Finite Group Databasebeta

← C22×C3×#3798 #13181535

Subgroups — C22×C3×#3798 #13181535

Order 1728.

4322 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×C3⋊D3C211yesmaximal, characteristic, complemented
2C22×C32×C4×D3C212yesmaximal, complemented
3C22×C3×#627C211yesmaximal, characteristic, complemented
4C22×C3×#626C212yesmaximal, complemented
5C22×C3×#625C211yesmaximal, characteristic, complemented
6C2×C3×#3798C2124yesmaximal, direct product factor
7C22×C3×#235—32nomaximal
8C22×#3798C311yesmaximal, characteristic, direct product factor
9C23×C3×C3⋊D3C2211yescharacteristic
10C23×C32×D3C2212yes—
11C22×C33×C4C2211yescharacteristic
12C22×C3×C3⋊Dic3C2211yescharacteristic
13C2×C3×C4×C3⋊D3C22112yescomplemented
14C22×C32×Dic3C2212yes—
15C2×C32×C4×D3C22124yescomplemented
16C2×C3×#627C22112yescomplemented
17C2×C3×#626C22124yescomplemented
18C2×C3×#625C22112yescomplemented
19C3×#3798C22164yesdirect product factor
20C23×C32×C4—32no—
21C22×C4×C3⋊D3C2×C311yescharacteristic, complemented
22C22×C3×#67—32no—
23C22×C3×#67—32no—
24C2×C3×#235—348no—
25C22×C3×D12—32no—
26C22×C3×C4×D3—22no—
27C22×C3×C4×D3—32no—
28C22×C3×C4×D3—61no—
29C22×C3×C4×D3D312yescomplemented
30C22×C3×C4×D3C2×C312yescomplemented
31C22×C3×Dic6—32no—
32C22×#627C2×C311yescharacteristic, complemented
33C22×#626C2×C312yescomplemented
34C22×#625C2×C311yescharacteristic, complemented
35C2×#3798C2×C3124yesdirect product factor
36C23×C33C2311yescharacteristic
37C22×C3×C3⋊D3—24no—
38C22×C3×C3⋊D3C2316yes—
39C22×C32×D3—28no—
40C22×C32×D3C23112yes—
41C2×C33×C4C2316yes—
42C2×C3×C3⋊Dic3C2316yes—
43C3×C4×C3⋊D3C23116yescomplemented
44C2×C32×Dic3C23112yes—
45C32×C4×D3C23132yescomplemented
46C3×#627C23116yescomplemented
47C3×#626C23132yescomplemented
48C3×#625C23116yescomplemented
49C22×C3×#41—91no—
50C22×#235—32no—
51C24×C32—32no—
52C23×C3⋊D3C22×C311yescharacteristic
53C23×C3×D3—22no—
54C23×C3×D3—32no—
55C23×C3×D3—61no—
56C23×C3×D3C2×D312yes—
57C23×C3×D3C22×C312yes—
58C22×C32×C4—21no—
59C22×C32×C4—22no—
60C22×C32×C4—32no—
61C22×C32×C4—324no—
62C22×C32×C4—41no—
63C22×C32×C4C2×D312yes—
64C22×C32×C4C22×C311yescharacteristic
65C22×C3⋊Dic3C22×C311yescharacteristic
66C2×C4×C3⋊D3C22×C3112yescomplemented
67C2×C3×#67—324no—
68C2×C3×#67—324no—
69C22×C3×Dic3—22no—
70C22×C3×Dic3—32no—
71C22×C3×Dic3—61no—
72C22×C3×Dic3C2×D312yes—
73C22×C3×Dic3C22×C312yes—
74C3×#235—3128no—
75C2×C3×D12—324no—
76C2×C3×C4×D3—224no—
77C2×C3×C4×D3—324no—
78C2×C3×C4×D3—612no—
79C2×C3×C4×D3C2×D3124yescomplemented
80C2×C3×C4×D3C22×C3124yescomplemented
81C2×C3×Dic6—324no—
82C2×#627C22×C3112yescomplemented
83C2×#626C22×C3124yescomplemented
84C2×#625C22×C3112yescomplemented
85#3798C22×C3164yesdirect product factor
86C22×C33#4114yescomplemented
87C22×C33C2413yes—
88C2×C3×C3⋊D3—212no—
89C2×C3×C3⋊D3C2414yes—
90C2×C32×D3—224no—
91C2×C32×D3C2418yes—
92C33×C4C2414yes—
93C3×C3⋊Dic3C2414yes—
94C32×Dic3C2418yes—
95C2×C3×#41—924no—
96C22×C3×Q8—91no—
97C22×C3×D4—92no—
98C22×C3×D4—91no—
99C23×C3×C4—32no—
100C23×C3×C4—32no—
101C23×C3×C4—62no—
102C23×C3×C4—91no—
103C22×#67—32no—
104C22×#67—32no—
105C2×#235—348no—
