Finite Group Databasebeta

← C22×C3×D3×C3⋊C8 #13174407

Subgroups — C22×C3×D3×C3⋊C8 #13174407

Order 1728.

1794 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C32×C4×D3C211yesmaximal, characteristic
2C22×C3×C3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×C3×D3×C3⋊C8C2112yesmaximal, direct product factor
5C23×C3×C3⋊C8—31nomaximal
6C22×C3×D3×C8—31nomaximal
7C22×D3×C3⋊C8C311yesmaximal, characteristic, direct product factor
8C23×C32×D3C411yescharacteristic
9C22×C33×C4C2211yescharacteristic
10C22×C32×Dic3C411yescharacteristic
11C2×C32×C4×D3C416yes—
12C2×C32×C4×D3C2216yes—
13C2×C3×C3⋊(C3⋊C8)C2216yescomplemented
14C2×C32×C3⋊C8C2216yescomplemented
15C3×D3×C3⋊C8C22116yesdirect product factor
16C23×C32×C4—31no—
17C22×C3×C4×D3—21no—
18C22×C3×C4×D3D311yescharacteristic
19C22×C3×C4×D3C2×C311yescharacteristic
20C22×C32×C8—31no—
21C22×C3⋊(C3⋊C8)C2×C311yescharacteristic, complemented
22C22×C3×C3⋊C8—21no—
23C22×C3×C3⋊C8—31no—
24C22×C3×C3⋊C8—31no—
25C22×C3×C3⋊C8—312no—
26C22×C3×C3⋊C8—61no—
27C22×C3×C3⋊C8D311yescharacteristic, direct product factor
28C22×C3×C3⋊C8C2×C311yescharacteristic, complemented
29C2×C3×D3×C8—312no—
30C2×D3×C3⋊C8C2×C3112yesdirect product factor
31C23×C33C2×C411yescharacteristic
32C22×C32×D3C818yescomplemented
33C22×C32×D3C2×C416yes—
34C2×C33×C4C2×C413yes—
35C2×C33×C4C2313yes—
36C2×C32×Dic3C2×C416yes—
37C32×C4×D3C2×C4112yes—
38C32×C4×D3C2314yes—
39C3×C3⋊(C3⋊C8)C2314yescomplemented
40C32×C3⋊C8C2314yescomplemented
41C23×C3×C8—91no—
42C23×C3⋊C8—31no—
43C22×D3×C8—31no—
44C24×C32—31no—
45C23×C3×D3—21no—
46C23×C3×D3Dic311yescharacteristic
47C23×C3×D3C3×C411yescharacteristic
48C22×C32×C4—21no—
49C22×C32×C4—21no—
50C22×C32×C4—21no—
51C22×C32×C4—31no—
52C22×C32×C4—36no—
53C22×C32×C4—36no—
54C22×C32×C4—41no—
55C22×C32×C4C2×D311yescharacteristic
56C22×C32×C4C2×D311yescharacteristic
57C22×C32×C4C22×C311yescharacteristic
58C22×C3×Dic3—21no—
59C22×C3×Dic3Dic311yescharacteristic
60C22×C3×Dic3C3×C411yescharacteristic
61C2×C3×C4×D3—26no—
62C2×C3×C4×D3—26no—
63C2×C3×C4×D3Dic316yes—
64C2×C3×C4×D3C3×C416yes—
65C2×C3×C4×D3C2×D316yes—
66C2×C3×C4×D3C22×C316yes—
67C2×C32×C8—36no—
68C2×C3⋊(C3⋊C8)C22×C316yescomplemented
69C2×C3×C3⋊C8—26no—
70C2×C3×C3⋊C8—36no—
71C2×C3×C3⋊C8—36no—
72C2×C3×C3⋊C8—316no—
73C2×C3×C3⋊C8—66no—
74C2×C3×C3⋊C8C2×D316yesdirect product factor
75C2×C3×C3⋊C8C22×C316yescomplemented
76C3×D3×C8—316no—
77D3×C3⋊C8C22×C3116yesdirect product factor
