Finite Group Databasebeta

← #77668

Subgroups — #77668

Order 384.

814 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#894C211yesmaximal, characteristic, complemented
2C2×#882C211yesmaximal, characteristic
3C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C2×D3×C22⋊C4C211yesmaximal, characteristic, complemented
5C3×#397C211yesmaximal, characteristic, complemented
6#4987C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4952C211yesmaximal, characteristic, complemented
10#4668C211yesmaximal, characteristic, complemented
11#4667C211yesmaximal, characteristic, complemented
12#4665C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4664C211yesmaximal, characteristic, complemented
16#2412—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
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, complemented
38C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
39C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
40C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
41C2×C4⋊Dic3C2211yescharacteristic, complemented
42C2×Dic3⋊C4C2211yescharacteristic, complemented
43C2×Dic3⋊C4C2211yescharacteristic, complemented
44#894—24no—
45#882—24no—
46D3×C22⋊C4—24no—
47D3×C22⋊C4—24no—
48C3×#95C2211yescharacteristic, complemented
49C3×#95C2211yescharacteristic, complemented
50C3×#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C3⋊#95C2211yescharacteristic, complemented
55C3⋊#95C2211yescharacteristic, complemented
56C2×#123—31no—
57C2×#120—31no—
58C22×C22⋊C4—31no—
59C22×C22⋊C4—31no—
60#397—31no—
61#397D311yescharacteristic, complemented
62#394—31no—
63#394—31no—
64#386—31no—
65#386—31no—
66#386—31no—
67#385—31no—
68#381—31no—
69#380—31no—
70#380—31no—
71C24×C3C2311yescharacteristic
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—22no—
75C23×D3—22no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3D414yes—
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C23×D3C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C2×#67—22no—
89C2×#67—41no—
90C2×#67—41no—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12—22no—
101C2×D12—22no—
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—41no—
111C2×C4×D3—41no—
112C2×C4×D3—41no—
113C2×C4×D3—41no—
114C3×C4⋊C4—22no—
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118#217—22no—
119#217—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4D414yescomplemented
130C4⋊Dic3—41no—
131Dic3⋊C4—22no—
132Dic3⋊C4—41no—
133C25—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4—31no—
140C23×C4—31no—
141#123—64no—
142#120—64no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4C2×D311yescharacteristic, complemented
147C2×C22⋊C4—31no—
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
170#95C2×D311yescharacteristic, complemented
171#95C2×D311yescharacteristic, complemented
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3C2×D412yes—
179C23×C3C2411yescharacteristic
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—28no—
189C22×D3—28no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—44no—
198C22×D3—44no—
199C22×D3—44no—
200C22×D3—44no—
201C22×D3—44no—
202C22×D3C2×D412yes—
203C22×D3C2×D412yes—
204C22×D3C2×D412yes—
205C22×D3#4112yes—
206C22×D3#4112yes—
207C22×D3#4112yes—
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×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4C2×D412yes—
226#67—44no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3C2×D412yes—
242D12—44no—
243D12—44no—
244C4×D3—44no—
245C4×D3—44no—
246C4×D3—44no—
247C4×D3—44no—
248C24—31no—
249C24—31no—
250C24—34no—
251C24—31no—
252C24—62no—
253C24—62no—
254C24—62no—
255C24—62no—
256C24—121no—
257C24—121no—
258C24C22×D311yescharacteristic
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—62no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—121no—
267C2×D4—121no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—121no—
284C22×C4—121no—
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D311yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—62no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—34no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—62no—