106C22×D12—32no—
107C22×C4×D3—32no—
108C22×C4×D3—61no—
109C22×C4×D3C3×D312yescomplemented
110C22×Dic6—32no—
111C23×C32—21no—
112C23×C32—22no—
113C23×C32—312no—
114C23×C32—41no—
115C23×C32—68no—
116C23×C32C22×D312yes—
117C23×C32C23×C311yescharacteristic
118C22×C3⋊D3—24no—
119C22×C3⋊D3C23×C316yes—
120C22×C3×D3—216no—
121C22×C3×D3—28no—
122C22×C3×D3—28no—
123C22×C3×D3—212no—
124C22×C3×D3—312no—
125C22×C3×D3—66no—
126C22×C3×D3—68no—
127C22×C3×D3—68no—
128C22×C3×D3C22×D3112yes—
129C22×C3×D3C23×C3112yes—
130C2×C32×C4—26no—
131C2×C32×C4—212no—
132C2×C32×C4—312no—
133C2×C32×C4—332no—
134C2×C32×C4—46no—
135C2×C32×C4C22×D3112yes—
136C2×C32×C4C23×C316yes—
137C2×C3⋊Dic3C23×C316yes—
138C4×C3⋊D3C23×C3116yescomplemented
139C3×#67—332no—
140C3×#67—332no—
141C2×C3×Dic3—212no—
142C2×C3×Dic3—312no—
143C2×C3×Dic3—66no—
144C2×C3×Dic3C22×D3112yes—
145C2×C3×Dic3C23×C3112yes—
146C3×D12—332no—
147C3×C4×D3—232no—
148C3×C4×D3—332no—
149C3×C4×D3—616no—
150C3×C4×D3C22×D3132yescomplemented
151C3×C4×D3C23×C3132yescomplemented
152C3×Dic6—332no—
153#627C23×C3116yescomplemented
154#626C23×C3132yescomplemented
155#625C23×C3116yescomplemented
156C22×#41—91noSylow
157C2×C33C2×#4116yescomplemented
158C2×C33C2511yescharacteristic
159C3×C3⋊D3—24no—
160C32×D3—28no—
161C24×C3—32no—
162C24×C3—32no—
163C24×C3—62no—
164C24×C3—91no—
165C23×D3—32no—
166C23×D3—61no—
167C23×D3C2×C3×D312yes—
168C3×#41—964no—
169C2×C3×Q8—912no—
170C2×C3×D4—924no—
171C2×C3×D4—912no—
172C22×C3×C4—21no—
173C22×C3×C4—22no—
174C22×C3×C4—32no—
175C22×C3×C4—32no—
176C22×C3×C4—324no—
177C22×C3×C4—324no—
178C22×C3×C4—41no—
179C22×C3×C4—624no—
180C22×C3×C4—62no—
181C22×C3×C4—91no—
182C22×C3×C4—912no—
183C22×C3×C4D3211yescharacteristic
184C22×C3×C4C2×C3×D312yes—
185C2×#67—324no—
186C2×#67—324no—
187C22×Dic3—32no—
188C22×Dic3—61no—
189C22×Dic3C2×C3×D312yes—
190#235—3128no—
191C2×D12—324no—
192C2×C4×D3—324no—
193C2×C4×D3—612no—
194C2×C4×D3C2×C3×D3124yescomplemented
195C2×Dic6—324no—
196C22×C32—23no—
197C22×C32—26no—
198C22×C32—24no—
199C22×C32—28no—
200C22×C32—38no—
201C22×C32—44no—
202C22×C32—43no—
203C22×C32—624no—
204C22×C32#23518yescomplemented
205C22×C32C3×#4114yescomplemented
206C22×C32C23×D316yes—
207C22×C32C24×C313yes—
208C2×C3⋊D3—212no—
209C2×C3⋊D3C24×C314yes—
210C2×C3×D3—248no—
211C2×C3×D3—224no—
212C2×C3×D3—224no—
213C2×C3×D3—28no—
214C2×C3×D3—38no—
215C2×C3×D3—64no—
216C2×C3×D3—624no—
217C2×C3×D3—624no—
218C2×C3×D3C23×D318yes—
219C2×C3×D3C24×C318yes—
220C32×C4—24no—
221C32×C4—28no—
222C32×C4—38no—
223C32×C4—44no—
224C32×C4C23×D318yes—
225C32×C4C24×C314yes—
226C3⋊Dic3C24×C314yes—
227C3×Dic3—28no—
228C3×Dic3—38no—
229C3×Dic3—64no—
230C3×Dic3C23×D318yes—
231C3×Dic3C24×C318yes—
232C2×#41—924no—
233C22×Q8—91no—
234C22×D4—92no—
235C22×D4—91no—
236C23×C4—32no—
237C23×C4—91no—
238C33C22×#4111yescharacteristic, complemented, Sylow
239C23×C3—22no—
240C23×C3—21no—
241C23×C3—312no—
242C23×C3—312no—
243C23×C3—41no—
244C23×C3—612no—
245C23×C3—616no—
246C23×C3—68no—
247C23×C3—68no—
248C23×C3—96no—
249C23×C3—184no—
250C23×C3C2×D3211yescharacteristic
251C23×C3C22×C3×D312yes—
