Finite Group Databasebeta

← C22×C3×#896 #11900319

Subgroups — C22×C3×#896 #11900319

Order 1152.

3889 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C3×C4×D3C211yesmaximal, characteristic
2C23×C3×Dic6C211yesmaximal, characteristic, complemented
3C22×C32×C4⋊C4C211yesmaximal, characteristic, complemented
4C22×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
5C22×C3×(C2×D3)⋊C4C211yesmaximal, characteristic
6C22×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
7C22×C3×Dic3⋊C4C211yesmaximal, characteristic, complemented
8C2×C3×#896C2124yesmaximal, direct product factor
9C22×C3×C22⋊Q8—31nomaximal
10C22×#896C311yesmaximal, characteristic, direct product factor
11C24×C3×D3C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C23×C32×C4C2211yescharacteristic
14C23×C32×C4C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C23×C3×Dic3C2211yescharacteristic
18C22×C3×C4×D3—28no—
19C22×C3×C4×D3C22112yes—
20C22×C3×Dic6—28no—
21C22×C3×Dic6C22112yescomplemented
22C2×C32×C4⋊C4C22112yescomplemented
23C2×C3×(C2×D3)⋊C4C22112yes—
24C2×C3×(C2×D3)⋊C4C22112yes—
25C2×C3×Dic3⋊C4C22112yescomplemented
26C2×C3×Dic3⋊C4C22112yescomplemented
27C3×#896C22164yesdirect product factor
28C23×C3×Q8—31no—
29C24×C3×C4—31no—
30C23×C4×D3C2×C311yescharacteristic
31C23×Dic6C2×C311yescharacteristic, complemented
32C2×C3×C22⋊Q8—324no—
33C22×C3×C4⋊C4—21no—
34C22×C3×C4⋊C4—31no—
35C22×C3×C4⋊C4—31no—
36C22×C3×C4⋊C4D311yescharacteristic, complemented
37C22×C3×C4⋊C4C2×C311yescharacteristic, complemented
38C22×C3×C22⋊C4—31no—
39C22×C3×C22⋊C4—31no—
40C22×(C2×D3)⋊C4C2×C311yescharacteristic
41C22×(C2×D3)⋊C4C2×C311yescharacteristic
42C22×Dic3⋊C4C2×C311yescharacteristic, complemented
43C22×Dic3⋊C4C2×C311yescharacteristic, complemented
44C2×#896C2×C3124yesdirect product factor
45C24×C32C2311yescharacteristic
46C23×C3×D3—24no—
47C23×C3×D3—24no—
48C23×C3×D3Q818yescomplemented
49C23×C3×D3C2316yes—
50C22×C32×C4—24no—
51C22×C32×C4—24no—
52C22×C32×C4—24no—
53C22×C32×C4C2316yes—
54C22×C32×C4C2316yes—
55C22×C32×C4C2316yes—
56C22×C3×Dic3—24no—
57C22×C3×Dic3—24no—
58C22×C3×Dic3D418yes—
59C22×C3×Dic3C2316yes—
60C22×C3×Dic3C2316yes—
61C22×C3×Dic3C2316yes—
62C2×C3×C4×D3—248no—
63C2×C3×C4×D3C23116yes—
64C2×C3×Dic6—248no—
65C2×C3×Dic6C23116yescomplemented
66C32×C4⋊C4C23116yescomplemented
67C3×(C2×D3)⋊C4C23116yes—
68C3×(C2×D3)⋊C4C23116yes—
69C3×Dic3⋊C4C23116yescomplemented
70C3×Dic3⋊C4C23116yescomplemented
71C22×C22⋊Q8—31noSylow
