Finite Group Databasebeta

← C3×#10592431 #13162770

Subgroups — C3×#10592431 #13162770

Order 1728.

1388 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1388 conjugacy classes of subgroups — too many to draw. Only its 222 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
1C22×C3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
2C3×#64139C211yesmaximal, characteristic, complemented
3C2×C3×#3745C212yesmaximal, complemented
4C33×C22⋊C8C211yesmaximal, characteristic, complemented
5C3×#63648C212yesmaximal, complemented
6#10592431C311yesmaximal, characteristic, direct product factor
7C3×#4732—34nomaximal
8C23×C3×C3⋊D3—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×#638—21no—
11C2×C3×C4⋊(C3⋊D3)—21no—
12C2×C3×C4⋊(C3⋊D3)—24no—
13C2×C3×C4⋊(C3⋊D3)C2212yes—
14C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
15C2×C33×C8C2212yescomplemented
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C3×#3745—24no—
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic, complemented
20C22×C3×D12—34no—
21#64139C2×C311yescharacteristic, complemented
22C2×#3745C2×C312yescomplemented
23C3×#930—34no—
24C2×C3×D24—38no—
25C32×C22⋊C8—24no—
26C32×C22⋊C8D314yescomplemented
27C32×C22⋊C8C2×C311yescharacteristic, complemented
28#63648C2×C312yescomplemented
29C3×#821—38no—
30C23×C33D411yescharacteristic
31C22×C3×C3⋊D3—24no—
32C22×C3×C3⋊D3—22no—
33C22×C3×C3⋊D3—21no—
34C22×C3×C3⋊D3—42no—
35C22×C3×C3⋊D3—42no—
36C2×C33×C4—22no—
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C3×#638—42no—
40C2×C3×C3⋊Dic3—21no—
41C3×C4⋊(C3⋊D3)—24no—
42C3×C4⋊(C3⋊D3)—22no—
43C3×C4⋊(C3⋊D3)—22no—
44C3×C4⋊(C3⋊D3)D414yescomplemented
45C33×C8—22no—
46C3×#447—91no—
47#4732—34no—
48C23×C3⋊D3—21no—
49C23×C3×D3—64no—
50C22×C32×C4—24no—
51C22×C32×C4C2×D314yes—
52C22×C32×C4C22×C311yescharacteristic
53C2×#638—21no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—24no—
56C2×C4⋊(C3⋊D3)C22×C312yes—
57C2×C4⋊(C3⋊D3)C22×C311yescharacteristic, complemented
58C2×C3×#67—64no—
59C2×C3×D12—38no—
60C2×C3×D12—34no—
61C2×C3×D12—616no—
62C2×C3×D12—64no—
63C2×C32×C8—28no—
64C2×C32×C8C2×D318yescomplemented
65C2×C32×C8C22×C312yescomplemented
66#3752—21no—
67C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
68#3745—24no—
69C3×(C2×D3)⋊C4—64no—
70C3×C4⋊Dic3—34no—
71C3×D24—616no—
72C22×C33—21no—
73C22×C33—21no—
74C22×C33D812yescomplemented
75C22×C33C2×D411yescharacteristic
76C2×C3×C3⋊D3—24no—
77C2×C3×C3⋊D3—48no—
78C2×C3×C3⋊D3—42no—
79C2×C3×C3⋊D3—42no—
80C2×C3×C3⋊D3—41no—
81C2×C3×C3⋊D3—41no—
82C2×C3×C3⋊D3—41no—
83C33×C4—21no—
84C33×C4C2×D412yes—
85C3×C3⋊Dic3—41no—
86C22×C3×D4—91no—
87C22×D12—34no—
88C2×C3×D8—92no—
89C3×#121—91no—
90#930—34no—
91C2×D24—38no—
92C3×#102—92no—
93C3×C22⋊C8—24no—
94C3×C22⋊C8C3×D314yescomplemented
95C3×C22⋊C8C3⋊D311yescharacteristic, complemented
96#821—38no—
97C23×C32—24no—
98C23×C32D1214yes—
99C23×C32C3×D411yescharacteristic
100C22×C3⋊D3—24no—
101C22×C3⋊D3—22no—
102C22×C3⋊D3—21no—
103C22×C3⋊D3—42no—
104C22×C3⋊D3—42no—
105C22×C3×D3—616no—
106C22×C3×D3—68no—
107C22×C3×D3—64no—
108C22×C3×D3—128no—
109C22×C3×D3—128no—
110C2×C32×C4—22no—
111C2×C32×C4—24no—
112C2×C32×C4—24no—
113C2×C32×C4—28no—
114C2×C32×C4—48no—
115C2×C32×C4D1214yes—