252C22×D3—28no—
253C22×D3—312no—
254C22×D3—66no—
255C22×D3—68no—
256C22×D3—68no—
257C22×D3C22×C3×D3112yes—
258C3×Q8—916no—
259C3×D4—932no—
260C3×D4—916no—
261C2×C3×C4—212no—
262C2×C3×C4—26no—
263C2×C3×C4—312no—
264C2×C3×C4—312no—
265C2×C3×C4—332no—
266C2×C3×C4—332no—
267C2×C3×C4—46no—
268C2×C3×C4—632no—
269C2×C3×C4—612no—
270C2×C3×C4—96no—
271C2×C3×C4—916no—
272C2×C3×C4C2×D3216yes—
273C2×C3×C4C22×C3×D3112yes—
274#67—332no—
275#67—332no—
276C2×Dic3—312no—
277C2×Dic3—66no—
278C2×Dic3C22×C3×D3112yes—
279D12—332no—
280C4×D3—332no—
281C4×D3—616no—
282C4×D3C22×C3×D3132yescomplemented
283Dic6—332no—
284C2×C32—26no—
285C2×C32—212no—
286C2×C32—21no—
287C2×C32—22no—
288C2×C32—46no—
289C2×C32—41no—
290C2×C32—68no—
291C2×C32C2×#235112yescomplemented
292C2×C32C2×C3×#4116yescomplemented
293C2×C32C24×D312yes—
294C2×C32C25×C311yescharacteristic
295C3⋊D3—24no—
296C3×D3—216no—
297C3×D3—28no—
298C3×D3—28no—
299C3×D3—68no—
300C3×D3—68no—
301C24—32no—
302C24—91no—
303#41—964no—
304C2×Q8—912no—
305C2×D4—924no—
306C2×D4—912no—
307C22×C4—32no—
308C22×C4—324no—
309C22×C4—91no—
310C22×C4—912no—
311C22×C4C3×D3211yescharacteristic
312C22×C3—23no—
313C22×C3—26no—
314C22×C3—24no—
315C22×C3—28no—
316C22×C3—38no—
317C22×C3—38no—
318C22×C3—43no—
319C22×C3—44no—
320C22×C3—68no—
321C22×C3—648no—
322C22×C3—624no—
323C22×C3—624no—
324C22×C3—94no—
325C22×C3—1812no—
326C22×C3#379814yesdirect product factor
327C22×C3C3×#23518yescomplemented
328C22×C3C22×D3213yes—
329C22×C3C23×C3×D316yes—
330C2×D3—224no—
331C2×D3—38no—
332C2×D3—64no—
333C2×D3—624no—
334C2×D3—624no—
335C2×D3C23×C3×D318yes—
336C3×C4—24no—
337C3×C4—28no—
338C3×C4—38no—
339C3×C4—38no—
340C3×C4—44no—
341C3×C4—68no—
342C3×C4—94no—
343C3×C4C22×D3214yes—
344C3×C4C23×C3×D318yes—
345Dic3—38no—
346Dic3—64no—
347Dic3C23×C3×D318yes—
348C32—21no—
349C32—22no—
350C32—41no—
351C32C22×#23512yescomplemented
352C32C22×C3×#4111yescharacteristic, complemented
353C23—312no—
354C23—68no—
355C23—96no—
356C23—184no—
357C23C2×C3×D3211yescharacteristic
358Q8—916no—
359D4—932no—
360D4—916no—
361C2×C4—312no—
362C2×C4—332no—
363C2×C4—96no—
364C2×C4—916no—
365C2×C4C2×C3×D3216yes—
366C2×C3—21no—
367C2×C3—22no—
368C2×C3—26no—
369C2×C3—212no—
370C2×C3—41no—
371C2×C3—46no—
372C2×C3—616no—
373C2×C3—68no—
374C2×C3—68no—
375C2×C3—184no—
376C2×C3C2×#379816yesdirect product factor
377C2×C3C2×C3×#235112yescomplemented
378C2×C3C23×D3211yescharacteristic
379C2×C3C24×C3×D312yes—
380D3—28no—
381D3—68no—
382D3—68no—
383C22—38no—
384C22—624no—
385C22—94no—
386C22—1812no—
387C22C3×#379814yesdirect product factor
388C22C22×C3×D3213yes—
389C4—38no—
390C4—94no—
391C4C22×C3×D3214yes—
392C3—21no—
393C3—22no—
394C3—41no—
395C3C22×#379811yescharacteristic, direct product factor
396C3C22×C3×#23512yescomplemented
397C2—68no—
398C2—184no—
399C2C2×C3×#379816yesdirect product factor
400C2C23×C3×D3211yescharacteristic
subgroupG/H
center, hypercenterC22×C3×C4D32
commutator, cosocleC2×C32C25×C3
FrattiniC2C23×C3×D32
FittingC22×C33×C4C22
socleC23×C33C23
hypocenterC32C22×C3×#41
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3×C4

