Finite Group Databasebeta

← #13139951

Subgroups — #13139951

Order 1728.

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

normal subgroups only

This group has 601 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
1C22×#634⋊C3C211yesmaximal, characteristic
2#11721826C211yesmaximal, characteristic
3C2×#3748⋊C3C211yesmaximal, characteristic
4#11720309C214yesmaximal, complemented
5#10591107—31nomaximal
6#10591099—31nomaximal
7C22×C4×H(3,𝔽3)C2211yescharacteristic
8C22×(C3⋊Dic3)⋊C3—21no—
9C2×#634⋊C3—21no—
10C2×#634⋊C3—22no—
11C2×#634⋊C3C414yescomplemented
12C2×#634⋊C3C2212yes—
13#88552—21no—
14C4⋊#1084C2212yes—
15C4×#1084—21no—
16#3748⋊C3—22no—
17C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
18C22×#634D311yescharacteristic
19C22×C3×Dic6—31no—
20C2×#3748D311yescharacteristic
21C3×#926—31no—
22C3×#926—31no—
23C3×#926—61no—
24C2×C3×#208—31no—
25#63567—34no—
26#63565—34no—
27C23×H(3,𝔽3)D411yescharacteristic
28C2×C4×H(3,𝔽3)C2×C412yes—
29C2×C4×H(3,𝔽3)D411yescharacteristic
30C2×C4×H(3,𝔽3)D412yes—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—22no—
33C2×(C3⋊Dic3)⋊C3—24no—
34C2×(C3⋊Dic3)⋊C3—22no—
35#634⋊C3—24no—
36#634⋊C3—22no—
37#634⋊C3C2×C414yes—
38(C3⋊(C3⋊C8))⋊C3—22no—
39#5225—91no—
40#5120—91no—
41#5004—91no—
42C22×C32×C4—21no—
43C22×C32×C4C2×D311yescharacteristic
44C22×C32×C4C2×D311yescharacteristic
45C22×C3⋊Dic3—21no—
46C2×#634—21no—
47C2×#634—22no—
48C2×#634Dic314yescomplemented
49C2×#634C2×D312yes—
50C22×C3×Dic3—61no—
51C2×C3×Dic6—34no—
52C2×C3×Dic6—32no—
53C2×C3×Dic6—61no—
54C2×C3×Dic6—62no—
55#3748—22no—
56C2×C3⋊(C3⋊C8)C2×D312yes—
57C3×#217—62no—
58C3×#217—61no—
59C3×#217—61no—
60C3×C4⋊Dic3—32no—
61C3×C4⋊Dic3—32no—
62C3×C4⋊Dic3—62no—
63C3×C4×Dic3—62no—
64C3×C4×Dic3—61no—
65C3×C4×Dic3—61no—
66C3×#208—62no—
67C2×C3×C3⋊C8—32no—
68C22×H(3,𝔽3)—22no—
69C22×H(3,𝔽3)C22⋊C412yes—
70C22×H(3,𝔽3)C2×D411yescharacteristic
71C4×H(3,𝔽3)C22⋊C412yes—
72C4×H(3,𝔽3)C22×C411yescharacteristic
73C4×H(3,𝔽3)C2×D411yescharacteristic
74#1084—42no—
75(C3⋊Dic3)⋊C3—24no—
76C22×C3×Q8—91no—
77C22×Dic6—31no—
78C22×Dic6—91no—
79C2×C3×M16—91no—
80C3×#117—91no—
81#926—31no—
82#926—31no—
83#926—61no—
84C2×#208—31no—
85C2×#208—91no—
86#835—94no—
87#816—94no—
88#810—94no—
89C23×C32—21no—
90C23×C32D1211yescharacteristic
91C23×C32#6711yescharacteristic
92C2×C32×C4—22no—
93C2×C32×C4—22no—
94C2×C32×C4—21no—
95C2×C32×C4—21no—
96C2×C32×C4C4×D312yes—
97C2×C32×C4D1211yescharacteristic
98C2×C32×C4C2×Dic312yes—
99C2×C32×C4#6712yes—
100C2×C32×C4#6712yes—
