Finite Group Databasebeta

← #13161105

Subgroups — #13161105

Order 1728.

1576 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1576 conjugacy classes of subgroups — too many to draw. Only its 144 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
1#11724679C211yesmaximal, characteristic
2C32×#936C211yesmaximal, characteristic, complemented
3#11723381C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5#11722048C212yesmaximal, complemented
6#10592760—31nomaximal
7#10591514—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6312—22no—
10#89078—22no—
11D4×C3⋊(C3⋊D3)—22no—
12#89076C2212yes—
13C2×C32×Dic6C2211yescharacteristic, complemented
14C3×#3748C2212yescomplemented
15C32×#217—22no—
16C32×C4×Dic3C2211yescharacteristic, complemented
17#88799—22no—
18#88795—22no—
19#64361—31no—
20#64361—34no—
21#64361—64no—
22C32×#124—31no—
23#64151—32no—
24C3×#936D314yescomplemented
25#63833—38no—
26#63660D311yescharacteristic, complemented
27#63632—32no—
28C3×#833—34no—
29#63570—38no—
30C23×C33D412yes—
31C22×C3⋊(C3⋊D3)—22no—
32C33×D4—22no—
33C2×C33×C4C2311yescharacteristic
34#6312—24no—
35#6312—22no—
36C2×C3⋊(C3⋊Dic3)—22no—
37C4⋊(C3⋊(C3⋊D3))D412yes—
38C4×C3⋊(C3⋊D3)—22no—
39#6308D412yes—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×Dic6—22no—
43C3×C3⋊(C3⋊C8)—22no—
44#5061—94no—
45#4746—91no—
46C2×C32×Q8—31no—
47C2×C32×D4—24no—
48C2×C32×D4C2×D311yescharacteristic
49C2×C32×D4C2×D314yes—
50C2×#638—616no—
51C2×#638—62no—
52C2×#638—68no—
53#3830—62no—
54#3830—68no—
55#3830—616no—
56D4×C3⋊D3—62no—
57D4×C3⋊D3—68no—
58D4×C3⋊D3—616no—
59#3828—32no—
60#3828—38no—
61#3828—68no—
62C2×C3×Dic6C2×D314yescomplemented
63C32×C22⋊C4—62no—
64C32×C42—31no—
65#3756—62no—
66#3755—62no—
67#3748C2×D312yescomplemented
68C3×#217—28no—
69C3×C4×Dic3C2×D314yescomplemented
70C3×#208—38no—
71#3719—68no—
72#3717—68no—
73C22×C33—22no—
74C22×C33—42no—
75C22×C33C2×D411yescharacteristic
76C2×C3⋊(C3⋊D3)—22no—
77C2×C3⋊(C3⋊D3)—82no—
78C33×C4C2×D412yes—
79C3⋊(C3⋊Dic3)—22no—
80C32×Dic3—41no—
81C32×Dic3—41no—
82#1004—94no—
83#1004—91no—
84#1004—184no—
85C3×#124—34no—
86#942—98no—
87#936C3⋊D311yescharacteristic, complemented
88#909—92no—
89C3×#100—91no—
90#833—34no—
91#825—92no—
92#805—98no—
93C23×C32—28no—
94C23×C32D1212yes—
95C23×C32#6718yes—
96C22×C3⋊D3—616no—
97C22×C3⋊D3—62no—
98C22×C3⋊D3—68no—
99C32×Q8—62no—
100C32×D4—216no—
101C32×D4—28no—
102C32×D4—22no—
103C2×C32×C4—24no—
104C2×C32×C4—61no—
105C2×C32×C4—61no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108#638—616no—
109#638—632no—
110#638—64no—
111#638—616no—
112#638—62no—
113#638—68no—
114C2×C3⋊Dic3—616no—
115C2×C3⋊Dic3—62no—
116C2×C3⋊Dic3—68no—
117C4⋊(C3⋊D3)—32no—
118C4⋊(C3⋊D3)—38no—
119C4⋊(C3⋊D3)—68no—
120C4×C3⋊D3—616no—
121C4×C3⋊D3—68no—
122C4×C3⋊D3—62no—
123#634—32no—
124#634—38no—
125#634—68no—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—24no—
128C3×Dic6—28no—
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—68no—
131#455—271noSylow
132C2×C33—21no—
133C2×C33—42no—
134C2×C33#12011yescharacteristic
135C3⋊(C3⋊D3)—42no—
136C2×C3×Q8—34no—
137C2×C3×D4—24no—
138C2×C3×D4D3214yes—
139C2×C3×D4C2×C3⋊D311yescharacteristic
140C2×#67—1816no—
141C2×#67—188no—
142C2×#67—182no—
143#237—188no—
144#237—182no—
145#237—1816no—
146D3×D4—188no—
147D3×D4—182no—
148D3×D4—1816no—
149#235—98no—
150#235—92no—
151#235—188no—
152C2×Dic6C2×C3⋊D311yescharacteristic, complemented
153C3×M16—92no—
154C3×C22⋊C4—68no—
155C3×C42—34no—
156#217—22no—
157#216—188no—
158#215—188no—
159C4×Dic3C2×C3⋊D311yescharacteristic, complemented
160#208—38no—
161Dic12—182no—
162#204—182no—
163C22×C32—216no—
164C22×C32—22no—
165C22×C32—28no—
166C22×C32—24no—
167C22×C32—42no—
168C22×C32—48no—
