊C3
hypocenterC32C3×#458
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
2. #64144⋊C3C2semidirect
3. #63661⋊C3C2semidirect
4. #10592942C3semidirect
9. C22×(C3⋊Dic3)⋊C3C22semidirect
11. #3757⋊C3C22semidirect
16. #64144C2×C3semidirect
18. #63661C2×C3semidirect
42. C22×C3⋊Dic3C22×C3semidirect
45. #3757C22×C3semidirect
149. H(3,𝔽3)#458semidirect
235. C32#5243semidirect
236. C32C3×#458semidirect

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
#64144⋊C32
#63661⋊C32
#105929423
C3×#52433