Finite Group Databasebeta

← C3×#10592940 #13163280

Subgroups — C3×#10592940 #13163280

Order 1728.

1672 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 1672 conjugacy classes of subgroups — too many to draw. Only its 214 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
1C33×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C3×#64158C212yesmaximal, complemented
3C3×#64149C211yesmaximal, characteristic, complemented
4C3×#63664C212yesmaximal, complemented
5C3×#63660C211yesmaximal, characteristic, complemented
6#10592940C311yesmaximal, characteristic, direct product factor
7C3×#5241—34nomaximal
8C33×#41—22no—
9C33×#41C2212yes—
10C2×C33×D4—22no—
11C2×C33×D4—22no—
12C2×C33×D4C2211yescharacteristic
13C2×C3×#638—22no—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
15C3×#3755—22no—
16C3×#3753—22no—
17C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
18C3×#3748C2212yescomplemented
19C32×H(5,𝔽2)—24no—
20C32×H(5,𝔽2)D314yescomplemented
21C32×H(5,𝔽2)C2×C311yescharacteristic, complemented
22#64158C2×C312yescomplemented
23#64149C2×C311yescharacteristic, complemented
24C3×#949—38no—
25C3×#940—34no—
26#63664C2×C312yescomplemented
27#63660C2×C311yescharacteristic, complemented
28C3×#837—38no—
29C3×#833—34no—
30C23×C33—22no—
31C23×C33D412yes—
32C22×C3×C3⋊D3—21no—
33C33×Q8D412yescomplemented
34C33×D4—24no—
35C33×D4—22no—
36C33×D4—22no—
37C33×D4D412yescomplemented
38C2×C33×C4—22no—
39C2×C33×C4—22no—
40C2×C33×C4C2311yescharacteristic
41C3×#638—44no—
42C2×C3×C3⋊Dic3—21no—
43C3×C4⋊(C3⋊D3)—22no—
44C3×C3⋊(C3⋊C8)—22no—
45C3×#453—91no—
46#5241—34no—
47C32×#41—216no—
48C32×#41—28no—
49C32×#41—28no—
50C32×#41—22no—
51C32×#41C2×D318yes—
52C32×#41C22×C312yes—
53C2×C32×D4—216no—
54C2×C32×D4—216no—
55C2×C32×D4—24no—
56C2×C32×D4—28no—
57C2×C32×D4—22no—
58C2×C32×D4—28no—
59C2×C32×D4—22no—
60C2×C32×D4C2×D314yes—
61C2×C32×D4C22×C311yescharacteristic
62C2×#638—22no—
63C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
64C2×C3×#67—68no—
65C2×C3×D12—34no—
66#3755—22no—
67#3753—22no—
68C4×C3⋊Dic3C22×C311yescharacteristic, complemented
69#3748C22×C312yescomplemented
70C3×#215—68no—
71C3×#213—68no—
72C3×C4×Dic3—34no—
73C3×#208—38no—
74C22×C33—22no—
75C22×C33—22no—
76C22×C33—42no—
77C22×C33—82no—
78C22×C33C2×D411yescharacteristic
79C2×C3×C3⋊D3—42no—
80C2×C3×C3⋊D3—41no—
81C33×C4—22no—
82C33×C4C2×D412yes—
83C3×C3⋊Dic3—22no—
84C3×H(5,𝔽2)—24no—
85C3×H(5,𝔽2)C3×D314yescomplemented
86C3×H(5,𝔽2)C3⋊D311yescharacteristic, complemented
87C3×AGL(1,ℤ/8)—92no—
88C3×#127—91no—
89#949—38no—
90#940—34no—
91C3×#104—92no—
92C3×#100—91no—
93#837—38no—
94#833—34no—
95C23×C32—216no—
96C23×C32—28no—
97C23×C32—22no—
98C23×C32—28no—
99C23×C32#6718yes—
100C23×C32C3×D412yes—
101C22×C3⋊D3—21no—
102C22×C3×D3—64no—
103C32×Q8—28no—
104C32×Q8#6718yescomplemented
105C32×Q8C3×D412yescomplemented
106C32×D4—232no—
107C32×D4—216no—
108C32×D4—24no—
109C32×D4—216no—
110C32×D4—216no—
111C32×D4—28no—
112C32×D4—22no—
113C32×D4—28no—
114C32×D4—22no—
115C32×D4—28no—
116C32×D4#6718yescomplemented
117C32×D4C3×D412yescomplemented
118C2×C32×C4—216no—
119C2×C32×C4—28no—
120C2×C32×C4—22no—
121C2×C32×C4—216no—
122C2×C32×C4—28no—
123C2×C32×C4—22no—
124C2×C32×C4—24no—
125C2×C32×C4C22×D314yes—
126C2×C32×C4C23×C311yescharacteristic
127#638—44no—
128C2×C3⋊Dic3—21no—
129C4⋊(C3⋊D3)—22no—
130C3×#67—1216no—
131C2×C3×Dic3—64no—
132C3×D12—68no—
133C3⋊(C3⋊C8)—22no—
134C3×C3⋊C8—68no—
135#453—91noSylow
136C2×C33—21no—
137C2×C33—42no—
138C2×C33—42no—
139C2×C33#12011yescharacteristic
140C3×C3⋊D3—81no—
141C3×#41—216no—
142C3×#41—28no—
143C3×#41—22no—
144C3×#41—28no—
145C3×#41C2×C3×D318yes—
146C3×#41C2×C3⋊D312yes—
147C2×C3×D4—216no—
148C2×C3×D4—216no—
149C2×C3×D4—28no—
150C2×C3×D4—22no—
151C2×C3×D4—28no—
152C2×C3×D4—22no—
153C2×C3×D4—24no—
154C2×C3×D4—91no—
155C2×C3×D4—182no—