101C2×C32×C4#6711yescharacteristic
102C2×C32×C4C22×D311yescharacteristic
103C2×C32×C4C22×D311yescharacteristic
104C2×C3⋊Dic3—24no—
105C2×C3⋊Dic3—22no—
106#634—24no—
107#634—22no—
108#634C2×Dic314yes—
109C2×C3×Dic3—64no—
110C2×C3×Dic3—62no—
111C2×C3×Dic3—64no—
112C2×C3×Dic3—62no—
113C2×C3×Dic3—62no—
114C3×Dic6—34no—
115C3×Dic6—64no—
116C3×Dic6—62no—
117C3⋊(C3⋊C8)—22no—
118C3×C3⋊C8—62no—
119#419—271noSylow
120C2×H(3,𝔽3)—22no—
121C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
122C2×H(3,𝔽3)#13712yes—
123C2×C3×Q8—92no—
124C2×C3×Q8—94no—
125C2×C3×Q8—182no—
126C2×C3×Q8—181no—
127C22×C3×C4—31no—
128C22×C3×C4—31no—
129C22×C3×C4—61no—
130C22×C3×C4—181no—
131C22×C3×C4D3211yescharacteristic
132C22×Dic3—61no—
133C22×Dic3—181no—
134C2×Dic6—34no—
135C2×Dic6—32no—
136C2×Dic6—62no—
137C2×Dic6—61no—
138C2×Dic6—92no—
139C2×Dic6—94no—
140C2×Dic6—182no—
141C2×Dic6—181no—
142C3×M16—182no—
143C2×C3×C8—92no—
144C3×C4⋊C4—92no—
145C3×C22⋊C4—181no—
146C3×C42—181no—
147#217—61no—
148#217—61no—
149#217—62no—
150C4⋊Dic3—32no—
151C4⋊Dic3—32no—
152C4⋊Dic3—62no—
153C4×Dic3—61no—
154C4×Dic3—61no—
155C4×Dic3—62no—
156#208—62no—
157#208—182no—
158C2×C3⋊C8—32no—
159C2×C3⋊C8—92no—
160C22×C32—22no—
161C22×C32—22no—
162C22×C32—22no—
163C22×C32—21no—
164C22×C32—42no—
165C22×C32(C2×D3)⋊C412yes—
166C22×C32#21712yes—
167C22×C32C2×D1211yescharacteristic
168C22×C32C2×#6711yescharacteristic
169C32×C4—22no—
170C32×C4—21no—
171C32×C4—21no—
172C32×C4(C2×D3)⋊C412yes—
173C32×C4#21712yes—
174C32×C4C2×C4×D311yescharacteristic
175C32×C4C22×Dic311yescharacteristic
176C32×C4C2×#6711yescharacteristic
177C32×C4C2×#6711yescharacteristic
178C3⋊Dic3—24no—
179C3×Dic3—64no—
180C3×Dic3—124no—
181C3×Dic3—122no—
182C3×Dic3—122no—
183C22×Q8—91no—
184C2×M16—91no—
185#117—91no—
186#103—274no—
187H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
188C23×C3—31no—
189C23×C3—31no—
190C23×C3—61no—
191C23×C3#62611yescharacteristic
192C3×Q8—94no—
193C3×Q8—182no—
194C3×Q8—184no—
195C2×C3×C4—32no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—32no—
199C2×C3×C4—31no—
200C2×C3×C4—31no—
201C2×C3×C4—31no—
202C2×C3×C4—31no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—61no—
206C2×C3×C4—61no—
207C2×C3×C4—182no—
208C2×C3×C4—184no—
209C2×C3×C4—182no—
210C2×C3×C4D3×Dic312yes—
211C2×C3×C4#62512yes—
212C2×C3×C4#62611yescharacteristic
213C2×C3×C4C2×D3211yescharacteristic
214C2×Dic3—64no—
215C2×Dic3—62no—
216C2×Dic3—62no—
217C2×Dic3—62no—
218C2×Dic3—64no—
219C2×Dic3—182no—
220C2×Dic3—184no—
