Finite Group Databasebeta

← #10594308

Subgroups — #10594308

Order 576.

551 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#64093C211yesmaximal, characteristic, complemented
2C3×#886C211yesmaximal, characteristic, complemented
3C3×#884C211yesmaximal, characteristic, complemented
4#63983C211yesmaximal, characteristic, complemented
5#63979C211yesmaximal, characteristic, complemented
6#63978C211yesmaximal, characteristic, complemented
7#63978C211yesmaximal, characteristic, complemented
8#63923C211yesmaximal, characteristic, complemented
9#63922C211yesmaximal, characteristic, complemented
10#63916C211yesmaximal, characteristic, complemented
11#63909C211yesmaximal, characteristic, complemented
12#63905C211yesmaximal, characteristic, complemented
13#63858C211yesmaximal, characteristic, complemented
14#63853C211yesmaximal, characteristic, complemented
15#63848C211yesmaximal, characteristic, complemented
16#5666—31nomaximal
17#5620—31nomaximal
18C22×D32C2211yescharacteristic
19C2×#638C2211yescharacteristic
20C2×#638C2211yescharacteristic, complemented
21C2×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C4×C3⋊D3C2211yescharacteristic, complemented
23C2×C3×#67C2211yescharacteristic
24C2×C3×#67C2211yescharacteristic
25C2×C3×#67C2211yescharacteristic, complemented
26C22×C3×Dic3C2211yescharacteristic, complemented
27C2×C3×D12C2211yescharacteristic
28C2×C3×C4×D3C2211yescharacteristic, complemented
29C2×#626C2211yescharacteristic
30C2×#626C2211yescharacteristic
31C2×#626C2211yescharacteristic
32C2×#626C2211yescharacteristic, complemented
33C2×#625C2211yescharacteristic
34C32×C22⋊C4C2211yescharacteristic, complemented
35#3752C2211yescharacteristic
36#3750C2211yescharacteristic, complemented
37C3×(C2×D3)⋊C4C2211yescharacteristic
38C3×(C2×D3)⋊C4C2211yescharacteristic
39C3×(C2×D3)⋊C4C2211yescharacteristic
40C3×C4⋊Dic3C2211yescharacteristic, complemented
41C3×C4⋊Dic3C2211yescharacteristic, complemented
42C3×Dic3⋊C4C2211yescharacteristic, complemented
43Dic3⋊Dic3C2211yescharacteristic, complemented
44Dic3⋊Dic3C2211yescharacteristic, complemented
45#3722C2211yescharacteristic
46#3722C2211yescharacteristic
47#3722C2211yescharacteristic, complemented
48(C2×D3)⋊Dic3C2211yescharacteristic
49(C2×D3)⋊Dic3C2211yescharacteristic
50(C2×D3)⋊Dic3C2211yescharacteristic
51(C2×D3)⋊Dic3C2211yescharacteristic, complemented
52Dic32C2211yescharacteristic, complemented
53C3×#123—31no—
54C3×#121—31no—
55#940—31no—
56#940—31no—
57#939—31no—
58#939—31no—
59#938—31no—
60#937—31no—
61#937—31no—
62#937—31no—
63#930—31no—
64#930—31no—
65#930—31no—
66#929—31no—
67#898—31no—
68#898—31no—
69#898—31no—
70#895—31no—
71#895—31no—
72#894—31no—
73#886D311yescharacteristic, complemented
74#884—31no—
75#884—31no—
76#884—61no—
77#884D311yescharacteristic, complemented
78#883—31no—
79#883—31no—
80#882—31no—
81#882—31no—
82#882—31no—
83#879—31no—
84C23×C32C2311yescharacteristic
85C22×C3⋊D3C2311yescharacteristic
86C22×C3⋊D3C2311yescharacteristic
87C22×C3×D3C2311yescharacteristic
88C22×C3×D3C2311yescharacteristic
89C22×C3×D3C2311yescharacteristic
