Finite Group Databasebeta

← #13137326

Subgroups — #13137326

Order 1728.

481 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 95 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#11721715C211yesmaximal, characteristic, complemented
2#11721385C211yesmaximal, characteristic
3C4×(C3⋊(C3⋊C8))⋊C3C211yesmaximal, characteristic, complemented
4#11720308C214yesmaximal, complemented
5#10590281—31nomaximal
6#10590277—31nomaximal
7C2×#6229—21no—
8C2×C4⋊((C3⋊D3)⋊C3)—21no—
9C2×C4⋊((C3⋊D3)⋊C3)—21no—
10C2×C4⋊((C3⋊D3)⋊C3)C2212yes—
11C42×H(3,𝔽3)C2211yescharacteristic
12C4⋊#1084—21no—
13C4⋊#1084C2212yes—
14C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
15#64083D311yescharacteristic
16C3×#874—31no—
17C3×#869—31no—
18C3×#869—31no—
19C3×#869—61no—
20C4×C3⋊(C3⋊C8)D311yescharacteristic, complemented
21C3×C4×C3⋊C8—31no—
22#63564—34no—
23#63562—34no—
24C22×(C3⋊D3)⋊C3—22no—
25C2×C4×H(3,𝔽3)D411yescharacteristic
26C2×C4×H(3,𝔽3)D411yescharacteristic
27C2×C4×H(3,𝔽3)C2311yescharacteristic
28C2×#1084—22no—
29#6229—42no—
30C4⋊((C3⋊D3)⋊C3)—24no—
31C4⋊((C3⋊D3)⋊C3)—42no—
32C4⋊((C3⋊D3)⋊C3)—42no—
33(C3⋊(C3⋊C8))⋊C3—22no—
34#5088—91no—
35#5070—91no—
36#4694—91no—
37C2×C4⋊(C3⋊D3)—21no—
38C2×C4⋊(C3⋊D3)—21no—
39C2×C4⋊(C3⋊D3)C2×D312yes—
40C2×C3×D12—32no—
41C2×C3×D12—61no—
42C2×C3×D12—61no—
43C2×C3×Dic6—62no—
44C2×C3×Dic6—61no—
45C2×C3×Dic6—61no—
46C32×C42—21no—
47C32×C42C2×D311yescharacteristic
48C32×C42C2×D311yescharacteristic
49C2×C3⋊(C3⋊C8)C2×D312yes—
50C3×C4⋊Dic3—32no—
51C3×C4⋊Dic3—32no—
52C3×C4⋊Dic3—62no—
53C3×C4⋊Dic3—62no—
54C3×C4⋊Dic3—61no—
55C3×C4⋊Dic3—61no—
56C2×C3×C3⋊C8—32no—
57C22×H(3,𝔽3)C2×D411yescharacteristic
58C2×(C3⋊D3)⋊C3—42no—
59C2×(C3⋊D3)⋊C3—42no—
60C2×(C3⋊D3)⋊C3—42no—
61C4×H(3,𝔽3)D812yes—
62C4×H(3,𝔽3)SD1612yes—
63C4×H(3,𝔽3)#4112yes—
64#1084—42no—
65C3×C4⋊Q8—91no—
66C3×#127—91no—
67#874—31no—
68#874—91no—
69#869—31no—
70#869—31no—
71#869—61no—
72C3×C4×C8—91no—
73#832—94no—
74#821—94no—
75#809—94no—
76C4×C3⋊C8—31no—
77C4×C3⋊C8—91no—
78C22×C3⋊D3—22no—
79C22×C3×D3—62no—
80C2×C32×C4—21no—
81C2×C32×C4—21no—
82C2×C32×C4—21no—
83C2×C32×C4D1211yescharacteristic
84C2×C32×C4D1211yescharacteristic
85C2×C32×C4#6711yescharacteristic
86C2×C32×C4#6711yescharacteristic
87C2×C32×C4C22×D311yescharacteristic
88C2×C32×C4C22×D311yescharacteristic
89C4⋊(C3⋊D3)—24no—
90C4⋊(C3⋊D3)—42no—
91C4⋊(C3⋊D3)—42no—
92C2×C3×Dic3—64no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C3×D12—64no—
96C3×D12—122no—
97C3×D12—122no—
98C3×Dic6—124no—
99C3×Dic6—122no—
100C3×Dic6—122no—
101C3⋊(C3⋊C8)—22no—
102C3×C3⋊C8—62no—
103#486—271noSylow
104C2×H(3,𝔽3)#12411yescharacteristic
105C2×H(3,𝔽3)C2×D811yescharacteristic
106C2×H(3,𝔽3)C2×SD1611yescharacteristic
107(C3⋊D3)⋊C3—82no—
108C2×C3×Q8—181no—
109C2×C3×D4—92no—
110C2×C3×D4—181no—
111C2×C3×D4—181no—
112C2×D12—32no—
113C2×D12—61no—
114C2×D12—61no—
115C2×D12—92no—
116C2×D12—181no—
117C2×D12—181no—
118C2×Dic6—61no—
119C2×Dic6—61no—
120C2×Dic6—62no—
121C2×C3×C8—92no—
122C3×C4⋊C4—92no—
123C3×C4⋊C4—181no—
124C3×C42—31no—
125C3×C42—31no—
126C3×C42—61no—
127C3×C42D3211yescharacteristic
128C4⋊Dic3—32no—
129C4⋊Dic3—32no—
130C4⋊Dic3—62no—
131C4⋊Dic3—61no—
132C4⋊Dic3—61no—
133C4⋊Dic3—62no—
134C2×C3⋊C8—32no—
135C2×C3⋊C8—92no—
136C22×C32—21no—
137C22×C32C2×D1211yescharacteristic
138C22×C32C2×#6711yescharacteristic
139C2×C3⋊D3—42no—
140C2×C3⋊D3—42no—
141C2×C3⋊D3—42no—
142C2×C3×D3—122no—
143C2×C3×D3—122no—
144C2×C3×D3—122no—
145C32×C4—22no—
146C32×C4—22no—
