Finite Group Databasebeta

← #13161093

Subgroups — #13161093

Order 1728.

1440 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1440 conjugacy classes of subgroups — too many to draw. Only its 164 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
1C32×#1004C211yesmaximal, characteristic, complemented
2#11723981C211yesmaximal, characteristic, complemented
3#11723367C212yesmaximal, complemented
4C3×#63660C211yesmaximal, characteristic, complemented
5#11722043C212yesmaximal, complemented
6#10592940—31nomaximal
7#10591510—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6312—22no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
11C2×C32×#67—22no—
12C32×#237—22no—
13C32×D3×D4—22no—
14C32×#235C2212yes—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C3×#3748C2212yescomplemented
17#88795—22no—
18#88790—22no—
19C32×H(5,𝔽2)—31no—
20C3×#1004D314yescomplemented
21#64158—32no—
22#64149—31no—
23#64149—34no—
24#64149—64no—
25#63825—38no—
26#63664—32no—
27#63660D311yescharacteristic, complemented
28C3×#833—34no—
29#63568—38no—
30C23×C33D412yes—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C32×D3—22no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35#6312—44no—
36C2×C3⋊(C3⋊Dic3)—21no—
37C4⋊(C3⋊(C3⋊D3))—22no—
38C32×#67—24no—
39C32×#67—22no—
40C2×C32×Dic3—22no—
41C32×D12D412yescomplemented
42C32×C4×D3—22no—
43C32×Dic6D412yescomplemented
44C3×C3⋊(C3⋊C8)—22no—
45#5241—94no—
46#4747—91no—
47C32×#41—32no—
48C32×#41—62no—
49C2×C32×D4—24no—
50C2×C32×D4—62no—
51C2×C32×D4—62no—
52C2×C32×D4C2×D311yescharacteristic
53C2×C32×D4C2×D314yes—
54C2×#638—616no—
55C2×#638—62no—
56C2×#638—68no—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C3×#67—28no—
61C3×#237—28no—
62C3×D3×D4—28no—
63C3×#235C2×D318yes—
64#3755—62no—
65#3753—62no—
66C4×C3⋊Dic3—31no—
67C4×C3⋊Dic3—34no—
68C4×C3⋊Dic3—64no—
69#3748C2×D312yescomplemented
70C3×#208—38no—
71#3717—68no—
72#3714—68no—
73C22×C33—22no—
74C22×C33—42no—
75C22×C33C2×D411yescharacteristic
76C2×C3⋊(C3⋊D3)—42no—
77C2×C3⋊(C3⋊D3)—41no—
78C2×C32×D3—22no—
79C2×C32×D3—82no—
80C33×C4C2×D412yes—
81C3⋊(C3⋊Dic3)—22no—
82C32×Dic3—22no—
83C3×H(5,𝔽2)—34no—
84#1004C3⋊D311yescharacteristic, complemented
85#949—98no—
86#940—94no—
87#940—91no—
88#940—184no—
89#908—92no—
90C3×#100—91no—
91#837—98no—
92#833—34no—
93#825—92no—
94C23×C32—28no—
95C23×C32—62no—
96C23×C32D1212yes—
97C23×C32#6718yes—
98C22×C3⋊D3—68no—
99C22×C3⋊D3—61no—
100C22×C3⋊D3—64no—
101C22×C3×D3—28no—
102C32×Q8—32no—
103C32×D4—22no—
104C32×D4—28no—
105C32×D4—32no—
106C32×D4—48no—
107C32×D4—64no—
108C32×D4—62no—
109C2×C32×C4—24no—
110C2×C32×C4—62no—
111C2×C32×C4—62no—
112C2×C32×C4C22×D314yes—
113C2×C32×C4C22×D311yescharacteristic
114#638—124no—
115#638—1216no—
116#638—1232no—
117C2×C3⋊Dic3—68no—
118C2×C3⋊Dic3—61no—
119C2×C3⋊Dic3—64no—
120C4⋊(C3⋊D3)—62no—
121C4⋊(C3⋊D3)—68no—
122C4⋊(C3⋊D3)—128no—
123C3×#67—216no—
124C3×#67—28no—
125C2×C3×Dic3—28no—
126C3×D12#6718yescomplemented
127C3×C4×D3—28no—
128C3×Dic6#6718yescomplemented
129C3⋊(C3⋊C8)—22no—
130C3×C3⋊C8—68no—
131#453—271noSylow
132C2×C33—21no—
133C2×C33—42no—
134C2×C33#12011yescharacteristic
135C3⋊(C3⋊D3)—81no—
136C32×D3—42no—
137C3×#41—38no—
138C3×#41—68no—
139C2×C3×D4—24no—
140C2×C3×D4—68no—
141C2×C3×D4—68no—
142C2×C3×D4D3214yes—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C2×#67—22no—
145C2×#67—1816no—
146C2×#67—188no—
147C2×#67—182no—
148#237—22no—
149D3×D4—22no—
150#235C2×C3⋊D312yes—
151C2×D12—94no—
152C2×D12—91no—
153C2×D12—184no—
154C3×M16—92no—
155#215—188no—
156#213—188no—
157C4×Dic3—94no—
158C4×Dic3—91no—
159C4×Dic3—184no—
160#208—38no—
161D24—182no—
162#204—182no—
163C22×C32—22no—
164C22×C32—24no—
165C22×C32—28no—
166C22×C32—48no—
167C22×C32—42no—
168C22×C32—48no—
169C22×C32—416no—
170C22×C32—62no—
171C22×C32—242no—
172C22×C32C2×D1211yescharacteristic
