Finite Group Databasebeta

← C3×#10592430 #13162769

Subgroups — C3×#10592430 #13162769

Order 1728.

1328 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1328 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
2C3×#64133C211yesmaximal, characteristic, complemented
3C2×C3×#3744C212yesmaximal, complemented
4C33×C22⋊C8C211yesmaximal, characteristic, complemented
5C3×#63648C212yesmaximal, complemented
6#10592430C311yesmaximal, characteristic, direct product factor
7C3×#4731—34nomaximal
8C23×C3×C3⋊D3—21no—
9C22×C33×C4C2211yescharacteristic
10C2×C3×C4⋊(C3⋊D3)—21no—
11C2×C3×C4⋊(C3⋊D3)—24no—
12C2×C3×C4⋊(C3⋊D3)C2212yes—
13C2×C3×#634C2211yescharacteristic, complemented
14C2×C33×C8C2212yescomplemented
15C3×#3757—21no—
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
17C3×#3750—21no—
18C3×#3744—24no—
19C22×C4⋊(C3⋊D3)C2×C311yescharacteristic
20C22×C3×D12—34no—
21#64133C2×C311yescharacteristic, complemented
22C2×#3744C2×C312yescomplemented
23C3×#924—34no—
24C2×C3×#204—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—42no—
34C22×C3×C3⋊D3—42no—
35C2×C33×C4—22no—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4C2311yescharacteristic
38C2×C3×C3⋊Dic3—21no—
39C2×C3×C3⋊Dic3—21no—
40C3×C4⋊(C3⋊D3)—24no—
41C3×C4⋊(C3⋊D3)—22no—
42C3×C4⋊(C3⋊D3)D414yes—
43C3×#634—22no—
44C33×C8—22no—
45C3×#450—91no—
46#4731—34no—
47C23×C3⋊D3—21no—
48C23×C3×D3—64no—
49C22×C32×C4—24no—
50C22×C32×C4C2×D314yes—
51C22×C32×C4C22×C311yescharacteristic
52C2×C4⋊(C3⋊D3)—21no—
53C2×C4⋊(C3⋊D3)—24no—
54C2×C4⋊(C3⋊D3)C22×C312yes—
55C2×#634C22×C311yescharacteristic, complemented
56C2×C3×D12—38no—
57C2×C3×D12—616no—
58C2×C3×D12—64no—
59C2×C3×Dic6—34no—
60C2×C32×C8—28no—
61C2×C32×C8C2×D318yescomplemented
62C2×C32×C8C22×C312yescomplemented
63#3757—21no—
64C4⋊(C3⋊Dic3)C22×C311yescharacteristic, complemented
65#3750—21no—
66#3744—24no—
67C3×#217—64no—
68C3×C4⋊Dic3—34no—
69C3×Dic3⋊C4—64no—
70C3×#204—616no—
71C22×C33—21no—
72C22×C33—21no—
73C22×C33SD1612yescomplemented
74C22×C33C2×D411yescharacteristic
75C2×C3×C3⋊D3—24no—
76C2×C3×C3⋊D3—48no—
77C2×C3×C3⋊D3—42no—
78C2×C3×C3⋊D3—42no—
79C33×C4—21no—
80C33×C4C2×D412yes—
81C3×C3⋊Dic3—41no—
82C3×C3⋊Dic3—41no—
83C22×C3×D4—91no—
84C22×D12—34no—
85C2×C3×SD16—92no—
86C3×C22⋊Q8—91no—
87#924—34no—
88C2×#204—38no—
89C3×#102—92no—
90C3×C22⋊C8—24no—
91C3×C22⋊C8C3×D314yescomplemented
92C3×C22⋊C8C3⋊D311yescharacteristic, complemented
93#821—38no—
94C23×C32—24no—
95C23×C32D1214yes—
96C23×C32C3×D411yescharacteristic
97C22×C3⋊D3—24no—
98C22×C3⋊D3—22no—
99C22×C3⋊D3—42no—
100C22×C3⋊D3—42no—
101C22×C3×D3—616no—
102C22×C3×D3—68no—
103C22×C3×D3—128no—
104C22×C3×D3—128no—
105C2×C32×C4—22no—
106C2×C32×C4—24no—
107C2×C32×C4—24no—
108C2×C32×C4—28no—
109C2×C32×C4—48no—
110C2×C32×C4D1214yes—
111C2×C32×C4C3×D411yescharacteristic
112C2×C32×C4C22×D314yes—
113C2×C32×C4C23×C311yescharacteristic
114C2×C3⋊Dic3—21no—
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—24no—
117C4⋊(C3⋊D3)—22no—
118C4⋊(C3⋊D3)C3×D414yes—
