Finite Group Databasebeta

← C3×#10596201 #13179481

Subgroups — C3×#10596201 #13179481

Order 1728.

1640 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C33×C4⋊Q8C211yesmaximal, characteristic, complemented
2C3×#64155C214yesmaximal, complemented
3C3×#64106C212yesmaximal, complemented
4C3×#64102C212yesmaximal
5C3×C4⋊C4×C3⋊D3C212yesmaximal, complemented
6C3×#64099C212yesmaximal, complemented
7C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
8C3×C4×#634C211yesmaximal, characteristic, complemented
9#10596201C311yesmaximal, characteristic, direct product factor
10C3×#5730—34nomaximal
11C2×C33×Q8C2212yes—
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×C4×C3⋊D3C2214yes—
15C2×C3×#634C2211yescharacteristic
16C33×C4⋊C4C2212yes—
17C33×C4⋊C4C2212yes—
18C33×C42C2211yescharacteristic
19C3×#3752C2212yes—
20C3×#3752C2214yes—
21C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
22C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C3×C4⋊(C3⋊Dic3)C2214yes—
24C3×#3750C2212yes—
25C3×#3750C2214yes—
26C3×C4×C3⋊Dic3C2212yes—
27C32×C4⋊Q8—24no—
28C32×C4⋊Q8D314yescomplemented
29C32×C4⋊Q8C2×C311yescharacteristic, complemented
30#64155C2×C314yescomplemented
31#64106C2×C312yescomplemented
32#64102C2×C312yes—
33C4⋊C4×C3⋊D3C2×C312yescomplemented
34#64099C2×C312yescomplemented
35C4×C4⋊(C3⋊D3)C2×C311yescharacteristic
36C4×#634C2×C311yescharacteristic, complemented
37C3×#946—316no—
38C3×#897—38no—
39C3×#893—38no—
40C3×D3×C4⋊C4—38no—
41C3×#890—38no—
42C3×C4×D12—34no—
43C3×C4×Dic6—34no—
44C22×C3×C3⋊D3C2312yes—
45C33×Q8—42no—
46C2×C33×C4C2311yescharacteristic
47C2×C33×C4C2314yes—
48C2×C33×C4C2311yescharacteristic
49C2×C33×C4C2311yescharacteristic
50C2×C3×C3⋊Dic3C2312yes—
51C2×C3×C3⋊Dic3C2312yes—
52C2×C3×C3⋊Dic3C2312yes—
53C3×C4⋊(C3⋊D3)Q814yescomplemented
54C3×C4×C3⋊D3—24no—
55C3×C4×C3⋊D3—44no—
56C3×#634—22no—
57C3×#554—91no—
58#5730—34no—
59C2×C32×Q8—28no—
60C2×C32×Q8C2×D318yes—
61C2×C32×Q8C22×C312yes—
62C2×C4⋊(C3⋊D3)C22×C311yescharacteristic
63C2×C4×C3⋊D3C22×C312yes—
64C2×C4×C3⋊D3C22×C314yes—
65C2×#634C22×C311yescharacteristic
66C2×C3×D12—34no—
67C2×C3×C4×D3—38no—
68C2×C3×C4×D3—316no—
69C2×C3×Dic6—34no—
70C32×C4⋊C4—28no—
71C32×C4⋊C4—28no—
72C32×C4⋊C4C2×D318yes—
73C32×C4⋊C4C2×D318yes—
74C32×C4⋊C4C22×C312yes—
75C32×C4⋊C4C22×C312yes—
76C32×C42—24no—
77C32×C42C2×D314yes—
78C32×C42C22×C311yescharacteristic
79#3752C22×C312yes—
