Finite Group Databasebeta

← #75840

Subgroups — #75840

Order 384.

688 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#882C211yesmaximal, characteristic
2C3×#388C211yesmaximal, characteristic, complemented
3#4990C211yesmaximal, characteristic, complemented
4#4988C211yesmaximal, characteristic, complemented
5#4957C211yesmaximal, characteristic, complemented
6#4956C211yesmaximal, characteristic, complemented
7#4955C211yesmaximal, characteristic, complemented
8#4955C211yesmaximal, characteristic, complemented
9#4951C211yesmaximal, characteristic, complemented
10#4940C211yesmaximal, characteristic, complemented
11#4939C211yesmaximal, characteristic, complemented
12#4670C211yesmaximal, characteristic, complemented
13#4669C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4665C211yesmaximal, characteristic, complemented
16#2375—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic, complemented
22C2×C3×C4⋊C4C2211yescharacteristic, complemented
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C42C2211yescharacteristic, complemented
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
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, complemented
40C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
42C2×C4⋊Dic3C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×C4×Dic3C2211yescharacteristic, complemented
45C2×C4×Dic3C2211yescharacteristic, complemented
46#882C418yescomplemented
47C3×#95C2211yescharacteristic, complemented
48C3×#95C2211yescharacteristic, complemented
49C3⋊#95C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C2×#120—31no—
54#390—31no—
55#390—31no—
56#390—31no—
57#388—31no—
58#388—31no—
59#388—31no—
60#388—31no—
61#388—31no—
62#388D311yescharacteristic, complemented
63#386—31no—
64#386—31no—
65#386—31no—
66#379—31no—
67#379—31no—
68C24×C3C2311yescharacteristic
69C23×D3—41no—
70C23×D3—41no—
71C23×D3—41no—
72C23×D3—41no—
73C23×D3—41no—
74C23×D3—41no—
75C23×D3C2×C414yescomplemented
76C23×D3C2311yescharacteristic
77C23×D3C2311yescharacteristic
78C23×D3C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—41no—
86C2×#67—41no—
87C2×#67C2×C414yes—
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×D12—41no—
94C2×D12—41no—
95C2×D12—41no—
96C2×D12—41no—
97C2×D12C2×C414yes—
98C2×D12C2×C414yes—
99C2×C4×D3—41no—
100C2×C4×D3—41no—
101C2×C4×D3—41no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4C2×C414yescomplemented
106C3×C42—41no—
107#217—41no—
108#217—41no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4—41no—
117(C2×D3)⋊C4—41no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4C2×C414yescomplemented
121(C2×D3)⋊C4C2×C414yescomplemented
122C4⋊Dic3—41no—
123Dic3⋊C4—41no—
124C4×Dic3—41no—
125C4×Dic3—41no—
126C25—31no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C23×C4—31no—
131#120—38no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4C2×D311yescharacteristic, complemented
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—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4C2×D311yescharacteristic
151C2×C22⋊C4C2×D311yescharacteristic
152C2×C22⋊C4C2×D311yescharacteristic
153C2×C42—31no—
154C2×C42—31no—
155C2×C42C2×D311yescharacteristic, complemented
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95C2×D311yescharacteristic, complemented
161#95C2×D311yescharacteristic, complemented
162C23×C3—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3C22×C412yes—
169C23×C3C2411yescharacteristic
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—44no—
189C22×D3—44no—
190C22×D3—44no—
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—44no—
194C22×D3—44no—
195C22×D3C22×C412yes—
196C22×D3C22×C412yes—
197C22×D3C22×C412yes—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4C22×C412yes—
215C2×C3×C4C22×C412yes—
216#67—44no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3C22×C412yes—
232D12—44no—
233D12—44no—
234C4×D3—44no—
235C24—31no—
236C24—31no—
237C24—31no—
238C24—34no—
239C24—121no—
240C24—121no—
241C24—121no—
242C24—121no—
243C24—121no—
244C24—121no—
245C24C22×D311yescharacteristic
246C2×D4—34no—