78C22×C33C2×C814yescomplemented
79C22×C33C22×C413yes—
80C2×C32×D3C2×C8124yescomplemented
81C2×C32×D3C22×C414yes—
82C33×C4C22×C413yes—
83C33×C4C2411yescharacteristic
84C32×Dic3C22×C414yes—
85C23×C3×C4—31no—
86C23×C3×C4—31no—
87C23×C3×C4—61no—
88C22×C4×D3C3×D311yescharacteristic
89C22×C3×C8—31no—
90C22×C3×C8—31no—
91C22×C3×C8—61no—
92C22×C3×C8—912no—
93C22×C3×C8—91no—
94C22×C3⋊C8—31no—
95C22×C3⋊C8—31no—
96C22×C3⋊C8—312no—
97C22×C3⋊C8—61no—
98C22×C3⋊C8C3×D311yescharacteristic, direct product factor
99C2×D3×C8—312no—
100C23×C32—21no—
101C23×C32—21no—
102C23×C32—21no—
103C23×C32—38no—
104C23×C32—36no—
105C23×C32—41no—
106C23×C32C4×D311yescharacteristic
107C23×C32C2×Dic311yescharacteristic
108C23×C32C2×C3×C411yescharacteristic
109C22×C3×D3—28no—
110C22×C3×D3—26no—
111C22×C3×D3C3⋊C818yesdirect product factor
112C22×C3×D3C3×C818yescomplemented
113C22×C3×D3C2×Dic316yes—
114C22×C3×D3C2×C3×C416yes—
115C2×C32×C4—23no—
116C2×C32×C4—23no—
117C2×C32×C4—23no—
118C2×C32×C4—23no—
119C2×C32×C4—23no—
120C2×C32×C4—23no—
121C2×C32×C4—36no—
122C2×C32×C4—312no—
123C2×C32×C4—34no—
124C2×C32×C4—43no—
125C2×C32×C4—43no—
126C2×C32×C4C4×D313yes—
127C2×C32×C4C2×Dic313yes—
128C2×C32×C4C2×C3×C413yes—
129C2×C32×C4C22×D313yes—
130C2×C32×C4C22×D313yes—
131C2×C32×C4C23×C313yes—
132C2×C3×Dic3—26no—
133C2×C3×Dic3C2×Dic316yes—
134C2×C3×Dic3C2×C3×C416yes—
135C3×C4×D3—212no—
136C3×C4×D3—24no—
137C3×C4×D3C2×Dic3112yes—
138C3×C4×D3C2×C3×C4112yes—
139C3×C4×D3C22×D314yes—
140C3×C4×D3C23×C314yes—
141C32×C8—34no—
142C3⋊(C3⋊C8)C23×C314yescomplemented
143C3×C3⋊C8—24no—
144C3×C3⋊C8—34no—
145C3×C3⋊C8—34no—
146C3×C3⋊C8—64no—
147C3×C3⋊C8C22×D314yesdirect product factor
148C3×C3⋊C8C23×C314yescomplemented
149C23×C8—91noSylow
150C2×C33C22×C816yescomplemented
151C2×C33C23×C411yescharacteristic
152C32×D3C22×C818yescomplemented
153C24×C3—31no—
154C24×C3—31no—
155C24×C3—61no—
156C23×D3C3×Dic311yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—31no—
161C22×C3×C4—31no—
162C22×C3×C4—36no—
163C22×C3×C4—36no—
164C22×C3×C4—36no—
165C22×C3×C4—36no—
166C22×C3×C4—41no—
167C22×C3×C4—66no—
168C22×C3×C4—66no—
169C22×C3×C4—61no—
170C22×C3×C4D3211yescharacteristic
171C22×C3×C4C2×C3×D311yescharacteristic
172C22×C3×C4C2×C3×D311yescharacteristic
173C22×Dic3C3×Dic311yescharacteristic
174C2×C4×D3C3×Dic316yes—
175C2×C4×D3C2×C3×D316yes—
176C2×C3×C8—36no—
177C2×C3×C8—36no—
178C2×C3×C8—66no—
179C2×C3×C8—916no—
180C2×C3×C8—96no—
181C2×C3⋊C8—36no—