119#634—22no—
120C2×C3×Dic3—64no—
121C2×C3×Dic3—64no—
122C3×D12—316no—
123C3×D12—616no—
124C3×D12—68no—
125C3×Dic6—68no—
126C32×C8—22no—
127C32×C8—28no—
128C32×C8—48no—
129#450—91noSylow
130C2×C33—22no—
131C2×C33#12011yescharacteristic
132C2×C33C2×SD1611yescharacteristic
133C2×C33AGL(1,ℤ/8)11yescharacteristic
134C3×C3⋊D3—44no—
135C24×C3—181no—
136C23×D3—64no—
137C2×C3×Q8—91no—
138C2×C3×D4—92no—
139C2×C3×D4—184no—
140C2×C3×D4—181no—
141C22×C3×C4—24no—
142C22×C3×C4C2×C3×D314yes—
143C22×C3×C4C2×C3⋊D311yescharacteristic
144C2×D12—38no—
145C2×D12—616no—
146C2×D12—64no—
147C2×Dic6—34no—
148C3×SD16—184no—
149C2×C3×C8—28no—
150C2×C3×C8C2×C3×D318yescomplemented
151C2×C3×C8C2×C3⋊D312yescomplemented
152C3×C4⋊C4—91no—
153C3×C4⋊C4—181no—
154C3×C22⋊C4—181no—
155#217—64no—
156C4⋊Dic3—34no—
157Dic3⋊C4—64no—
158#204—616no—
159C22×C32—21no—
160C22×C32—21no—
161C22×C32—28no—
162C22×C32—24no—
163C22×C32—24no—
164C22×C32—24no—
165C22×C32—44no—
166C22×C32—44no—
167C22×C32#20418yescomplemented
168C22×C32C3×SD1612yescomplemented
169C22×C32C2×D1214yes—
170C22×C32C2×C3×D411yescharacteristic
171C2×C3⋊D3—24no—
172C2×C3⋊D3—48no—
173C2×C3⋊D3—42no—
174C2×C3⋊D3—42no—
175C2×C3×D3—616no—
176C2×C3×D3—1232no—
177C2×C3×D3—128no—
178C2×C3×D3—128no—
179C32×C4—21no—
180C32×C4—28no—
181C32×C4—24no—
182C32×C4—44no—
183C32×C4D3×D418yes—
184C32×C4C2×C3×D412yes—
185C3⋊Dic3—41no—
186C3⋊Dic3—41no—
187C3×Dic3—124no—
188C3×Dic3—124no—
189C22×D4—91no—
190C2×SD16—92no—
191C22⋊Q8—91no—
192#102—92no—
193C22⋊C8C3×C3⋊D311yescharacteristic, complemented
194C33#45011yescharacteristic, complemented, Sylow
195C23×C3—24no—
196C23×C3—184no—
197C23×C3—182no—
198C23×C3—362no—
199C23×C3—362no—
200C23×C3C3×D1214yes—
201C23×C3C4⋊(C3⋊D3)11yescharacteristic
202C22×D3—616no—
203C22×D3—68no—
204C22×D3—128no—
205C22×D3—128no—
206C3×Q8—182no—
207C3×D4—94no—
208C3×D4—184no—
209C3×D4—182no—
210C2×C3×C4—28no—
211C2×C3×C4—22no—
212C2×C3×C4—24no—
213C2×C3×C4—24no—
214C2×C3×C4—48no—
215C2×C3×C4—181no—
216C2×C3×C4—181no—
217C2×C3×C4C3×D1214yes—
218C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
219C2×C3×C4C22×C3×D314yes—
220C2×C3×C4C22×C3⋊D311yescharacteristic
221C2×Dic3—64no—
222C2×Dic3—64no—
223D12—316no—
224D12—616no—
225D12—68no—
226Dic6—68no—
227C3×C8—28no—
228C3×C8—22no—
229C3×C8—48no—
230C2×C32—22no—
231C2×C32—24no—
232C2×C32—24no—
233C2×C32—24no—
234C2×C32—28no—
235C2×C32—48no—
236C2×C32#88214yes—
237C2×C32C2×#20414yes—
238C2×C32#90814yes—
239C2×C32C3×#12011yescharacteristic
240C2×C32C2×C3×SD1611yescharacteristic
241C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
242C3⋊D3—44no—
243C3×D3—1216no—
244C24—181no—
245C2×Q8—91no—
246C2×D4—92no—
247C2×D4—184no—
248C2×D4—181no—
249C22×C4C2×C3×C3⋊D311yescharacteristic
250SD16—184no—
251C2×C8C2×C3×C3⋊D312yescomplemented
252C4⋊C4—91no—
253C4⋊C4—181no—
254C22⋊C4—181no—
255C22×C3—24no—
256C22×C3—24no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—28no—