247C2×D4—34no—
248C2×D4—34no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4—121no—
263C22×C4C22×D311yescharacteristic
264C22×C4C22×D311yescharacteristic
265C22×C4C22×D311yescharacteristic
266C22×C4C22×D311yescharacteristic
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C4⋊C4—41no—
270C4⋊C4—121no—
271C4⋊C4—121no—
272C22⋊C4—34no—
273C22⋊C4—34no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—121no—
283C22⋊C4—121no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4C4×D314yescomplemented
290C42—41no—
291C42—121no—
292C42—121no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3C23×C411yescharacteristic
301C22×C3H(5,𝔽2)11yescharacteristic
302C22×C3H(5,𝔽2)11yescharacteristic
303C22×C3H(5,𝔽2)11yescharacteristic
304C22×C3H(5,𝔽2)11yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C22×C3H(5,𝔽2)11yescharacteristic
307C2×D3—22no—
308C2×D3—22no—
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—22no—
313C2×D3—22no—
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—416no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335Dic3—22no—
336Dic3—41no—
337Dic3—41no—
338Dic3—41no—
339Dic3—41no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—32no—
347C23—32no—
348C23—32no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—124no—
368C23—124no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23C2×C4×D312yes—
375C23C23×D311yescharacteristic
376D4—124no—
377D4—124no—
378D4—124no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—32no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—124no—
411C2×C4C2×C4×D312yes—
412C2×C4C2×C4×D312yes—
413C2×C3—42no—
414C2×C3#51811yescharacteristic
415C2×C3#51811yescharacteristic
416C2×C3#51811yescharacteristic
417C2×C3#56011yescharacteristic
418C2×C3#56011yescharacteristic
419C2×C3#56011yescharacteristic
420C2×C3H(3,𝔽4)11yescharacteristic
421D3—42no—
422D3—42no—
423D3—42no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—1216no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22—121no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22C22×C4×D311yescharacteristic
454C22#100411yescharacteristic
455C22#100411yescharacteristic
456C22#100911yescharacteristic
457C22#100911yescharacteristic
458C22#100911yescharacteristic
459C22#100911yescharacteristic
460C4—22no—
461C4—22no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—41no—
466C4—62no—
467C4—121no—
468C4—121no—
469C4—121no—
470C4—121no—
471C3#237511yescharacteristic, complemented, Sylow
472C2—42no—
473C2—122no—
474C2—122no—
475C2—122no—
476C2#552111yescharacteristic
477C2#554511yescharacteristic
478C2#554511yescharacteristic
479C2#562011yescharacteristic
480C2#566611yescharacteristic
481C2#568311yescharacteristic
482C2#571111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#388C2
socle, cosocleC23×C3C24
prime radicalC2×#882C2
hypercenter#388D3
hypocenterC3#2375
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C3×#388C2semidirect
3. #4990C2semidirect
4. #4988C2semidirect
5. #4957C2semidirect
6. #4956C2semidirect
7. #4955C2semidirect
8. #4955C2semidirect
9. #4951C2semidirect
10. #4940C2semidirect
11. #4939C2semidirect
12. #4670C2semidirect
13. #4669C2semidirect
14. #4665C2semidirect
15. #4665C2semidirect
21. C22×C4×D3C22semidirect
22. C2×C3×C4⋊C4C22semidirect
26. C2×C3×C42C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×(C2×D3)⋊C4C22semidirect
42. C2×C4⋊Dic3C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×C4×Dic3C22semidirect
45. C2×C4×Dic3C22semidirect
46. #882C4semidirect
47. C3×#95C22semidirect
48. C3×#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
62. #388D3semidirect
75. C23×D3C2×C4semidirect
105. C3×C22⋊C4C2×C4semidirect
120. (C2×D3)⋊C4C2×C4semidirect
121. (C2×D3)⋊C4C2×C4semidirect
134. C2×C4⋊C4C2×D3semidirect
155. C2×C42C2×D3semidirect
160. #95C2×D3semidirect
161. #95C2×D3semidirect
289. C22⋊C4C4×D3semidirect
471. C3#2375semidirect

Maximal subgroups

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

subgroupindex
C2×#8822
C3×#3882
#49902
#49882
#49572
#49562
#49552
#49512
#49402
#49392
#46702
#46692
#46652
#23753