305C22⋊C4—62no—
306C22⋊C4—62no—
307C22⋊C4—62no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3C22×D411yescharacteristic
318C22×C3C22×D411yescharacteristic
319C22×C3C2×#4111yescharacteristic
320C22×C3C2×#4111yescharacteristic
321C22×C3C2×#4111yescharacteristic
322C22×C3H(5,𝔽2)11yescharacteristic
323C22×C3H(5,𝔽2)11yescharacteristic
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—22no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—22no—
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—416no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3#12114yes—
345C2×D3#12314yescomplemented
346C3×C4—22no—
347C3×C4—41no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—41no—
351C3×C4—41no—
352Dic3—22no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356Dic3—41no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—32no—
364C23—32no—
365C23—32no—
366C23—32no—
367C23—32no—
368C23—32no—
369C23—68no—
370C23—68no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23—61no—
379C23—61no—
380C23—61no—
381C23—61no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—124no—
387C23—124no—
388C23—124no—
389C23—124no—
390C23—124no—
391C23D3×D412yes—
392C23C23×D311yescharacteristic
393D4—124no—
394D4—124no—
395D4—124no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—32no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—61no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—124no—
429C2×C4—124no—
430C2×C4—124no—
431C2×C4—124no—
432C2×C4D3×D412yes—
433C2×C3—42no—
434C2×C3C2×#12111yescharacteristic
435C2×C3C2×#12311yescharacteristic
436C2×C3#53411yescharacteristic
437C2×C3#54611yescharacteristic
438C2×C3#54611yescharacteristic
439C2×C3#55111yescharacteristic
440C2×C3#55111yescharacteristic
441D3—24no—
442D3—24no—
443D3—81no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—34no—
448C22—34no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—62no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—62no—
458C22—62no—
459C22—62no—
460C22—62no—
461C22—62no—
462C22—62no—
463C22—62no—
464C22—62no—
465C22—1216no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22—121no—
471C22—121no—
472C22—121no—
473C22C2×D3×D411yescharacteristic
474C22C2×D3×D411yescharacteristic
475C22D3×#4111yescharacteristic
476C22D3×#4111yescharacteristic
477C22D3×#4111yescharacteristic
478C22#100911yescharacteristic
479C22#100911yescharacteristic
480C4—22no—
481C4—41no—
482C4—41no—
483C4—41no—
484C4—41no—
485C4—41no—
486C4—62no—
487C4—121no—
488C4—121no—
489C4—121no—
490C4—121no—
491C3#241211yescharacteristic, complemented, Sylow
492C2—42no—
493C2—64no—
494C2—64no—
495C2—241no—
496C2D3×#12111yescharacteristic
497C2D3×#12311yescharacteristic
498C2#565311yescharacteristic
499C2#565411yescharacteristic
500C2#567611yescharacteristic
501C2#570511yescharacteristic
502C2#570511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#397C2
prime radicalC2×#882C2
hypercenter#397D3
hypocenterC3#2412
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C2×#894C2semidirect
3. C2×D3×C22⋊C4C2semidirect
4. C2×D3×C22⋊C4C2semidirect
5. C3×#397C2semidirect
6. #4987C2semidirect
7. #4961C2semidirect
8. #4959C2semidirect
9. #4952C2semidirect
10. #4668C2semidirect
11. #4667C2semidirect
12. #4665C2semidirect
13. #4665C2semidirect
14. #4665C2semidirect
15. #4664C2semidirect
25. C2×C3×C4⋊C4C22semidirect
37. C2×(C2×D3)⋊C4C22semidirect
38. C2×(C2×D3)⋊C4C22semidirect
39. C2×(C2×D3)⋊C4C22semidirect
40. C2×(C2×D3)⋊C4C22semidirect
41. C2×C4⋊Dic3C22semidirect
42. C2×Dic3⋊C4C22semidirect
43. C2×Dic3⋊C4C22semidirect
48. C3×#95C22semidirect
49. C3×#95C22semidirect
50. C3×#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
54. C3⋊#95C22semidirect
55. C3⋊#95C22semidirect
61. #397D3semidirect
129. (C2×D3)⋊C4D4semidirect
146. C2×C4⋊C4C2×D3semidirect
169. #95C2×D3semidirect
170. #95C2×D3semidirect
171. #95C2×D3semidirect
345. C2×D3#123semidirect
491. C3#2412semidirect

Maximal subgroups

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

subgroupindex
C2×#8942
C2×#8822
C2×D3×C22⋊C42
C3×#3972
#49872
#49612
#49592
#49522
#46682
#46672
#46652
#46642
#24123