Finite Group Databasebeta

← C3×#73645 #11806029

Subgroups — C3×#73645 #11806029

Order 1152.

1217 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D12C211yesmaximal, characteristic
2C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
3C3×#4990C214yesmaximal, complemented
4C3×#4939C212yesmaximal
5C3×#4936C211yesmaximal, characteristic, complemented
6C3×C4×C4⋊Dic3C212yesmaximal, complemented
7C3×#4666C212yesmaximal, complemented
8C3×#4664C212yesmaximal, complemented
9C3×#2357—31nomaximal
10#73645C311yesmaximal, characteristic, direct product factor
11C22×C3×D12C2211yescharacteristic
12C22×C3×C4×D3C2212yes—
13C2×C32×C4⋊C4C2212yes—
14C2×C32×C42C2212yes—
15C2×C32×C42C2211yescharacteristic
16C2×C3×(C2×D3)⋊C4C2214yes—
17C2×C3×(C2×D3)⋊C4C2214yes—
18C2×C3×(C2×D3)⋊C4C2212yes—
19C2×C3×C4⋊Dic3C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×C4×Dic3C2214yes—
22C3×C4×D12C418yescomplemented
23C32×#95C2211yescharacteristic
24C32×#95C2211yescharacteristic
25C3×C3⋊#95C2212yes—
26C3×C3⋊#95C2214yes—
27C3×C3⋊#95C2212yes—
28C2×C3×C4×D4—31no—
29C2×C4×D12C2×C311yescharacteristic
30C3×#390—32no—
31C3×#388—34no—
32C3×#382—31no—
33C3×#381—32no—
34C3×C4×C4⋊C4—21no—
35C3×C4×C4⋊C4—32no—
36C3×C4×C4⋊C4D311yescharacteristic, complemented
37C3×C4×C4⋊C4C2×C311yescharacteristic, complemented
38C3×C4×C22⋊C4—32no—
39#4990C2×C314yescomplemented
40#4939C2×C312yes—
41#4936C2×C311yescharacteristic, complemented
42C4×C4⋊Dic3C2×C312yescomplemented
43#4666C2×C312yescomplemented
44#4664C2×C312yescomplemented
45C23×C3×D3C2312yes—
46C22×C32×C4C2312yes—
47C22×C32×C4C2311yescharacteristic
48C22×C32×C4C2311yescharacteristic
49C22×C32×C4C2312yes—
50C22×C32×C4C2311yescharacteristic
51C22×C3×Dic3C2314yes—
52C22×C3×Dic3C2312yes—
53C2×C3×D12—22no—
54C2×C3×D12—22no—
55C2×C3×D12C2×C414yes—
56C2×C3×C4×D3—44no—
57C2×C3×C4×D3C2×C418yes—
58C32×C4⋊C4—24no—
59C32×C42—24no—
60C32×C42C2×C414yes—
61C3×(C2×D3)⋊C4—28no—
62C3×(C2×D3)⋊C4—44no—
63C3×(C2×D3)⋊C4C2×C418yes—
64C3×C4⋊Dic3—24no—
65C3×C4⋊Dic3C2×C414yes—
66C3×C4×Dic3—44no—
67#2357—31noSylow
68C22×C3×D4—31no—
69C23×C3×C4—32no—
70C22×D12C22×C311yescharacteristic
71C22×C4×D3C22×C312yes—
72C3×C4×D4—38no—
73C2×C3×C4⋊C4—22no—
74C2×C3×C4⋊C4—31no—
75C2×C3×C4⋊C4—32no—
76C2×C3×C4⋊C4C2×D312yes—
77C2×C3×C4⋊C4C22×C312yes—
78C2×C3×C22⋊C4—32no—
79C2×C3×C22⋊C4—34no—
80C2×C3×C22⋊C4—34no—
81C2×C3×C42—21no—
82C2×C3×C42—22no—
83C2×C3×C42—34no—
84C2×C3×C42C2×D312yes—
85C2×C3×C42C2×D311yescharacteristic
86C2×C3×C42C22×C312yes—
87C2×C3×C42C22×C311yescharacteristic
88C2×(C2×D3)⋊C4C22×C314yes—
89C2×(C2×D3)⋊C4C22×C314yes—