72C25×C3—31no—
73C24×D3C22×C311yescharacteristic
74C22×C3×Q8—312no—
75C22×C3×Q8—68no—
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C23×C3×C4—31no—
80C23×C3×C4—31no—
81C23×C3×C4—31no—
82C23×C3×C4—312no—
83C23×C3×C4—68no—
84C23×C3×C4C2×D311yescharacteristic
85C23×C3×C4C2×D311yescharacteristic
86C23×C3×C4C2×D311yescharacteristic
87C23×C3×C4C22×C311yescharacteristic
88C23×C3×C4C22×C311yescharacteristic
89C23×C3×C4C22×C311yescharacteristic
90C23×Dic3C22×C311yescharacteristic
91C23×Dic3C22×C311yescharacteristic
92C23×Dic3C22×C311yescharacteristic
93C22×C4×D3—28no—
94C22×C4×D3C22×C3112yes—
95C22×Dic6—28no—
96C22×Dic6C22×C3112yescomplemented
97C3×C22⋊Q8—364no—
98C2×C3×C4⋊C4—212no—
99C2×C3×C4⋊C4—312no—
100C2×C3×C4⋊C4—312no—
101C2×C3×C4⋊C4C2×D3112yescomplemented
102C2×C3×C4⋊C4C22×C3112yescomplemented
103C2×C3×C22⋊C4—312no—
104C2×C3×C22⋊C4—312no—
105C2×(C2×D3)⋊C4C22×C3112yes—
106C2×(C2×D3)⋊C4C22×C3112yes—
107C2×Dic3⋊C4C22×C3112yescomplemented
108C2×Dic3⋊C4C22×C3112yescomplemented
109#896C22×C3164yesdirect product factor
110C23×C32C2×D414yes—
111C23×C32C2×Q814yes—
112C23×C32#4114yes—
113C23×C32C2413yes—
114C22×C3×D3—232no—
115C22×C3×D3—212no—
116C22×C3×D3—212no—
117C22×C3×D3C2×Q8124yescomplemented
118C22×C3×D3C2414yes—
119C2×C32×C4—212no—
120C2×C32×C4—212no—
121C2×C32×C4—212no—
122C2×C32×C4C2414yes—
123C2×C32×C4C2414yes—
124C2×C32×C4C2414yes—
125C2×C3×Dic3—212no—
126C2×C3×Dic3—212no—
127C2×C3×Dic3C2×D4124yes—
128C2×C3×Dic3C2414yes—
129C2×C3×Dic3C2414yes—
130C2×C3×Dic3C2414yes—
131C3×C4×D3—232no—
132C3×Dic6—232no—
133C23×Q8—31no—
134C24×C4—31no—
135C2×C22⋊Q8—324no—
136C22×C4⋊C4—31no—
137C22×C4⋊C4—31no—
138C22×C4⋊C4C3×D311yescharacteristic, complemented
139C22×C22⋊C4—31no—
140C22×C22⋊C4—31no—
141C24×C3—21no—
142C24×C3—38no—
143C24×C3—36no—
144C24×C3—64no—
145C24×C3—64no—
146C24×C3C22×D311yescharacteristic
147C24×C3C23×C311yescharacteristic
148C23×D3—24no—
149C23×D3—24no—
150C23×D3C3×Q818yescomplemented
151C23×D3C23×C316yes—
152C2×C3×Q8—316no—
153C2×C3×Q8—648no—
154C22×C3×C4—28no—
155C22×C3×C4—24no—
156C22×C3×C4—24no—
157C22×C3×C4—24no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—24no—
161C22×C3×C4—26no—
162C22×C3×C4—26no—
163C22×C3×C4—26no—
164C22×C3×C4—38no—
165C22×C3×C4—36no—
166C22×C3×C4—36no—
167C22×C3×C4—36no—
168C22×C3×C4—316no—
169C22×C3×C4—44no—
170C22×C3×C4—44no—
171C22×C3×C4—648no—
172C22×C3×C4—64no—
173C22×C3×C4—64no—
174C22×C3×C4C22×D316yes—
175C22×C3×C4C22×D316yes—
