Finite Group Databasebeta

← #13140945

Subgroups — #13140945

Order 1728.

681 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 681 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×Q8×#272C211yesmaximal, characteristic
2#11721089C211yesmaximal, characteristic
3C2×#88435C211yesmaximal, characteristic
4#63662⋊C3C212yesmaximal, complemented
5#11720326C212yesmaximal, complemented
6#10591557—32nomaximal
7C2×Q8×H(3,𝔽3)C2211yescharacteristic
8Q8×#272—22no—
9Q8×#272C414yescomplemented
10C2×C4×#272—21no—
11C2×C4×#272C2211yescharacteristic
12C2×#6229—21no—
13C2×#6229C2211yescharacteristic
14C4⋊((C3⋊Dic3)⋊C3)C2212yes—
15C2×(C3⋊(C3⋊C8))⋊C3C2212yes—
16#88451—21no—
17#88447—21no—
18#88435—22no—
19C2×C3×D3×Q8—32no—
20C2×C3×D3×Q8—61no—
21#63841—32no—
22C2×#3711—32no—
23#63662D312yescomplemented
24C3×#835—32no—
25#63566—32no—
26#63565—32no—
27C22×#272D411yescharacteristic
28Q8×H(3,𝔽3)—21no—
29Q8×H(3,𝔽3)C2×C412yes—
30C2×C4×H(3,𝔽3)—21no—
31C2×C4×H(3,𝔽3)C2311yescharacteristic
32C2×#1084—21no—
33C2×#1084D411yescharacteristic
34C4×#272—24no—
35C4×#272C2×C412yes—
36C4×#272D412yes—
37#6229—21no—
38#6229—24no—
39#6229C2×C412yes—
40C2×(C3⋊Dic3)⋊C3—22no—
41(C3⋊(C3⋊C8))⋊C3—22no—
42#5225—91no—
43#4802—92no—
44C2×C32×Q8—21no—
45C2×C32×Q8C2×D312yes—
46C3×D3×Q8—38no—
47C3×D3×Q8—64no—
48C3×D3×Q8—64no—
49C3×D3×Q8—64no—
50C2×C3×C4×D3—32no—
51C2×C3×C4×D3—62no—
52C2×C3×C4×D3—61no—
53C2×C3×C4×D3—62no—
54C2×C3×Dic6—32no—
55C2×C3×Dic6—62no—
56C2×C3×Dic6—61no—
57C2×C3×Dic6—62no—
58C4⋊(C3⋊Dic3)C2×D312yes—
59C2×C3⋊(C3⋊C8)C2×D312yes—
60C3×C4⋊Dic3—32no—
61C2×C3×C3⋊C8—32no—
62(C2×D3)⋊Dic3—62no—
63Dic32—62no—
64#3711—64no—
65C22×H(3,𝔽3)C2×D411yescharacteristic
66C2×#272—22no—
67C2×#272C22⋊C412yes—
68C4×H(3,𝔽3)—22no—
69C4×H(3,𝔽3)C22×C411yescharacteristic
70C4×H(3,𝔽3)C2×D411yescharacteristic
71#1084—22no—
72#1084C22⋊C412yes—
73(C3⋊Dic3)⋊C3—42no—
74C22×C3×Q8—91no—
75C2×D3×Q8—32no—
76C2×D3×Q8—61no—
77#926—91no—
78C2×#208—91no—
79C2×ΓL(1,𝔽25)—92no—
80#892—92no—
81C3×#103—92no—
82#835—32no—
83#835—92no—
84#835—92no—
85#816—92no—
86#810—92no—
87C22×C3×D3—32no—
88C22×C3×D3—61no—
89C32×Q8—22no—
90C32×Q8—22no—
91C32×Q8—22no—
92C32×Q8C4×D314yes—
93C2×C32×C4—22no—
94C2×C32×C4—22no—
95C2×C32×C4—21no—
96C2×C32×C4C22×D312yes—
97C2×C3⋊Dic3—22no—
98C2×C3×Dic3—32no—
99C2×C3×Dic3—61no—
100C2×C3×Dic3—62no—
101C2×C3×Dic3—62no—
102C2×C3×Dic3—62no—
103C3×C4×D3—34no—
104C3×C4×D3—34no—
105C3×C4×D3—62no—
106C3×C4×D3—62no—
107C3×C4×D3—68no—
108C3×C4×D3—68no—
109C3×Dic6—34no—
110C3×Dic6—62no—
111C3×Dic6—68no—
112C3×Dic6—68no—
113C3×Dic6—62no—
114C3×Dic6—62no—
115C3⋊(C3⋊C8)—22no—
116C3×C3⋊C8—62no—
117#419—271noSylow
118C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
119C2×H(3,𝔽3)#13712yes—
120#272—22no—
121C2×C3×Q8—32no—
122C2×C3×Q8—61no—
123C2×C3×Q8—94no—
124C2×C3×Q8—91no—
125C2×C3×Q8—181no—
126C2×C3×Q8—182no—
127C2×C3×Q8D3211yescharacteristic
128C22×C3×C4—91no—
129C22×C3×C4—181no—
130D3×Q8—38no—
131D3×Q8—64no—
132D3×Q8—64no—
133D3×Q8—64no—
134C2×C4×D3—32no—
135C2×C4×D3—61no—
136C2×C4×D3—62no—
137C2×C4×D3—62no—
138C2×Dic6—32no—
139C2×Dic6—61no—
140C2×Dic6—62no—
141C2×Dic6—62no—
142C2×C3×C8—92no—
143C3×C4⋊C4—92no—
144#217—181no—
145(C2×D3)⋊C4—182no—
146C4⋊Dic3—32no—
147C4⋊Dic3—92no—
148C4×Dic3—182no—
149C4×Dic3—181no—
150#208—182no—
151C2×C3⋊C8—32no—
152C2×C3⋊C8—92no—
153ΓL(1,𝔽25)—184no—
154C22×C32—21no—
155C22×C32D3×D412yes—