90C2×(C2×D3)⋊C4C22×C312yes—
91C2×C4⋊Dic3C22×C312yes—
92C2×C4⋊Dic3C22×C311yescharacteristic
93C2×C4×Dic3C22×C314yes—
94C4×D12C3×C418yescomplemented
95C3×#95—21no—
96C3×#95—21no—
97C3×#95—32no—
98C3×#95—34no—
99C3×#95—32no—
100C3×#95C2×D311yescharacteristic
101C3×#95C2×D311yescharacteristic
102C3×#95C22×C311yescharacteristic
103C3×#95C22×C311yescharacteristic
104C3⋊#95C22×C312yes—
105C3⋊#95C22×C314yes—
106C3⋊#95C22×C312yes—
107C23×C32C2411yescharacteristic
108C22×C3×D3—24no—
109C22×C3×D3—22no—
110C22×C3×D3—22no—
111C22×C3×D3—24no—
112C22×C3×D3C22×C414yes—
113C2×C32×C4—22no—
114C2×C32×C4—22no—
115C2×C32×C4—22no—
116C2×C32×C4—22no—
117C2×C32×C4—22no—
118C2×C32×C4—22no—
119C2×C32×C4—22no—
120C2×C32×C4C22×C412yes—
121C2×C32×C4C22×C412yes—
122C2×C32×C4C22×C412yes—
123C2×C32×C4#4114yes—
124C2×C32×C4#4112yes—
125C2×C32×C4#4112yes—
126C2×C3×Dic3—24no—
127C2×C3×Dic3—24no—
128C2×C3×Dic3—24no—
129C2×C3×Dic3—24no—
130C2×C3×Dic3C22×C414yes—
131C3×D12—28no—
132C3×C4×D3—48no—
133C2×C4×D4—31no—
134#390—32no—
135#388—34no—
136#382—31no—
137#381—32no—
138C4×C4⋊C4—32no—
139C4×C4⋊C4C3×D311yescharacteristic, complemented
140C4×C22⋊C4—32no—
141C24×C3—32no—
142C23×D3C23×C312yes—
143C2×C3×D4—34no—
144C2×C3×D4—62no—
145C2×C3×D4—62no—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—32no—
152C22×C3×C4—38no—
153C22×C3×C4—34no—
154C22×C3×C4—124no—
155C22×C3×C4C22×D312yes—
156C22×C3×C4C22×D311yescharacteristic
157C22×C3×C4C22×D311yescharacteristic
158C22×C3×C4C22×D312yes—
159C22×C3×C4C22×D311yescharacteristic
160C22×C3×C4C23×C312yes—
161C22×C3×C4C23×C311yescharacteristic
162C22×C3×C4C23×C311yescharacteristic
163C22×C3×C4C23×C312yes—
164C22×C3×C4C23×C311yescharacteristic
165C22×Dic3C23×C314yes—
166C22×Dic3C23×C312yes—
167C2×D12—22no—
168C2×D12—22no—
169C2×D12C2×C3×C414yes—
170C2×C4×D3—44no—
171C2×C4×D3C2×C3×C418yes—
172C3×C4⋊C4—28no—
173C3×C4⋊C4—24no—
174C3×C4⋊C4—24no—
175C3×C4⋊C4—34no—
176C3×C4⋊C4—64no—
177C3×C22⋊C4—38no—
178C3×C22⋊C4—68no—
179C3×C22⋊C4—124no—
180C3×C42—24no—
181C3×C42—24no—
182C3×C42—24no—
183C3×C42—44no—
184C3×C42—124no—
185C3×C42C4×D314yes—
186C3×C42C2×C3×C414yes—
187(C2×D3)⋊C4—28no—
188(C2×D3)⋊C4—44no—
189(C2×D3)⋊C4C2×C3×C418yes—
190C4⋊Dic3—24no—
191C4⋊Dic3C2×C3×C414yes—
192C4×Dic3—44no—
193C22×C32C23×C411yescharacteristic
194C22×C32C2×#4112yes—
195C22×C32C2×#4111yescharacteristic
196C22×C32C2×#4111yescharacteristic
197C22×C32H(5,𝔽2)12yes—
198C2×C3×D3—24no—
199C2×C3×D3—24no—
200C2×C3×D3—24no—
201C2×C3×D3—44no—
202C2×C3×D3—44no—
203C32×C4—22no—
204C32×C4—22no—
205C32×C4—24no—
206C32×C4—42no—
207C32×C4#11714yes—
208C3×Dic3—24no—
