Finite Group Databasebeta

← #10765243

Subgroups — #10765243

Order 768.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2993C211yesmaximal, characteristic, complemented
2#83141C211yesmaximal, characteristic
3#79286C211yesmaximal, characteristic, complemented
4#71750C211yesmaximal, characteristic, complemented
5#70954C211yesmaximal, characteristic, complemented
6#70953C211yesmaximal, characteristic, complemented
7#70937C211yesmaximal, characteristic, complemented
8C8×#215C211yesmaximal, characteristic, complemented
9#70863C211yesmaximal, characteristic, complemented
10#70862C211yesmaximal, characteristic, complemented
11#70840C211yesmaximal, characteristic, complemented
12C8×#213C211yesmaximal, characteristic, complemented
13#70744C211yesmaximal, characteristic, complemented
14#70496C211yesmaximal, characteristic, complemented
15#70495C211yesmaximal, characteristic, complemented
16#17918—31nomaximal, Sylow
17C3×#529C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C3×#446C2211yescharacteristic, complemented
20C3×C8×Q8C2211yescharacteristic, complemented
21C3×#436—21no—
22C3×#436C2211yescharacteristic, complemented
23C3×#435—21no—
24C3×C8×D4C2211yescharacteristic, complemented
25C3×#433—21no—
26C3×#431C2211yescharacteristic
27C3×#424—21no—
28C3×#423C2211yescharacteristic
29#5066C411yescharacteristic
30#5066C411yescharacteristic
31#5057C411yescharacteristic
32#5057C411yescharacteristic
33#5046C411yescharacteristic
34#5044C411yescharacteristic
35#5036C411yescharacteristic
36#5034C411yescharacteristic
37C4×#215C2211yescharacteristic
38C4×#215C2211yescharacteristic
39C4×#213C2211yescharacteristic
40C4×#213C2211yescharacteristic
41#4925C2211yescharacteristic
42#4925C2211yescharacteristic
43#4925C2211yescharacteristic
44#4834—21no—
45#4834—21no—
46D3×C22⋊C8—21no—
47D3×C22⋊C8—21no—
48C8×D12C2211yescharacteristic
49C8×D12C2211yescharacteristic
50C8×D12C2211yescharacteristic
51C8×D12C2211yescharacteristic
52#4535C2211yescharacteristic, complemented
53#4535C2211yescharacteristic, complemented
54D4⋊(C3⋊C8)C2211yescharacteristic, complemented
55D4⋊(C3⋊C8)C2211yescharacteristic, complemented
56#4483C2211yescharacteristic
57#4483C2211yescharacteristic
58#4483C2211yescharacteristic
59#4483C2211yescharacteristic
60#4480C2211yescharacteristic, complemented
61#4480C2211yescharacteristic, complemented
62#4479C2211yescharacteristic, complemented
63#4479C2211yescharacteristic, complemented
64#4455C2211yescharacteristic, complemented
65#4454C2211yescharacteristic, complemented
66#4454C2211yescharacteristic, complemented
67C8×C3⋊C8C2211yescharacteristic, complemented
68#3105—31no—
69#2993D311yescharacteristic, complemented
70#2972—31no—
71#1586—31no—
72#1585—31no—
73#1582—31no—
74#1581—31no—
75#1580—31no—
76SD16⋊C8—31no—
77C8×SD16—31no—
78C8×D8—31no—
79#1574—31no—
80#1486—31no—
81#1482—31no—
82#1454—31no—
83C22×D12C2×C411yescharacteristic
84C22×C4×D3—21no—
85C2×C3×M16—21no—
86C2×C3×M16—21no—
87C22×C3×C8—21no—
88C3×#126—21no—
89C3×#125C2×C411yescharacteristic
90C3×#124C2×C411yescharacteristic
91C3×#123—21no—
92C3×C22⋊Q8C2×C411yescharacteristic
93C3×#121C2×C411yescharacteristic
94C3×C4×Q8C2311yescharacteristic
95C3×C4×D4—21no—
96C3×C4×D4C2311yescharacteristic
97C3×#117—21no—
98C2×C3×C42C2311yescharacteristic
99C2×#215C2×C411yescharacteristic