260C22×C3—24no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—184no—
264C22×C3—368no—
265C22×C3—362no—
266C22×C3—362no—
267C22×C3C3×#20418yescomplemented
268C22×C3#374412yescomplemented
269C22×C3C2×C3×D1214yes—
270C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
271C2×D3—616no—
272C2×D3—1232no—
273C2×D3—128no—
274C2×D3—128no—
275C3×C4—24no—
276C3×C4—21no—
277C3×C4—28no—
278C3×C4—44no—
279C3×C4—361no—
280C3×C4—361no—
281C3×C4C3×D3×D418yes—
282C3×C4D4×C3⋊D312yes—
283Dic3—124no—
284Dic3—124no—
285C32—24no—
286C32#473114yescomplemented
287C32C3×#45011yescharacteristic, complemented
288C23—184no—
289C23—182no—
290C23—362no—
291C23—362no—
292C23C3×C4⋊(C3⋊D3)11yescharacteristic
293Q8—182no—
294D4—94no—
295D4—184no—
296D4—182no—
297C2×C4—22no—
298C2×C4—181no—
299C2×C4—181no—
300C2×C4C3×C4⋊(C3⋊D3)11yescharacteristic
301C2×C4C22×C3×C3⋊D311yescharacteristic
302C8—22no—
303C2×C3—28no—
304C2×C3—22no—
305C2×C3—24no—
306C2×C3—24no—
307C2×C3—24no—
308C2×C3—48no—
309C2×C3—364no—
310C2×C3C3×#88214yes—
311C2×C3C2×C3×#20414yes—
312C2×C3C3×#90814yes—
313C2×C3#6409111yescharacteristic
314C2×C3C2×#374411yescharacteristic
315C2×C3#6411711yescharacteristic
316D3—1216no—
317C22—21no—
318C22—21no—
319C22—184no—
320C22—368no—
321C22—362no—
322C22—362no—
323C22C3×#374412yescomplemented
324C22C2×C3×C4⋊(C3⋊D3)11yescharacteristic
325C4—21no—
326C4—361no—
327C4—361no—
328C4C3×D4×C3⋊D312yes—
329C3—24no—
330C3C3×#473114yescomplemented
331C3#1059243011yescharacteristic, direct product factor
332C2—22no—
333C2—364no—
334C2C3×#6409111yescharacteristic
335C2C2×C3×#374411yescharacteristic
336C2C3×#6411711yescharacteristic
subgroupG/H
centerC22×C3C2×C4⋊(C3⋊D3)
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C22×C3×C3⋊D3
FittingC33×C22⋊C8C2
socleC22×C33C2×D4
prime radicalC22×C3×C4⋊(C3⋊D3)C2
hypercenterC3×C22⋊C8C3⋊D3
hypocenterC32C3×#450
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
2. C3×#64133C2semidirect
3. C2×C3×#3744C2semidirect
4. C33×C22⋊C8C2semidirect
5. C3×#63648C2semidirect
6. #10592430C3direct (N × H)
13. C2×C3×#634C22semidirect
14. C2×C33×C8C22semidirect
16. C3×C4⋊(C3⋊Dic3)C22semidirect
21. #64133C2×C3semidirect
22. C2×#3744C2×C3semidirect
26. C32×C22⋊C8D3semidirect
27. C32×C22⋊C8C2×C3semidirect
28. #63648C2×C3semidirect
55. C2×#634C22×C3semidirect
61. C2×C32×C8C2×D3semidirect
62. C2×C32×C8C22×C3semidirect
64. C4⋊(C3⋊Dic3)C22×C3semidirect
73. C22×C33SD16semidirect
91. C3×C22⋊C8C3×D3semidirect
92. C3×C22⋊C8C3⋊D3semidirect
150. C2×C3×C8C2×C3×D3semidirect
151. C2×C3×C8C2×C3⋊D3semidirect
167. C22×C32#204semidirect
168. C22×C32C3×SD16semidirect
193. C22⋊C8C3×C3⋊D3semidirect
194. C33#450semidirect
251. C2×C8C2×C3×C3⋊D3semidirect
267. C22×C3C3×#204semidirect
268. C22×C3#3744semidirect
286. C32#4731semidirect
287. C32C3×#450semidirect
323. C22C3×#3744semidirect
330. C3C3×#4731semidirect
331. C3#10592430direct (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×#641332
C2×C3×#37442
C33×C22⋊C82
C3×#636482
#105924303
C3×#47313