116C2×C32×C4C3×D411yescharacteristic
117C2×C32×C4C22×D314yes—
118C2×C32×C4C23×C311yescharacteristic
119#638—42no—
120C2×C3⋊Dic3—21no—
121C4⋊(C3⋊D3)—24no—
122C4⋊(C3⋊D3)—22no—
123C4⋊(C3⋊D3)—22no—
124C4⋊(C3⋊D3)C3×D414yescomplemented
125C3×#67—128no—
126C2×C3×Dic3—64no—
127C3×D12—316no—
128C3×D12—616no—
129C3×D12—68no—
130C3×D12—68no—
131C32×C8—22no—
132C32×C8—28no—
133C32×C8—48no—
134#447—91noSylow
135C2×C33—22no—
136C2×C33#12011yescharacteristic
137C2×C33C2×D811yescharacteristic
138C2×C33AGL(1,ℤ/8)11yescharacteristic
139C3×C3⋊D3—44no—
140C3×C3⋊D3—81no—
141C24×C3—181no—
142C23×D3—64no—
143C2×C3×D4—92no—
144C2×C3×D4—91no—
145C2×C3×D4—184no—
146C2×C3×D4—181no—
147C2×C3×D4—181no—
148C22×C3×C4—24no—
149C22×C3×C4C2×C3×D314yes—
150C22×C3×C4C2×C3⋊D311yescharacteristic
151C2×#67—64no—
152C2×D12—38no—
153C2×D12—34no—
154C2×D12—616no—
155C2×D12—64no—
156C3×D8—184no—
157C2×C3×C8—28no—
158C2×C3×C8C2×C3×D318yescomplemented
159C2×C3×C8C2×C3⋊D312yescomplemented
160C3×C4⋊C4—91no—
161C3×C22⋊C4—181no—
162(C2×D3)⋊C4—64no—
163C4⋊Dic3—34no—
164D24—616no—
165C22×C32—21no—
166C22×C32—21no—
167C22×C32—28no—
168C22×C32—24no—
169C22×C32—24no—
170C22×C32—24no—
171C22×C32—44no—
172C22×C32—44no—
173C22×C32D2418yescomplemented
174C22×C32C3×D812yescomplemented
175C22×C32C2×D1214yes—
176C22×C32C2×C3×D411yescharacteristic
177C2×C3⋊D3—24no—
178C2×C3⋊D3—48no—
179C2×C3⋊D3—42no—
180C2×C3⋊D3—42no—
181C2×C3⋊D3—41no—
182C2×C3⋊D3—41no—
183C2×C3⋊D3—41no—
184C2×C3×D3—616no—
185C2×C3×D3—1232no—
186C2×C3×D3—128no—
187C2×C3×D3—128no—
188C2×C3×D3—124no—
189C2×C3×D3—124no—
190C2×C3×D3—124no—
191C32×C4—21no—
192C32×C4—28no—
193C32×C4—24no—
194C32×C4—44no—
195C32×C4D3×D418yes—
196C32×C4C2×C3×D412yes—
197C3⋊Dic3—41no—
198C3×Dic3—124no—
199C22×D4—91no—
200C2×D8—92no—
201#121—91no—
202#102—92no—
203C22⋊C8C3×C3⋊D311yescharacteristic, complemented
204C33#44711yescharacteristic, complemented, Sylow
205C23×C3—24no—
206C23×C3—184no—
207C23×C3—182no—
208C23×C3—181no—
209C23×C3—362no—
210C23×C3—362no—
211C23×C3C3×D1214yes—
212C23×C3C4⋊(C3⋊D3)11yescharacteristic
213C22×D3—616no—
214C22×D3—68no—
215C22×D3—64no—
216C22×D3—128no—
217C22×D3—128no—
218C3×D4—94no—
219C3×D4—184no—
220C3×D4—182no—
221C3×D4—182no—
222C3×D4—362no—
223C2×C3×C4—28no—
224C2×C3×C4—22no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—48no—
228C2×C3×C4—181no—
229C2×C3×C4C3×D1214yes—
230C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
231C2×C3×C4C22×C3×D314yes—
232C2×C3×C4C22×C3⋊D311yescharacteristic
233#67—128no—
234C2×Dic3—64no—
235D12—316no—
236D12—616no—
237D12—68no—
238D12—68no—
239C3×C8—28no—
240C3×C8—22no—
241C3×C8—48no—
242C2×C32—22no—
243C2×C32—24no—
244C2×C32—24no—
245C2×C32—24no—
246C2×C32—28no—
247C2×C32—48no—
248C2×C32#88214yes—
249C2×C32C2×D2414yes—
250C2×C32#90814yes—
251C2×C32C3×#12011yescharacteristic
252C2×C32C2×C3×D811yescharacteristic
253C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
254C3⋊D3—44no—
255C3⋊D3—81no—
256C3×D3—1216no—
257C3×D3—244no—
258C24—181no—
259C2×D4—92no—
260C2×D4—91no—
261C2×D4—184no—
262C2×D4—181no—
263C2×D4—181no—
264C22×C4C2×C3×C3⋊D311yescharacteristic
265D8—184no—
266C2×C8C2×C3×C3⋊D312yescomplemented
267C4⋊C4—91no—
268C22⋊C4—181no—
269C22×C3—24no—
