Finite Group Databasebeta

← #13161090

Subgroups — #13161090

Order 1728.

1440 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 1440 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
2#11723981C211yesmaximal, characteristic, complemented
3#11723371C212yesmaximal, complemented
4C32×#833C211yesmaximal, characteristic, complemented
5#11722045C212yesmaximal, complemented
6#10592446—31nomaximal
7#10591510—34nomaximal
8C2×C33×D4C2211yescharacteristic
9C2×#6312—22no—
10C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic, complemented
11C2×C3×#638—22no—
12C3×#3830—22no—
13C3×D4×C3⋊D3—22no—
14C3×#3828C2212yes—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic, complemented
16C32×#208C2212yescomplemented
17#88797—22no—
18#88791—22no—
19#64361D311yescharacteristic, complemented
20C3×#1004—34no—
21#64149—31no—
22#64149—34no—
23#64149—64no—
24#64117—32no—
25#63825—38no—
26C32×#100—31no—
27#63652—32no—
28C3×#833D314yescomplemented
29#63568—38no—
30C23×C33D412yes—
31C22×C3⋊(C3⋊D3)—21no—
32C22×C3×C3⋊D3—22no—
33C33×D4—22no—
34C2×C33×C4C2311yescharacteristic
35#6312—44no—
36C2×C3⋊(C3⋊Dic3)—21no—
37C4⋊(C3⋊(C3⋊D3))—22no—
38C3×#638—24no—
39C3×#638—22no—
40C2×C3×C3⋊Dic3—22no—
41C3×C4⋊(C3⋊D3)D412yescomplemented
42C3×C4×C3⋊D3—22no—
43C3×#634D412yescomplemented
44C32×C3⋊C8—22no—
45#5241—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—22no—
55D4×C3⋊D3—22no—
56#3828C2×D312yes—
57C2×C4⋊(C3⋊D3)—31no—
58C2×C4⋊(C3⋊D3)—34no—
59C2×C4⋊(C3⋊D3)—64no—
60C2×C3×#67—68no—
61C3×#237—68no—
62C3×D3×D4—68no—
63C3×#235—38no—
64C32×M16—32no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—64no—
68#3745—62no—
69#3744—62no—
70C3×#208C2×D318yescomplemented
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×C3×C3⋊D3—22no—
79C2×C3×C3⋊D3—82no—
80C33×C4C2×D412yes—
81C3⋊(C3⋊Dic3)—22no—
82C3×C3⋊Dic3—22no—
83C3×H(5,𝔽2)—91no—
84#1004—34no—
85#949—92no—
86#940—94no—
87#940—91no—
88#940—184no—
89#908—98no—
90C3×#100—34no—
91#837—92no—
92#833C3⋊D311yescharacteristic, complemented
93#825—98no—
94C23×C32—28no—
95C23×C32D1218yes—
96C23×C32#6712yes—
97C22×C3⋊D3—22no—
98C22×C3⋊D3—68no—
99C22×C3⋊D3—64no—
100C22×C3⋊D3—61no—
101C22×C3×D3—68no—
102C32×D4—22no—
103C32×D4—28no—
104C32×D4—48no—
105C2×C32×C4—24no—
106C2×C32×C4C22×D314yes—
107C2×C32×C4C22×D311yescharacteristic
108#638—24no—
109#638—22no—
110#638—1216no—
111#638—124no—
112#638—1232no—
113C2×C3⋊Dic3—22no—
114C2×C3⋊Dic3—68no—
115C2×C3⋊Dic3—64no—
116C2×C3⋊Dic3—61no—
117C4⋊(C3⋊D3)—62no—
118C4⋊(C3⋊D3)—68no—
119C4⋊(C3⋊D3)—128no—
120C4⋊(C3⋊D3)#6712yescomplemented
121C4×C3⋊D3—22no—
122#634#6712yescomplemented
123C3×#67—616no—
124C3×#67—68no—
125C2×C3×Dic3—68no—
126C3×D12—38no—
127C3×C4×D3—68no—
128C3×Dic6—38no—
129C32×C8—62no—
130C3×C3⋊C8—28no—
131#453—271noSylow
132C2×C33—21no—
133C2×C33—42no—
134C2×C33#12011yescharacteristic
135C3⋊(C3⋊D3)—81no—
136C3×C3⋊D3—42no—
137C3×#41—92no—
138C3×#41—182no—
139C2×C3×D4—24no—
140C2×C3×D4—182no—
141C2×C3×D4—182no—
142C2×C3×D4D3214yes—
143C2×C3×D4C2×C3⋊D311yescharacteristic
144C2×#67—68no—
145C2×#67—1816no—
146C2×#67—188no—
147C2×#67—182no—
148#237—68no—
149D3×D4—68no—
150#235—38no—
151C2×D12—94no—
152C2×D12—91no—
153C2×D12—184no—
154C3×M16—38no—
155#215—182no—
156#213—182no—
157C4×Dic3—94no—
158C4×Dic3—91no—
159C4×Dic3—184no—
160#208C2×C3⋊D312yescomplemented
161D24—188no—
162#204—188no—
163C22×C32—22no—
164C22×C32—24no—
165C22×C32—28no—
166C22×C32—48no—
167C22×C32—48no—
168C22×C32—42no—
169C22×C32—416no—
170C22×C32C2×D1214yes—
171C22×C32C2×#6711yescharacteristic
172C2×C3⋊D3—22no—
173C2×C3⋊D3—82no—
174C2×C3⋊D3—128no—
175C2×C3⋊D3—1216no—