209C3×Dic3—44no—
210C22×D4—31no—
211C23×C4—32no—
212C4×D4—38no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—32no—
215C2×C4⋊C4C2×C3×D312yes—
216C2×C22⋊C4—32no—
217C2×C22⋊C4—34no—
218C2×C22⋊C4—34no—
219C2×C42—34no—
220C2×C42C2×C3×D312yes—
221C2×C42C2×C3×D311yescharacteristic
222#95—32no—
223#95—34no—
224#95—32no—
225#95C2×C3×D311yescharacteristic
226#95C2×C3×D311yescharacteristic
227C23×C3—21no—
228C23×C3—34no—
229C23×C3—64no—
230C23×C3—62no—
231C23×C3—62no—
232C23×C3—64no—
233C23×C3C23×D311yescharacteristic
234C23×C3C24×C311yescharacteristic
235C22×D3—24no—
236C22×D3—22no—
237C22×D3—22no—
238C22×D3—24no—
239C22×D3C22×C3×C414yes—
240C3×D4—68no—
241C2×C3×C4—24no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—24no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
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—24no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—34no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—64no—
270C2×C3×C4—64no—
271C2×C3×C4—64no—
272C2×C3×C4—64no—
273C2×C3×C4—128no—
274C2×C3×C4C2×C4×D312yes—
275C2×C3×C4C2×C4×D312yes—
276C2×C3×C4C2×C4×D312yes—
277C2×C3×C4#23514yes—
278C2×C3×C4#23712yes—
279C2×C3×C4#23912yes—
280C2×C3×C4C22×C3×C412yes—
281C2×C3×C4C22×C3×C412yes—
282C2×C3×C4C22×C3×C412yes—
283C2×C3×C4C3×#4114yes—
284C2×C3×C4C3×#4112yes—
285C2×C3×C4C3×#4112yes—
286C2×Dic3—24no—
287C2×Dic3—24no—
288C2×Dic3—24no—
289C2×Dic3—24no—
290C2×Dic3C22×C3×C414yes—
291D12—28no—
292C4×D3—48no—
293C2×C32C2×#11711yescharacteristic
294C2×C32C4×#4111yescharacteristic
295C2×C32#51811yescharacteristic
296C2×C32#55312yes—
297C2×C32#55912yes—
298C3×D3—44no—
299C24—32no—
300C2×D4—34no—
301C2×D4—62no—
302C2×D4—62no—
303C22×C4—32no—
304C22×C4—38no—
305C22×C4—34no—
306C22×C4—124no—
307C22×C4C22×C3×D312yes—
308C22×C4C22×C3×D311yescharacteristic
309C22×C4C22×C3×D311yescharacteristic
310C22×C4C22×C3×D312yes—
311C22×C4C22×C3×D311yescharacteristic
312C4⋊C4—24no—
313C4⋊C4—34no—
314C4⋊C4—64no—
315C22⋊C4—38no—
316C22⋊C4—68no—
317C22⋊C4—124no—
318C42—24no—
319C42—124no—
320C42C3×C4×D314yes—
321C22×C3—21no—
322C22×C3—22no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—64no—
327C22×C3—64no—
328C22×C3—64no—
329C22×C3—124no—
330C22×C3—124no—
331C22×C3C22×C4×D311yescharacteristic
332C22×C3C2×#23512yes—
333C22×C3C2×#23711yescharacteristic
334C22×C3C2×#23911yescharacteristic
335C22×C3#100912yes—
336C22×C3C23×C3×C411yescharacteristic
337C22×C3C2×C3×#4112yes—
338C22×C3C2×C3×#4111yescharacteristic
339C22×C3C2×C3×#4111yescharacteristic
340C22×C3C3×H(5,𝔽2)12yes—
341C2×D3—24no—
342C2×D3—24no—
343C2×D3—24no—