90C2×D32—41no—
91C2×D32—41no—
92C2×D32—41no—
93C2×C32×C4C2311yescharacteristic
94C2×C32×C4C2311yescharacteristic
95#638—41no—
96#638—41no—
97C2×C3⋊Dic3C2311yescharacteristic
98C2×C3⋊Dic3C2311yescharacteristic
99C4⋊(C3⋊D3)—41no—
100C4×C3⋊D3—41no—
101C3×#67—41no—
102C3×#67—41no—
103C3×#67—41no—
104C2×C3×Dic3—41no—
105C2×C3×Dic3C2311yescharacteristic
106C2×C3×Dic3C2311yescharacteristic
107C2×C3×Dic3C2311yescharacteristic
108C2×C3×Dic3C2311yescharacteristic
109C2×C3×Dic3C2311yescharacteristic
110C3×D12—41no—
111C3×C4×D3—41no—
112#626—41no—
113#626—41no—
114#626—41no—
115#626—41no—
116#625—41no—
117#560—91noSylow
118C23×D3—31no—
119C23×D3—31no—
120C2×C3×D4—31no—
121C2×C3×D4—31no—
122C2×C3×D4—31no—
123C2×C3×D4—31no—
124C22×C3×C4—31no—
125C22×C3×C4—31no—
126C2×#67—31no—
127C2×#67—31no—
128C2×#67—31no—
129C2×#67—31no—
130C2×#67—31no—
131C2×#67—31no—
132C2×#67—31no—
133C2×#67—31no—
134C2×#67—31no—
135C2×#67—31no—
136C2×#67—61no—
137C2×#67—61no—
138C2×#67C2×D311yescharacteristic
139C2×#67C2×D311yescharacteristic
140C2×#67C2×D311yescharacteristic, complemented
141C22×Dic3C2×D311yescharacteristic, complemented
142C2×D12—31no—
143C2×D12—31no—
144C2×D12—31no—
145C2×D12—31no—
146C2×D12—31no—
147C2×D12—31no—
148C2×D12—61no—
149C2×D12C2×D311yescharacteristic
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—61no—
153C2×C4×D3C2×D311yescharacteristic, complemented
154C3×C4⋊C4—31no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—31no—
159C3×C22⋊C4—31no—
160C3×C22⋊C4—31no—
161C3×C22⋊C4C2×D311yescharacteristic, complemented
162C3×C22⋊C4C2×D311yescharacteristic, complemented
163#217—31no—
164#217—31no—
165#217—31no—
166#217—31no—
167(C2×D3)⋊C4—31no—
168(C2×D3)⋊C4—31no—
169(C2×D3)⋊C4—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4—31no—
173(C2×D3)⋊C4—31no—
174(C2×D3)⋊C4—31no—
175(C2×D3)⋊C4—31no—
176(C2×D3)⋊C4—31no—
177(C2×D3)⋊C4—31no—
178(C2×D3)⋊C4—31no—
179(C2×D3)⋊C4—61no—
180(C2×D3)⋊C4C2×D311yescharacteristic
181(C2×D3)⋊C4C2×D311yescharacteristic
182(C2×D3)⋊C4C2×D311yescharacteristic
183C4⋊Dic3—31no—
184C4⋊Dic3—31no—
185C4⋊Dic3C2×D311yescharacteristic, complemented
186C4⋊Dic3C2×D311yescharacteristic, complemented
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4—61no—
192Dic3⋊C4C2×D311yescharacteristic, complemented
193C4×Dic3—31no—
194C4×Dic3—31no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32C2411yescharacteristic
199C2×C3⋊D3—21no—
200C2×C3⋊D3—21no—
201C2×C3⋊D3—21no—
202C2×C3⋊D3—21no—
203C2×C3⋊D3—21no—
204C2×C3⋊D3—21no—
205C2×C3×D3—21no—
206C2×C3×D3—21no—
207C2×C3×D3—21no—
208C2×C3×D3—21no—
209C2×C3×D3—21no—
210C2×C3×D3—21no—
211C2×C3×D3—21no—
212C2×C3×D3—21no—
213C2×C3×D3—21no—
214D32—44no—
215C32×C4—21no—
216C32×C4—21no—
217C3⋊Dic3—21no—
218C3⋊Dic3—21no—
219C3×Dic3—21no—
220C3×Dic3—21no—
221C3×Dic3—21no—
222C3×Dic3—21no—
223C3×Dic3—21no—
224#127—91no—
225#126—91no—
226#126—91no—
227#123—31no—
228#123—91no—
229#123—91no—
230#121—31no—
231#121—91no—
232#121—91no—
233#121—91no—
234#121—91no—
235#121—91no—
236#120—91no—
237#120—91no—
238#120—91no—
239C23×C3—21no—
240C23×C3—31no—
241C23×C3—31no—
