Finite Group Databasebeta

← #13161095

Subgroups — #13161095

Order 1728.

1672 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1672 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, complemented
2C3×#64149C211yesmaximal, characteristic, complemented
3#11723370C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722047C212yesmaximal, complemented
6#10592446—31nomaximal
7#10591511—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6312—22no—
10#89078—22no—
11D4×C3⋊(C3⋊D3)—22no—
12#89076C2212yes—
13C2×C3×#638—22no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
15C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
16C32×#208C2212yescomplemented
17#88796—22no—
18#88791—22no—
19#64361—31no—
20#64361—34no—
21#64361—64no—
22#64149D311yescharacteristic, complemented
23#64117—32no—
24C3×#940—34no—
25#63830—38no—
26C32×#100—31no—
27#63652—32no—
28C3×#833D314yescomplemented
29#63570—38no—
30C23×C33D412yes—
31C22×C3⋊(C3⋊D3)—22no—
32C22×C3×C3⋊D3—21no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35#6312—24no—
36#6312—22no—
37C2×C3⋊(C3⋊Dic3)—22no—
38C4⋊(C3⋊(C3⋊D3))D412yescomplemented
39C4×C3⋊(C3⋊D3)—22no—
40#6308D412yescomplemented
41C3×#638—44no—
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—22no—
44C32×C3⋊C8—22no—
45#5077—91no—
46#4747—94no—
47C2×C32×D4—24no—
48C2×C32×D4C2×D314yes—
49C2×C32×D4C2×D311yescharacteristic
50C2×#638—22no—
51C2×#638—616no—
52C2×#638—68no—
53C2×#638—62no—
54#3830—62no—
55#3830—68no—
56#3830—616no—
57D4×C3⋊D3—62no—
58D4×C3⋊D3—68no—
59D4×C3⋊D3—616no—
60#3828—32no—
61#3828—38no—
62#3828—68no—
63C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
64C2×C3×#67—68no—
65C2×C3×D12—34no—
66C32×M16—32no—
67C4×C3⋊Dic3C2×D311yescharacteristic, complemented
68#3745—62no—
69#3744—62no—
70C3×C4×Dic3—34no—
71C3×#208C2×D318yescomplemented
72#3716—68no—
73#3714—68no—
74C22×C33—22no—
75C22×C33—42no—
76C22×C33C2×D411yescharacteristic
77C2×C3⋊(C3⋊D3)—22no—
78C2×C3⋊(C3⋊D3)—82no—
79C2×C3×C3⋊D3—42no—
80C2×C3×C3⋊D3—41no—
81C33×C4C2×D412yes—
82C3⋊(C3⋊Dic3)—22no—
83C3×C3⋊Dic3—22no—
84#1004—94no—
85#1004—91no—
86#1004—184no—
87C3×#127—91no—
88#940—34no—
89#932—92no—
90#908—98no—
91C3×#100—34no—
92#833C3⋊D311yescharacteristic, complemented
93#825—98no—
94#805—92no—
95C23×C32—28no—
96C23×C32D1218yes—
97C23×C32#6712yes—
98C22×C3⋊D3—21no—
99C22×C3⋊D3—616no—
100C22×C3⋊D3—68no—
101C22×C3⋊D3—62no—
102C22×C3×D3—64no—
103C32×D4—216no—
104C32×D4—28no—
105C32×D4—22no—
106C2×C32×C4—24no—
107C2×C32×C4C22×D314yes—
108C2×C32×C4C22×D311yescharacteristic
109#638—44no—
110#638—616no—
111#638—632no—
112#638—616no—
113#638—64no—
114#638—68no—
115#638—62no—
116C2×C3⋊Dic3—21no—
117C2×C3⋊Dic3—616no—
118C2×C3⋊Dic3—68no—
119C2×C3⋊Dic3—62no—
120C4⋊(C3⋊D3)—22no—
121C4⋊(C3⋊D3)—32no—
122C4⋊(C3⋊D3)—38no—
123C4⋊(C3⋊D3)—68no—
124C4×C3⋊D3—616no—
125C4×C3⋊D3—68no—
126C4×C3⋊D3—62no—
127#634—32no—
128#634—38no—
129#634—68no—
130C3×#67—1216no—
131C2×C3×Dic3—64no—
132C3×D12—68no—
133C32×C8—62no—
134C3×C3⋊C8—28no—
135#453—271noSylow
136C2×C33—21no—
137C2×C33—42no—
138C2×C33#12011yescharacteristic
139C3⋊(C3⋊D3)—42no—
140C3×C3⋊D3—81no—
141C2×C3×D4—24no—
142C2×C3×D4—91no—
143C2×C3×D4—182no—
144C2×C3×D4D3214yes—
145C2×C3×D4C2×C3⋊D311yescharacteristic
146C2×#67—68no—
147C2×#67—1816no—
148C2×#67—188no—
149C2×#67—182no—
150#237—188no—
151#237—182no—
152#237—1816no—
153D3×D4—188no—
154D3×D4—182no—
155D3×D4—1816no—
156#235—98no—
157#235—92no—
158#235—188no—
159C2×D12—34no—
160C3×M16—38no—
161C3×C42—91no—
162#214—182no—
163#213—182no—
164C4×Dic3—34no—
165#208C2×C3⋊D312yescomplemented
166D24—188no—
167#204—188no—
168C22×C32—216no—
169C22×C32—28no—
170C22×C32—22no—
171C22×C32—24no—
172C22×C32—42no—
173C22×C32—48no—
174C22×C32—88no—
175C22×C32C2×D1214yes—
176C22×C32C2×#6711yescharacteristic
177C2×C3⋊D3—42no—