344C2×D3—44no—
345C2×D3—44no—
346C3×C4—24no—
347C3×C4—22no—
348C3×C4—22no—
349C3×C4—24no—
350C3×C4—22no—
351C3×C4—22no—
352C3×C4—24no—
353C3×C4—24no—
354C3×C4—42no—
355C3×C4—44no—
356C3×C4—44no—
357C3×C4—42no—
358C3×C4—42no—
359C3×C4—64no—
360C3×C4—124no—
361C3×C4#89214yes—
362C3×C4C3×#11714yes—
363Dic3—24no—
364Dic3—44no—
365C32#235711yescharacteristic, complemented, Sylow
366C23—34no—
367C23—64no—
368C23—62no—
369C23—62no—
370C23—64no—
371C23C23×C3×D311yescharacteristic
372D4—68no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—34no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—64no—
384C2×C4—64no—
385C2×C4—128no—
386C2×C4C2×C3×C4×D312yes—
387C2×C4C2×C3×C4×D312yes—
388C2×C4C2×C3×C4×D312yes—
389C2×C4C3×#23514yes—
390C2×C4C3×#23712yes—
391C2×C4C3×#23912yes—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—22no—
395C2×C3—22no—
396C2×C3—21no—
397C2×C3—124no—
398C2×C3C4×#23511yescharacteristic
399C2×C3C2×#89211yescharacteristic
400C2×C3#552111yescharacteristic
401C2×C3#556512yes—
402C2×C3#558512yes—
403C2×C3C2×C3×#11711yescharacteristic
404C2×C3C3×C4×#4111yescharacteristic
405C2×C3C3×#51811yescharacteristic
406C2×C3C3×#55312yes—
407C2×C3C3×#55912yes—
408D3—44no—
409C22—64no—
410C22—64no—
411C22—64no—
412C22—124no—
413C22—124no—
414C22C22×C3×C4×D311yescharacteristic
415C22C2×C3×#23512yes—
416C22C2×C3×#23711yescharacteristic
417C22C2×C3×#23911yescharacteristic
418C22C3×#100912yes—
419C4—22no—
420C4—24no—
421C4—22no—
422C4—42no—
423C4—64no—
424C4—124no—
425C4C3×#89214yes—
426C3—21no—
427C3#7364511yescharacteristic, direct product factor
428C3C3×#235711yescharacteristic, complemented
429C2—124no—
430C2C3×C4×#23511yescharacteristic
431C2C2×C3×#89211yescharacteristic
432C2C3×#552111yescharacteristic
433C2C3×#556512yes—
434C2C3×#558512yes—
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×#2357
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
2. C32×C4×C4⋊C4C2semidirect
3. C3×#4990C2semidirect
5. C3×#4936C2semidirect
6. C3×C4×C4⋊Dic3C2semidirect
7. C3×#4666C2semidirect
8. C3×#4664C2semidirect
10. #73645C3direct (N × H)
22. C3×C4×D12C4semidirect
36. C3×C4×C4⋊C4D3semidirect
37. C3×C4×C4⋊C4C2×C3semidirect
39. #4990C2×C3semidirect
41. #4936C2×C3semidirect
42. C4×C4⋊Dic3C2×C3semidirect
43. #4666C2×C3semidirect
44. #4664C2×C3semidirect
94. C4×D12C3×C4semidirect
139. C4×C4⋊C4C3×D3semidirect
365. C32#2357semidirect
427. C3#73645direct (N × H)
428. C3C3×#2357semidirect

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×#49902
C3×#49392
C3×#49362
C3×C4×C4⋊Dic32
C3×#46662
C3×#46642
C3×#23573
#736453