80#3752C22×C314yes—
81C4⋊(C3⋊Dic3)C22×C311yescharacteristic
82C4⋊(C3⋊Dic3)C22×C311yescharacteristic
83C4⋊(C3⋊Dic3)C22×C314yes—
84#3750C22×C312yes—
85#3750C22×C314yes—
86C4×C3⋊Dic3C22×C312yes—
87C3×(C2×D3)⋊C4—316no—
88C3×(C2×D3)⋊C4—38no—
89C3×C4⋊Dic3—34no—
90C3×C4⋊Dic3—316no—
91C3×C4⋊Dic3—34no—
92C3×Dic3⋊C4—316no—
93C3×Dic3⋊C4—38no—
94C3×C4×Dic3—38no—
95C22×C33C2411yescharacteristic
96C2×C3×C3⋊D3—22no—
97C2×C3×C3⋊D3—22no—
98C2×C3×C3⋊D3C2×Q814yes—
99C33×C4—21no—
100C33×C4—24no—
101C33×C4C2×Q812yes—
102C33×C4#4112yes—
103C3×C3⋊Dic3—22no—
104C3×C3⋊Dic3—22no—
105C3×C3⋊Dic3—22no—
106C3×C4⋊Q8—24no—
107C3×C4⋊Q8C3×D314yescomplemented
108C3×C4⋊Q8C3⋊D311yescharacteristic, complemented
109C3×#125—92no—
110C3×C22⋊Q8—94no—
111C3×C22⋊Q8—92no—
112C3×C4×Q8—91no—
113C3×C4×D4—92no—
114C3×C4×D4—91no—
115C2×C3×C4⋊C4—92no—
116#946—316no—
117#897—38no—
118#893—38no—
119D3×C4⋊C4—38no—
120#890—38no—
121C4×D12—34no—
122C4×Dic6—34no—
123C22×C3⋊D3C23×C312yes—
124C22×C3×D3—38no—
125C32×Q8—48no—
126C32×Q8—42no—
127C32×Q8—416no—
128C2×C32×C4—24no—
129C2×C32×C4—24no—
130C2×C32×C4—24no—
131C2×C32×C4—216no—
132C2×C32×C4C22×D314yes—
133C2×C32×C4C22×D3116yes—
134C2×C32×C4C22×D314yes—
135C2×C32×C4C22×D314yes—
136C2×C32×C4C23×C311yescharacteristic
137C2×C32×C4C23×C314yes—
138C2×C32×C4C23×C311yescharacteristic
139C2×C32×C4C23×C311yescharacteristic
140C2×C3⋊Dic3C23×C312yes—
141C2×C3⋊Dic3C23×C312yes—
142C2×C3⋊Dic3C23×C312yes—
143C4⋊(C3⋊D3)C3×Q814yescomplemented
144C4×C3⋊D3—24no—
145C4×C3⋊D3—44no—
146#634—22no—
147C2×C3×Dic3—38no—
148C2×C3×Dic3—38no—
149C2×C3×Dic3—38no—
150C3×D12—316no—
151C3×C4×D3—616no—
152C3×C4×D3—1216no—
153C3×Dic6—68no—
154#554—91noSylow
155C2×C33C22×Q811yescharacteristic
156C2×C33C2×#4111yescharacteristic
157C2×C33H(5,𝔽2)11yescharacteristic
158C3×C3⋊D3—24no—
159C2×C3×Q8—28no—
160C2×C3×Q8—91no—
161C2×C3×Q8C2×C3×D318yes—
162C2×C3×Q8C2×C3⋊D312yes—
163C2×C3×D4—91no—
164C22×C3×C4—92no—
165C22×C3×C4—94no—
166C2×D12—34no—
167C2×C4×D3—38no—
168C2×C4×D3—316no—
169C2×Dic6—34no—
170C3×C4⋊C4—28no—
171C3×C4⋊C4—28no—
172C3×C4⋊C4—94no—
173C3×C4⋊C4—92no—
174C3×C4⋊C4—91no—
175C3×C4⋊C4—91no—
176C3×C4⋊C4—94no—
177C3×C4⋊C4C2×C3×D318yes—
178C3×C4⋊C4C2×C3×D318yes—
179C3×C4⋊C4C2×C3⋊D312yes—
180C3×C4⋊C4C2×C3⋊D312yes—
181C3×C22⋊C4—94no—
182C3×C22⋊C4—92no—
183C3×C42—24no—
184C3×C42—92no—
185C3×C42C2×C3×D314yes—
186C3×C42C2×C3⋊D311yescharacteristic
