Finite Group Databasebeta

← #77158

Subgroups — #77158

Order 384.

712 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic, complemented
2C2×#882C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4991C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4960C211yesmaximal, characteristic, complemented
10#4960C211yesmaximal, characteristic, complemented
11#4959C211yesmaximal, characteristic, complemented
12#4955C211yesmaximal, characteristic, complemented
13#4952C211yesmaximal, characteristic, complemented
14#4674C211yesmaximal, characteristic, complemented
15#4672C211yesmaximal, characteristic, complemented
16#2812—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic, complemented
23C22×D12C2211yescharacteristic
24C22×D12C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic, complemented
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic, complemented
29C2×C3×C22⋊C4C2211yescharacteristic, complemented
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic, complemented
33#937—24no—
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×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
43C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
44C2×(C2×D3)⋊C4C2211yescharacteristic, complemented
45C2×C4⋊Dic3C2211yescharacteristic, complemented
46C2×Dic3⋊C4C2211yescharacteristic, complemented
47#882—24no—
48C3×#95C2211yescharacteristic, complemented
49C3⋊#95C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#120—31no—
55C2×#120—31no—
56#397—31no—
57#397—31no—
58#397—31no—
59#394—31no—
60#394—31no—
61#394—31no—
62#394—31no—
63#394—31no—
64#394D311yescharacteristic, complemented
65#392—31no—
66#392—31no—
67#392—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C23×D3—41no—
72C23×D3—41no—
73C23×D3—41no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3D414yescomplemented
78C23×D3C2311yescharacteristic
79C23×D3C2311yescharacteristic
80C23×D3C2311yescharacteristic
81C2×C3×D4—22no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C2×#67—41no—
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C2×D12—22no—
107C2×D12—22no—
108C2×D12—41no—
109C2×D12—41no—
110C2×D12—41no—
111C2×D12—41no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—41no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117#217—22no—
118#217—41no—
119#217—41no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—41no—
125(C2×D3)⋊C4—41no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128(C2×D3)⋊C4—41no—
129(C2×D3)⋊C4—41no—
130(C2×D3)⋊C4—41no—
131C4⋊Dic3—41no—
132Dic3⋊C4—41no—
133C25—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C22×D4—31no—
139C22×D4—31no—
140C22×D4C2×D311yescharacteristic
141#120—64no—
142#120—64no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4C2×D311yescharacteristic, complemented
146C2×C22⋊C4—31no—
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—31no—
160C2×C22⋊C4C2×D311yescharacteristic
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic, complemented
163C2×C22⋊C4C2×D311yescharacteristic, complemented
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×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3C2×D412yes—
184C23×C3C2411yescharacteristic
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C22×D3—21no—
202C22×D3—21no—
203C22×D3—21no—
204C22×D3—44no—
205C22×D3—44no—
206C22×D3—44no—
207C22×D3—44no—
208C22×D3—44no—
209C22×D3—44no—
210C22×D3—44no—
211C22×D3C2×D412yes—
212C22×D3C2×D412yes—
213C3×D4—44no—
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×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—21no—
229#67—44no—
230#67—44no—
231#67—44no—
232#67—44no—
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×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248D12—44no—
249D12—44no—
250C24—31no—
251C24—31no—
252C24—34no—
253C24—31no—
254C24—121no—
255C24—121no—
256C24—121no—
257C24—121no—
258C24—121no—
259C24—121no—
260C24C22×D311yescharacteristic
261C24C22×D311yescharacteristic
262C2×D4—22no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—62no—
266C2×D4—62no—
267C2×D4—62no—
268C2×D4—62no—
269C2×D4—62no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—121no—
280C2×D4—121no—
281C2×D4—121no—
