Finite Group Databasebeta

← #77418

Subgroups — #77418

Order 384.

780 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#897C211yesmaximal, characteristic, complemented
2C2×#896C211yesmaximal, characteristic, complemented
3C2×D3×C22⋊C4C211yesmaximal, characteristic
4C2×D3×C22⋊C4C211yesmaximal, characteristic
5C3×#396C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4987C211yesmaximal, characteristic, complemented
10#4955C211yesmaximal, characteristic
11#4949C211yesmaximal, characteristic, complemented
12#4948C211yesmaximal, characteristic, complemented
13#4947C211yesmaximal, characteristic, complemented
14#4668C211yesmaximal, characteristic, complemented
15#4667C211yesmaximal, characteristic, complemented
16#2437—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic, complemented
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C4⋊C4C2211yescharacteristic, complemented
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic, complemented
41C2×C4⋊Dic3C2211yescharacteristic, complemented
42C2×Dic3⋊C4C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44C2×Dic3⋊C4C2211yescharacteristic, complemented
45C2×Dic3⋊C4C2211yescharacteristic, complemented
46#897—24no—
47#896—24no—
48D3×C22⋊C4—24no—
49D3×C22⋊C4—24no—
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×C22⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C22×C22⋊C4—31no—
59C22×C22⋊C4—31no—
60#399—31no—
61#396—31no—
62#396D311yescharacteristic, complemented
63#393—31no—
64#385—31no—
65#385—31no—
66#385—31no—
67#385—31no—
68#380—31no—
69#380—31no—
70#379—31no—
71C24×C3C2311yescharacteristic
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3Q814yescomplemented
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×C4×D3—22no—
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—22no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C2×C4×D3—41no—
104C2×C4×D3—41no—
105C2×Dic6—22no—
106C2×Dic6—22no—
107C2×Dic6—41no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—41no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114#217—22no—
115#217—22no—
116#217—41no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4D414yes—
124(C2×D3)⋊C4D414yes—
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127Dic3⋊C4—22no—
128Dic3⋊C4—22no—
129Dic3⋊C4—41no—
130Dic3⋊C4—41no—
131C25—31no—
132C22×Q8—31no—
133C23×C4—31no—
134C23×C4—31no—
135C23×C4—31no—
136C23×C4—31no—
137C22⋊Q8—64no—
138C22⋊Q8—64no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4C2×D311yescharacteristic, complemented
146C2×C4⋊C4C2×D311yescharacteristic, complemented
147C2×C4⋊C4C2×D311yescharacteristic, complemented
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—64no—
160C2×C22⋊C4—64no—
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic
163C2×C22⋊C4C2×D311yescharacteristic
164#95—31no—
165#95—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95C2×D311yescharacteristic, complemented
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3C2×Q812yes—
177C23×C3C2411yescharacteristic
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—28no—
187C22×D3—28no—
188C22×D3—44no—
189C22×D3—44no—
190C22×D3—44no—
191C22×D3—44no—
192C22×D3—44no—
193C22×D3C2×D412yes—
194C22×D3C2×D412yes—
195C22×D3C2×Q812yes—
196C22×D3C2×Q812yes—
197C22×D3#4112yes—
198C22×D3#4112yes—
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×C4—21no—
215C2×C3×C4C2×D412yes—
216C2×C3×C4C2×D412yes—
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×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3C2×D412yes—
234C2×Dic3C2×D412yes—
235C4×D3—44no—
236C4×D3—44no—
237C4×D3—44no—
238C4×D3—44no—
239Dic6—44no—
240C24—31no—
241C24—31no—
242C24—34no—
243C24—62no—
244C24—62no—
245C24—62no—
246C24—62no—
247C24—121no—
248C24—121no—
249C24C22×D311yescharacteristic
250C2×Q8—62no—
251C2×Q8—62no—
252C2×Q8—121no—
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—62no—
