Finite Group Databasebeta

← C3×#73770 #11805497

Subgroups — C3×#73770 #11805497

Order 1152.

1505 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C212yesmaximal
2C2×C3×C4×D12C211yesmaximal, characteristic
3C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
4C3×#4989C214yesmaximal, complemented
5C3×#4936C211yesmaximal, characteristic, complemented
6C3×#4928C212yesmaximal, complemented
7C3×#4667C212yesmaximal, complemented
8C3×D3×#95C212yesmaximal, complemented
9C3×#2349—31nomaximal
10#73770C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2214yes—
13C22×C3×C4×D3C2212yes—
14C2×C32×C4⋊C4C2212yes—
15C2×C32×C42C2212yes—
16C2×C32×C42C2211yescharacteristic
17C2×C3×(C2×D3)⋊C4C2214yes—
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×C4⋊Dic3C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4C2214yes—
22C3×C4×D12—28no—
23C3×C4×D12C418yescomplemented
24C32×#95C2211yescharacteristic
25C32×#95C2211yescharacteristic
26C3×C3⋊#95C2212yes—
27C3×C3⋊#95C2214yes—
28C3×C3⋊#95C2212yes—
29C2×C3×C4×D4—31no—
30C2×C3×C4×D4—32no—
31C2×C4×D12C2×C312yes—
32C2×C4×D12C2×C311yescharacteristic
33C3×#389—32no—
34C3×#385—34no—
35C3×#382—31no—
36C3×#380—32no—
37C3×C4×C4⋊C4—21no—
38C3×C4×C4⋊C4D311yescharacteristic, complemented
39C3×C4×C4⋊C4C2×C311yescharacteristic, complemented
40C2×C3×#95—32no—
41#4989C2×C314yescomplemented
42#4936C2×C311yescharacteristic, complemented
43#4928C2×C312yescomplemented
44#4667C2×C312yescomplemented
45D3×#95C2×C312yescomplemented
46C23×C3×D3C2312yes—
47C22×C32×C4C2312yes—
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2311yescharacteristic
50C22×C32×C4C2312yes—
51C22×C32×C4C2311yescharacteristic
52C22×C3×Dic3C2314yes—
53C22×C3×Dic3C2312yes—
54C2×C3×D12C2×C414yes—
55C2×C3×D12D414yes—
56C2×C3×D12Q814yes—
57C2×C3×C4×D3—28no—
58C2×C3×C4×D3—28no—
59C2×C3×C4×D3—44no—
60C2×C3×C4×D3—44no—
61C2×C3×C4×D3C2×C418yescomplemented
62C32×C4⋊C4—24no—
63C32×C42—24no—
64C32×C42C2×C414yescomplemented
65C3×(C2×D3)⋊C4—28no—
66C3×(C2×D3)⋊C4C2×C418yescomplemented
67C3×C4⋊Dic3—24no—
68C3×C4⋊Dic3C2×C414yescomplemented
69C3×Dic3⋊C4—44no—
70#2349—31noSylow
71C22×C3×D4—31no—
72C23×C3×C4—32no—
73C23×C3×C4—34no—
74C22×D12C22×C311yescharacteristic
75C22×C4×D3C22×C314yes—
76C22×C4×D3C22×C312yes—
77C3×C4×D4—38no—
78C3×C4×D4—68no—
79C2×C3×C4⋊C4—22no—
80C2×C3×C4⋊C4—31no—
81C2×C3×C4⋊C4—32no—
82C2×C3×C4⋊C4—34no—
83C2×C3×C4⋊C4C2×D312yes—
84C2×C3×C4⋊C4C22×C312yes—
85C2×C3×C22⋊C4—32no—
86C2×C3×C22⋊C4—34no—
87C2×C3×C42—21no—
88C2×C3×C42—22no—
89C2×C3×C42C2×D312yes—
90C2×C3×C42C2×D311yescharacteristic
91C2×C3×C42C22×C312yes—
92C2×C3×C42C22×C311yescharacteristic
93C2×(C2×D3)⋊C4C22×C314yes—
94C2×(C2×D3)⋊C4C22×C312yes—
95C2×C4⋊Dic3C22×C312yes—
96C2×C4⋊Dic3C22×C311yescharacteristic
97C2×Dic3⋊C4C22×C314yes—
98C4×D12—28no—
99C4×D12C3×C418yescomplemented