156C2×C3×D4C2×C3×D314yes—
157C2×C3×D4C2×C3⋊D311yescharacteristic
158C2×#67—68no—
159C2×D12—34no—
160C3×SD16—182no—
161C3×D8—182no—
162C3×M16—92no—
163C3×C42—91no—
164#215—68no—
165#213—68no—
166C4×Dic3—34no—
167#208—38no—
168C22×C32—216no—
169C22×C32—216no—
170C22×C32—28no—
171C22×C32—28no—
172C22×C32—22no—
173C22×C32—22no—
174C22×C32—24no—
175C22×C32—42no—
176C22×C32—48no—
177C22×C32—88no—
178C22×C32—816no—
179C22×C32—88no—
180C22×C32—82no—
181C22×C32C2×#6714yes—
182C22×C32C2×C3×D411yescharacteristic
183C2×C3⋊D3—42no—
184C2×C3⋊D3—41no—
185C2×C3×D3—128no—
186C2×C3×D3—124no—
187C32×C4—216no—
188C32×C4—28no—
189C32×C4—22no—
190C32×C4—28no—
191C32×C4C2×#6718yes—
192C32×C4C2×C3×D412yes—
193C3⋊Dic3—22no—
194C3×Dic3—68no—
195H(5,𝔽2)C3×C3⋊D311yescharacteristic, complemented
196AGL(1,ℤ/8)—92no—
197#127—91no—
198#104—92no—
199#100—91no—
200C33#45311yescharacteristic, complemented, Sylow
201C23×C3—216no—
202C23×C3—28no—
203C23×C3—22no—
204C23×C3—28no—
205C23×C3—181no—
206C23×C3C3×#6718yes—
207C23×C3#63812yes—
208C22×D3—64no—
209C3×Q8—28no—
210C3×Q8C3×#6718yescomplemented
211C3×Q8#63812yescomplemented
212C3×D4—232no—
213C3×D4—216no—
214C3×D4—24no—
215C3×D4—216no—
216C3×D4—216no—
217C3×D4—28no—
218C3×D4—22no—
219C3×D4—28no—
220C3×D4—22no—
221C3×D4—28no—
222C3×D4—182no—
223C3×D4—364no—
224C3×D4C3×#6718yescomplemented
225C3×D4#63812yescomplemented
226C2×C3×C4—216no—
227C2×C3×C4—28no—
228C2×C3×C4—22no—
229C2×C3×C4—216no—
230C2×C3×C4—28no—
231C2×C3×C4—22no—
232C2×C3×C4—24no—
233C2×C3×C4—181no—
234C2×C3×C4C22×C3×D314yes—
235C2×C3×C4C22×C3⋊D311yescharacteristic
236#67—1216no—
237C2×Dic3—64no—
238D12—68no—
239C3×C8—182no—
240C3⋊C8—68no—
241C2×C32—21no—
242C2×C32—24no—
243C2×C32—24no—
244C2×C32—44no—
245C2×C32—416no—
246C2×C32—416no—
247C2×C32—48no—
248C2×C32—48no—
249C2×C32—42no—
250C2×C32—42no—
251C2×C32#95314yes—
252C2×C32C3×#12011yescharacteristic
253C3⋊D3—81no—
254C3×D3—244no—
255#41—22no—
256#41C2×C3×C3⋊D312yes—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—91no—
260C2×D4—182no—
261C2×D4C2×C3×C3⋊D311yescharacteristic
262SD16—182no—
263D8—182no—
264M16—92no—
265C42—91no—
266C22×C3—216no—
267C22×C3—216no—
268C22×C3—28no—
269C22×C3—28no—
270C22×C3—22no—
271C22×C3—22no—
272C22×C3—24no—
273C22×C3—48no—
274C22×C3—42no—
275C22×C3—88no—
276C22×C3—816no—
277C22×C3—88no—
278C22×C3—82no—
279C22×C3—362no—
280C22×C3—361no—
281C22×C3C2×C3×#6714yes—
282C22×C3C2×#63811yescharacteristic
283C2×D3—128no—
284C2×D3—124no—
285C3×C4—216no—
286C3×C4—28no—
287C3×C4—22no—
288C3×C4—28no—
289C3×C4—182no—
290C3×C4C2×C3×#6718yes—
291C3×C4C2×#63812yes—
292Dic3—68no—
293C32—24no—
294C32#524114yescomplemented
295C32C3×#45311yescharacteristic, complemented
296C23—22no—
297C23—181no—
298C23C3×#63812yes—
299Q8C3×#63812yescomplemented
300D4—24no—
301D4—22no—
302D4—22no—
303D4—182no—
304D4—364no—
305D4C3×#63812yescomplemented
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—181no—
309C2×C4C22×C3×C3⋊D311yescharacteristic
310C8—182no—
311C2×C3—24no—
312C2×C3—21no—
313C2×C3—24no—
314C2×C3—44no—
315C2×C3—416no—
316C2×C3—416no—
317C2×C3—48no—
318C2×C3—48no—
319C2×C3—42no—
320C2×C3—42no—
321C2×C3—721no—
322C2×C3C3×#95314yes—
323C2×C3#6416211yescharacteristic
324D3—244no—
325C22—22no—
326C22—22no—
327C22—42no—
328C22—82no—
329C22—362no—
330C22—361no—
331C22C2×C3×#63811yescharacteristic
332C4—22no—
333C4—182no—
334C4C2×C3×#63812yes—
335C3—24no—
336C3C3×#524114yescomplemented
337C3#1059294011yescharacteristic, direct product factor
338C2—21no—
339C2—42no—
340C2—42no—
341C2—721no—
342C2C3×#6416211yescharacteristic
subgroupG/H
centerC2×C3#64162
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×H(5,𝔽2)C2
socleC2×C33#120
hypercenterC3×H(5,𝔽2)C3⋊D3
hypocenterC32C3×#453
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C3×H(5,𝔽2)

