Finite Group Databasebeta

← #13140588

Subgroups — #13140588

Order 1728.

929 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 929 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#272C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721159C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720324C211yesmaximal, characteristic, complemented
7#11720323C211yesmaximal, characteristic, complemented
8#10591448—31nomaximal
9#10591447—31nomaximal
10C23×#272—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#6235—21no—
13D4×#272—22no—
14D4×#272—22no—
15C2×C4⋊#272C2211yescharacteristic
16C2×C4×#272C2211yescharacteristic
17C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
18C2×#6200—21no—
19#3753⋊C3—22no—
20C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
21C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88450—21no—
24#88436—22no—
25C2×C3×D3×D4—31no—
26C2×C3×D3×D4—31no—
27C2×C3×D3×D4—61no—
28C2×#3753D311yescharacteristic, complemented
29C2×C3×#213—31no—
30#63909—31no—
31#63908—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63659D311yescharacteristic, complemented
35C3×#832—31no—
36#63563—31no—
37#63562—31no—
38(C2×D3)⋊(C3⋊C8)—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×#272—24no—
42C22×#272—21no—
43C22×#272—42no—
44C22×#272—42no—
45C22×#272D411yescharacteristic
46C22×(C3⋊D3)⋊C3—21no—
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)D412yescomplemented
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50#6235—24no—
51C2×#1084D411yescharacteristic
52C4⋊#272—21no—
53C4⋊#272D412yescomplemented
54C4×#272—21no—
55C4×#272—21no—
56C2×(C3⋊Dic3)⋊C3—21no—
57C4⋊((C3⋊D3)⋊C3)—21no—
58C4⋊((C3⋊D3)⋊C3)—21no—
59#6200—42no—
60(C3⋊(C3⋊C8))⋊C3—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62#5217—91no—
63#5156—91no—
64#4773—91no—
65C23×C3×D3—61no—
66C23×C3×D3—61no—
67C23×C3×D3—62no—
68C2×C32×D4—21no—
69C2×C32×D4C2×D311yescharacteristic
70C2×C32×D4C2×D311yescharacteristic
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
72C2×C3×#67—61no—
73C2×C3×#67—61no—
74C2×C3×#67—62no—
75C3×D3×D4—62no—
76C3×D3×D4—62no—
77C3×D3×D4—64no—
78C3×D3×D4—64no—
79C3×D3×D4—62no—
80C3×D3×D4—62no—
81C2×C3×D12—31no—
82C2×C3×D12—31no—
83C2×C3×D12—31no—
84C2×C3×D12—61no—
85C2×C3×C4×D3—31no—
86C2×C3×C4×D3—31no—
87C2×C3×C4×D3—61no—
88C2×#626—61no—
89C2×#625—61no—
90#3753—22no—
91C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
94C3×#213—62no—
95C3×C4⋊Dic3—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98#3722—61no—
99(C2×D3)⋊Dic3—61no—
100#3714—62no—
101#3713—62no—
102C22×H(3,𝔽3)—22no—
103C22×H(3,𝔽3)—41no—
104C22×H(3,𝔽3)—41no—
105C22×H(3,𝔽3)C2×D411yescharacteristic
106C2×#272—21no—
107C2×#272—21no—
108C2×#272—22no—
109C2×#272—48no—
110C2×#272—41no—
111C2×#272—41no—
112C2×#272D812yescomplemented
113C2×(C3⋊D3)⋊C3—41no—
114C2×(C3⋊D3)⋊C3—41no—
115C2×(C3⋊D3)⋊C3—41no—
116C4×H(3,𝔽3)C2×D411yescharacteristic
117C4×H(3,𝔽3)C2×D411yescharacteristic
118#1084—21no—
119(C3⋊Dic3)⋊C3—41no—
120C22×C3×D4—91no—
121C2×D3×D4—31no—
122C2×D3×D4—31no—
123C2×D3×D4—61no—