100C3×#95—21no—
101C3×#95—21no—
102C3×#95—32no—
103C3×#95—34no—
104C3×#95—32no—
105C3×#95C2×D311yescharacteristic
106C3×#95C2×D311yescharacteristic
107C3×#95C22×C311yescharacteristic
108C3×#95C22×C311yescharacteristic
109C3⋊#95C22×C312yes—
110C3⋊#95C22×C314yes—
111C3⋊#95C22×C312yes—
112C23×C32C2411yescharacteristic
113C22×C3×D3—24no—
114C22×C3×D3—24no—
115C22×C3×D3C22×C414yes—
116C22×C3×D3C2×D414yes—
117C22×C3×D3C2×Q814yes—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4—22no—
121C2×C32×C4—22no—
122C2×C32×C4—22no—
123C2×C32×C4—22no—
124C2×C32×C4—22no—
125C2×C32×C4C22×C412yes—
126C2×C32×C4C22×C412yes—
127C2×C32×C4C22×C412yes—
128C2×C32×C4C2×D412yes—
129C2×C32×C4C2×Q812yes—
130C2×C32×C4#4114yes—
131C2×C3×Dic3—24no—
132C2×C3×Dic3—24no—
133C2×C3×Dic3—24no—
134C2×C3×Dic3—24no—
135C2×C3×Dic3C22×C414yes—
136C3×D12C4⋊C4116yescomplemented
137C3×C4×D3—48no—
138C3×C4×D3—416no—
139C2×C4×D4—31no—
140C2×C4×D4—32no—
141#389—32no—
142#385—34no—
143#382—31no—
144#380—32no—
145C4×C4⋊C4C3×D311yescharacteristic, complemented
146C2×#95—32no—
147C24×C3—32no—
148C23×D3C23×C312yes—
149C2×C3×D4—34no—
150C2×C3×D4—34no—
151C2×C3×D4—34no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—32no—
158C22×C3×C4—38no—
159C22×C3×C4—34no—
160C22×C3×C4—68no—
161C22×C3×C4—68no—
162C22×C3×C4—124no—
163C22×C3×C4—124no—
164C22×C3×C4C22×D312yes—
165C22×C3×C4C22×D311yescharacteristic
166C22×C3×C4C22×D311yescharacteristic
167C22×C3×C4C22×D312yes—
168C22×C3×C4C22×D311yescharacteristic
169C22×C3×C4C23×C312yes—
170C22×C3×C4C23×C311yescharacteristic
171C22×C3×C4C23×C311yescharacteristic
172C22×C3×C4C23×C312yes—
173C22×C3×C4C23×C311yescharacteristic
174C22×Dic3C23×C314yes—
175C22×Dic3C23×C312yes—
176C2×D12C2×C3×C414yes—
177C2×D12C3×D414yes—
178C2×D12C3×Q814yes—
179C2×C4×D3—28no—
180C2×C4×D3—28no—
181C2×C4×D3—44no—
182C2×C4×D3—44no—
183C2×C4×D3C2×C3×C418yescomplemented
184C3×C4⋊C4—28no—
185C3×C4⋊C4—24no—
186C3×C4⋊C4—24no—
187C3×C4⋊C4—34no—
188C3×C4⋊C4—64no—
189C3×C4⋊C4—124no—
190C3×C22⋊C4—38no—
191C3×C22⋊C4—68no—
192C3×C42—24no—
193C3×C42—24no—
194C3×C42—24no—
195C3×C42—44no—
196C3×C42C4×D314yescomplemented
197C3×C42C2×C3×C414yescomplemented
198(C2×D3)⋊C4—28no—
199(C2×D3)⋊C4C2×C3×C418yescomplemented
200C4⋊Dic3—24no—
201C4⋊Dic3C2×C3×C414yescomplemented
202Dic3⋊C4—44no—
203C22×C32C23×C411yescharacteristic
204C22×C32C22×D411yescharacteristic
205C22×C32C22×Q811yescharacteristic
206C22×C32C2×#4112yes—
207C22×C32H(5,𝔽2)12yes—
208C2×C3×D3—24no—
209C2×C3×D3—24no—
210C2×C3×D3—28no—
211C2×C3×D3—28no—
212C2×C3×D3C2×C4⋊C418yes—
213C32×C4—22no—
214C32×C4—22no—
215C32×C4—24no—
216C32×C4—42no—
217C32×C4C2×C4⋊C414yes—
218C3×Dic3—24no—
219C3×Dic3—44no—
220C22×D4—31no—
221C23×C4—32no—
222C23×C4—34no—
223C4×D4—38no—
224C4×D4—68no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—32no—