282C2×D4—121no—
283C22×C4—31no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4—31no—
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D311yescharacteristic
291C22×C4C22×D311yescharacteristic
292C22×C4C22×D311yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—22no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—62no—
304C22⋊C4—62no—
305C22⋊C4—121no—
306C22⋊C4—121no—
307C22⋊C4—121no—
308C22⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3C22×D411yescharacteristic
329C22×C3H(5,𝔽2)11yescharacteristic
330C22×C3H(5,𝔽2)11yescharacteristic
331C22×C3H(5,𝔽2)11yescharacteristic
332C22×C3H(5,𝔽2)11yescharacteristic
333C22×C3H(5,𝔽2)11yescharacteristic
334C22×C3H(5,𝔽2)11yescharacteristic
335C2×D3—22no—
336C2×D3—22no—
337C2×D3—22no—
338C2×D3—22no—
339C2×D3—22no—
340C2×D3—22no—
341C2×D3—416no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C2×D3—41no—
348C2×D3—41no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—41no—
352C2×D3—41no—
353C2×D3—41no—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4—41no—
362Dic3—41no—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366Dic3—41no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—32no—
381C23—32no—
382C23—61no—
383C23—61no—
384C23—61no—
385C23—61no—
386C23—61no—
387C23—61no—
388C23—61no—
389C23—61no—
390C23—61no—
391C23—61no—
392C23—61no—
393C23—61no—
394C23—61no—
395C23—61no—
396C23—61no—
397C23—61no—
398C23—61no—
399C23—61no—
400C23—61no—
401C23—124no—
402C23—124no—
403C23—124no—
404C23—124no—
405C23—124no—
406C23—124no—
407C23—124no—
408C23D3×D412yes—
409C23C23×D311yescharacteristic
410D4—44no—
411D4—124no—
412D4—124no—
413D4—124no—
414D4—124no—
415D4—124no—
416D4—124no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C3—42no—
448C2×C3—81no—
449C2×C3#53411yescharacteristic
450C2×C3#53411yescharacteristic
451C2×C3#53411yescharacteristic
452C2×C3#56011yescharacteristic
453C2×C3#56011yescharacteristic
454C2×C3#56011yescharacteristic
455C2×C3H(3,𝔽4)11yescharacteristic
456D3—42no—
457D3—42no—
458D3—81no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—62no—
474C22—62no—
475C22—62no—
476C22—62no—
477C22—62no—
478C22—62no—
479C22—1216no—
480C22—121no—
481C22—121no—
482C22—121no—
483C22—121no—
484C22—121no—
485C22—121no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—121no—
491C22—121no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22C2×D3×D411yescharacteristic
496C22#100411yescharacteristic
497C22#100411yescharacteristic
498C22#100411yescharacteristic
499C22#100911yescharacteristic
500C22#100911yescharacteristic
501C22#100911yescharacteristic
502C4—41no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—121no—
508C4—121no—
509C4—121no—
510C4—121no—
511C4—121no—
512C3#281211yescharacteristic, complemented, Sylow
513C2—42no—
514C2—81no—
515C2—122no—
516C2—122no—
517C2—241no—
518C2#558911yescharacteristic
519C2#560511yescharacteristic
520C2#562011yescharacteristic
521C2#562011yescharacteristic
522C2#565311yescharacteristic
523C2#566611yescharacteristic
524C2#568311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
hypercenter#394D3
hypocenterC3#2812
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#937C2semidirect
2. C2×#882C2semidirect
3. C3×#394C2semidirect
4. #5223C2semidirect
5. #4992C2semidirect
6. #4991C2semidirect
7. #4961C2semidirect
8. #4961C2semidirect
9. #4960C2semidirect
10. #4960C2semidirect
11. #4959C2semidirect
12. #4955C2semidirect
13. #4952C2semidirect
14. #4674C2semidirect
15. #4672C2semidirect
22. C22×#67C22semidirect
25. C2×C3×C4⋊C4C22semidirect
28. C2×C3×C22⋊C4C22semidirect
29. C2×C3×C22⋊C4C22semidirect
32. C2×#217C22semidirect
42. C2×(C2×D3)⋊C4C22semidirect
43. C2×(C2×D3)⋊C4C22semidirect
44. C2×(C2×D3)⋊C4C22semidirect
45. C2×C4⋊Dic3C22semidirect
46. C2×Dic3⋊C4C22semidirect
48. C3×#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
64. #394D3semidirect
77. C23×D3D4semidirect
145. C2×C4⋊C4C2×D3semidirect
162. C2×C22⋊C4C2×D3semidirect
163. C2×C22⋊C4C2×D3semidirect
169. #95C2×D3semidirect
512. C3#2812semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8822
C3×#3942
#52232
#49922
#49912
#49612
#49602
#49592
#49552
#49522
#46742
#46722
#28123