260C22×C4—62no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C22×C4C22×D311yescharacteristic
276C22×C4C22×D311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—121no—
285C4⋊C4—121no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—34no—
289C22⋊C4—34no—
290C22⋊C4—41no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—121no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3C22×D411yescharacteristic
308C22×C3C22×D411yescharacteristic
309C22×C3C22×Q811yescharacteristic
310C22×C3C2×#4111yescharacteristic
311C22×C3C2×#4111yescharacteristic
312C22×C3H(5,𝔽2)11yescharacteristic
313C22×C3H(5,𝔽2)11yescharacteristic
314C2×D3—22no—
315C2×D3—22no—
316C2×D3—22no—
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—416no—
327C2×D3C22⋊Q814yescomplemented
328C2×D3C22⋊Q814yescomplemented
329C3×C4—22no—
330C3×C4—22no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335Dic3—22no—
336Dic3—22no—
337Dic3—41no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—32no—
348C23—32no—
349C23—32no—
350C23—32no—
351C23—32no—
352C23—32no—
353C23—68no—
354C23—68no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—124no—
364C23—124no—
365C23—124no—
366C23—124no—
367C23—124no—
368C23D3×Q812yes—
369C23C23×D311yescharacteristic
370Q8—124no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
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—32no—
388C2×C4—32no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
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—124no—
406C2×C4—124no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4D3×D412yes—
410C2×C4D3×D412yes—
411C2×C3—42no—
412C2×C3C2×C22⋊Q811yescharacteristic
413C2×C3C2×C22⋊Q811yescharacteristic
414C2×C3#54311yescharacteristic
415C2×C3D4211yescharacteristic
416C2×C3#54611yescharacteristic
417C2×C3#54611yescharacteristic
418C2×C3#55111yescharacteristic
419D3—24no—
420D3—24no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—34no—
425C22—34no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—62no—
442C22—1216no—
443C22C2×D3×D411yescharacteristic
444C22C2×D3×D411yescharacteristic
445C22#100411yescharacteristic
446C22C2×D3×Q811yescharacteristic
447C22D3×#4111yescharacteristic
448C22D3×#4111yescharacteristic
449C22#100911yescharacteristic
450C4—22no—
451C4—22no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—62no—
457C4—62no—
458C4—121no—
459C4—121no—
460C4—121no—
461C4—121no—
462C3#243711yescharacteristic, complemented, Sylow
463C2—42no—
464C2—64no—
465C2—64no—
466C2#560911yescharacteristic
467C2#561011yescharacteristic
468C2#561211yescharacteristic
469C2D3×C22⋊Q811yescharacteristic
470C2D3×C22⋊Q811yescharacteristic
471C2#563411yescharacteristic
472C2#570511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
prime radicalC24×D3C22
hypercenter#396D3
hypocenterC3#2437
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#897C2semidirect
2. C2×#896C2semidirect
5. C3×#396C2semidirect
6. #4989C2semidirect
7. #4989C2semidirect
8. #4989C2semidirect
9. #4987C2semidirect
11. #4949C2semidirect
12. #4948C2semidirect
13. #4947C2semidirect
14. #4668C2semidirect
15. #4667C2semidirect
22. C22×Dic6C22semidirect
23. C2×C3×C4⋊C4C22semidirect
24. C2×C3×C4⋊C4C22semidirect
25. C2×C3×C4⋊C4C22semidirect
40. C2×C4⋊Dic3C22semidirect
41. C2×C4⋊Dic3C22semidirect
42. C2×Dic3⋊C4C22semidirect
43. C2×Dic3⋊C4C22semidirect
44. C2×Dic3⋊C4C22semidirect
45. C2×Dic3⋊C4C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
62. #396D3semidirect
78. C23×D3Q8semidirect
145. C2×C4⋊C4C2×D3semidirect
146. C2×C4⋊C4C2×D3semidirect
147. C2×C4⋊C4C2×D3semidirect
169. #95C2×D3semidirect
327. C2×D3C22⋊Q8semidirect
328. C2×D3C22⋊Q8semidirect
462. C3#2437semidirect

Maximal subgroups

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

subgroupindex
C2×#8972
C2×#8962
C2×D3×C22⋊C42
C3×#3962
#49892
#49872
#49552
#49492
#49482
#49472
#46682
#46672
#24373