124C2×C3×D8—91no—
125#938—91no—
126C2×#213—31no—
127C2×#213—91no—
128C2×#213—91no—
129C2×#213—91no—
130#930—91no—
131C2×D24—91no—
132#895—91no—
133C3×#102—91no—
134#832—31no—
135#832—91no—
136#832—91no—
137#830—91no—
138#821—91no—
139(C2×D3)⋊C8—91no—
140(C2×D3)⋊C8—91no—
141#809—91no—
142C23×C32—22no—
143C23×C32—21no—
144C23×C32—21no—
145C22×C3⋊D3—21no—
146C22×C3×D3—31no—
147C22×C3×D3—31no—
148C22×C3×D3—61no—
149C22×C3×D3—62no—
150C22×C3×D3—68no—
151C22×C3×D3—64no—
152C22×C3×D3—64no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C22×C3×D3—122no—
157C22×C3×D3—122no—
158C22×C3×D3—124no—
159C22×C3×D3—124no—
160C22×C3×D3—122no—
161C22×C3×D3—122no—
162C32×D4—22no—
163C32×D4—21no—
164C32×D4—21no—
165C32×D4—22no—
166C32×D4D1212yescomplemented
167C32×D4#6712yescomplemented
168C2×C32×C4—21no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C3⋊Dic3—21no—
172C4⋊(C3⋊D3)—21no—
173C4⋊(C3⋊D3)—21no—
174C3×#67—68no—
175C3×#67—64no—
176C3×#67—64no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C3×D12—32no—
182C3×D12—32no—
183C3×D12—62no—
184C3×D12—62no—
185C3×D12—61no—
186C3×D12—61no—
187C3×D12—61no—
188C3×D12—61no—
189C3×C4×D3—62no—
190C3×C4×D3—62no—
191C3×C4×D3—61no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195#626—122no—
196#625—122no—
197C3⋊(C3⋊C8)—21no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—61no—
201#447—271noSylow
202C2×H(3,𝔽3)—42no—
203C2×H(3,𝔽3)#12011yescharacteristic
204C2×H(3,𝔽3)C2×D811yescharacteristic
205C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
206#272—22no—
207#272—42no—
208(C3⋊D3)⋊C3—81no—
209C24×C3—181no—
210C23×D3—61no—
211C23×D3—61no—
212C23×D3—62no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C2×C3×D4—91no—
217C2×C3×D4—91no—
218C2×C3×D4—181no—
219C2×C3×D4—182no—
220C2×C3×D4—182no—
221C2×C3×D4D3211yescharacteristic
222C22×C3×C4—91no—
223C2×#67—61no—
224C2×#67—61no—
225C2×#67—62no—
226C2×#67—181no—
227C2×#67—181no—
228D3×D4—62no—
229D3×D4—62no—
230D3×D4—62no—
231D3×D4—62no—
232D3×D4—64no—
233D3×D4—64no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12—91no—
239C2×D12—181no—
240C2×C4×D3—31no—
241C2×C4×D3—31no—
242C2×C4×D3—61no—
243C3×D8—182no—
244C2×C3×C8—91no—
245C2×C3×C8—91no—
246C3×C4⋊C4—91no—
247#217—181no—
248#213—62no—
249#213—182no—
250#213—182no—
251#213—182no—
252(C2×D3)⋊C4—181no—
253(C2×D3)⋊C4—181no—
254C4⋊Dic3—31no—
255C4⋊Dic3—91no—
256C2×C3⋊C8—31no—
257C2×C3⋊C8—31no—
258C2×C3⋊C8—91no—
259C2×C3⋊C8—91no—
260D24—182no—
261C22×C32—24no—
262C22×C32—22no—
263C22×C32—22no—
264C22×C32—21no—
265C22×C32—42no—
266C22×C32—42no—
267C22×C32—41no—
268C22×C32—41no—
269C22×C32—41no—
270C22×C32—41no—
271C22×C32D3×D411yescharacteristic
272C22×C32D3×D411yescharacteristic
273C2×C3⋊D3—41no—
274C2×C3⋊D3—41no—
275C2×C3⋊D3—41no—
276C2×C3×D3—32no—
277C2×C3×D3—32no—
278C2×C3×D3—62no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—61no—
285C2×C3×D3—64no—
286C2×C3×D3—62no—
287C2×C3×D3—62no—
288C2×C3×D3—1216no—
289C2×C3×D3—128no—
290C2×C3×D3—128no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—121no—