221Dic6—34no—
222Dic6—64no—
223Dic6—62no—
224Dic6—94no—
225Dic6—182no—
226Dic6—184no—
227C3×C8—182no—
228C3⋊C8—62no—
229C3⋊C8—182no—
230C2×C32—22no—
231C2×C32—22no—
232C2×C32—22no—
233C2×C32—21no—
234C2×C32—42no—
235C2×C32#90912yes—
236C2×C32C2×(C2×D3)⋊C411yescharacteristic
237C2×C32#94212yes—
238C2×C32C2×#21711yescharacteristic
239C2×Q8—94no—
240C2×Q8—92no—
241C2×Q8—182no—
242C2×Q8—181no—
243C22×C4—181no—
244C22×C4((C3⋊D3)⋊C3)⋊C211yescharacteristic
245M16—182no—
246C2×C8—92no—
247C4⋊C4—92no—
248C22⋊C4—181no—
249C42—181no—
250C22×C3—22no—
251C22×C3—31no—
252C22×C3—31no—
253C22×C3—32no—
254C22×C3—32no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3—61no—
258C22×C3—62no—
259C22×C3—122no—
260C22×C3(C2×D3)⋊Dic312yes—
261C22×C3C2×#62611yescharacteristic
262C3×C4—31no—
263C3×C4—31no—
264C3×C4—31no—
265C3×C4—31no—
266C3×C4—32no—
267C3×C4—32no—
268C3×C4—61no—
269C3×C4—61no—
270C3×C4—62no—
271C3×C4—184no—
272C3×C4—362no—
273C3×C4(C2×D3)⋊Dic312yes—
274C3×C4C2×D3×Dic311yescharacteristic
275C3×C4C2×#62511yescharacteristic
276Dic3—64no—
277Dic3—122no—
278Dic3—122no—
279Dic3—124no—
280Dic3—184no—
281C32—21no—
282C32#512011yescharacteristic, complemented
283C32#522511yescharacteristic, complemented
284C23#620011yescharacteristic
285Q8—94no—
286Q8—182no—
287Q8—184no—
288C2×C4—182no—
289C2×C4—184no—
290C2×C4—182no—
291C2×C4((C3⋊D3)⋊C3)⋊C412yes—
292C2×C4#619812yes—
293C2×C4#620011yescharacteristic
294C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
295C8—182no—
296C2×C3—22no—
297C2×C3—31no—
298C2×C3—31no—
299C2×C3—32no—
300C2×C3—32no—
301C2×C3—62no—
302C2×C3—62no—
303C2×C3—61no—
304C2×C3—62no—
305C2×C3—122no—
306C2×C3#6383312yes—
307C2×C3C2×(C2×D3)⋊Dic311yescharacteristic
308C22—22no—
309C22#8845012yes—
310C22C2×#620011yescharacteristic
311C4—184no—
312C4—362no—
313C4#8845012yes—
314C4C2×((C3⋊D3)⋊C3)⋊C411yescharacteristic
315C4C2×#619811yescharacteristic
316C3—31no—
317C3—31no—
318C3—61no—
319C3#1059110011yescharacteristic
320C2—22no—
321C2#1172106912yes—
322C2C2×#8845011yescharacteristic
subgroupG/H
centerC22C2×#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)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
4. #11720309C2semidirect
11. C2×#634⋊C3C4semidirect
48. C2×#634Dic3semidirect
187. H(3,𝔽3)#419semidirect
282. C32#5120semidirect
283. C32#5225semidirect

Maximal subgroups

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

subgroupindex
C22×#634⋊C32
#117218262
C2×#3748⋊C32
#117203092
#105911073
#105910993