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C33×H(5,𝔽2)C2semidirect
2. C3×#64158C2semidirect
3. C3×#64149C2semidirect
4. C3×#63664C2semidirect
5. C3×#63660C2semidirect
6. #10592940C3direct (N × H)
14. C2×C3×C4⋊(C3⋊D3)C22semidirect
17. C3×C4×C3⋊Dic3C22semidirect
18. C3×#3748C22semidirect
20. C32×H(5,𝔽2)D3semidirect
21. C32×H(5,𝔽2)C2×C3semidirect
22. #64158C2×C3semidirect
23. #64149C2×C3semidirect
26. #63664C2×C3semidirect
27. #63660C2×C3semidirect
33. C33×Q8D4semidirect
37. C33×D4D4semidirect
63. C2×C4⋊(C3⋊D3)C22×C3semidirect
68. C4×C3⋊Dic3C22×C3semidirect
69. #3748C22×C3semidirect
85. C3×H(5,𝔽2)C3×D3semidirect
86. C3×H(5,𝔽2)C3⋊D3semidirect
104. C32×Q8#67semidirect
105. C32×Q8C3×D4semidirect
116. C32×D4#67semidirect
117. C32×D4C3×D4semidirect
195. H(5,𝔽2)C3×C3⋊D3semidirect
200. C33#453semidirect
210. C3×Q8C3×#67semidirect
211. C3×Q8#638semidirect
224. C3×D4C3×#67semidirect
225. C3×D4#638semidirect
294. C32#5241semidirect
295. C32C3×#453semidirect
299. Q8C3×#638semidirect
305. D4C3×#638semidirect
336. C3C3×#5241semidirect
337. C3#10592940direct (N × H)

Maximal subgroups

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

subgroupindex
C33×H(5,𝔽2)2
C3×#641582
C3×#641492
C3×#636642
C3×#636602
#105929403
C3×#52413