Finite Group Databasebeta

← #13161275

Subgroups — #13161275

Order 1728.

1052 conjugacy classes of subgroups; only the 236 normal subgroups are shown.

normal subgroups only

This group has 1052 conjugacy classes of subgroups — too many to draw. Only its 236 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×C32×#239C211yesmaximal, characteristic
2D3×#3748C212yesmaximal, complemented
3#11722206C211yesmaximal, characteristic, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722025C212yesmaximal, complemented
6#10592936—31nomaximal
7#10591637—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C32×#239—22no—
10C32×#239—22no—
11C2×C32×D12C412yes—
12C2×C32×D12C2211yescharacteristic
13C2×C32×C4×D3C412yes—
14C2×C32×C4×D3C2211yescharacteristic
15#88855C2212yes—
16C3×#3748C2212yes—
17#88786—22no—
18D3×C3⋊(C3⋊C8)—22no—
19C2×C32×#41—31no—
20C2×C3×#239D314yes—
21C2×#3748—32no—
22D3×#208—38no—
23#63663—31no—
24#63663—34no—
25#63663—64no—
26#63663D311yescharacteristic, complemented
27#63660—32no—
28C3×#836—34no—
29#63558—38no—
30C22×C32×D3C2×C412yes—
31C22×C32×D3C2×C411yescharacteristic
32C33×Q8—22no—
33C2×C33×C4C2×C412yes—
34C2×C33×C4C2311yescharacteristic
35C2×C32×Dic3C2×C411yescharacteristic
36C32×D12—24no—
37C32×D12—22no—
38C32×C4×D3—22no—
39C32×C4×D3—22no—
40C32×C4×D3D412yes—
41C32×C4×D3D412yes—
42C3⋊(C3⋊(C3⋊C8))—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5237—94no—
45#4751—91no—
46C32×#41—62no—
47C32×#41—62no—
48C2×C32×Q8—24no—
49C2×C32×Q8C2×D311yescharacteristic
50C2×C32×Q8C2×D314yes—
51C2×C32×D4—32no—
52C2×C32×D4—31no—
53C22×C32×C4—31no—
54C22×C32×C4—32no—
55C3×#239—28no—
56C3×#239—28no—
57C2×C3×D12Dic318yes—
58C2×C3×D12C2×D314yes—
59C2×C3×C4×D3Dic318yes—
60C2×C3×C4×D3C2×D314yes—
61#3748—32no—
62#3748—38no—
63#3748—62no—
64#3748—68no—
65#3748C2×D312yes—
66C2×C3⋊(C3⋊C8)—62no—
67C3×#208—38no—
68#3711—68no—
69D3×C3⋊C8—68no—
70C22×C33C22×C411yescharacteristic
71C2×C32×D3—22no—
72C2×C32×D3—41no—
73C2×C32×D3—42no—
74C2×C32×D3C22⋊C412yes—
75C33×C4—22no—
76C33×C4C2×D412yes—
77C32×Dic3C22⋊C411yescharacteristic
78C32×Dic3C22⋊C411yescharacteristic
79C2×C3×#41—34no—
80C2×#239C3⋊D311yescharacteristic
81C2×#208—98no—
82D3×M16—92no—
83C3×#101—91no—
84#836—34no—
85#836—94no—
86#836—91no—
87#836—184no—
88#833—98no—
89#823—92no—
90C23×C32—31no—
91C23×C32—32no—
92C22×C3×D3C2×Dic318yes—
93C22×C3×D3C2×Dic314yes—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—48no—
97C32×D4—64no—
98C32×D4—62no—
99C2×C32×C4—28no—
100C2×C32×C4—24no—
101C2×C32×C4—31no—
102C2×C32×C4—32no—
103C2×C32×C4—32no—
104C2×C32×C4—62no—
105C2×C32×C4—62no—
106C2×C32×C4C4×D312yes—
107C2×C32×C4C2×Dic318yes—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3×Dic3C2×Dic314yes—
111C3×D12—216no—
112C3×D12—28no—
113C3×C4×D3—28no—
114C3×C4×D3—28no—
115C3×C4×D3#6718yes—
116C3×C4×D3#6718yes—
117C3⋊(C3⋊C8)—22no—
118C3⋊(C3⋊C8)—62no—
119C3⋊(C3⋊C8)—68no—
120C3⋊(C3⋊C8)—128no—
121C3×C3⋊C8—68no—
122#413—271noSylow
123C2×C33—21no—
124C2×C33C2×C22⋊C411yescharacteristic
125C32×D3—22no—
126C32×D3—42no—
127C3×#41—68no—
128C3×#41—68no—
129C2×C3×Q8—24no—
130C2×C3×Q8D3214yes—
131C2×C3×Q8C2×C3⋊D311yescharacteristic
132C2×C3×D4—38no—
133C2×C3×D4—34no—
134C22×C3×C4—34no—
135C22×C3×C4—38no—
136#239—22no—
137#239—22no—
138C2×D12C3⋊Dic312yes—
139C2×D12C2×C3⋊D311yescharacteristic
140C2×C4×D3C3⋊Dic312yes—
141C2×C4×D3C2×C3⋊D311yescharacteristic
142C3×M16—92no—
143#208—38no—
144#208—98no—
145#208—92no—
146#208—188no—
147#208—188no—
