Finite Group Databasebeta

← C3×#73669 #11808841

Subgroups — C3×#73669 #11808841

Order 1152.

1105 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×Dic6C212yesmaximal
2C32×C4×C4⋊C4C211yesmaximal, characteristic
3C3×#4982C212yesmaximal
4C3×#4979C212yesmaximal
5C3×C4×C4⋊Dic3C211yesmaximal, characteristic
6C3×#4932C211yesmaximal, characteristic
7C3×C4×Dic3⋊C4C212yesmaximal
8C3×#4657C212yesmaximal
9C3×#4656C212yesmaximal
10C3×#2399—31nomaximal
11#73669C311yesmaximal, characteristic, direct product factor
12C22×C3×Dic6C2212yes—
13C2×C32×C4⋊C4C2212yes—
14C2×C32×C42C2212yes—
15C2×C32×C42C2211yescharacteristic
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×C4⋊Dic3C2212yes—
18C2×C3×Dic3⋊C4C2212yes—
19C2×C3×Dic3⋊C4C2212yes—
20C2×C3×Dic3⋊C4C2212yes—
21C2×C3×Dic3⋊C4C2212yes—
22C2×C3×C4×Dic3C2212yes—
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C3×C4×Dic6—28no—
26C32×#95C2211yescharacteristic
27C32×#95C2211yescharacteristic
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30C3×C3⋊#95C2212yes—
31C3×C3⋊#95C2212yes—
32C3×C3⋊#95C2211yescharacteristic
33C3×C3⋊#95C2211yescharacteristic
34C2×C3×C4×Q8—32no—
35C2×C4×Dic6C2×C312yes—
36C3×#398—32no—
37C3×#398—32no—
38C3×#395—32no—
39C3×#395—32no—
40C3×C4×C4⋊C4—21no—
41C3×C4×C4⋊C4—32no—
42C3×C4×C4⋊C4—31no—
43C3×C4×C4⋊C4D311yescharacteristic
44C3×C4×C4⋊C4C2×C311yescharacteristic
45C3×#376—31no—
46#4982C2×C312yes—
47#4979C2×C312yes—
48C4×C4⋊Dic3C2×C311yescharacteristic
49#4932C2×C311yescharacteristic
50C4×Dic3⋊C4C2×C312yes—
51#4657C2×C312yes—
52#4656C2×C312yes—
53C22×C32×C4C2312yes—
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2311yescharacteristic
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2312yes—
61C22×C3×Dic3C2312yes—
62C2×C3×Dic6—24no—
63C2×C3×Dic6—24no—
64C2×C3×Dic6—24no—
65C32×C4⋊C4—24no—
66C32×C42—24no—
67C32×C42Q814yescomplemented
68C3×C4⋊Dic3—24no—
69C3×C4⋊Dic3—24no—
70C3×Dic3⋊C4—24no—
71C3×Dic3⋊C4—24no—
72C3×Dic3⋊C4—24no—
73C3×Dic3⋊C4—24no—
74C3×C4×Dic3—24no—
75C3×C4×Dic3D414yes—
76C3×C4×Dic3Q814yescomplemented
77#2399—31noSylow
78C22×C3×Q8—32no—
79C22×Dic6C22×C312yes—
80C3×C4×Q8—68no—
81C2×C3×C4⋊C4—22no—
82C2×C3×C4⋊C4—32no—
83C2×C3×C4⋊C4—32no—
84C2×C3×C4⋊C4—32no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4C2×D312yes—
89C2×C3×C4⋊C4C22×C312yes—
90C2×C3×C42—21no—
91C2×C3×C42—22no—
92C2×C3×C42—31no—
93C2×C3×C42—31no—
94C2×C3×C42—32no—
95C2×C3×C42C2×D312yes—
96C2×C3×C42C2×D311yescharacteristic
97C2×C3×C42C22×C312yes—
98C2×C3×C42C22×C311yescharacteristic
99C2×C4⋊Dic3C22×C312yes—
100C2×C4⋊Dic3C22×C312yes—
101C2×Dic3⋊C4C22×C312yes—
102C2×Dic3⋊C4C22×C312yes—
103C2×Dic3⋊C4C22×C312yes—
104C2×Dic3⋊C4C22×C312yes—
105C2×C4×Dic3C22×C312yes—
106C2×C4×Dic3C22×C311yescharacteristic
107C2×C4×Dic3C22×C311yescharacteristic
108C4×Dic6—28no—
109C3×#95—21no—
110C3×#95—21no—
111C3×#95—32no—
112C3×#95—32no—
113C3×#95—32no—
114C3×#95—31no—
115C3×#95—32no—
116C3×#95—31no—
117C3×#95C2×D311yescharacteristic
118C3×#95C2×D311yescharacteristic
119C3×#95C22×C311yescharacteristic
120C3×#95C22×C311yescharacteristic
121C3⋊#95C22×C312yes—
122C3⋊#95C22×C312yes—
123C3⋊#95C22×C312yes—
124C3⋊#95C22×C312yes—
125C3⋊#95C22×C311yescharacteristic
126C3⋊#95C22×C311yescharacteristic
127C23×C32C2411yescharacteristic
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—22no—
131C2×C32×C4—22no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4C2×D412yes—