294C2×C3×D3—121no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4C2×D1211yescharacteristic
303C32×C4D3×D411yescharacteristic
304C32×C4D3×D411yescharacteristic
305C32×C4C2×#6711yescharacteristic
306C3⋊Dic3—41no—
307C3×Dic3—62no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—121no—
311C22×D4—91no—
312C2×D8—91no—
313C2×D8—271no—
314#121—271no—
315#102—91no—
316#102—271no—
317C22⋊C8—271no—
318H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
319C23×C3—21no—
320C23×C3—61no—
321C23×C3—61no—
322C23×C3—62no—
323C23×C3—91no—
324C23×C3—184no—
325C23×C3—181no—
326C23×C3—181no—
327C23×C3—362no—
328C23×C3—362no—
329C22×D3—31no—
330C22×D3—31no—
331C22×D3—61no—
332C22×D3—61no—
333C22×D3—61no—
334C22×D3—62no—
335C22×D3—64no—
336C22×D3—64no—
337C22×D3—68no—
338C22×D3—61no—
339C22×D3—122no—
340C22×D3—122no—
341C22×D3—124no—
342C22×D3—122no—
343C22×D3—122no—
344C22×D3—124no—
345C22×D3—181no—
346C3×D4—21no—
347C3×D4—32no—
348C3×D4—32no—
349C3×D4—61no—
350C3×D4—61no—
351C3×D4—62no—
352C3×D4—62no—
353C3×D4—92no—
354C3×D4—181no—
355C3×D4—181no—
356C3×D4—184no—
357C3×D4—181no—
358C3×D4#62612yes—
359C2×C3×C4—31no—
360C2×C3×C4—31no—
361C2×C3×C4—61no—
362C2×C3×C4—91no—
363C2×C3×C4—181no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4C2×D3211yescharacteristic
367#67—64no—
368#67—64no—
369#67—68no—
370#67—362no—
371#67—362no—
372C2×Dic3—31no—
373C2×Dic3—31no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—181no—
377D12—32no—
378D12—32no—
379D12—62no—
380D12—61no—
381D12—61no—
382D12—62no—
383D12—61no—
384D12—61no—
385D12—181no—
386D12—181no—
387D12—362no—
388C4×D3—61no—
389C4×D3—61no—
390C4×D3—61no—
391C4×D3—61no—
392C4×D3—62no—
393C4×D3—62no—
394C3×C8—181no—
395C3×C8—181no—
396C3⋊C8—61no—
397C3⋊C8—61no—
398C3⋊C8—181no—
399C3⋊C8—181no—
400C2×C32—21no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—44no—
404C2×C32—42no—
405C2×C32—42no—
406C2×C32#88211yescharacteristic
407C2×C32D3×D811yescharacteristic
408C2×C32D3×D811yescharacteristic
409C2×C32#91111yescharacteristic
410C2×C32#91411yescharacteristic
411C2×C32#93711yescharacteristic
412C3⋊D3—81no—
413C3×D3—64no—
414C3×D3—62no—
415C3×D3—62no—
416C3×D3—124no—
417C3×D3—122no—
418C3×D3—122no—
419C3×D3—241no—
420C24—181no—
421C2×D4—91no—
422C2×D4—91no—
423C2×D4—181no—
424C2×D4—182no—
425C2×D4—182no—
426C2×D4—541no—
427C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C22×C4—91no—
429D8—182no—
430D8—542no—
431C2×C8—91no—
432C2×C8—91no—
433C4⋊C4—91no—
434C22⋊C4—541no—
435C22×C3—22no—
436C22×C3—31no—
437C22×C3—31no—
438C22×C3—41no—
439C22×C3—41no—
440C22×C3—62no—
441C22×C3—62no—
442C22×C3—64no—
443C22×C3—61no—
444C22×C3—92no—
445C22×C3—121no—
446C22×C3—121no—
447C22×C3—121no—
448C22×C3—121no—
449C22×C3—122no—
450C22×C3—122no—
451C22×C3—181no—
452C22×C3—181no—
453C22×C3—182no—
454C22×C3—361no—
455C22×C3—361no—
456C22×C3—361no—
457C22×C3—368no—
458C22×C3—361no—
459C22×C3—361no—
460C22×C3#381111yescharacteristic
461C2×D3—32no—
462C2×D3—32no—
463C2×D3—62no—
464C2×D3—61no—
465C2×D3—61no—
466C2×D3—61no—
467C2×D3—61no—
468C2×D3—62no—