100C2×#215C2×C411yescharacteristic
101C2×#213C2×C411yescharacteristic
102C2×#213C2×C411yescharacteristic
103C2×(C2×D3)⋊C4—21no—
104C2×C4⋊Dic3C2×C411yescharacteristic
105C2×D3×C8—21no—
106C2×D3×C8—21no—
107C2×D3×C8—21no—
108C2×D3×C8—21no—
109C4×D12—21no—
110C4×D12—21no—
111C4×D12C2×C411yescharacteristic
112C4×D12C2×C411yescharacteristic
113C4×D12C2311yescharacteristic
114C4×D12C2311yescharacteristic
115C3×C4⋊C8—21no—
116C3×C4⋊C8—21no—
117C3×C4⋊C8—21no—
118C3×C4⋊C8D411yescharacteristic
119C3×C4⋊C8D411yescharacteristic
120C3×C4⋊C8C2311yescharacteristic
121C3×C4⋊C8C2311yescharacteristic
122C3×C22⋊C8—21no—
123C3×C22⋊C8—21no—
124C3×C22⋊C8D411yescharacteristic
125C3×C22⋊C8D411yescharacteristic
126C3×#97—21no—
127C3×C4×C8—21no—
128C3×C4×C8C2311yescharacteristic
129C3×C4×C8C2311yescharacteristic
130#835C2×C411yescharacteristic
131#835C2×C411yescharacteristic
132#832C2×C411yescharacteristic
133#832C2×C411yescharacteristic
134#830—21no—
135#830—21no—
136#830C2×C411yescharacteristic
137#830C2×C411yescharacteristic
138(C2×D3)⋊C8—21no—
139(C2×D3)⋊C8—21no—
140(C2×D3)⋊C8—21no—
141(C2×D3)⋊C8—21no—
142#809C2×C411yescharacteristic
143#809C2×C411yescharacteristic
144#809C2×C411yescharacteristic
145#809C2×C411yescharacteristic
146#808C2×C411yescharacteristic
147#808C2×C411yescharacteristic
148#807C2×C411yescharacteristic
149#807C2×C411yescharacteristic
150C4⋊(C3⋊C8)—21no—
151C4⋊(C3⋊C8)C2×C411yescharacteristic
152C4⋊(C3⋊C8)C2×C411yescharacteristic
153C4⋊(C3⋊C8)C2311yescharacteristic
154C4⋊(C3⋊C8)C2311yescharacteristic
155C4×C3⋊C8—21no—
156C4×C3⋊C8C2311yescharacteristic
157C4×C3⋊C8C2311yescharacteristic
158C4×C3⋊C8C2311yescharacteristic
159C4×C3⋊C8C2311yescharacteristic
160#529C2×D311yescharacteristic
161C2×C4×D4—31no—
162#484—31no—
163#481—31no—
164#478—31no—
165#478—31no—
166#463—31no—
167#463—31no—
168#450—31no—
169#447—31no—
170#446C2×D311yescharacteristic, complemented
171C8×Q8C2×D311yescharacteristic, complemented
172C4×SD16—31no—
173C4×SD16—31no—
174C4×D8—31no—
175C4×D8—31no—
176#436—21no—
177#436C2×D311yescharacteristic, complemented
178#435—21no—
179C8×D4—31no—
180C8×D4—31no—
181C8×D4—31no—
182C8×D4—31no—
183C8×D4—61no—
184C8×D4—61no—
185C8×D4C2×D311yescharacteristic, complemented
186#433—21no—
187#431—31no—
188#431—31no—
189#431—31no—
190#431C2×D311yescharacteristic
191#424—21no—
192#423C2×D311yescharacteristic
193C2×C22⋊C8—61no—
194C2×C22⋊C8—61no—
195#335—31no—
196#335—31no—
197#334—31no—
198#334—31no—
199Q8⋊C8—31no—
200Q8⋊C8—31no—
201D4⋊C8—31no—
202D4⋊C8—31no—
203D4⋊C8—31no—
204D4⋊C8—31no—
205D4⋊C8—31no—
206D4⋊C8—31no—
207#322—31no—
208#322—31no—
209#322—31no—
210C82—31no—
211C23×D3—21no—
212C2×C3×Q8C22×C411yescharacteristic
213C2×C3×D4—21no—
214C2×C3×D4C22×C411yescharacteristic
215C22×C3×C4—21no—
216C22×C3×C4C22×C411yescharacteristic
217C22×C3×C4C2×D411yescharacteristic
218C22×C3×C4#4111yescharacteristic
219C22×Dic3—21no—
220C2×D12—21no—
221C2×D12—22no—
222C2×D12—22no—
223C2×D12C22×C411yescharacteristic
224C2×D12C22×C411yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—81no—
230C2×C4×D3—81no—
231C3×M16—81no—
232C3×M16—81no—
233C2×C3×C8—21no—
234C2×C3×C8—21no—
235C2×C3×C8—81no—
236C2×C3×C8C2×D411yescharacteristic
237C2×C3×C8C2×D411yescharacteristic
238C2×C3×C8C2×D411yescharacteristic
239C2×C3×C8C2×D411yescharacteristic