136C2×C32×C4C2×Q812yes—
137C2×C32×C4C2×Q812yes—
138C2×C32×C4C2×Q812yes—
139C2×C32×C4C2×Q812yes—
140C2×C32×C4#4114yes—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3C2×D414yes—
151C2×C3×Dic3C2×Q814yes—
152C2×C3×Dic3#4114yes—
153C3×Dic6—216no—
154C2×C4×Q8—32no—
155#398—32no—
156#398—32no—
157#395—32no—
158#395—32no—
159C4×C4⋊C4—32no—
160C4×C4⋊C4—31no—
161C4×C4⋊C4C3×D311yescharacteristic
162#376—31no—
163C2×C3×Q8—64no—
164C2×C3×Q8—64no—
165C2×C3×Q8—64no—
166C22×C3×C4—21no—
167C22×C3×C4—21no—
168C22×C3×C4—21no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4C22×D312yes—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C22×C3×C4C22×D312yes—
179C22×C3×C4C22×D311yescharacteristic
180C22×C3×C4C23×C312yes—
181C22×C3×C4C23×C311yescharacteristic
182C22×C3×C4C23×C311yescharacteristic
183C22×C3×C4C23×C312yes—
184C22×C3×C4C23×C311yescharacteristic
185C22×Dic3C23×C312yes—
186C22×Dic3C23×C312yes—
187C22×Dic3C23×C312yes—
188C22×Dic3C23×C312yes—
189C2×Dic6—24no—
190C2×Dic6—24no—
191C2×Dic6—24no—
192C3×C4⋊C4—28no—
193C3×C4⋊C4—24no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—64no—
196C3×C4⋊C4—64no—
197C3×C4⋊C4—64no—
198C3×C4⋊C4—64no—
199C3×C4⋊C4—64no—
200C3×C4⋊C4—64no—
201C3×C42—24no—
202C3×C42—24no—
203C3×C42—24no—
204C3×C42—34no—
205C3×C42—34no—
206C3×C42—44no—
207C3×C42—64no—
208C3×C42Dic614yescomplemented
209C3×C42C3×Q814yescomplemented
210C4⋊Dic3—24no—
211C4⋊Dic3—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—24no—
215Dic3⋊C4—24no—
216C4×Dic3—24no—
217C4×Dic3C3×D414yes—
218C4×Dic3C3×Q814yescomplemented
219C22×C32C22×D411yescharacteristic
220C22×C32C22×Q811yescharacteristic
221C22×C32C22×Q811yescharacteristic
222C22×C32C2×#4112yes—
223C22×C32C2×#4112yes—
224C32×C4—22no—
225C32×C4—24no—
226C32×C4—22no—
227C32×C4—42no—
228C32×C4C4⋊Q814yes—
229C3×Dic3—24no—
230C3×Dic3—24no—
231C3×Dic3—24no—
232C3×Dic3—42no—
233C22×Q8—32no—
234C4×Q8—68no—
235C2×C4⋊C4—32no—
236C2×C4⋊C4—32no—
237C2×C4⋊C4—32no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4C2×C3×D312yes—
242C2×C42—31no—
243C2×C42—31no—
244C2×C42—32no—
245C2×C42C2×C3×D312yes—
246C2×C42C2×C3×D311yescharacteristic
247#95—32no—
248#95—32no—
249#95—32no—
250#95—31no—
251#95—32no—
252#95—31no—
253#95C2×C3×D311yescharacteristic
254#95C2×C3×D311yescharacteristic
255C23×C3—21no—
256C23×C3C23×D311yescharacteristic
257C23×C3C24×C311yescharacteristic
258C3×Q8—616no—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
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—24no—
269C2×C3×C4—22no—
270C2×C3×C4—22no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—34no—
282C2×C3×C4—34no—
283C2×C3×C4—34no—
284C2×C3×C4—42no—
285C2×C3×C4—42no—
286C2×C3×C4—42no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4—62no—
296C2×C3×C4—62no—
297C2×C3×C4—62no—
298C2×C3×C4C2×Dic612yes—
299C2×C3×C4C2×Dic612yes—
300C2×C3×C4C2×Dic612yes—
301C2×C3×C4#23514yes—
302C2×C3×C4D3×D412yes—
303C2×C3×C4D3×Q812yes—
304C2×C3×C4C2×C3×D412yes—
305C2×C3×C4C2×C3×Q812yes—
306C2×C3×C4C2×C3×Q812yes—
307C2×C3×C4C2×C3×Q812yes—
308C2×C3×C4C2×C3×Q812yes—
309C2×C3×C4C3×#4114yes—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3C2×C3×D414yes—
320C2×Dic3C2×C3×Q814yes—
321C2×Dic3C3×#4114yes—
322Dic6—216no—
323C2×C32#52512yes—
324C2×C32C2×C4⋊Q811yescharacteristic
325C2×C32#53311yescharacteristic