469C2×D3—62no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—64no—
473C2×D3—128no—
474C2×D3—128no—
475C2×D3—1216no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—122no—
481C2×D3—122no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C2×D3—361no—
486C2×D3—361no—
487C2×D3—361no—
488C3×C4—31no—
489C3×C4—31no—
490C3×C4—31no—
491C3×C4—31no—
492C3×C4—61no—
493C3×C4—61no—
494C3×C4—181no—
495C3×C4—361no—
496C3×C4C2×#62611yescharacteristic
497C3×C4#381111yescharacteristic
498Dic3—61no—
499Dic3—61no—
500Dic3—62no—
501Dic3—121no—
502Dic3—361no—
503C32—21no—
504C32#477311yescharacteristic, complemented
505C32#515611yescharacteristic, complemented
506C23—21no—
507C23—91no—
508C23—184no—
509C23—181no—
510C23—181no—
511C23—362no—
512C23—362no—
513D4—21no—
514D4—92no—
515D4—184no—
516D4—181no—
517D4—181no—
518D4—181no—
519D4—1082no—
520D4#620012yescomplemented
521C2×C4—91no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4—181no—
525C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C8—181no—
527C8—181no—
528C2×C3—31no—
529C2×C3—31no—
530C2×C3—31no—
531C2×C3—31no—
532C2×C3—31no—
533C2×C3—31no—
534C2×C3—42no—
535C2×C3—61no—
536C2×C3—61no—
537C2×C3—61no—
538C2×C3—122no—
539C2×C3—122no—
540C2×C3—124no—
541C2×C3—182no—
542C2×C3—362no—
543C2×C3—721no—
544C2×C3#6392611yescharacteristic
545C2×C3#6393711yescharacteristic
546C2×C3#6398111yescharacteristic
547D3—62no—
548D3—62no—
549D3—64no—
550D3—122no—
551D3—122no—
552D3—124no—
553D3—241no—
554D3—721no—
555C22—22no—
556C22—41no—
557C22—41no—
558C22—92no—
559C22—181no—
560C22—181no—
561C22—182no—
562C22—368no—
563C22—361no—
564C22—361no—
565C22—361no—
566C22—361no—
567C22—361no—
568C22#8867711yescharacteristic
569C4—181no—
570C4—361no—
571C4C2×#620011yescharacteristic
572C4#8867711yescharacteristic
573C3—31no—
574C3—31no—
575C3—61no—
576C3#1059145611yescharacteristic
577C2—42no—
578C2—182no—
579C2—362no—
580C2—721no—
581C2#1172119011yescharacteristic
582C2#1172119311yescharacteristic
583C2#1172132311yescharacteristic
subgroupG/H
centerC22#88677
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3#3811
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×#272C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721159C2semidirect
4. C2×#88436C2semidirect
5. #63659⋊C3C2semidirect
6. #11720324C2semidirect
7. #11720323C2semidirect
17. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
20. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
21. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
28. C2×#3753D3semidirect
34. #63659D3semidirect
48. D4×H(3,𝔽3)D4semidirect
53. C4⋊#272D4semidirect
71. C2×C4⋊(C3⋊D3)C2×D3semidirect
91. C4⋊(C3⋊Dic3)C2×D3semidirect
92. C2×C3⋊(C3⋊C8)C2×D3semidirect
93. C2×C3⋊(C3⋊C8)C2×D3semidirect
112. C2×#272D8semidirect
166. C32×D4D12semidirect
167. C32×D4#67semidirect
318. H(3,𝔽3)#447semidirect
504. C32#4773semidirect
505. C32#5156semidirect
520. D4#6200semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#2722
C2×#3753⋊C32
#117211592
C2×#884362
#63659⋊C32
#117203242
#117203232
#105914483
#105914473