270C22×C3—24no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—28no—
274C22×C3—24no—
275C22×C3—44no—
276C22×C3—44no—
277C22×C3—184no—
278C22×C3—368no—
279C22×C3—362no—
280C22×C3—362no—
281C22×C3—361no—
282C22×C3—361no—
283C22×C3—361no—
284C22×C3C3×D2418yescomplemented
285C22×C3#374512yescomplemented
286C22×C3C2×C3×D1214yes—
287C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
288C2×D3—616no—
289C2×D3—1232no—
290C2×D3—128no—
291C2×D3—128no—
292C2×D3—124no—
293C2×D3—124no—
294C2×D3—124no—
295C3×C4—24no—
296C3×C4—21no—
297C3×C4—28no—
298C3×C4—44no—
299C3×C4—361no—
300C3×C4C3×D3×D418yes—
301C3×C4D4×C3⋊D312yes—
302Dic3—124no—
303C32—24no—
304C32#473214yescomplemented
305C32C3×#44711yescharacteristic, complemented
306C23—184no—
307C23—182no—
308C23—181no—
309C23—362no—
310C23—362no—
311C23C3×C4⋊(C3⋊D3)11yescharacteristic
312D4—94no—
313D4—184no—
314D4—182no—
315D4—182no—
316D4—362no—
317C2×C4—22no—
318C2×C4—181no—
319C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
320C2×C4C22×C3×C3⋊D311yescharacteristic
321C8—22no—
322C2×C3—28no—
323C2×C3—22no—
324C2×C3—24no—
325C2×C3—24no—
326C2×C3—24no—
327C2×C3—48no—
328C2×C3—364no—
329C2×C3—721no—
330C2×C3C3×#88214yes—
331C2×C3C2×C3×D2414yes—
332C2×C3C3×#90814yes—
333C2×C3#6409111yescharacteristic
334C2×C3C2×#374511yescharacteristic
335C2×C3#6411711yescharacteristic
336D3—1216no—
337D3—244no—
338C22—21no—
339C22—21no—
340C22—184no—
341C22—368no—
342C22—362no—
343C22—362no—
344C22—361no—
345C22—361no—
346C22—361no—
347C22C3×#374512yescomplemented
348C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
349C4—21no—
350C4—361no—
351C4C3×D4×C3⋊D312yes—
352C3—24no—
353C3C3×#473214yescomplemented
354C3#1059243111yescharacteristic, direct product factor
355C2—22no—
356C2—364no—
357C2—721no—
358C2C3×#6409111yescharacteristic
359C2C2×C3×#374511yescharacteristic
360C2C3×#6411711yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#447
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4⋊(C3⋊D3)C2semidirect
2. C3×#64139C2semidirect
3. C2×C3×#3745C2semidirect
4. C33×C22⋊C8C2semidirect
5. C3×#63648C2semidirect
6. #10592431C3direct (N × H)
14. C2×C3×C4⋊(C3⋊D3)C22semidirect
15. C2×C33×C8C22semidirect
17. C3×C4⋊(C3⋊Dic3)C22semidirect
19. C22×C4⋊(C3⋊D3)C2×C3semidirect
21. #64139C2×C3semidirect
22. C2×#3745C2×C3semidirect
26. C32×C22⋊C8D3semidirect
27. C32×C22⋊C8C2×C3semidirect
28. #63648C2×C3semidirect
44. C3×C4⋊(C3⋊D3)D4semidirect
57. C2×C4⋊(C3⋊D3)C22×C3semidirect
64. C2×C32×C8C2×D3semidirect
65. C2×C32×C8C22×C3semidirect
67. C4⋊(C3⋊Dic3)C22×C3semidirect
74. C22×C33D8semidirect
94. C3×C22⋊C8C3×D3semidirect
95. C3×C22⋊C8C3⋊D3semidirect
124. C4⋊(C3⋊D3)C3×D4semidirect
158. C2×C3×C8C2×C3×D3semidirect
159. C2×C3×C8C2×C3⋊D3semidirect
173. C22×C32D24semidirect
174. C22×C32C3×D8semidirect
203. C22⋊C8C3×C3⋊D3semidirect
204. C33#447semidirect
266. C2×C8C2×C3×C3⋊D3semidirect
284. C22×C3C3×D24semidirect
285. C22×C3#3745semidirect
304. C32#4732semidirect
305. C32C3×#447semidirect
347. C22C3×#3745semidirect
353. C3C3×#4732semidirect
354. C3#10592431direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4⋊(C3⋊D3)2
C3×#641392
C2×C3×#37452
C33×C22⋊C82
C3×#636482
#105924313
C3×#47323