326C2×C32#53311yescharacteristic
327C2×C32#53312yes—
328C2×Q8—64no—
329C2×Q8—64no—
330C2×Q8—64no—
331C22×C4—32no—
332C22×C4—32no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4C22×C3×D312yes—
336C22×C4C22×C3×D311yescharacteristic
337C22×C4C22×C3×D311yescharacteristic
338C22×C4C22×C3×D312yes—
339C22×C4C22×C3×D311yescharacteristic
340C4⋊C4—24no—
341C4⋊C4—64no—
342C4⋊C4—64no—
343C4⋊C4—64no—
344C4⋊C4—64no—
345C4⋊C4—64no—
346C4⋊C4—64no—
347C42—24no—
348C42—34no—
349C42—34no—
350C42—64no—
351C42C3×Dic614yescomplemented
352C22×C3—21no—
353C22×C3—22no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—22no—
357C22×C3C22×Dic611yescharacteristic
358C22×C3C2×#23512yes—
359C22×C3C2×D3×D411yescharacteristic
360C22×C3C2×D3×Q811yescharacteristic
361C22×C3D3×#4112yes—
362C22×C3C22×C3×D411yescharacteristic
363C22×C3C22×C3×Q811yescharacteristic
364C22×C3C22×C3×Q811yescharacteristic
365C22×C3C2×C3×#4112yes—
366C22×C3C2×C3×#4112yes—
367C3×C4—24no—
368C3×C4—22no—
369C3×C4—22no—
370C3×C4—24no—
371C3×C4—24no—
372C3×C4—22no—
373C3×C4—22no—
374C3×C4—24no—
375C3×C4—42no—
376C3×C4—44no—
377C3×C4—44no—
378C3×C4—42no—
379C3×C4—42no—
380C3×C4—64no—
381C3×C4—64no—
382C3×C4—64no—
383C3×C4—122no—
384C3×C4#88814yes—
385C3×C4C3×C4⋊Q814yes—
386Dic3—24no—
387Dic3—24no—
388Dic3—24no—
389Dic3—42no—
390C32#239911yescharacteristic, complemented, Sylow
391C23C23×C3×D311yescharacteristic
392Q8—616no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—34no—
401C2×C4—34no—
402C2×C4—34no—
403C2×C4—62no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4C2×C3×Dic612yes—
413C2×C4C2×C3×Dic612yes—
414C2×C4C2×C3×Dic612yes—
415C2×C4C3×#23514yes—
416C2×C4C3×D3×D412yes—
417C2×C4C3×D3×Q812yes—
418C2×C3—21no—
419C2×C3—21no—
420C2×C3—22no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3#547011yescharacteristic
424C2×C3C2×#88811yescharacteristic
425C2×C3#551711yescharacteristic
426C2×C3#554712yes—
427C2×C3#557812yes—
428C2×C3C3×#52512yes—
429C2×C3C2×C3×C4⋊Q811yescharacteristic
430C2×C3C3×#53311yescharacteristic
431C2×C3C3×#53312yes—
432C2×C3C3×#53311yescharacteristic
433C22C22×C3×Dic611yescharacteristic
434C22C2×C3×#23512yes—
435C22C2×C3×D3×D411yescharacteristic
436C22C2×C3×D3×Q811yescharacteristic
437C22C3×D3×#4112yes—
438C4—22no—
439C4—24no—
440C4—22no—
441C4—42no—
442C4—64no—
443C4—64no—
444C4—64no—
445C4—122no—
446C4C3×#88814yes—
447C3—21no—
448C3#7366911yescharacteristic, direct product factor
449C3C3×#239911yescharacteristic, complemented
450C2C3×#547011yescharacteristic
451C2C2×C3×#88811yescharacteristic
452C2C3×#551711yescharacteristic
453C2C3×#554712yes—
454C2C3×#557812yes—
subgroupG/H
centerC22×C3×C4C22×D3
commutator, cosocleC23×C3C24×C3
FrattiniC23C23×C3×D3
FittingC32×C4×C4⋊C4C2
socle, prime radicalC23×C32C24
hypercenterC3×C4×C4⋊C4D3
hypocenterC3C3×#2399
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C3×C4 ◁ C3×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
11. #73669C3direct (N × H)
67. C32×C42Q8semidirect
76. C3×C4×Dic3Q8semidirect
208. C3×C42Dic6semidirect
209. C3×C42C3×Q8semidirect
218. C4×Dic3C3×Q8semidirect
351. C42C3×Dic6semidirect
390. C32#2399semidirect
448. C3#73669direct (N × H)
449. C3C3×#2399semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×Dic62
C32×C4×C4⋊C42
C3×#49822
C3×#49792
C3×C4×C4⋊Dic32
C3×#49322
C3×C4×Dic3⋊C42
C3×#46572
C3×#46562
C3×#23993
#736693