176C2×C3⋊D3—122no—
177C2×C3⋊D3—128no—
178C2×C3⋊D3—124no—
179C2×C3⋊D3—121no—
180C2×C3×D3—68no—
181C2×C3×D3—248no—
182C32×C4—28no—
183C32×C4D3×D418yes—
184C32×C4C2×#6712yes—
185C3⋊Dic3—22no—
186C3⋊Dic3—616no—
187C3⋊Dic3—62no—
188C3⋊Dic3—68no—
189C3×Dic3—68no—
190H(5,𝔽2)—91no—
191AGL(1,ℤ/8)—272no—
192#127—271no—
193#104—272no—
194#100—31no—
195C33#45311yescharacteristic, complemented, Sylow
196C23×C3—28no—
197C23×C3—182no—
198C23×C3#62618yes—
199C23×C3C4⋊(C3⋊D3)12yes—
200C22×D3—68no—
201C22×D3—188no—
202C22×D3—184no—
203C22×D3—181no—
204C3×Q8—92no—
205C3×D4—28no—
206C3×D4—22no—
207C3×D4—48no—
208C3×D4—92no—
209C3×D4—184no—
210C3×D4—182no—
211C2×C3×C4—24no—
212C2×C3×C4—182no—
213C2×C3×C4—182no—
214C2×C3×C4C2×D3214yes—
215C2×C3×C4C22×C3⋊D311yescharacteristic
216#67—616no—
217#67—68no—
218#67—3616no—
219#67—364no—
220#67—3632no—
221C2×Dic3—68no—
222C2×Dic3—188no—
223C2×Dic3—184no—
224C2×Dic3—181no—
225D12—38no—
226D12—188no—
227D12—182no—
228D12—368no—
229C4×D3—68no—
230Dic6—38no—
231C3×C8—68no—
232C3⋊C8—22no—
233C2×C32—28no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—24no—
237C2×C32—42no—
238C2×C32—48no—
239C2×C32—88no—
240C2×C32#88214yes—
241C2×C32#95311yescharacteristic
242C3⋊D3—42no—
243C3⋊D3—248no—
244C3⋊D3—241no—
245C3⋊D3—244no—
246C3×D3—128no—
247#41—92no—
248#41—182no—
249C2×D4—182no—
250C2×D4—182no—
251C2×D4—271no—
252C2×D4—542no—
253C2×D4D3×C3⋊D311yescharacteristic
254SD16—542no—
255D8—542no—
256M16—32no—
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—182no—
266C22×C3—722no—
267C22×C3C2×#62614yes—
268C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
269C2×D3—68no—
270C2×D3—248no—
271C2×D3—368no—
272C2×D3—3616no—
273C2×D3—368no—
274C2×D3—362no—
275C2×D3—361no—
276C2×D3—364no—
277C3×C4—28no—
278C3×C4—182no—
279C3×C4D3×#6718yes—
280C3×C4D4×C3⋊D312yes—
281Dic3—68no—
282Dic3—1816no—
283Dic3—188no—
284Dic3—182no—
285C32—24no—
286C32#474714yescomplemented
287C32#524111yescharacteristic, complemented
288C23—182no—
289C23—541no—
290C23#629212yes—
291Q8—92no—
292D4—22no—
293D4—92no—
294D4—184no—
295D4—182no—
296D4—542no—
297D4—1084no—
298C2×C4—182no—
299C2×C4—182no—
300C2×C4—541no—
301C2×C4C2×D3×C3⋊D311yescharacteristic
302C8—62no—
303C2×C3—28no—
304C2×C3—24no—
305C2×C3—21no—
306C2×C3—24no—
307C2×C3—48no—
308C2×C3—42no—
309C2×C3—88no—
310C2×C3—362no—
311C2×C3#6397714yes—
312C2×C3#6409111yescharacteristic
313D3—128no—
314D3—728no—
315D3—724no—
316D3—721no—
317C22—22no—
318C22—42no—
319C22—182no—
320C22—722no—
321C22—1082no—
322C22—1081no—
323C22C2×#629211yescharacteristic
324C4—182no—
325C4—542no—
326C4C3⋊D3×#6712yes—
327C3—24no—
328C3#1059151014yescomplemented
329C3#1059244611yescharacteristic, complemented
330C2—21no—
331C2—42no—
332C2—362no—
333C2—2161no—
334C2#1172363611yescharacteristic
subgroupG/H
centerC2#11723636
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. #11723981C2semidirect
3. #11723371C2semidirect
4. C32×#833C2semidirect
5. #11722045C2semidirect
10. C2×C4⋊(C3⋊(C3⋊D3))C22semidirect
15. C4×C3⋊(C3⋊Dic3)C22semidirect
16. C32×#208C22semidirect
19. #64361D3semidirect
28. C3×#833D3semidirect
41. C3×C4⋊(C3⋊D3)D4semidirect
43. C3×#634D4semidirect
70. C3×#208C2×D3semidirect
92. #833C3⋊D3semidirect
120. C4⋊(C3⋊D3)#67semidirect
122. #634#67semidirect
160. #208C2×C3⋊D3semidirect
195. C33#453semidirect
286. C32#4747semidirect
287. C32#5241semidirect
328. C3#10591510semidirect
329. C3#10592446semidirect

Maximal subgroups

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

subgroupindex
C3×#643612
#117239812
#117233712
C32×#8332
#117220452
#105924463
#105915103