242C23×C3—31no—
243C23×C3C22×D311yescharacteristic
244C23×C3C22×D311yescharacteristic
245C22×D3—31no—
246C22×D3—31no—
247C22×D3—31no—
248C22×D3—31no—
249C22×D3—61no—
250C22×D3—61no—
251C22×D3—121no—
252C22×D3—121no—
253C22×D3—121no—
254C22×D3—121no—
255C22×D3—121no—
256C22×D3—121no—
257C22×D3C22×D311yescharacteristic
258C22×D3C22×D311yescharacteristic
259C22×D3C22×D311yescharacteristic
260C3×D4—121no—
261C3×D4—121no—
262C3×D4—121no—
263C3×D4—121no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—31no—
271C2×C3×C4—121no—
272C2×C3×C4—121no—
273C2×C3×C4C22×D311yescharacteristic
274C2×C3×C4C22×D311yescharacteristic
275C2×C3×C4C22×D311yescharacteristic
276C2×C3×C4C22×D311yescharacteristic
277#67—41no—
278#67—41no—
279#67—41no—
280#67—121no—
281#67—121no—
282#67—121no—
283#67—121no—
284#67—121no—
285#67—121no—
286#67—121no—
287#67—121no—
288#67—121no—
289#67—121no—
290#67—122no—
291#67—122no—
292C2×Dic3—31no—
293C2×Dic3—31no—
294C2×Dic3—31no—
295C2×Dic3—31no—
296C2×Dic3—41no—
297C2×Dic3—61no—
298C2×Dic3—61no—
299C2×Dic3C22×D311yescharacteristic
300C2×Dic3C22×D311yescharacteristic
301C2×Dic3C22×D311yescharacteristic
302C2×Dic3C22×D311yescharacteristic
303C2×Dic3C22×D311yescharacteristic
304D12—41no—
305D12—121no—
306D12—121no—
307D12—122no—
308D12—121no—
309D12—121no—
310D12—121no—
311D12—121no—
312C4×D3—41no—
313C4×D3—121no—
314C4×D3—121no—
315C4×D3—122no—
316C2×C32—41no—
317C2×C32H(5,𝔽2)11yescharacteristic
318C2×C32H(5,𝔽2)11yescharacteristic
319C2×C32H(5,𝔽2)11yescharacteristic
320C3⋊D3—41no—
321C3⋊D3—41no—
322C3×D3—41no—
323C3×D3—41no—
324C3×D3—41no—
325C24—91no—
326C2×D4—31no—
327C2×D4—31no—
328C2×D4—31no—
329C2×D4—31no—
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—91no—
336C2×D4—91no—
337C2×D4—91no—
338C22×C4—31no—
339C22×C4—31no—
340C22×C4—91no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—91no—
345C4⋊C4—91no—
346C4⋊C4—91no—
347C22⋊C4—31no—
348C22⋊C4—31no—
349C22⋊C4—31no—
350C22⋊C4—91no—
351C22⋊C4—91no—
352C22⋊C4—91no—
353C22⋊C4—91no—
354C22⋊C4—91no—
355C22⋊C4—91no—
356C22⋊C4—91no—
357C22⋊C4—91no—
358C22⋊C4D3211yescharacteristic, complemented
359C42—91no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—61no—
371C22×C3—61no—
372C22×C3—61no—
373C22×C3—61no—
374C22×C3—61no—
375C22×C3—61no—
376C22×C3—61no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3C23×D311yescharacteristic
380C22×C3C23×D311yescharacteristic
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—21no—
390C2×D3—62no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C2×D3—61no—
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—61no—
398C2×D3—61no—
399C2×D3—61no—
400C2×D3—61no—
401C2×D3—61no—
402C2×D3—61no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—121no—
407C2×D3—121no—
408C2×D3—124no—
409C2×D3—124no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—61no—
420C3×C4—61no—
421Dic3—21no—
422Dic3—21no—
423Dic3—21no—
424Dic3—21no—
425Dic3—21no—
426Dic3—62no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic3—61no—
431Dic3—121no—
432C32#56011yescharacteristic, complemented, Sylow
433C23—31no—