176C22×C3×C4C22×D316yes—
177C22×C3×C4C23×C316yes—
178C22×C3×C4C23×C316yes—
179C22×C3×C4C23×C316yes—
180C22×Dic3—24no—
181C22×Dic3—24no—
182C22×Dic3C3×D418yes—
183C22×Dic3C23×C316yes—
184C22×Dic3C23×C316yes—
185C22×Dic3C23×C316yes—
186C2×C4×D3—248no—
187C2×C4×D3C23×C3116yes—
188C2×Dic6—248no—
189C2×Dic6C23×C3116yescomplemented
190C3×C4⋊C4—216no—
191C3×C4⋊C4—316no—
192C3×C4⋊C4—316no—
193C3×C4⋊C4C22×D3116yescomplemented
194C3×C4⋊C4C23×C3116yescomplemented
195C3×C22⋊C4—316no—
196C3×C22⋊C4—316no—
197(C2×D3)⋊C4C23×C3116yes—
198(C2×D3)⋊C4C23×C3116yes—
199Dic3⋊C4C23×C3116yescomplemented
200Dic3⋊C4C23×C3116yescomplemented
201C22×C32C22⋊Q8116yescomplemented
202C22×C32C22×D416yes—
203C22×C32C22×Q816yes—
204C22×C32C2×#4116yes—
205C22×C32C2511yescharacteristic
206C2×C3×D3—248no—
207C2×C3×D3—24no—
208C2×C3×D3—24no—
209C2×C3×D3C22×Q818yescomplemented
210C32×C4—24no—
211C32×C4—24no—
212C32×C4—24no—
213C3×Dic3—24no—
214C3×Dic3—24no—
215C3×Dic3C22×D418yes—
216C25—31no—
217C22×Q8—312no—
218C22×Q8—68no—
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—31no—
222C23×C4—312no—
223C23×C4—68no—
224C23×C4C2×C3×D311yescharacteristic
225C23×C4C2×C3×D311yescharacteristic
226C23×C4C2×C3×D311yescharacteristic
227C22⋊Q8—364no—
228C2×C4⋊C4—312no—
229C2×C4⋊C4—312no—
230C2×C4⋊C4C2×C3×D3112yescomplemented
231C2×C22⋊C4—312no—
232C2×C22⋊C4—312no—
233C23×C3—23no—
234C23×C3—24no—
235C23×C3—24no—
236C23×C3—24no—
237C23×C3—324no—
238C23×C3—34no—
239C23×C3—632no—
240C23×C3—612no—
241C23×C3—612no—
242C23×C3#23514yes—
243C23×C3D3×D414yes—
244C23×C3D3×Q814yes—
245C23×C3C2×C3×D414yes—
246C23×C3C2×C3×Q814yes—
247C23×C3C3×#4114yes—
248C23×C3C23×D313yes—
249C23×C3C24×C313yes—
250C22×D3—232no—
251C22×D3—212no—
252C22×D3—212no—
253C22×D3C2×C3×Q8124yescomplemented
254C22×D3C24×C314yes—
255C3×Q8—632no—
256C2×C3×C4—224no—
257C2×C3×C4—212no—
258C2×C3×C4—212no—
259C2×C3×C4—212no—
260C2×C3×C4—212no—
261C2×C3×C4—212no—
262C2×C3×C4—212no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—324no—
267C2×C3×C4—34no—
268C2×C3×C4—34no—
269C2×C3×C4—34no—
270C2×C3×C4—412no—
271C2×C3×C4—412no—
272C2×C3×C4—632no—
273C2×C3×C4—612no—
274C2×C3×C4—612no—
275C2×C3×C4C23×D314yes—
276C2×C3×C4C23×D314yes—
277C2×C3×C4C23×D314yes—
278C2×C3×C4C24×C314yes—
279C2×C3×C4C24×C314yes—
280C2×C3×C4C24×C314yes—
281C2×Dic3—212no—
282C2×Dic3—212no—
283C2×Dic3C2×C3×D4124yes—
284C2×Dic3C24×C314yes—
285C2×Dic3C24×C314yes—
286C2×Dic3C24×C314yes—
287C4×D3—232no—