187(C2×D3)⋊C4—38no—
188(C2×D3)⋊C4—316no—
189C4⋊Dic3—34no—
190C4⋊Dic3—34no—
191C4⋊Dic3—316no—
192Dic3⋊C4—38no—
193Dic3⋊C4—316no—
194C4×Dic3—38no—
195C22×C32—24no—
196C22×C32C23×D314yes—
197C22×C32C24×C311yescharacteristic
198C2×C3⋊D3—22no—
199C2×C3⋊D3—22no—
200C2×C3⋊D3C2×C3×Q814yes—
201C2×C3×D3—316no—
202C2×C3×D3—68no—
203C2×C3×D3—68no—
204C32×C4—28no—
205C32×C4—24no—
206C32×C4—24no—
207C32×C4—21no—
208C32×C4—28no—
209C32×C4—28no—
210C32×C4—216no—
211C32×C4—416no—
212C32×C4D3×Q818yes—
213C32×C4#23918yes—
214C32×C4C2×C3×Q812yes—
215C32×C4C3×#4112yes—
216C3⋊Dic3—22no—
217C3⋊Dic3—22no—
218C3⋊Dic3—22no—
219C3×Dic3—68no—
220C3×Dic3—68no—
221C3×Dic3—68no—
222C4⋊Q8C3×C3⋊D311yescharacteristic, complemented
223#125—92no—
224C22⋊Q8—94no—
225C22⋊Q8—92no—
226C4×Q8—91no—
227C4×D4—92no—
228C4×D4—91no—
229C2×C4⋊C4—92no—
230C33#55411yescharacteristic, complemented, Sylow
231C23×C3—92no—
232C22×D3—38no—
233C3×Q8—48no—
234C3×Q8—42no—
235C3×Q8—416no—
236C3×Q8—182no—
237C3×D4—94no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—216no—
242C2×C3×C4—92no—
243C2×C3×C4—92no—
244C2×C3×C4—92no—
245C2×C3×C4—184no—
246C2×C3×C4—364no—
247C2×C3×C4C22×C3×D314yes—
248C2×C3×C4C22×C3×D3116yes—
249C2×C3×C4C22×C3×D314yes—
250C2×C3×C4C22×C3×D314yes—
251C2×C3×C4C22×C3⋊D311yescharacteristic
252C2×C3×C4C22×C3⋊D314yes—
253C2×C3×C4C22×C3⋊D311yescharacteristic
254C2×C3×C4C22×C3⋊D311yescharacteristic
255C2×Dic3—38no—
256C2×Dic3—38no—
257C2×Dic3—38no—
258D12—316no—
259C4×D3—616no—
260C4×D3—1216no—
261Dic6—68no—
262C2×C32—24no—
263C2×C32—24no—
264C2×C32—24no—
265C2×C32#100414yes—
266C2×C32C2×D3×Q814yes—
267C2×C32C2×#23914yes—
268C2×C32C22×C3×Q811yescharacteristic
269C2×C32C2×C3×#4111yescharacteristic
270C2×C32C3×H(5,𝔽2)11yescharacteristic
271C3⋊D3—24no—
272C3×D3—616no—
273C2×Q8—91no—
274C2×Q8C2×C3×C3⋊D312yes—
275C2×D4—91no—
276C22×C4—92no—
277C22×C4—94no—
278C4⋊C4—91no—
279C4⋊C4—94no—
280C4⋊C4—91no—
281C4⋊C4—92no—
282C4⋊C4—94no—
283C4⋊C4C2×C3×C3⋊D312yes—
284C4⋊C4C2×C3×C3⋊D312yes—
285C22⋊C4—94no—
286C22⋊C4—92no—
287C42—92no—
288C42C2×C3×C3⋊D311yescharacteristic
289C22×C3—24no—
290C22×C3—94no—
291C22×C3—182no—
292C22×C3—182no—
293C22×C3C23×C3×D314yes—
294C22×C3C23×C3⋊D311yescharacteristic
295C2×D3—316no—
296C2×D3—68no—
297C2×D3—68no—
298C3×C4—28no—
299C3×C4—216no—
300C3×C4—24no—
301C3×C4—24no—
302C3×C4—21no—
303C3×C4—28no—
304C3×C4—28no—
