Finite Group Databasebeta

← #13161088

Subgroups — #13161088

Order 1728.

1192 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1192 conjugacy classes of subgroups — too many to draw. Only its 148 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
1C3×#64361C211yesmaximal, characteristic, complemented
2C32×#940C211yesmaximal, characteristic, complemented
3#11723364C212yesmaximal, complemented
4#11722203C211yesmaximal, characteristic, complemented
5#11722046C212yesmaximal, complemented
6#10592776—31nomaximal
7#10591509—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C3×#3830—22no—
11C3×D4×C3⋊D3—22no—
12C3×#3828C2212yes—
13C2×C32×#67—22no—
14C2×C32×D12C2211yescharacteristic, complemented
15#88855C2212yescomplemented
16C32×C4×Dic3C2211yescharacteristic, complemented
17#88792—22no—
18#88789—22no—
19#64361D311yescharacteristic, complemented
20C3×#1004—34no—
21C32×#127—31no—
22#64141—32no—
23C3×#940D314yescomplemented
24#63823—38no—
25#63660—31no—
26#63660—34no—
27#63660—64no—
28#63632—32no—
29#63569—38no—
30C23×C33D412yes—
31C22×C3×C3⋊D3—22no—
32C22×C32×D3—21no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35C3×#638—24no—
36C3×#638—22no—
37C2×C3×C3⋊Dic3—22no—
38C3×C4⋊(C3⋊D3)D412yescomplemented
39C3×C4×C3⋊D3—22no—
40C3×#634D412yescomplemented
41C32×#67—44no—
42C2×C32×Dic3—21no—
43C32×D12—22no—
44C3⋊(C3⋊(C3⋊C8))—22no—
45#5241—91no—
46#5077—94no—
47C2×C32×D4—24no—
48C2×C32×D4—31no—
49C2×C32×D4—62no—
50C2×C32×D4C2×D314yes—
51C2×C32×D4C2×D311yescharacteristic
52C2×#638—22no—
53#3830—22no—
54D4×C3⋊D3—22no—
55#3828C2×D312yes—
56C2×C3×#67—28no—
57C2×C3×#67—68no—
58C3×#237—68no—
59C3×D3×D4—68no—
60C3×#235—38no—
61C2×C3×D12C2×D314yescomplemented
62C32×C42—31no—
63#3754—62no—
64#3753—62no—
65#3748—32no—
66#3748—38no—
67#3748—68no—
68C3×C4×Dic3C2×D314yescomplemented
69#3715—68no—
70#3713—68no—
71C22×C33—22no—
72C22×C33—42no—
73C22×C33C2×D411yescharacteristic
74C2×C3×C3⋊D3—22no—
75C2×C3×C3⋊D3—82no—
76C2×C32×D3—42no—
77C2×C32×D3—41no—
78C33×C4C2×D412yes—
79C3×C3⋊Dic3—22no—
80C32×Dic3—22no—
81C3×H(5,𝔽2)—91no—
82#1004—34no—
83C3×#127—34no—
84#949—92no—
85#940C3⋊D311yescharacteristic, complemented
86#932—98no—
87#837—92no—
88#833—94no—
89#833—91no—
90#833—184no—
91#805—98no—
92C23×C32—28no—
93C23×C32—61no—
94C23×C32#6718yes—
95C23×C32#6712yes—
96C22×C3⋊D3—22no—
97C22×C3×D3—24no—
98C22×C3×D3—68no—
99C32×D4—22no—
100C32×D4—28no—
101C32×D4—48no—
102C32×D4—62no—
103C32×D4—124no—
104C2×C32×C4—24no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108#638—24no—
109#638—22no—
110C2×C3⋊Dic3—22no—
111C4⋊(C3⋊D3)#6712yescomplemented
112C4×C3⋊D3—22no—
113#634#6712yescomplemented
114C3×#67—416no—
115C3×#67—616no—
116C3×#67—68no—
117C2×C3×Dic3—24no—
118C2×C3×Dic3—68no—
119C3×D12—28no—
120C3×D12—38no—
121C3×C4×D3—68no—
122C3×Dic6—38no—
123C3⋊(C3⋊C8)—62no—
124C3⋊(C3⋊C8)—68no—
125C3⋊(C3⋊C8)—128no—
126#453—271noSylow
127C2×C33—21no—
128C2×C33—42no—
129C2×C33#12011yescharacteristic
130C3×C3⋊D3—42no—
131C32×D3—81no—
132C3×#41—92no—
133C3×#41—182no—
134C2×C3×D4—24no—
135C2×C3×D4—34no—
136C2×C3×D4—68no—
137C2×C3×D4—182no—
138C2×C3×D4—182no—
139C2×C3×D4D3214yes—
140C2×C3×D4C2×C3⋊D311yescharacteristic
141C2×#67—22no—
142C2×#67—68no—
143#237—68no—
144D3×D4—68no—
145#235—38no—
146C2×D12C2×C3⋊D311yescharacteristic, complemented
147C3×C42—34no—
148#215—182no—
149#214—188no—
150#213—182no—
151#213—188no—
152C4×Dic3C2×C3⋊D311yescharacteristic, complemented
153#208—98no—
154#208—92no—
155#208—188no—
156C22×C32—22no—
157C22×C32—24no—
158C22×C32—28no—
159C22×C32—48no—
160C22×C32—42no—
161C22×C32—48no—
162C22×C32—88no—
163C22×C32—122no—
164C22×C32—121no—