227C2×C4⋊C4—34no—
228C2×C4⋊C4C2×C3×D312yes—
229C2×C22⋊C4—32no—
230C2×C22⋊C4—34no—
231C2×C42C2×C3×D312yes—
232C2×C42C2×C3×D311yescharacteristic
233#95—32no—
234#95—34no—
235#95—32no—
236#95C2×C3×D311yescharacteristic
237#95C2×C3×D311yescharacteristic
238C23×C3—21no—
239C23×C3—34no—
240C23×C3—34no—
241C23×C3—34no—
242C23×C3—64no—
243C23×C3—64no—
244C23×C3C23×D311yescharacteristic
245C23×C3C24×C311yescharacteristic
246C22×D3—24no—
247C22×D3—24no—
248C22×D3C22×C3×C414yes—
249C22×D3C2×C3×D414yes—
250C22×D3C2×C3×Q814yes—
251C3×D4—316no—
252C2×C3×C4—24no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—24no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—34no—
275C2×C3×C4—42no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—64no—
281C2×C3×C4—64no—
282C2×C3×C4—64no—
283C2×C3×C4—64no—
284C2×C3×C4—128no—
285C2×C3×C4—1216no—
286C2×C3×C4C2×C4×D312yes—
287C2×C3×C4C2×C4×D312yes—
288C2×C3×C4C2×C4×D312yes—
289C2×C3×C4#23514yes—
290C2×C3×C4D3×D412yes—
291C2×C3×C4D3×Q812yes—
292C2×C3×C4C22×C3×C412yes—
293C2×C3×C4C22×C3×C412yes—
294C2×C3×C4C22×C3×C412yes—
295C2×C3×C4C2×C3×D412yes—
296C2×C3×C4C2×C3×Q812yes—
297C2×C3×C4C3×#4114yes—
298C2×Dic3—24no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—24no—
302C2×Dic3C22×C3×C414yes—
303D12C3×C4⋊C4116yescomplemented
304C4×D3—48no—
305C4×D3—416no—
306C2×C32C22×C4⋊C411yescharacteristic
307C2×C32C4×#4111yescharacteristic
308C2×C32#51811yescharacteristic
309C2×C32#54612yes—
310C2×C32#55412yes—
311C3×D3—28no—
312C24—32no—
313C2×D4—34no—
314C2×D4—34no—
315C2×D4—34no—
316C22×C4—32no—
317C22×C4—38no—
318C22×C4—34no—
319C22×C4—68no—
320C22×C4—68no—
321C22×C4—124no—
322C22×C4—124no—
323C22×C4C22×C3×D312yes—
324C22×C4C22×C3×D311yescharacteristic
325C22×C4C22×C3×D311yescharacteristic
326C22×C4C22×C3×D312yes—
327C22×C4C22×C3×D311yescharacteristic
328C4⋊C4—24no—
329C4⋊C4—34no—
330C4⋊C4—64no—
331C4⋊C4—124no—
332C22⋊C4—38no—
333C22⋊C4—68no—
334C42—24no—
335C42C3×C4×D314yescomplemented
336C22×C3—21no—
337C22×C3—22no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—22no—
341C22×C3—38no—
342C22×C3—64no—
343C22×C3—64no—
344C22×C3—68no—
345C22×C3—68no—
346C22×C3C22×C4×D311yescharacteristic
347C22×C3C2×#23512yes—
348C22×C3C2×D3×D411yescharacteristic
349C22×C3C2×D3×Q811yescharacteristic
350C22×C3#100912yes—
351C22×C3C23×C3×C411yescharacteristic
352C22×C3C22×C3×D411yescharacteristic
353C22×C3C22×C3×Q811yescharacteristic
354C22×C3C2×C3×#4112yes—
355C22×C3C3×H(5,𝔽2)12yes—
356C2×D3—24no—
357C2×D3—24no—
358C2×D3—28no—
359C2×D3—28no—
360C2×D3C2×C3×C4⋊C418yes—
361C3×C4—24no—
362C3×C4—22no—
363C3×C4—22no—
364C3×C4—24no—
365C3×C4—22no—
366C3×C4—22no—
367C3×C4—24no—
368C3×C4—24no—
369C3×C4—42no—
370C3×C4—44no—
371C3×C4—44no—
372C3×C4—42no—
373C3×C4—42no—
374C3×C4—64no—
375C3×C4—124no—
376C3×C4D3×C4⋊C414yes—
377C3×C4C2×C3×C4⋊C414yes—