288Dic6—232no—
289C2×C32C2×C22⋊Q8112yescomplemented
290C2×C32C23×D411yescharacteristic
291C2×C32C23×Q811yescharacteristic
292C2×C32C22×#4111yescharacteristic
293C3×D3—28no—
294C24—36no—
295C24—38no—
296C24—64no—
297C24—64no—
298C24C22×C3×D311yescharacteristic
299C2×Q8—316no—
300C2×Q8—648no—
301C22×C4—24no—
302C22×C4—24no—
303C22×C4—24no—
304C22×C4—36no—
305C22×C4—38no—
306C22×C4—36no—
307C22×C4—36no—
308C22×C4—316no—
309C22×C4—648no—
310C22×C4—64no—
311C22×C4—64no—
312C22×C4C22×C3×D316yes—
313C22×C4C22×C3×D316yes—
314C22×C4C22×C3×D316yes—
315C4⋊C4—316no—
316C4⋊C4—316no—
317C4⋊C4C22×C3×D3116yescomplemented
318C22⋊C4—316no—
319C22⋊C4—316no—
320C22×C3—216no—
321C22×C3—26no—
322C22×C3—26no—
323C22×C3—26no—
324C22×C3—21no—
325C22×C3—38no—
326C22×C3—64no—
327C22×C3—64no—
328C22×C3—648no—
329C22×C3#896116yesdirect product factor
330C22×C3C3×C22⋊Q8116yescomplemented
331C22×C3C2×#23516yes—
332C22×C3C2×D3×D416yes—
333C22×C3C2×D3×Q816yes—
334C22×C3C22×C3×D416yes—
335C22×C3C22×C3×Q816yes—
336C22×C3C2×C3×#4116yes—
337C22×C3C24×D311yescharacteristic
338C22×C3C25×C311yescharacteristic
339C2×D3—24no—
340C2×D3—24no—
341C2×D3—248no—
342C2×D3C22×C3×Q818yescomplemented
343C3×C4—28no—
344C3×C4—24no—
345C3×C4—24no—
346C3×C4—24no—
347C3×C4—24no—
348C3×C4—24no—
349C3×C4—24no—
350C3×C4—38no—
351C3×C4—44no—
352C3×C4—44no—
353C3×C4—64no—
354C3×C4—64no—
355Dic3—24no—
356Dic3—24no—
357Dic3C22×C3×D418yes—
358C32C22×C22⋊Q811yescharacteristic, complemented, Sylow
359C23—324no—
360C23—34no—
361C23—632no—
362C23—612no—
363C23—612no—
364C23C3×#23514yes—
365C23C3×D3×D414yes—
366C23C3×D3×Q814yes—
367C23C23×C3×D313yes—
368Q8—632no—
369C2×C4—212no—
370C2×C4—212no—
371C2×C4—212no—
372C2×C4—324no—
373C2×C4—34no—
374C2×C4—34no—
375C2×C4—34no—
376C2×C4—632no—
377C2×C4—612no—
378C2×C4—612no—
379C2×C4C23×C3×D314yes—
380C2×C4C23×C3×D314yes—
381C2×C4C23×C3×D314yes—
382C2×C3—212no—
383C2×C3—21no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—68no—
387C2×C3C2×#896112yesdirect product factor
388C2×C3C2×C3×C22⋊Q8112yescomplemented
389C2×C3C22×#23511yescharacteristic
390C2×C3C22×D3×D411yescharacteristic
391C2×C3C22×D3×Q811yescharacteristic
392C2×C3C23×C3×D411yescharacteristic
393C2×C3C23×C3×Q811yescharacteristic
394C2×C3C22×C3×#4111yescharacteristic
395D3—28no—
396C22—38no—
397C22—648no—
398C22—64no—
399C22—64no—
400C22C3×#896116yesdirect product factor
401C22C2×C3×#23516yes—
402C22C2×C3×D3×D416yes—
403C22C2×C3×D3×Q816yes—