147C32×C4—22no—
148C32×C4#20412yes—
149C32×C4D2412yes—
150C32×C4#21312yes—
151C32×C4#21412yes—
152C32×C4#23512yes—
153C32×C4#23712yes—
154C3×Dic3—124no—
155C3×Dic3—122no—
156C3×Dic3—122no—
157C4⋊Q8—91no—
158#127—91no—
159#102—274no—
160C4×C8—91no—
161H(3,𝔽3)#48611yescharacteristic, complemented, Sylow
162C23×C3—182no—
163C22×D3—62no—
164C22×D3—182no—
165C3×Q8—362no—
166C3×D4—184no—
167C3×D4—362no—
168C3×D4—362no—
169C2×C3×C4—31no—
170C2×C3×C4—31no—
171C2×C3×C4—31no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—61no—
176C2×C3×C4—61no—
177C2×C3×C4—61no—
178C2×C3×C4—182no—
179C2×C3×C4#62611yescharacteristic
180C2×C3×C4#62611yescharacteristic
181C2×C3×C4C2×D3211yescharacteristic
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3—64no—
185D12—64no—
186D12—122no—
187D12—122no—
188D12—184no—
189D12—362no—
190D12—362no—
191Dic6—122no—
192Dic6—122no—
193Dic6—124no—
194C3×C8—182no—
195C3⋊C8—62no—
196C3⋊C8—182no—
197C2×C32—21no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32#87511yescharacteristic
201C2×C32C2×#20411yescharacteristic
202C2×C32C2×D2411yescharacteristic
203C2×C32C2×#21311yescharacteristic
204C2×C32C2×#21411yescharacteristic
205C2×C32#93611yescharacteristic
206C3⋊D3—82no—
207C3×D3—242no—
208C2×Q8—181no—
209C2×D4—92no—
210C2×D4—181no—
211C2×D4—181no—
212C2×C8—92no—
213C4⋊C4—92no—
214C4⋊C4—181no—
215C42((C3⋊D3)⋊C3)⋊C211yescharacteristic
216C22×C3—31no—
217C22×C3—31no—
218C22×C3—61no—
219C22×C3—362no—
220C22×C3—362no—
221C22×C3—362no—
222C22×C3C2×#62611yescharacteristic
223C2×D3—122no—
224C2×D3—122no—
225C2×D3—122no—
226C2×D3—362no—
227C2×D3—362no—
228C2×D3—362no—
229C3×C4—32no—
230C3×C4—32no—
231C3×C4—32no—
232C3×C4—32no—
233C3×C4—32no—
234C3×C4—32no—
235C3×C4—62no—
236C3×C4—62no—
237C3×C4—62no—
238C3×C4—362no—
239C3×C4#371412yes—
240C3×C4#371612yes—
241C3×C4#379512yes—
242Dic3—122no—
243Dic3—122no—
244Dic3—124no—
245C32—21no—
246C32#469411yescharacteristic, complemented
247C32#507011yescharacteristic, complemented
248C23—182no—
249Q8—362no—
250D4—184no—
251D4—362no—
252D4—362no—
253C2×C4—182no—
254C2×C4#620011yescharacteristic
255C2×C4#620011yescharacteristic
256C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
257C8—182no—
258C2×C3—31no—
259C2×C3—31no—
260C2×C3—31no—
261C2×C3—31no—
262C2×C3—31no—
263C2×C3—31no—
264C2×C3—61no—
265C2×C3—61no—
266C2×C3—61no—
267C2×C3—722no—
268C2×C3C2×#371411yescharacteristic
269C2×C3C2×#371611yescharacteristic
270C2×C3#6386011yescharacteristic
271D3—242no—
272D3—722no—
273C22—362no—
274C22—362no—
275C22—362no—
276C22C2×#620011yescharacteristic
277C4—362no—
278C4#8843612yes—
279C4#8843812yes—
280C4#8864812yes—
281C3—31no—
282C3—31no—
283C3—61no—
284C3#1059027311yescharacteristic
285C2—722no—
286C2C2×#8843611yescharacteristic
287C2C2×#8843811yescharacteristic
288C2#1172111011yescharacteristic
subgroupG/H
centerC22C2×#6200
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC42×H(3,𝔽3)C22
socleC22×C3C2×#626
prime radical#11721385C2
hypercenterC42((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#486
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11721715C2semidirect
3. C4×(C3⋊(C3⋊C8))⋊C3C2semidirect
4. #11720308C2semidirect
20. C4×C3⋊(C3⋊C8)D3semidirect
161. H(3,𝔽3)#486semidirect
246. C32#4694semidirect
247. C32#5070semidirect

Maximal subgroups

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

subgroupindex
#117217152
#117213852
C4×(C3⋊(C3⋊C8))⋊C32
#117203082
#105902813
#105902773