182C2×C3⋊C8—36no—
183C2×C3⋊C8—316no—
184C2×C3⋊C8—66no—
185C2×C3⋊C8C2×C3×D316yesdirect product factor
186D3×C8—316no—
187C22×C32—24no—
188C22×C32—24no—
189C22×C32—24no—
190C22×C32—23no—
191C22×C32—23no—
192C22×C32—23no—
193C22×C32—324no—
194C22×C32—34no—
195C22×C32—44no—
196C22×C32—43no—
197C22×C32D3×C814yescomplemented
198C22×C32C2×C3⋊C814yescomplemented
199C22×C32C2×C3×C814yescomplemented
200C22×C32C2×C4×D313yes—
201C22×C32C22×Dic313yes—
202C22×C32C22×C3×C413yes—
203C2×C3×D3—224no—
204C2×C3×D3—24no—
205C2×C3×D3C2×C3⋊C8124yesdirect product factor
206C2×C3×D3C2×C3×C8124yescomplemented
207C2×C3×D3C22×Dic314yes—
208C2×C3×D3C22×C3×C414yes—
209C32×C4—21no—
210C32×C4—21no—
211C32×C4—21no—
212C32×C4—23no—
213C32×C4—23no—
214C32×C4—23no—
215C32×C4—34no—
216C32×C4—43no—
217C32×C4—41no—
218C32×C4C2×C4×D313yes—
219C32×C4C22×Dic313yes—
220C32×C4C22×C3×C413yes—
221C32×C4C23×D311yescharacteristic
222C32×C4C23×D311yescharacteristic
223C32×C4C24×C311yescharacteristic
224C3×Dic3—24no—
225C3×Dic3C22×Dic314yes—
226C3×Dic3C22×C3×C414yes—
227C23×C4—31no—
228C22×C8—31no—
229C22×C8—912no—
230C22×C8—91no—
231C33C23×C811yescharacteristic, complemented, Sylow
232C23×C3—21no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—38no—
236C23×C3—38no—
237C23×C3—36no—
238C23×C3—36no—
239C23×C3—41no—
240C23×C3—66no—
241C23×C3—68no—
242C23×C3D3×Dic311yescharacteristic
243C23×C3C3×C4×D311yescharacteristic
244C23×C3C2×C3×Dic311yescharacteristic
245C22×D3C3×C3⋊C818yesdirect product factor
246C22×D3C2×C3×Dic316yes—
247C2×C3×C4—23no—
248C2×C3×C4—23no—
249C2×C3×C4—23no—
250C2×C3×C4—23no—
251C2×C3×C4—23no—
252C2×C3×C4—23no—
253C2×C3×C4—36no—
254C2×C3×C4—36no—
255C2×C3×C4—34no—
256C2×C3×C4—34no—
257C2×C3×C4—312no—
258C2×C3×C4—312no—
259C2×C3×C4—43no—
260C2×C3×C4—43no—
261C2×C3×C4—64no—
262C2×C3×C4—612no—
263C2×C3×C4—66no—
264C2×C3×C4D3×Dic313yes—
265C2×C3×C4C3×C4×D313yes—
266C2×C3×C4C2×C3×Dic313yes—
267C2×C3×C4C2×D3213yes—
268C2×C3×C4C22×C3×D313yes—
269C2×C3×C4C22×C3×D313yes—
270C2×Dic3C2×C3×Dic316yes—
271C4×D3C2×C3×Dic3112yes—
272C4×D3C22×C3×D314yes—
273C3×C8—34no—
274C3×C8—34no—
275C3×C8—64no—
276C3×C8—94no—
277C3⋊C8—34no—
278C3⋊C8—34no—
279C3⋊C8—64no—
280C3⋊C8C22×C3×D314yesdirect product factor
281C2×C32—26no—
282C2×C32—26no—
283C2×C32—21no—
284C2×C32—26no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—38no—
288C2×C32—46no—
289C2×C32—41no—
290C2×C32C2×D3×C816yescomplemented