404C22C24×C3×D311yescharacteristic
405C4—24no—
406C4—24no—
407C4—24no—
408C4—38no—
409C4—64no—
410C4—64no—
411C3—21no—
412C3C22×#89611yescharacteristic, direct product factor
413C3C22×C3×C22⋊Q811yescharacteristic, complemented
414C2—68no—
415C2C2×C3×#896112yesdirect product factor
416C2C22×C3×#23511yescharacteristic
417C2C22×C3×D3×D411yescharacteristic
418C2C22×C3×D3×Q811yescharacteristic
subgroupG/H
centerC24×C3C22×D3
commutator, cosocleC22×C3C25×C3
FrattiniC22C24×C3×D3
FittingC22×C32×C4⋊C4C2
socleC24×C32C23
prime radicalC24×C3×D3C22
hypercenterC22×C3×C4⋊C4D3
hypocenterC3C22×C3×C22⋊Q8
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C24×C3 ◁ C22×C3×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C23×C3×Dic6C2semidirect
3. C22×C32×C4⋊C4C2semidirect
6. C22×C3×Dic3⋊C4C2semidirect
7. C22×C3×Dic3⋊C4C2semidirect
8. C2×C3×#896C2direct (N × H)
10. C22×#896C3direct (N × H)
21. C22×C3×Dic6C22semidirect
22. C2×C32×C4⋊C4C22semidirect
25. C2×C3×Dic3⋊C4C22semidirect
26. C2×C3×Dic3⋊C4C22semidirect
27. C3×#896C22direct (N × H)
31. C23×Dic6C2×C3semidirect
36. C22×C3×C4⋊C4D3semidirect
37. C22×C3×C4⋊C4C2×C3semidirect
42. C22×Dic3⋊C4C2×C3semidirect
43. C22×Dic3⋊C4C2×C3semidirect
44. C2×#896C2×C3direct (N × H)
48. C23×C3×D3Q8semidirect
65. C2×C3×Dic6C23semidirect
66. C32×C4⋊C4C23semidirect
69. C3×Dic3⋊C4C23semidirect
70. C3×Dic3⋊C4C23semidirect
96. C22×Dic6C22×C3semidirect
101. C2×C3×C4⋊C4C2×D3semidirect
102. C2×C3×C4⋊C4C22×C3semidirect
107. C2×Dic3⋊C4C22×C3semidirect
108. C2×Dic3⋊C4C22×C3semidirect
109. #896C22×C3direct (N × H)
117. C22×C3×D3C2×Q8semidirect
138. C22×C4⋊C4C3×D3semidirect
150. C23×D3C3×Q8semidirect
189. C2×Dic6C23×C3semidirect
193. C3×C4⋊C4C22×D3semidirect
194. C3×C4⋊C4C23×C3semidirect
199. Dic3⋊C4C23×C3semidirect
200. Dic3⋊C4C23×C3semidirect
201. C22×C32C22⋊Q8semidirect
209. C2×C3×D3C22×Q8semidirect
230. C2×C4⋊C4C2×C3×D3semidirect
253. C22×D3C2×C3×Q8semidirect
289. C2×C32C2×C22⋊Q8semidirect
317. C4⋊C4C22×C3×D3semidirect
329. C22×C3#896direct (N × H)
330. C22×C3C3×C22⋊Q8semidirect
342. C2×D3C22×C3×Q8semidirect
358. C32C22×C22⋊Q8semidirect
387. C2×C3C2×#896direct (N × H)
388. C2×C3C2×C3×C22⋊Q8semidirect
400. C22C3×#896direct (N × H)
412. C3C22×#896direct (N × H)
413. C3C22×C3×C22⋊Q8semidirect
415. C2C2×C3×#896direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×C4×D32
C23×C3×Dic62
C22×C32×C4⋊C42
C22×C3×(C2×D3)⋊C42
C22×C3×Dic3⋊C42
C2×C3×#8962
C22×C3×C22⋊Q83
C22×#8963