Finite Group Databasebeta

← #83081

Subgroups — #83081

Order 384.

664 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#126C211yesmaximal, characteristic, complemented
2C2×D3×C22⋊C4C211yesmaximal, characteristic
3#4994C211yesmaximal, characteristic, complemented
4#4992C211yesmaximal, characteristic, complemented
5#4990C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4987C211yesmaximal, characteristic, complemented
9#4955C211yesmaximal, characteristic
10#4955C211yesmaximal, characteristic
11#4952C211yesmaximal, characteristic, complemented
12#4952C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4940C211yesmaximal, characteristic, complemented
15C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
16#2480—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C3×#126—24no—
21C2×C3×C4⋊C4C2211yescharacteristic, complemented
22C2×C3×C4⋊C4C2211yescharacteristic, complemented
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C42C2211yescharacteristic, complemented
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×C4⋊Dic3C2211yescharacteristic, complemented
42C2×Dic3⋊C4C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic3C2211yescharacteristic, complemented
45D3×C22⋊C4—24no—
46C3⋊#95C2211yescharacteristic, complemented
47C3⋊#95C2211yescharacteristic, complemented
48C3⋊#95C2211yescharacteristic, complemented
49C3⋊#95C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#126D311yescharacteristic, complemented
55C22×C22⋊C4—31no—
56#397—31no—
57#397—31no—
58#397—31no—
59#396—31no—
60#388—31no—
61#388—31no—
62#388—31no—
63#385—31no—
64#385—31no—
65#380—31no—
66#379—31no—
67#379—31no—
68C4×C22⋊C4—31no—
69C24×C3C2311yescharacteristic
70C23×D3—22no—
71C23×D3—22no—
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3C2311yescharacteristic
78C23×D3C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C2×C4×D3—22no—
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C3×C4⋊C4—22no—
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—22no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4D414yes—
103C3×C42—22no—
104#217—22no—
105#217—41no—
106#217—41no—
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4—22no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117C4⋊Dic3—41no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—41no—
121C25—31no—
122C23×C4—31no—
123C23×C4—31no—
124#126—24no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4C2×D311yescharacteristic, complemented
129C2×C4⋊C4C2×D311yescharacteristic, complemented
130C2×C4⋊C4C2×D311yescharacteristic, complemented
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—64no—
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C22⋊C4C2×D311yescharacteristic
148C2×C42—31no—
149C2×C42C2×D311yescharacteristic, complemented
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95—31no—
157#95—31no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3C2×D412yes—
165C23×C3C2411yescharacteristic
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—28no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—44no—
178C22×D3—44no—
179C22×D3—44no—
180C22×D3—44no—
181C22×D3—44no—
182C22×D3—44no—
183C22×D3#4112yes—
184C22×D3#4112yes—
185C22×D3#4112yes—
186C22×D3#4112yes—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4C2×D412yes—
204C2×C3×C4C2×D412yes—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C4×D3—44no—
224C4×D3—44no—
225C24—31no—
226C24—31no—
227C24—62no—
228C24—62no—
229C24—62no—
230C24—62no—
231C24—62no—
232C24—121no—
233C24—121no—
234C24C22×D311yescharacteristic
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—62no—
242C22×C4—62no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—121no—
246C22×C4—121no—
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C4⋊C4—121no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—62no—
262C22⋊C4—62no—
263C22⋊C4—62no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4#6714yes—
275C42—22no—
276C42—121no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—22no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C22×C3C2×#4111yescharacteristic
286C22×C3C2×#4111yescharacteristic
287C22×C3C2×#4111yescharacteristic
288C22×C3C2×#4111yescharacteristic
289C22×C3H(5,𝔽2)11yescharacteristic
290C22×C3H(5,𝔽2)11yescharacteristic
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3—22no—
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—416no—
301C2×D3—41no—
302C2×D3—41no—
303C2×D3—41no—
304C2×D3—41no—
305C2×D3#12614yescomplemented
306C3×C4—22no—
307C3×C4—22no—
308C3×C4—41no—
309C3×C4—41no—
310C3×C4—41no—
311C3×C4—41no—
312Dic3—41no—
313Dic3—41no—
314Dic3—41no—
315Dic3—41no—
316Dic3—41no—
317Dic3—41no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—32no—
325C23—32no—
326C23—32no—
327C23—32no—
328C23—68no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—124no—
340C23—124no—
341C23—124no—
342C23—124no—
343C23—124no—
344C23—124no—
345C23C2×#6712yes—
346C23C23×D311yescharacteristic
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—124no—
382C2×C4—124no—
383C2×C4C2×#6712yes—
384C2×C4C2×#6712yes—
385C2×C3—42no—
386C2×C3#52511yescharacteristic
387C2×C3C2×#12611yescharacteristic
388C2×C3#53811yescharacteristic
389C2×C3#54611yescharacteristic
390C2×C3#54611yescharacteristic
391C2×C3#55111yescharacteristic
392C2×C3#55111yescharacteristic
393D3—24no—
394D3—42no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—34no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—62no—
404C22—62no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—62no—
412C22—1216no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22D3×#4111yescharacteristic
418C22D3×#4111yescharacteristic
419C22D3×#4111yescharacteristic
420C22D3×#4111yescharacteristic
421C22#100911yescharacteristic
422C22#100911yescharacteristic
423C22C22×#6711yescharacteristic
424C4—22no—
425C4—22no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—121no—
431C4—121no—
432C4—121no—
433C4—121no—
434C4—121no—
435C4—121no—
436C3#248011yescharacteristic, complemented, Sylow
437C2—42no—
438C2—64no—
439C2—122no—
440C2D3×#12611yescharacteristic
441C2#570411yescharacteristic
442C2#570511yescharacteristic
443C2#570511yescharacteristic
444C2#582811yescharacteristic
445C2#582911yescharacteristic
446C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#126C2
prime radicalC24×D3C22
hypercenterC2×#126D3
hypocenterC3#2480
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#126

Complements

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

NH ≅ G/Ntype
1. C2×C3×#126C2semidirect
3. #4994C2semidirect
4. #4992C2semidirect
5. #4990C2semidirect
6. #4989C2semidirect
7. #4989C2semidirect
8. #4987C2semidirect
11. #4952C2semidirect
12. #4952C2semidirect
13. #4951C2semidirect
14. #4940C2semidirect
15. C4×(C2×D3)⋊C4C2semidirect
21. C2×C3×C4⋊C4C22semidirect
22. C2×C3×C4⋊C4C22semidirect
23. C2×C3×C4⋊C4C22semidirect
27. C2×C3×C42C22semidirect
41. C2×C4⋊Dic3C22semidirect
42. C2×Dic3⋊C4C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×C4×Dic3C22semidirect
46. C3⋊#95C22semidirect
47. C3⋊#95C22semidirect
48. C3⋊#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C2×#126D3semidirect
128. C2×C4⋊C4C2×D3semidirect
129. C2×C4⋊C4C2×D3semidirect
130. C2×C4⋊C4C2×D3semidirect
149. C2×C42C2×D3semidirect
305. C2×D3#126semidirect
436. C3#2480semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1262
C2×D3×C22⋊C42
#49942
#49922
#49902
#49892
#49872
#49552
#49522
#49512
#49402
C4×(C2×D3)⋊C42
#24803