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4C22×C411yescharacteristic
245C3×C4⋊C4C22×C411yescharacteristic
246C3×C22⋊C4—21no—
247C3×C22⋊C4—21no—
248C3×C22⋊C4—21no—
249C3×C22⋊C4—21no—
250C3×C22⋊C4—21no—
251C3×C42—21no—
252C3×C42C22×C411yescharacteristic
253C3×C42C2×D411yescharacteristic
254C3×C42#4111yescharacteristic
255C3×C42C2411yescharacteristic
256#215—22no—
257#215—22no—
258#213—22no—
259#213—22no—
260(C2×D3)⋊C4—21no—
261(C2×D3)⋊C4—21no—
262(C2×D3)⋊C4—21no—
263(C2×D3)⋊C4—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3C22×C411yescharacteristic
266C4⋊Dic3C22×C411yescharacteristic
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8C22×C411yescharacteristic
272C2×C3⋊C8C22×C411yescharacteristic
273C2×C3⋊C8C22×C411yescharacteristic
274C2×C3⋊C8C22×C411yescharacteristic
275D3×C8—81no—
276D3×C8—81no—
277D3×C8—81no—
278D3×C8—81no—
279C22×D4—31no—
280C23×C4—61no—
281C2×SD16—31no—
282C2×SD16—31no—
283C2×D8—31no—
284C2×D8—31no—
285C2×M16—21no—
286C2×M16—21no—
287C22×C8—21no—
288C22×C8—61no—
289C22×C8—61no—
290C22×C8—61no—
291C22×C8—61no—
292#126—21no—
293#125C4×D311yescharacteristic
294#124C4×D311yescharacteristic
295#123—21no—
296C22⋊Q8C4×D311yescharacteristic
297#121C4×D311yescharacteristic
298C4×Q8C22×D311yescharacteristic
299C4×D4—21no—
300C4×D4—31no—
301C4×D4—31no—
302C4×D4—31no—
303C4×D4—31no—
304C4×D4—61no—
305C4×D4—61no—
306C4×D4C22×D311yescharacteristic
307#117—21no—
308C2×C4⋊C4—31no—
309C2×C22⋊C4—61no—
310C2×C42C22×D311yescharacteristic
311#107—31no—
312#107—31no—
313#106—31no—
314#106—31no—
315C4⋊C8—21no—
316C4⋊C8—21no—
317C4⋊C8—21no—
318C4⋊C8—31no—
319C4⋊C8—31no—
320C4⋊C8—31no—
321C4⋊C8—31no—
322C4⋊C8—61no—
323C4⋊C8#6711yescharacteristic
324C4⋊C8#6711yescharacteristic
325C4⋊C8C22×D311yescharacteristic
326C4⋊C8C22×D311yescharacteristic
327#103—31no—
328#103—31no—
329#102—31no—
330#102—31no—
331#102—31no—
332#102—31no—
333#102—31no—
334#102—31no—
335C22⋊C8—21no—
336C22⋊C8—21no—
337C22⋊C8—31no—
338C22⋊C8—31no—
339C22⋊C8—61no—
340C22⋊C8—61no—
341C22⋊C8—61no—
342C22⋊C8—61no—
343C22⋊C8—61no—
344C22⋊C8—61no—
345C22⋊C8#6711yescharacteristic
346C22⋊C8#6711yescharacteristic
347#97—21no—
348C4×C8—21no—
349C4×C8—31no—
350C4×C8—31no—
351C4×C8—31no—
352C4×C8—31no—
353C4×C8—61no—
354C4×C8C22×D311yescharacteristic
355C4×C8C22×D311yescharacteristic
356C23×C3—21no—
357C23×C3C4×D411yescharacteristic
358C22×D3—21no—
359C22×D3—21no—
360C22×D3—24no—
361C22×D3—81no—
362C22×D3—81no—
363C3×Q8—21no—
364C3×Q8—21no—
365C3×D4—21no—
366C3×D4—21no—
367C3×D4—81no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—41no—
378C2×C3×C4C4×D411yescharacteristic
379C2×C3×C4C4×D411yescharacteristic
380C2×C3×C4C4×D411yescharacteristic
381C2×C3×C4C23×C411yescharacteristic
382C2×C3×C4C22×D411yescharacteristic
383C2×C3×C4C2×#4111yescharacteristic
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—81no—
387D12—24no—
388D12—21no—
389D12—21no—
390D12#13112yes—
391D12#13112yes—
392C4×D3—41no—
393C4×D3—41no—
394C4×D3—41no—
395C4×D3—41no—
396C4×D3—81no—
397C4×D3—81no—
398C3×C8—21no—
399C3×C8—21no—
400C3×C8—21no—
401C3×C8—21no—
402C3×C8—41no—
403C3×C8—41no—
404C3⋊C8—21no—
405C3⋊C8—21no—
406C3⋊C8—21no—
407C3⋊C8—21no—
408C3⋊C8—41no—
409C3⋊C8—41no—
410C3⋊C8—41no—
411C3⋊C8—41no—
412C24—61no—
413C2×Q8C2×C4×D311yescharacteristic
414C2×D4—21no—
415C2×D4—31no—
416C2×D4—31no—