291C2×C32C22×C3⋊C816yescomplemented
292C2×C32C22×C3×C816yescomplemented
293C2×C32C22×C4×D311yescharacteristic
294C2×C32C23×Dic311yescharacteristic
295C2×C32C23×C3×C411yescharacteristic
296C3×D3—28no—
297C3×D3C22×C3⋊C818yesdirect product factor
298C3×D3C22×C3×C818yescomplemented
299C24—31no—
300C22×C4—31no—
301C22×C4—36no—
302C22×C4—36no—
303C22×C4C3×D3211yescharacteristic
304C2×C8—36no—
305C2×C8—916no—
306C2×C8—96no—
307C22×C3—23no—
308C22×C3—23no—
309C22×C3—23no—
310C22×C3—24no—
311C22×C3—24no—
312C22×C3—24no—
313C22×C3—324no—
314C22×C3—34no—
315C22×C3—324no—
316C22×C3—34no—
317C22×C3—43no—
318C22×C3—44no—
319C22×C3—64no—
320C22×C3—624no—
321C22×C3D3×C3⋊C814yesdirect product factor
322C22×C3C3×D3×C814yescomplemented
323C22×C3C2×C3×C3⋊C814yescomplemented
324C22×C3C2×D3×Dic313yes—
325C22×C3C2×C3×C4×D313yes—
326C22×C3C22×C3×Dic313yes—
327C2×D3C2×C3×C3⋊C8124yesdirect product factor
328C2×D3C22×C3×Dic314yes—
329C3×C4—21no—
330C3×C4—21no—
331C3×C4—21no—
332C3×C4—23no—
333C3×C4—23no—
334C3×C4—23no—
335C3×C4—34no—
336C3×C4—34no—
337C3×C4—41no—
338C3×C4—43no—
339C3×C4—64no—
340C3×C4C2×D3×Dic313yes—
341C3×C4C2×C3×C4×D313yes—
342C3×C4C22×C3×Dic313yes—
343C3×C4C22×D3211yescharacteristic
344C3×C4C23×C3×D311yescharacteristic
345C3×C4C23×C3×D311yescharacteristic
346Dic3C22×C3×Dic314yes—
347C32—21no—
348C32—21no—
349C32—21no—
350C32—41no—
351C32C22×D3×C811yescharacteristic, complemented
352C32C23×C3⋊C811yescharacteristic, complemented
353C32C23×C3×C811yescharacteristic, complemented
354C23—36no—
355C23—38no—
356C23C3×D3×Dic311yescharacteristic
357C2×C4—36no—
358C2×C4—34no—
359C2×C4—312no—
360C2×C4C3×D3×Dic313yes—
361C2×C4C2×C3×D3213yes—
362C8—34no—
363C8—94no—
364C2×C3—21no—
365C2×C3—21no—
366C2×C3—21no—
367C2×C3—26no—
368C2×C3—26no—
369C2×C3—26no—
370C2×C3—38no—
371C2×C3—38no—
372C2×C3—41no—
373C2×C3—46no—
374C2×C3—68no—
375C2×C3C2×D3×C3⋊C816yesdirect product factor
376C2×C3C2×C3×D3×C816yescomplemented
377C2×C3C22×C3×C3⋊C816yescomplemented
378C2×C3C22×D3×Dic311yescharacteristic
379C2×C3C22×C3×C4×D311yescharacteristic
380C2×C3C23×C3×Dic311yescharacteristic
381D3C22×C3×C3⋊C818yesdirect product factor
382C22—34no—
383C22—324no—
384C22C3×D3×C3⋊C814yesdirect product factor
385C22C2×C3×D3×Dic313yes—
386C4—34no—
387C4C2×C3×D3×Dic313yes—
388C4C22×C3×D3211yescharacteristic
389C3—21no—
390C3—21no—
391C3—21no—
392C3—41no—
393C3C22×D3×C3⋊C811yescharacteristic, direct product factor