305C3×C4—416no—
306C3×C4—182no—
307C3×C4—182no—
308C3×C4—182no—
309C3×C4C3×D3×Q818yes—
310C3×C4C3×#23918yes—
311C3×C4Q8×C3⋊D312yes—
312C3×C4#383212yes—
313Dic3—68no—
314Dic3—68no—
315Dic3—68no—
316C32—24no—
317C32#573014yescomplemented
318C32C3×#55411yescharacteristic, complemented
319C23—92no—
320Q8—42no—
321Q8—182no—
322D4—94no—
323C2×C4—92no—
324C2×C4—92no—
325C2×C4—92no—
326C2×C4—184no—
327C2×C4—364no—
328C2×C4C22×C3×C3⋊D311yescharacteristic
329C2×C4C22×C3×C3⋊D314yes—
330C2×C4C22×C3×C3⋊D311yescharacteristic
331C2×C4C22×C3×C3⋊D311yescharacteristic
332C2×C3—24no—
333C2×C3—24no—
334C2×C3—24no—
335C2×C3—184no—
336C2×C3C3×#100414yes—
337C2×C3C2×C3×D3×Q814yes—
338C2×C3C2×C3×#23914yes—
339C2×C3#6436111yescharacteristic
340C2×C3C2×Q8×C3⋊D311yescharacteristic
341C2×C3C2×#383211yescharacteristic
342D3—616no—
343C22—94no—
344C22—182no—
345C22—182no—
346C22C23×C3×C3⋊D311yescharacteristic
347C4—24no—
348C4—21no—
349C4—182no—
350C4—182no—
351C4—182no—
352C4C3×Q8×C3⋊D312yes—
353C4C3×#383212yes—
354C3—24no—
355C3C3×#573014yescomplemented
356C3#1059620111yescharacteristic, direct product factor
357C2—184no—
358C2C3×#6436111yescharacteristic
359C2C2×C3×Q8×C3⋊D311yescharacteristic
360C2C2×C3×#383211yescharacteristic
subgroupG/H
centerC22×C3C23×C3⋊D3
commutator, cosocleC22×C32C24×C3
FrattiniC22C23×C3×C3⋊D3
FittingC33×C4⋊Q8C2
socleC22×C33C24
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC3×C4⋊Q8C3⋊D3
hypocenterC32C3×#554
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C33×C4⋊Q8C2semidirect
2. C3×#64155C2semidirect
3. C3×#64106C2semidirect
5. C3×C4⋊C4×C3⋊D3C2semidirect
6. C3×#64099C2semidirect
8. C3×C4×#634C2semidirect
9. #10596201C3direct (N × H)
28. C32×C4⋊Q8D3semidirect
29. C32×C4⋊Q8C2×C3semidirect
30. #64155C2×C3semidirect
31. #64106C2×C3semidirect
33. C4⋊C4×C3⋊D3C2×C3semidirect
34. #64099C2×C3semidirect
36. C4×#634C2×C3semidirect
53. C3×C4⋊(C3⋊D3)Q8semidirect
107. C3×C4⋊Q8C3×D3semidirect
108. C3×C4⋊Q8C3⋊D3semidirect
143. C4⋊(C3⋊D3)C3×Q8semidirect
222. C4⋊Q8C3×C3⋊D3semidirect
230. C33#554semidirect
317. C32#5730semidirect
318. C32C3×#554semidirect
355. C3C3×#5730semidirect
356. C3#10596201direct (N × H)

Maximal subgroups

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

subgroupindex
C33×C4⋊Q82
C3×#641552
C3×#641062
C3×#641022
C3×C4⋊C4×C3⋊D32
C3×#640992
C3×C4×C4⋊(C3⋊D3)2
C3×C4×#6342
#105962013
C3×#57303