148C2×C3⋊C8—188no—
149ΓL(1,𝔽25)—182no—
150D3×C8—182no—
151C22×C32—24no—
152C22×C32—32no—
153C22×C32—62no—
154C22×C32—121no—
155C22×C32—122no—
156C22×C32C2×C4×D311yescharacteristic
157C22×C32C22×Dic314yes—
158C2×C3×D3—28no—
159C2×C3×D3—44no—
160C2×C3×D3—48no—
161C2×C3×D3#21718yes—
162C32×C4—22no—
163C32×C4—28no—
164C32×C4—28no—
165C32×C4—31no—
166C32×C4—31no—
167C32×C4—48no—
168C32×C4D3×D412yes—
169C32×C4C2×#6718yes—
170C3×Dic3#21714yes—
171C3×Dic3#21714yes—
172C2×#41—31no—
173C2×M16—272no—
174#101—91no—
175#101—271no—
176#100—272no—
177C33#41311yescharacteristic, complemented, Sylow
178C23×C3—34no—
179C23×C3—38no—
180C22×D3C2×C3⋊Dic312yes—
181C22×D3C2×C3⋊Dic311yescharacteristic
182C3×Q8—28no—
183C3×Q8—22no—
184C3×Q8—48no—
185C3×D4—616no—
186C3×D4—68no—
187C2×C3×C4—28no—
188C2×C3×C4—24no—
189C2×C3×C4—34no—
190C2×C3×C4—38no—
191C2×C3×C4—38no—
192C2×C3×C4—68no—
193C2×C3×C4—68no—
194C2×C3×C4D3×Dic318yes—
195C2×C3×C4C2×C3⋊Dic312yes—
196C2×C3×C4C2×D3214yes—
197C2×C3×C4C22×C3⋊D311yescharacteristic
198C2×Dic3C2×C3⋊Dic311yescharacteristic
199D12—24no—
200D12—22no—
201C4×D3—22no—
202C4×D3—22no—
203C4×D3#63812yes—
204C4×D3#63812yes—
205C3×C8—182no—
206C3⋊C8—68no—
207C3⋊C8—188no—
208C3⋊C8—182no—
209C3⋊C8—368no—
210C2×C32—21no—
211C2×C32—24no—
212C2×C32—24no—
213C2×C32—44no—
214C2×C32—62no—
215C2×C32—122no—
216C2×C32D3×C22⋊C411yescharacteristic
217C2×C32C2×#21714yes—
218C3×D3—28no—
219C3×D3—48no—
220#41—62no—
221#41—62no—
222C2×Q8D3×C3⋊D311yescharacteristic
223C2×D4—32no—
224C2×D4—31no—
225C22×C4—31no—
226C22×C4—32no—
227M16—92no—
228M16—272no—
229M16—542no—
230C2×C8—542no—
231C22×C3—24no—
232C22×C3—38no—
233C22×C3—68no—
234C22×C3—124no—
235C22×C3—128no—
236C22×C3C2×D3×Dic314yes—
237C22×C3C22×C3⋊Dic311yescharacteristic
238C2×D3—22no—
239C2×D3—41no—
240C2×D3—42no—
241C2×D3#375712yes—
242C3×C4—22no—
243C3×C4—28no—
244C3×C4—28no—
245C3×C4—34no—
246C3×C4—34no—
247C3×C4—48no—
248C3×C4D3×#6718yes—
249C3×C4C2×#63812yes—
250Dic3#375711yescharacteristic
251Dic3#375711yescharacteristic
252C32—24no—
253C32#475111yescharacteristic, complemented
254C32#523714yescomplemented
255C23—31no—
256C23—32no—
257Q8—22no—
258D4—64no—
259D4—62no—
260C2×C4—31no—
261C2×C4—32no—
262C2×C4—32no—
263C2×C4—62no—
264C2×C4—62no—
265C2×C4D3×C3⋊Dic312yes—
266C2×C4C2×D3×C3⋊D311yescharacteristic
267C8—182no—
268C8—542no—
269C2×C3—24no—
270C2×C3—21no—
271C2×C3—24no—
272C2×C3—44no—
273C2×C3—68no—
274C2×C3—128no—
275C2×C3D3×#21714yes—
276C2×C3C2×#375711yescharacteristic
277D3—22no—
278D3—42no—
279C22—32no—
280C22—62no—
281C22—121no—
282C22—122no—
283C22C2×D3×C3⋊Dic311yescharacteristic
284C4—22no—
285C4—31no—
286C4—31no—
287C4D3×#63812yes—
288C3—24no—
289C3#1059163714yescomplemented
290C3#1059293611yescharacteristic, complemented
291C2—21no—
292C2—62no—
293C2—122no—
294C2D3×#375711yescharacteristic
subgroupG/H
centerC2D3×#3757
commutatorC22×C33C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33C2×C22⋊C4
prime radicalC2×C32×#239C2
cosocleC2×C33×C4C23
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#413
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. D3×#3748C2semidirect
3. #11722206C2semidirect
4. C3×#63663C2semidirect
5. #11722025C2semidirect
26. #63663D3semidirect
177. C33#413semidirect
253. C32#4751semidirect
254. C32#5237semidirect
289. C3#10591637semidirect
290. C3#10592936semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#2392
D3×#37482
#117222062
C3×#636632
#117220252
#105929363
#105916373