378Dic3—24no—
379Dic3—44no—
380C32#234911yescharacteristic, complemented, Sylow
381C23—34no—
382C23—34no—
383C23—34no—
384C23—64no—
385C23—64no—
386C23C23×C3×D311yescharacteristic
387D4—316no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—34no—
396C2×C4—64no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—64no—
400C2×C4—128no—
401C2×C4—1216no—
402C2×C4C2×C3×C4×D312yes—
403C2×C4C2×C3×C4×D312yes—
404C2×C4C2×C3×C4×D312yes—
405C2×C4C3×#23514yes—
406C2×C4C3×D3×D412yes—
407C2×C4C3×D3×Q812yes—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—21no—
413C2×C3—68no—
414C2×C3C4×#23511yescharacteristic
415C2×C3C2×D3×C4⋊C411yescharacteristic
416C2×C3#552111yescharacteristic
417C2×C3#555012yes—
418C2×C3#557912yes—
419C2×C3C22×C3×C4⋊C411yescharacteristic
420C2×C3C3×C4×#4111yescharacteristic
421C2×C3C3×#51811yescharacteristic
422C2×C3C3×#54612yes—
423C2×C3C3×#55412yes—
424D3—28no—
425C22—38no—
426C22—64no—
427C22—68no—
428C22—68no—
429C22—64no—
430C22C22×C3×C4×D311yescharacteristic
431C22C2×C3×#23512yes—
432C22C2×C3×D3×D411yescharacteristic
433C22C2×C3×D3×Q811yescharacteristic
434C22C3×#100912yes—
435C4—22no—
436C4—24no—
437C4—22no—
438C4—42no—
439C4—64no—
440C4—124no—
441C4C3×D3×C4⋊C414yes—
442C3—21no—
443C3#7377011yescharacteristic, direct product factor
444C3C3×#234911yescharacteristic, complemented
445C2—68no—
446C2C3×C4×#23511yescharacteristic
447C2C2×C3×D3×C4⋊C411yescharacteristic
448C2C3×#552111yescharacteristic
449C2C3×#555012yes—
450C2C3×#557912yes—
subgroupG/H
centerC23×C3C23×D3
commutatorC22×C3C23×C3×C4
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socleC23×C32C24
prime radicalC22×C3×D12C22
cosocleC23×C3C24×C3
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2349
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C32×C4×C4⋊C4C2semidirect
4. C3×#4989C2semidirect
5. C3×#4936C2semidirect
6. C3×#4928C2semidirect
7. C3×#4667C2semidirect
8. C3×D3×#95C2semidirect
10. #73770C3direct (N × H)
23. C3×C4×D12C4semidirect
38. C3×C4×C4⋊C4D3semidirect
39. C3×C4×C4⋊C4C2×C3semidirect
41. #4989C2×C3semidirect
42. #4936C2×C3semidirect
43. #4928C2×C3semidirect
44. #4667C2×C3semidirect
45. D3×#95C2×C3semidirect
61. C2×C3×C4×D3C2×C4semidirect
64. C32×C42C2×C4semidirect
66. C3×(C2×D3)⋊C4C2×C4semidirect
68. C3×C4⋊Dic3C2×C4semidirect
99. C4×D12C3×C4semidirect
136. C3×D12C4⋊C4semidirect
145. C4×C4⋊C4C3×D3semidirect
183. C2×C4×D3C2×C3×C4semidirect
196. C3×C42C4×D3semidirect
197. C3×C42C2×C3×C4semidirect
199. (C2×D3)⋊C4C2×C3×C4semidirect
201. C4⋊Dic3C2×C3×C4semidirect
303. D12C3×C4⋊C4semidirect
335. C42C3×C4×D3semidirect
380. C32#2349semidirect
443. C3#73770direct (N × H)
444. C3C3×#2349semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D122
C32×C4×C4⋊C42
C3×#49892
C3×#49362
C3×#49282
C3×#46672
C3×D3×#952
C3×#23493
#737703