178C2×C3⋊D3—41no—
179C2×C3⋊D3—616no—
180C2×C3⋊D3—68no—
181C2×C3⋊D3—62no—
182C2×C3⋊D3—2416no—
183C2×C3⋊D3—248no—
184C2×C3⋊D3—242no—
185C2×C3×D3—128no—
186C2×C3×D3—124no—
187C32×C4—28no—
188C32×C4D3×D418yes—
189C32×C4D3×D412yes—
190C3⋊Dic3—22no—
191C3⋊Dic3—616no—
192C3⋊Dic3—68no—
193C3⋊Dic3—62no—
194C3×Dic3—68no—
195H(5,𝔽2)—271no—
196AGL(1,ℤ/8)—272no—
197#127—91no—
198#104—272no—
199#100—31no—
200C33#45311yescharacteristic, complemented, Sylow
201C23×C3—28no—
202C23×C3—181no—
203C23×C3#62618yes—
204C23×C3C4⋊(C3⋊D3)12yes—
205C22×D3—64no—
206C22×D3—1816no—
207C22×D3—182no—
208C22×D3—188no—
209C3×D4—216no—
210C3×D4—28no—
211C3×D4—22no—
212C3×D4—182no—
213C3×D4—364no—
214C2×C3×C4—24no—
215C2×C3×C4—181no—
216C2×C3×C4C2×D3214yes—
217C2×C3×C4C22×C3⋊D311yescharacteristic
218#67—1216no—
219#67—1816no—
220#67—1832no—
221#67—184no—
222#67—1816no—
223#67—188no—
224#67—182no—
225C2×Dic3—64no—
226C2×Dic3—1816no—
227C2×Dic3—182no—
228C2×Dic3—188no—
229D12—68no—
230D12—98no—
231D12—92no—
232D12—188no—
233C4×D3—1816no—
234C4×D3—188no—
235C4×D3—182no—
236Dic6—98no—
237Dic6—92no—
238Dic6—188no—
239C3×C8—68no—
240C3⋊C8—22no—
241C2×C32—21no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—44no—
245C2×C32—416no—
246C2×C32—48no—
247C2×C32—42no—
248C2×C32#88214yes—
249C2×C32#93711yescharacteristic
250C3⋊D3—81no—
251C3⋊D3—1216no—
252C3⋊D3—128no—
253C3⋊D3—122no—
254C3×D3—244no—
255#41—272no—
256#41—542no—
257C2×D4—91no—
258C2×D4—182no—
259C2×D4—542no—
260C2×D4—542no—
261C2×D4D3×C3⋊D311yescharacteristic
262SD16—542no—
263D8—542no—
264M16—32no—
265C42—91no—
266C22×C3—216no—
267C22×C3—22no—
268C22×C3—28no—
269C22×C3—24no—
270C22×C3—48no—
271C22×C3—42no—
272C22×C3—88no—
273C22×C3—362no—
274C22×C3—361no—
275C22×C3C2×#62614yes—
276C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
277C2×D3—128no—
278C2×D3—124no—
279C2×D3—1816no—
280C2×D3—182no—
281C2×D3—188no—
282C2×D3—7216no—
283C2×D3—728no—
284C2×D3—722no—
285C3×C4—28no—
286C3×C4—182no—
287C3×C4#381118yes—
288C3×C4D4×C3⋊D312yes—
289Dic3—68no—
290Dic3—1816no—
291Dic3—182no—
292Dic3—188no—
293C32—24no—
294C32#474714yescomplemented
295C32#507711yescharacteristic, complemented
296C23—181no—
297C23—542no—
298C23#629212yes—
299Q8—272no—
300D4—22no—
301D4—182no—
302D4—272no—
303D4—364no—
304D4—544no—
305D4—542no—
306C2×C4—181no—
307C2×C4—542no—
308C2×C4—542no—
309C2×C4C2×D3×C3⋊D311yescharacteristic
310C8—62no—
311C2×C3—24no—
312C2×C3—21no—
313C2×C3—24no—
314C2×C3—44no—
315C2×C3—416no—
316C2×C3—48no—
317C2×C3—42no—
318C2×C3—721no—
319C2×C3#6398114yes—
320C2×C3#6409111yescharacteristic
321D3—244no—
322D3—3616no—
323D3—368no—
324D3—362no—
325C22—22no—
326C22—42no—
327C22—362no—
328C22—361no—
329C22—542no—
330C22—2162no—
331C22C2×#629211yescharacteristic
332C4—182no—
333C4—542no—
334C4#8903912yes—
335C3—24no—
336C3#1059151114yescomplemented
337C3#1059244611yescharacteristic, complemented
338C2—21no—
339C2—42no—
340C2—721no—
341C2—1082no—
342C2#1172363911yescharacteristic
subgroupG/H
centerC2#11723639
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. #11724679C2semidirect
2. C3×#64149C2semidirect
3. #11723370C2semidirect
4. C32×#833C2semidirect
5. #11722047C2semidirect
14. C2×C3×C4⋊(C3⋊D3)C22semidirect
15. C3×C4×C3⋊Dic3C22semidirect
16. C32×#208C22semidirect
22. #64149D3semidirect
28. C3×#833D3semidirect
38. C4⋊(C3⋊(C3⋊D3))D4semidirect
40. #6308D4semidirect
63. C2×C4⋊(C3⋊D3)C2×D3semidirect
67. C4×C3⋊Dic3C2×D3semidirect
71. C3×#208C2×D3semidirect
92. #833C3⋊D3semidirect
165. #208C2×C3⋊D3semidirect
200. C33#453semidirect
294. C32#4747semidirect
295. C32#5077semidirect
336. C3#10591511semidirect
337. C3#10592446semidirect

Maximal subgroups

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

subgroupindex
#117246792
C3×#641492
#117233702
C32×#8332
#117220472
#105924463
#105915113