394C3C22×C3×D3×C811yescharacteristic, complemented
395C3C23×C3×C3⋊C811yescharacteristic, complemented
396C2—38no—
397C2C2×C3×D3×C3⋊C816yesdirect product factor
398C2C22×C3×D3×Dic311yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, hypocenterC32C23×C3×C8
FrattiniC4C22×C3×D32
FittingC22×C33×C4C22
socleC23×C33C2×C4
prime radicalC23×C32×D3C4
cosocleC32×C4C24×C3
LCSG ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×C3⋊(C3⋊C8)C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C2×C3×D3×C3⋊C8C2direct (N × H)
7. C22×D3×C3⋊C8C3direct (N × H)
13. C2×C3×C3⋊(C3⋊C8)C22semidirect
14. C2×C32×C3⋊C8C22semidirect
15. C3×D3×C3⋊C8C22direct (N × H)
21. C22×C3⋊(C3⋊C8)C2×C3semidirect
27. C22×C3×C3⋊C8D3direct (N × H)
28. C22×C3×C3⋊C8C2×C3semidirect
30. C2×D3×C3⋊C8C2×C3direct (N × H)
32. C22×C32×D3C8semidirect
39. C3×C3⋊(C3⋊C8)C23semidirect
40. C32×C3⋊C8C23semidirect
68. C2×C3⋊(C3⋊C8)C22×C3semidirect
74. C2×C3×C3⋊C8C2×D3direct (N × H)
75. C2×C3×C3⋊C8C22×C3semidirect
77. D3×C3⋊C8C22×C3direct (N × H)
78. C22×C33C2×C8semidirect
80. C2×C32×D3C2×C8semidirect
98. C22×C3⋊C8C3×D3direct (N × H)
111. C22×C3×D3C3⋊C8direct (N × H)
112. C22×C3×D3C3×C8semidirect
142. C3⋊(C3⋊C8)C23×C3semidirect
147. C3×C3⋊C8C22×D3direct (N × H)
148. C3×C3⋊C8C23×C3semidirect
150. C2×C33C22×C8semidirect
152. C32×D3C22×C8semidirect
185. C2×C3⋊C8C2×C3×D3direct (N × H)
197. C22×C32D3×C8semidirect
198. C22×C32C2×C3⋊C8semidirect
199. C22×C32C2×C3×C8semidirect
205. C2×C3×D3C2×C3⋊C8direct (N × H)
206. C2×C3×D3C2×C3×C8semidirect
231. C33C23×C8semidirect
245. C22×D3C3×C3⋊C8direct (N × H)
280. C3⋊C8C22×C3×D3direct (N × H)
290. C2×C32C2×D3×C8semidirect
291. C2×C32C22×C3⋊C8semidirect
292. C2×C32C22×C3×C8semidirect
297. C3×D3C22×C3⋊C8direct (N × H)
298. C3×D3C22×C3×C8semidirect
321. C22×C3D3×C3⋊C8direct (N × H)
322. C22×C3C3×D3×C8semidirect
323. C22×C3C2×C3×C3⋊C8semidirect
327. C2×D3C2×C3×C3⋊C8direct (N × H)
351. C32C22×D3×C8semidirect
352. C32C23×C3⋊C8semidirect
353. C32C23×C3×C8semidirect
375. C2×C3C2×D3×C3⋊C8direct (N × H)
376. C2×C3C2×C3×D3×C8semidirect
377. C2×C3C22×C3×C3⋊C8semidirect
381. D3C22×C3×C3⋊C8direct (N × H)
384. C22C3×D3×C3⋊C8direct (N × H)
393. C3C22×D3×C3⋊C8direct (N × H)
394. C3C22×C3×D3×C8semidirect
395. C3C23×C3×C3⋊C8semidirect
397. C2C2×C3×D3×C3⋊C8direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C32×C4×D32
C22×C3×C3⋊(C3⋊C8)2
C22×C32×C3⋊C82
C2×C3×D3×C3⋊C82
C23×C3×C3⋊C83
C22×C3×D3×C83
C22×D3×C3⋊C83