Complements

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

NH ≅ G/Ntype
1. C22×C3×C4×C3⋊D3C2semidirect
2. C22×C32×C4×D3C2semidirect
3. C22×C3×#627C2semidirect
4. C22×C3×#626C2semidirect
5. C22×C3×#625C2semidirect
6. C2×C3×#3798C2direct (N × H)
8. C22×#3798C3direct (N × H)
13. C2×C3×C4×C3⋊D3C22semidirect
15. C2×C32×C4×D3C22semidirect
16. C2×C3×#627C22semidirect
17. C2×C3×#626C22semidirect
18. C2×C3×#625C22semidirect
19. C3×#3798C22direct (N × H)
21. C22×C4×C3⋊D3C2×C3semidirect
29. C22×C3×C4×D3D3semidirect
30. C22×C3×C4×D3C2×C3semidirect
32. C22×#627C2×C3semidirect
33. C22×#626C2×C3semidirect
34. C22×#625C2×C3semidirect
35. C2×#3798C2×C3direct (N × H)
43. C3×C4×C3⋊D3C23semidirect
45. C32×C4×D3C23semidirect
46. C3×#627C23semidirect
47. C3×#626C23semidirect
48. C3×#625C23semidirect
66. C2×C4×C3⋊D3C22×C3semidirect
79. C2×C3×C4×D3C2×D3semidirect
80. C2×C3×C4×D3C22×C3semidirect
82. C2×#627C22×C3semidirect
83. C2×#626C22×C3semidirect
84. C2×#625C22×C3semidirect
85. #3798C22×C3direct (N × H)
86. C22×C33#41semidirect
109. C22×C4×D3C3×D3semidirect
138. C4×C3⋊D3C23×C3semidirect
150. C3×C4×D3C22×D3semidirect
151. C3×C4×D3C23×C3semidirect
153. #627C23×C3semidirect
154. #626C23×C3semidirect
155. #625C23×C3semidirect
157. C2×C33C2×#41semidirect
194. C2×C4×D3C2×C3×D3semidirect
204. C22×C32#235semidirect
205. C22×C32C3×#41semidirect
238. C33C22×#41semidirect
282. C4×D3C22×C3×D3semidirect
291. C2×C32C2×#235semidirect
292. C2×C32C2×C3×#41semidirect
326. C22×C3#3798direct (N × H)
327. C22×C3C3×#235semidirect
351. C32C22×#235semidirect
352. C32C22×C3×#41semidirect
376. C2×C3C2×#3798direct (N × H)
377. C2×C3C2×C3×#235semidirect
387. C22C3×#3798direct (N × H)
395. C3C22×#3798direct (N × H)
396. C3C22×C3×#235semidirect
399. C2C2×C3×#3798direct (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⋊D32
C22×C32×C4×D32
C22×C3×#6272
C22×C3×#6262
C22×C3×#6252
C2×C3×#37982
C22×C3×#2353
C22×#37983