Finite Group Databasebeta

← #13161097

Subgroups — #13161097

Order 1728.

1096 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1096 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
2C32×#936C211yesmaximal, characteristic, complemented
3#11723375C212yesmaximal, complemented
4#11722203C211yesmaximal, characteristic, complemented
5#11722046C212yesmaximal, complemented
6#10592760—31nomaximal
7#10591512—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×C3×#638—22no—
10C3×#3830—22no—
11C3×D4×C3⋊D3—22no—
12C3×#3828C2212yes—
13C2×C32×Dic6C2211yescharacteristic, complemented
14#88855C2212yescomplemented
15C32×#217—22no—
16C32×C4×Dic3C2211yescharacteristic, complemented
17#88798—22no—
18#88793—22no—
19#64361D311yescharacteristic
20C3×#1004—34no—
21C32×#124—31no—
22#64151—32no—
23C3×#936D314yescomplemented
24#63827—38no—
25#63660—31no—
26#63660—34no—
27#63660—64no—
28#63632—32no—
29#63569—38no—
30C23×C33D412yes—
31C22×C3×C3⋊D3—22no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34C3×#638—24no—
35C3×#638—22no—
36C2×C3×C3⋊Dic3—22no—
37C3×C4⋊(C3⋊D3)D412yes—
38C3×C4×C3⋊D3—22no—
39C3×#634D412yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×Dic6—22no—
43C3⋊(C3⋊(C3⋊C8))—22no—
44#5242—91no—
45#5061—94no—
46C2×C32×Q8—31no—
47C2×C32×D4—24no—
48C2×C32×D4C2×D314yes—
49C2×C32×D4C2×D311yescharacteristic
50C2×#638—22no—
51#3830—22no—
52D4×C3⋊D3—22no—
53#3828C2×D312yes—
54C2×C3×#67—68no—
55C3×#237—68no—
56C3×D3×D4—68no—
57C3×#235—38no—
58C2×C3×Dic6C2×D314yescomplemented
59C32×C22⋊C4—62no—
60C32×C42—31no—
61#3756—62no—
62#3755—62no—
63#3748—32no—
64#3748—38no—
65#3748—68no—
66C3×#217—28no—
67C3×C4×Dic3C2×D314yescomplemented
68#3718—68no—
69#3715—68no—
70C22×C33—22no—
71C22×C33—42no—
72C22×C33C2×D411yescharacteristic
73C2×C3×C3⋊D3—22no—
74C2×C3×C3⋊D3—82no—
75C33×C4C2×D412yes—
76C3×C3⋊Dic3—22no—
77C32×Dic3—41no—
78C32×Dic3—41no—
79C3×H(5,𝔽2)—91no—
80#1004—34no—
81C3×#124—34no—
82#951—92no—
83#942—98no—
84#936C3⋊D311yescharacteristic, complemented
85#837—92no—
86#833—94no—
87#833—91no—
88#833—184no—
89#805—98no—
90C23×C32—28no—
91C23×C32#6718yes—
92C23×C32#6712yes—
93C22×C3⋊D3—22no—
94C22×C3×D3—68no—
95C32×Q8—62no—
96C32×D4—22no—
97C32×D4—28no—
98C32×D4—48no—
99C2×C32×C4—24no—
100C2×C32×C4—61no—
101C2×C32×C4—61no—
102C2×C32×C4C22×D314yes—
103C2×C32×C4C22×D311yescharacteristic
104#638—24no—
105#638—22no—
106C2×C3⋊Dic3—22no—
107C4⋊(C3⋊D3)#6712yes—
108C4×C3⋊D3—22no—
109#634#6712yes—
110C3×#67—616no—
111C3×#67—68no—
112C2×C3×Dic3—24no—
113C2×C3×Dic3—24no—
114C2×C3×Dic3—68no—
115C3×D12—38no—
116C3×C4×D3—68no—
117C3×Dic6—28no—
118C3×Dic6—38no—
119C3⋊(C3⋊C8)—62no—
120C3⋊(C3⋊C8)—68no—
121C3⋊(C3⋊C8)—128no—
122#455—271noSylow
123C2×C33—21no—
124C2×C33—42no—
125C2×C33#12011yescharacteristic
126C3×C3⋊D3—42no—
127C3×#41—92no—
128C3×#41—182no—
129C2×C3×Q8—34no—
130C2×C3×D4—24no—
131C2×C3×D4—182no—
132C2×C3×D4—182no—
133C2×C3×D4D3214yes—
134C2×C3×D4C2×C3⋊D311yescharacteristic
135C2×#67—68no—
136#237—68no—
137D3×D4—68no—
138#235—38no—
139C2×Dic6C2×C3⋊D311yescharacteristic, complemented
140C3×C22⋊C4—68no—
141C3×C42—34no—
142#217—22no—
143#216—182no—
144#216—188no—
145#215—188no—
146#214—182no—
147C4×Dic3C2×C3⋊D311yescharacteristic, complemented
148#208—98no—
149#208—92no—
150#208—188no—
151C22×C32—22no—
152C22×C32—24no—
153C22×C32—28no—
154C22×C32—48no—
155C22×C32—42no—
156C22×C32—48no—
157C22×C32—88no—