434C23—31no—
435C23—31no—
436C23—91no—
437C23—91no—
438C23—361no—
439C23—361no—
440C23—361no—
441C23C2×D3211yescharacteristic
442D4—121no—
443D4—121no—
444D4—121no—
445D4—121no—
446D4—361no—
447D4—361no—
448D4—361no—
449D4—361no—
450D4—361no—
451D4—361no—
452D4—361no—
453D4—361no—
454C2×C4—31no—
455C2×C4—31no—
456C2×C4—31no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—91no—
460C2×C4—91no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—361no—
464C2×C4C2×D3211yescharacteristic
465C2×C4C2×D3211yescharacteristic
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—81no—
472C2×C3—121no—
473C2×C3—121no—
474C2×C3—121no—
475C2×C3#100411yescharacteristic
476C2×C3#100411yescharacteristic
477C2×C3#100411yescharacteristic
478C2×C3#100911yescharacteristic
479C2×C3#100911yescharacteristic
480C2×C3#100911yescharacteristic
481D3—41no—
482D3—41no—
483D3—41no—
484D3—122no—
485D3—121no—
486D3—121no—
487D3—121no—
488D3—121no—
489D3—241no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—61no—
494C22—61no—
495C22—61no—
496C22—61no—
497C22—61no—
498C22—61no—
499C22—61no—
500C22—61no—
501C22—61no—
502C22—181no—
503C22—181no—
504C22—181no—
505C22—181no—
506C22—181no—
507C22—181no—
508C22—364no—
509C22C22×D3211yescharacteristic
510C4—21no—
511C4—21no—
512C4—61no—
513C4—61no—
514C4—61no—
515C4—61no—
516C4—61no—
517C4—181no—
518C4—181no—
519C3—21no—
520C3#562011yescharacteristic, complemented
521C3#566611yescharacteristic, complemented
522C2—41no—
523C2—121no—
524C2—121no—
525C2—121no—
526C2—361no—
527C2—361no—
528C2#6430711yescharacteristic
529C2#6431411yescharacteristic
530C2#6431411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#560
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64093C2semidirect
2. C3×#886C2semidirect
3. C3×#884C2semidirect
4. #63983C2semidirect
5. #63979C2semidirect
6. #63978C2semidirect
7. #63978C2semidirect
8. #63923C2semidirect
9. #63922C2semidirect
10. #63916C2semidirect
11. #63909C2semidirect
12. #63905C2semidirect
13. #63858C2semidirect
14. #63853C2semidirect
15. #63848C2semidirect
20. C2×#638C22semidirect
22. C2×C4×C3⋊D3C22semidirect
25. C2×C3×#67C22semidirect
26. C22×C3×Dic3C22semidirect
28. C2×C3×C4×D3C22semidirect
32. C2×#626C22semidirect
34. C32×C22⋊C4C22semidirect
36. #3750C22semidirect
40. C3×C4⋊Dic3C22semidirect
41. C3×C4⋊Dic3C22semidirect
42. C3×Dic3⋊C4C22semidirect
43. Dic3⋊Dic3C22semidirect
44. Dic3⋊Dic3C22semidirect
47. #3722C22semidirect
51. (C2×D3)⋊Dic3C22semidirect
52. Dic32C22semidirect
73. #886D3semidirect
77. #884D3semidirect
140. C2×#67C2×D3semidirect
141. C22×Dic3C2×D3semidirect
153. C2×C4×D3C2×D3semidirect
161. C3×C22⋊C4C2×D3semidirect
162. C3×C22⋊C4C2×D3semidirect
185. C4⋊Dic3C2×D3semidirect
186. C4⋊Dic3C2×D3semidirect
192. Dic3⋊C4C2×D3semidirect
358. C22⋊C4D32semidirect
432. C32#560semidirect
520. C3#5620semidirect
521. C3#5666semidirect

Maximal subgroups

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

subgroupindex
#640932
C3×#8862
C3×#8842
#639832
#639792
#639782
#639232
#639222
#639162
#639092
#639052
#638582
#638532
#638482
#56663
#56203