169C22×C32—88no—
170C22×C32C2×D1211yescharacteristic
171C22×C32C2×#6714yes—
172C2×C3⋊D3—616no—
173C2×C3⋊D3—62no—
174C2×C3⋊D3—68no—
175C2×C3⋊D3—2416no—
176C2×C3⋊D3—242no—
177C2×C3⋊D3—248no—
178C32×C4—28no—
179C32×C4—121no—
180C32×C4—121no—
181C32×C4D3×D418yes—
182C32×C4D3×D412yes—
183C3⋊Dic3—616no—
184C3⋊Dic3—62no—
185C3⋊Dic3—68no—
186C3×Dic3—44no—
187C3×Dic3—44no—
188H(5,𝔽2)—271no—
189#137—272no—
190#124—31no—
191#104—272no—
192#100—91no—
193C33#45511yescharacteristic, complemented, Sylow
194C23×C3—28no—
195C23×C3#62618yes—
196C23×C3#63812yes—
197C22×D3—1816no—
198C22×D3—182no—
199C22×D3—188no—
200C3×Q8—68no—
201C3×D4—216no—
202C3×D4—28no—
203C3×D4—22no—
204C2×C3×C4—24no—
205C2×C3×C4—64no—
206C2×C3×C4—64no—
207C2×C3×C4C2×D3214yes—
208C2×C3×C4C22×C3⋊D311yescharacteristic
209#67—1816no—
210#67—1832no—
211#67—184no—
212#67—1816no—
213#67—188no—
214#67—182no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—1816no—
218C2×Dic3—182no—
219C2×Dic3—188no—
220D12—98no—
221D12—92no—
222D12—188no—
223C4×D3—1816no—
224C4×D3—188no—
225C4×D3—182no—
226Dic6—22no—
227Dic6—98no—
228Dic6—92no—
229Dic6—188no—
230C3×C8—182no—
231C3⋊C8—68no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—24no—
235C2×C32—44no—
236C2×C32—416no—
237C2×C32—42no—
238C2×C32—48no—
239C2×C32#88211yescharacteristic
240C2×C32#93714yes—
241C3⋊D3—1216no—
242C3⋊D3—122no—
243C3⋊D3—128no—
244#41—272no—
245#41—542no—
246C2×Q8—31no—
247C2×D4—542no—
248C2×D4—542no—
249C2×D4D3×C3⋊D311yescharacteristic
250Q16—542no—
251SD16—542no—
252M16—92no—
253C22⋊C4—62no—
254C42—31no—
255C22×C3—216no—
256C22×C3—22no—
257C22×C3—28no—
258C22×C3—24no—
259C22×C3—48no—
260C22×C3—42no—
261C22×C3—88no—
262C22×C3C2×#62614yes—
263C22×C3C2×#63811yescharacteristic
264C2×D3—1816no—
265C2×D3—182no—
266C2×D3—188no—
267C2×D3—7216no—
268C2×D3—728no—
269C2×D3—722no—
270C3×C4—28no—
271C3×C4—124no—
272C3×C4—124no—
273C3×C4#381118yes—
274C3×C4D4×C3⋊D312yes—
275Dic3—41no—
276Dic3—41no—
277Dic3—1816no—
278Dic3—182no—
279Dic3—188no—
280C32—24no—
281C32#474611yescharacteristic, complemented
282C32#506114yescomplemented
283C23—542no—
284C23#629112yes—
285Q8—62no—
286Q8—272no—
287D4—22no—
288D4—272no—
289D4—544no—
290D4—542no—
291C2×C4—61no—
292C2×C4—61no—
293C2×C4—542no—
294C2×C4—542no—
295C2×C4C2×D3×C3⋊D311yescharacteristic
296C8—182no—
297C2×C3—24no—
298C2×C3—21no—
299C2×C3—24no—
300C2×C3—44no—
301C2×C3—416no—
302C2×C3—48no—
303C2×C3—42no—
304C2×C3#6398114yes—
305C2×C3#6414611yescharacteristic
306D3—3616no—
307D3—368no—
308D3—362no—
309C22—22no—
310C22—42no—
311C22—542no—
312C22—2162no—
313C22C2×#629111yescharacteristic
314C4—121no—
315C4—121no—
316C4—542no—
317C4#8903912yes—
318C3—24no—
319C3#1059151414yescomplemented
320C3#1059276011yescharacteristic, complemented
321C2—21no—
322C2—42no—
323C2—1082no—
324C2#1172363811yescharacteristic
subgroupG/H
centerC2#11723638
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×D4C22
socleC2×C33#120
prime radical#11724679C2
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. #11723381C2semidirect
4. C3×#63660C2semidirect
5. #11722048C2semidirect
13. C2×C32×Dic6C22semidirect
14. C3×#3748C22semidirect
16. C32×C4×Dic3C22semidirect
24. C3×#936D3semidirect
26. #63660D3semidirect
62. C2×C3×Dic6C2×D3semidirect
67. #3748C2×D3semidirect
69. C3×C4×Dic3C2×D3semidirect
87. #936C3⋊D3semidirect
152. C2×Dic6C2×C3⋊D3semidirect
159. C4×Dic3C2×C3⋊D3semidirect
193. C33#455semidirect
281. C32#4746semidirect
282. C32#5061semidirect
319. C3#10591514semidirect
320. C3#10592760semidirect

Maximal subgroups

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

subgroupindex
#117246792
C32×#9362
#117233812
C3×#636602
#117220482
#105927603
#105915143