173C22×C32C2×#6714yes—
174C2×C3⋊D3—128no—
175C2×C3⋊D3—1216no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—122no—
178C2×C3⋊D3—121no—
179C2×C3⋊D3—124no—
180C2×C3×D3—28no—
181C2×C3×D3—88no—
182C32×C4—28no—
183C32×C4—62no—
184C32×C4D3×D412yes—
185C32×C4C2×#6718yes—
186C3⋊Dic3—616no—
187C3⋊Dic3—68no—
188C3⋊Dic3—62no—
189C3×Dic3—28no—
190H(5,𝔽2)—31no—
191AGL(1,ℤ/8)—272no—
192#127—271no—
193#104—272no—
194#100—91no—
195C33#45311yescharacteristic, complemented, Sylow
196C23×C3—28no—
197C23×C3—68no—
198C23×C3#62618yes—
199C23×C3#63812yes—
200C22×D3—22no—
201C22×D3—188no—
202C22×D3—184no—
203C22×D3—181no—
204C3×Q8—38no—
205C3×D4—28no—
206C3×D4—22no—
207C3×D4—38no—
208C3×D4—48no—
209C3×D4—616no—
210C3×D4—68no—
211C2×C3×C4—24no—
212C2×C3×C4—68no—
213C2×C3×C4—68no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216#67—24no—
217#67—22no—
218#67—3616no—
219#67—364no—
220#67—3632no—
221C2×Dic3—22no—
222C2×Dic3—188no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225D12—188no—
226D12—182no—
227D12—368no—
228D12#63812yescomplemented
229C4×D3—22no—
230Dic6#63812yescomplemented
231C3×C8—182no—
232C3⋊C8—68no—
233C2×C32—28no—
234C2×C32—24no—
235C2×C32—21no—
236C2×C32—24no—
237C2×C32—42no—
238C2×C32—48no—
239C2×C32—88no—
240C2×C32—122no—
241C2×C32#88211yescharacteristic
242C2×C32#95314yes—
243C3⋊D3—248no—
244C3⋊D3—244no—
245C3⋊D3—241no—
246C3×D3—48no—
247#41—32no—
248#41—62no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—271no—
252C2×D4—542no—
253C2×D4D3×C3⋊D311yescharacteristic
254SD16—542no—
255D8—542no—
256M16—92no—
257C42—271no—
258C22×C3—22no—
259C22×C3—28no—
260C22×C3—24no—
261C22×C3—48no—
262C22×C3—48no—
263C22×C3—42no—
264C22×C3—416no—
265C22×C3—68no—
266C22×C3—248no—
267C22×C3C2×#62614yes—
268C22×C3C2×#63811yescharacteristic
269C2×D3—22no—
270C2×D3—82no—
271C2×D3—368no—
272C2×D3—3616no—
273C2×D3—368no—
274C2×D3—362no—
275C2×D3—361no—
276C2×D3—364no—
277C3×C4—28no—
278C3×C4—68no—
279C3×C4D3×#6718yes—
280C3×C4C2×#63812yes—
281Dic3—22no—
282Dic3—1816no—
283Dic3—188no—
284Dic3—182no—
285C32—24no—
286C32#474711yescharacteristic, complemented
287C32#524114yescomplemented
288C23—62no—
289C23—541no—
290C23#629112yes—
291Q8—32no—
292D4—22no—
293D4—32no—
294D4—64no—
295D4—62no—
296D4—542no—
297D4—1084no—
298C2×C4—62no—
299C2×C4—62no—
300C2×C4—541no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—182no—
303C2×C3—28no—
304C2×C3—24no—
305C2×C3—21no—
306C2×C3—24no—
307C2×C3—48no—
308C2×C3—42no—
309C2×C3—88no—
310C2×C3—128no—
311C2×C3#6397714yes—
312C2×C3#6416211yescharacteristic
313D3—42no—
314D3—728no—
315D3—724no—
316D3—721no—
317C22—22no—
318C22—42no—
319C22—62no—
320C22—242no—
321C22—1082no—
322C22—1081no—
323C22C2×#629111yescharacteristic
324C4—62no—
325C4—542no—
326C4D3×#63812yes—
327C3—24no—
328C3#1059151014yescomplemented
329C3#1059294011yescharacteristic, complemented
330C2—21no—
331C2—42no—
332C2—122no—
333C2—2161no—
334C2#1172363011yescharacteristic
subgroupG/H
centerC2#11723630
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. C32×#1004C2semidirect
2. #11723981C2semidirect
3. #11723367C2semidirect
4. C3×#63660C2semidirect
5. #11722043C2semidirect
10. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C4×C3⋊(C3⋊Dic3)C22semidirect
16. C3×#3748C22semidirect
20. C3×#1004D3semidirect
27. #63660D3semidirect
41. C32×D12D4semidirect
43. C32×Dic6D4semidirect
69. #3748C2×D3semidirect
84. #1004C3⋊D3semidirect
126. C3×D12#67semidirect
128. C3×Dic6#67semidirect
195. C33#453semidirect
228. D12#638semidirect
230. Dic6#638semidirect
286. C32#4747semidirect
287. C32#5241semidirect
328. C3#10591510semidirect
329. C3#10592940semidirect

Maximal subgroups

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

subgroupindex
C32×#10042
#117239812
#117233672
C3×#636602
#117220432
#105929403
#105915103