417C2×D4—62no—
418C2×D4—62no—
419C2×D4—61no—
420C2×D4C2×C4×D311yescharacteristic
421C22×C4—21no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—241no—
428C22×C4—241no—
429C22×C4C2×C4×D311yescharacteristic
430C22×C4#23511yescharacteristic
431C22×C4C2×#6711yescharacteristic
432SD16—62no—
433SD16—62no—
434D8—62no—
435D8—62no—
436M16—81no—
437M16—81no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—31no—
441C2×C8—31no—
442C2×C8—31no—
443C2×C8—31no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8—61no—
447C2×C8—61no—
448C2×C8—81no—
449C2×C8—241no—
450C2×C8—241no—
451C2×C8—241no—
452C2×C8—241no—
453C2×C8C2×#6711yescharacteristic
454C2×C8C2×#6711yescharacteristic
455C2×C8C2×#6711yescharacteristic
456C2×C8C2×#6711yescharacteristic
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—31no—
462C4⋊C4—31no—
463C4⋊C4—61no—
464C4⋊C4C2×C4×D311yescharacteristic
465C4⋊C4C2×C4×D311yescharacteristic
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—61no—
472C22⋊C4—61no—
473C22⋊C4—61no—
474C22⋊C4—61no—
475C42—21no—
476C42C2×C4×D311yescharacteristic
477C42#23511yescharacteristic
478C42C2×#6711yescharacteristic
479C42C23×D311yescharacteristic
480C22×C3—21no—
481C22×C3—21no—
482C22×C3—21no—
483C22×C3—41no—
484C22×C3—41no—
485C22×C3—41no—
486C22×C3C2×C4×D411yescharacteristic
487C2×D3—22no—
488C2×D3—22no—
489C2×D3—41no—
490C2×D3—41no—
491C2×D3—41no—
492C2×D3—41no—
493C2×D3—84no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4C2×#13111yescharacteristic
502C3×C4C2×#13111yescharacteristic
503C3×C4#57611yescharacteristic
504C3×C4#57711yescharacteristic
505Dic3—41no—
506Dic3—41no—
507C23—21no—
508C23—64no—
509C23—61no—
510C23—61no—
511C23—241no—
512C23—241no—
513C23C4×#6711yescharacteristic
514Q8—21no—
515Q8—21no—
516D4—21no—
517D4—21no—
518D4—32no—
519D4—32no—
520D4—64no—
521D4—61no—
522D4—61no—
523D4—81no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—241no—
541C2×C4—241no—
542C2×C4—241no—
543C2×C4C4×#6711yescharacteristic
544C2×C4C4×#6711yescharacteristic
545C2×C4C4×#6711yescharacteristic
546C2×C4C22×C4×D311yescharacteristic
547C2×C4C2×#23511yescharacteristic
548C2×C4C22×#6711yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—21no—
553C8—41no—
554C8—41no—
555C8—61no—
556C8—61no—
557C8—61no—
558C8—61no—
559C8—121no—
560C8—121no—
561C8—121no—
562C8—121no—
563C2×C3—41no—
564C2×C3—81no—
565C2×C3#292911yescharacteristic
566C2×C3#294611yescharacteristic
567C2×C3#295711yescharacteristic
568D3—42no—
569D3—42no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—62no—
577C22—62no—
578C22—121no—
579C22—121no—
580C22—121no—
581C22—121no—
582C22—244no—
583C22C2×C4×#6711yescharacteristic
584C4—21no—
585C4—21no—
586C4—21no—
587C4—21no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—121no—
592C4—121no—
593C4D3×#13111yescharacteristic
594C4D3×#13111yescharacteristic
595C4#583511yescharacteristic
596C4#583711yescharacteristic
597C3#1791811yescharacteristic, complemented, Sylow
598C2—41no—
599C2—81no—
600C2—122no—
601C2—122no—
602C2#8256411yescharacteristic
603C2#8379411yescharacteristic
604C2#8381211yescharacteristic
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#2993C2
socleC22×C3C2×C4×D4
prime radical#5036C4
cosocleC3×C42C24
hypercenter#2993D3
hypocenterC3#17918
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2993