156C2×C3×D3—34no—
157C2×C3×D3—62no—
158C2×C3×D3—64no—
159C2×C3×D3—62no—
160C2×C3×D3—62no—
161C32×C4—24no—
162C32×C4—24no—
163C32×C4—21no—
164C32×C4—21no—
165C32×C4C2×C4×D312yes—
166C32×C4D3×D412yes—
167C3⋊Dic3—42no—
168C3×Dic3—34no—
169C3×Dic3—62no—
170C3×Dic3—64no—
171C3×Dic3—64no—
172C3×Dic3—122no—
173C22×Q8—91no—
174C2×M16—271no—
175#117—271no—
176#103—92no—
177#103—272no—
178H(3,𝔽3)#41911yescharacteristic, complemented, Sylow
179C23×C3—91no—
180C22×D3—32no—
181C22×D3—61no—
182C3×Q8—21no—
183C3×Q8—34no—
184C3×Q8—62no—
185C3×Q8—62no—
186C3×Q8—62no—
187C3×Q8—92no—
188C3×Q8—184no—
189C3×Q8—181no—
190C3×Q8#62412yes—
191C2×C3×C4—21no—
192C2×C3×C4—32no—
193C2×C3×C4—62no—
194C2×C3×C4—62no—
195C2×C3×C4—61no—
196C2×C3×C4—91no—
197C2×C3×C4—92no—
198C2×C3×C4—92no—
199C2×C3×C4—184no—
200C2×C3×C4—181no—
201C2×C3×C4—182no—
202C2×C3×C4C2×D3211yescharacteristic
203C2×Dic3—32no—
204C2×Dic3—61no—
205C2×Dic3—62no—
206C2×Dic3—62no—
207C2×Dic3—62no—
208C2×Dic3—182no—
209C4×D3—34no—
210C4×D3—34no—
211C4×D3—62no—
212C4×D3—62no—
213C4×D3—68no—
214C4×D3—68no—
215Dic6—34no—
216Dic6—62no—
217Dic6—68no—
218Dic6—62no—
219Dic6—68no—
220Dic6—62no—
221C3×C8—182no—
222C3⋊C8—62no—
223C3⋊C8—182no—
224C2×C32—22no—
225C2×C32—21no—
226C2×C32D3×C22⋊C412yes—
227C2×C32#91512yes—
228C2×C32#91812yes—
229C3×D3—64no—
230C3×D3—64no—
231C2×Q8—94no—
232C2×Q8—91no—
233C2×Q8—181no—
234C2×Q8—182no—
235C2×Q8((C3⋊D3)⋊C3)⋊C211yescharacteristic
236C22×C4—91no—
237C22×C4—181no—
238M16—542no—
239C2×C8—92no—
240C4⋊C4—92no—
241C22⋊C4—541no—
242C42—541no—
243C22×C3—32no—
244C22×C3—61no—
245C22×C3—92no—
246C22×C3—182no—
247C22×C3#381111yescharacteristic
248C2×D3—34no—
249C2×D3—62no—
250C2×D3—62no—
251C2×D3—62no—
252C2×D3—64no—
253C3×C4—22no—
254C3×C4—32no—
255C3×C4—32no—
256C3×C4—64no—
257C3×C4—64no—
258C3×C4—61no—
259C3×C4—61no—
260C3×C4—92no—
261C3×C4—182no—
262C3×C4—362no—
263C3×C4C2×#62411yescharacteristic
264C3×C4#381111yescharacteristic
265Dic3—34no—
266Dic3—62no—
267Dic3—64no—
268Dic3—64no—
269Dic3—122no—
270Dic3—362no—
271C32—21no—
272C32#480212yescomplemented
273C23—91no—
274Q8—21no—
275Q8—92no—
276Q8—184no—
277Q8—181no—
278Q8#619912yes—
279C2×C4—21no—
280C2×C4—91no—
281C2×C4—92no—
282C2×C4—92no—
283C2×C4—184no—
284C2×C4—181no—
285C2×C4—182no—
286C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
287C8—182no—
288C2×C3—32no—
289C2×C3—32no—
290C2×C3—32no—
291C2×C3—62no—
292C2×C3—61no—
293C2×C3—182no—
294C2×C3#6395112yes—
295C2×C3#6397411yescharacteristic
296D3—64no—
297D3—64no—
298C22—92no—
299C22—182no—
300C22#8867711yescharacteristic
301C4—22no—
302C4—92no—
303C4—182no—
304C4—362no—
305C4C2×#619911yescharacteristic
306C4#8867711yescharacteristic
307C3—32no—
308C3—61no—
309C3#1059156211yescharacteristic
310C2—182no—
311C2#1172123112yes—
312C2#1172132011yescharacteristic
subgroupG/H
centerC22#88677
commutatorC4×H(3,𝔽3)C22×C4
FrattiniC2×C3×C4C2×D32
FittingC2×Q8×H(3,𝔽3)C22
socleC22×C3#3811
prime radicalC22×#272D4
cosocleC2×C4×H(3,𝔽3)C23
hypercenterC2×Q8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#419
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
4. #63662⋊C3C2semidirect
5. #11720326C2semidirect
9. Q8×#272C4semidirect
23. #63662D3semidirect
178. H(3,𝔽3)#419semidirect
272. C32#4802semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×#2722
#117210892
C2×#884352
#63662⋊C32
#117203262
#105915573