165C22×C32C2×#6714yes—
166C22×C32C2×#6711yescharacteristic
167C2×C3⋊D3—22no—
168C2×C3⋊D3—82no—
169C2×C3×D3—48no—
170C2×C3×D3—44no—
171C2×C3×D3—68no—
172C2×C3×D3—248no—
173C32×C4—28no—
174C32×C4—62no—
175C32×C4D3×D418yes—
176C32×C4C2×#6712yes—
177C3⋊Dic3—22no—
178C3×Dic3—28no—
179C3×Dic3—68no—
180H(5,𝔽2)—91no—
181AGL(1,ℤ/8)—272no—
182#127—31no—
183#104—272no—
184#100—271no—
185C33#45311yescharacteristic, complemented, Sylow
186C23×C3—28no—
187C23×C3—64no—
188C23×C3—182no—
189C23×C3#62518yes—
190C23×C3#63812yes—
191C22×D3—21no—
192C22×D3—68no—
193C3×Q8—92no—
194C3×D4—28no—
195C3×D4—22no—
196C3×D4—48no—
197C3×D4—68no—
198C3×D4—92no—
199C3×D4—1216no—
200C3×D4—184no—
201C3×D4—182no—
202C2×C3×C4—24no—
203C2×C3×C4—64no—
204C2×C3×C4—182no—
205C2×C3×C4—182no—
206C2×C3×C4C2×D3214yes—
207C2×C3×C4C22×C3⋊D311yescharacteristic
208#67—44no—
209#67—616no—
210#67—68no—
211C2×Dic3—21no—
212C2×Dic3—68no—
213D12—22no—
214D12—38no—
215C4×D3—68no—
216Dic6—38no—
217C3⋊C8—188no—
218C3⋊C8—182no—
219C3⋊C8—368no—
220C2×C32—21no—
221C2×C32—24no—
222C2×C32—24no—
223C2×C32—44no—
224C2×C32—42no—
225C2×C32—48no—
226C2×C32—88no—
227C2×C32—241no—
228C2×C32#93714yes—
229C2×C32#95311yescharacteristic
230C3⋊D3—42no—
231C3×D3—84no—
232C3×D3—128no—
233#41—92no—
234#41—182no—
235C2×D4—31no—
236C2×D4—62no—
237C2×D4—182no—
238C2×D4—182no—
239C2×D4D3×C3⋊D311yescharacteristic
240SD16—542no—
241D8—542no—
242M16—272no—
243C42—31no—
244C22×C3—22no—
245C22×C3—28no—
246C22×C3—24no—
247C22×C3—48no—
248C22×C3—42no—
249C22×C3—48no—
250C22×C3—88no—
251C22×C3—128no—
252C22×C3—124no—
253C22×C3—182no—
254C22×C3—722no—
255C22×C3C2×#62514yes—
256C22×C3C2×#63811yescharacteristic
257C2×D3—42no—
258C2×D3—41no—
259C2×D3—68no—
260C2×D3—248no—
261C3×C4—28no—
262C3×C4—68no—
263C3×C4—182no—
264C3×C4D3×#6718yes—
265C3×C4D4×C3⋊D312yes—
266Dic3—22no—
267Dic3—68no—
268C32—24no—
269C32#507714yescomplemented
270C32#524111yescharacteristic, complemented
271C23—61no—
272C23—182no—
273C23#629012yes—
274Q8—92no—
275D4—22no—
276D4—62no—
277D4—92no—
278D4—124no—
279D4—184no—
280D4—182no—
281C2×C4—61no—
282C2×C4—182no—
283C2×C4—182no—
284C2×C4C2×D3×C3⋊D311yescharacteristic
285C8—542no—
286C2×C3—21no—
287C2×C3—24no—
288C2×C3—24no—
289C2×C3—44no—
290C2×C3—42no—
291C2×C3—48no—
292C2×C3—88no—
293C2×C3—244no—
294C2×C3—362no—
295C2×C3#6397614yes—
296C2×C3#6414611yescharacteristic
297D3—81no—
298D3—128no—
299C22—22no—
300C22—42no—
301C22—122no—
302C22—121no—
303C22—182no—
304C22—722no—
305C22C2×#629011yescharacteristic
306C4—62no—
307C4—182no—
308C4C3⋊D3×#6712yes—
309C3—24no—
310C3#1059150914yescomplemented
311C3#1059277611yescharacteristic, complemented
312C2—21no—
313C2—42no—
314C2—241no—
315C2—362no—
316C2#1172363411yescharacteristic
subgroupG/H
centerC2#11723634
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#453
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C3×#64361C2semidirect
2. C32×#940C2semidirect
3. #11723364C2semidirect
4. #11722203C2semidirect
5. #11722046C2semidirect
14. C2×C32×D12C22semidirect
15. #88855C22semidirect
16. C32×C4×Dic3C22semidirect
19. #64361D3semidirect
23. C3×#940D3semidirect
38. C3×C4⋊(C3⋊D3)D4semidirect
40. C3×#634D4semidirect
61. C2×C3×D12C2×D3semidirect
68. C3×C4×Dic3C2×D3semidirect
85. #940C3⋊D3semidirect
111. C4⋊(C3⋊D3)#67semidirect
113. #634#67semidirect
146. C2×D12C2×C3⋊D3semidirect
152. C4×Dic3C2×C3⋊D3semidirect
185. C33#453semidirect
269. C32#5077semidirect
270. C32#5241semidirect
310. C3#10591509semidirect
311. C3#10592776semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
C32×#9402
#117233642
#117222032
#117220462
#105927763
#105915093