Complements

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

NH ≅ G/Ntype
1. C3×#2993C2semidirect
3. #79286C2semidirect
4. #71750C2semidirect
5. #70954C2semidirect
6. #70953C2semidirect
7. #70937C2semidirect
8. C8×#215C2semidirect
9. #70863C2semidirect
10. #70862C2semidirect
11. #70840C2semidirect
12. C8×#213C2semidirect
13. #70744C2semidirect
14. #70496C2semidirect
15. #70495C2semidirect
19. C3×#446C22semidirect
20. C3×C8×Q8C22semidirect
22. C3×#436C22semidirect
24. C3×C8×D4C22semidirect
52. #4535C22semidirect
53. #4535C22semidirect
54. D4⋊(C3⋊C8)C22semidirect
55. D4⋊(C3⋊C8)C22semidirect
60. #4480C22semidirect
61. #4480C22semidirect
62. #4479C22semidirect
63. #4479C22semidirect
64. #4455C22semidirect
65. #4454C22semidirect
66. #4454C22semidirect
67. C8×C3⋊C8C22semidirect
69. #2993D3semidirect
170. #446C2×D3semidirect
171. C8×Q8C2×D3semidirect
177. #436C2×D3semidirect
185. C8×D4C2×D3semidirect
597. C3#17918semidirect

Maximal subgroups

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

subgroupindex
C3×#29932
#831412
#792862
#717502
#709542
#709532
#709372
C8×#2152
#708632
#708622
#708402
C8×#2132
#707442
#704962
#704952
#179183