158C22×C32C2×#6714yes—
159C22×C32C2×#6711yescharacteristic
160C2×C3⋊D3—22no—
161C2×C3⋊D3—82no—
162C2×C3×D3—68no—
163C2×C3×D3—248no—
164C32×C4—28no—
165C32×C4—121no—
166C32×C4—121no—
167C32×C4D3×D418yes—
168C32×C4C2×#6712yes—
169C3⋊Dic3—22no—
170C3×Dic3—44no—
171C3×Dic3—44no—
172C3×Dic3—68no—
173H(5,𝔽2)—91no—
174#137—272no—
175#124—31no—
176#104—272no—
177#100—271no—
178C33#45511yescharacteristic, complemented, Sylow
179C23×C3—28no—
180C23×C3—182no—
181C23×C3#62518yes—
182C23×C3#63812yes—
183C22×D3—68no—
184C3×Q8—68no—
185C3×Q8—92no—
186C3×D4—28no—
187C3×D4—22no—
188C3×D4—48no—
189C3×D4—92no—
190C3×D4—184no—
191C3×D4—182no—
192C2×C3×C4—24no—
193C2×C3×C4—64no—
194C2×C3×C4—64no—
195C2×C3×C4—182no—
196C2×C3×C4—182no—
197C2×C3×C4C2×D3214yes—
198C2×C3×C4C22×C3⋊D311yescharacteristic
199#67—616no—
200#67—68no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—68no—
204D12—38no—
205C4×D3—68no—
206Dic6—22no—
207Dic6—38no—
208C3⋊C8—188no—
209C3⋊C8—182no—
210C3⋊C8—368no—
211C2×C32—21no—
212C2×C32—24no—
213C2×C32—24no—
214C2×C32—44no—
215C2×C32—42no—
216C2×C32—48no—
217C2×C32—88no—
218C2×C32#93714yes—
219C2×C32#95311yescharacteristic
220C3⋊D3—42no—
221C3×D3—128no—
222#41—92no—
223#41—182no—
224C2×Q8—31no—
225C2×D4—182no—
226C2×D4—182no—
227C2×D4D3×C3⋊D311yescharacteristic
228Q16—542no—
229SD16—542no—
230M16—272no—
231C22⋊C4—62no—
232C42—31no—
233C22×C3—22no—
234C22×C3—28no—
235C22×C3—24no—
236C22×C3—48no—
237C22×C3—48no—
238C22×C3—42no—
239C22×C3—88no—
240C22×C3—182no—
241C22×C3—722no—
242C22×C3C2×#62514yes—
243C22×C3C2×#63811yescharacteristic
244C2×D3—68no—
245C2×D3—248no—
246C3×C4—28no—
247C3×C4—124no—
248C3×C4—124no—
249C3×C4—182no—
250C3×C4D3×#6718yes—
251C3×C4D4×C3⋊D312yes—
252Dic3—41no—
253Dic3—41no—
254Dic3—68no—
255C32—24no—
256C32#506114yescomplemented
257C32#524211yescharacteristic, complemented
258C23—182no—
259C23#629012yes—
260Q8—62no—
261Q8—92no—
262D4—22no—
263D4—92no—
264D4—184no—
265D4—182no—
266C2×C4—61no—
267C2×C4—61no—
268C2×C4—182no—
269C2×C4—182no—
270C2×C4C2×D3×C3⋊D311yescharacteristic
271C8—542no—
272C2×C3—24no—
273C2×C3—21no—
274C2×C3—24no—
275C2×C3—44no—
276C2×C3—48no—
277C2×C3—42no—
278C2×C3—88no—
279C2×C3—362no—
280C2×C3#6397614yes—
281C2×C3#6414611yescharacteristic
282D3—128no—
283C22—22no—
284C22—42no—
285C22—182no—
286C22—722no—
287C22C2×#629011yescharacteristic
288C4—121no—
289C4—121no—
290C4—182no—
291C4C3⋊D3×#6712yes—
292C3—24no—
293C3#1059151214yescomplemented
294C3#1059276011yescharacteristic, complemented
295C2—21no—
296C2—42no—
297C2—362no—
298C2#1172363411yescharacteristic
subgroupG/H
centerC2#11723634
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radicalC3×#64361C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#455
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
2. C32×#936C2semidirect
3. #11723375C2semidirect
4. #11722203C2semidirect
5. #11722046C2semidirect
13. C2×C32×Dic6C22semidirect
14. #88855C22semidirect
16. C32×C4×Dic3C22semidirect
23. C3×#936D3semidirect
58. C2×C3×Dic6C2×D3semidirect
67. C3×C4×Dic3C2×D3semidirect
84. #936C3⋊D3semidirect
139. C2×Dic6C2×C3⋊D3semidirect
147. C4×Dic3C2×C3⋊D3semidirect
178. C33#455semidirect
256. C32#5061semidirect
257. C32#5242semidirect
293. C3#10591512semidirect
294. C3#10592760semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
C32×#9362
#117233752
#117222032
#117220462
#105927603
#105915123