Finite Group Databasebeta

← #10992718

Subgroups — #10992718

Order 768.

2146 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2357C211yesmaximal, characteristic, complemented
2#82416C211yesmaximal, characteristic, complemented
3C4×#940C211yesmaximal, characteristic, complemented
4#76476C212yesmaximal, complemented
5#76449C212yesmaximal, complemented
6#75553C214yesmaximal, complemented
7#75495C214yesmaximal, complemented
8#74239C211yesmaximal, characteristic, complemented
9#74220C211yesmaximal, characteristic, complemented
10#74066C214yesmaximal, complemented
11#73787C212yesmaximal, complemented
12#73753C212yesmaximal, complemented
13#73645C211yesmaximal, characteristic, complemented
14C4×#893C211yesmaximal, characteristic, complemented
15#73470C214yesmaximal, complemented
16#36211—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×#940C2211yescharacteristic
19C2×#938C2212yes—
20C2×C4×#67C2214yes—
21C2×#895C2212yes—
22C2×#893C2212yes—
23C2×#890C2212yes—
24C2×#884C2214yes—
25C2×#884C2214yes—
26C2×C4×D12C2211yescharacteristic
27C2×C4×D12C2211yescharacteristic
28C2×#872C2214yes—
29C2×C42×D3C2212yes—
30C3×#390C2212yes—
31C3×#388C2214yes—
32C3×#382C2211yescharacteristic
33C3×#381C2212yes—
34C3×C4×C4⋊C4C2212yes—
35C3×C4×C4⋊C4C2211yescharacteristic
36C3×C4×C22⋊C4C2212yes—
37#5221C2211yescharacteristic
38#5213C2214yes—
39#4990C2214yes—
40#4988C2211yescharacteristic
41#4987C2212yes—
42#4987C2212yes—
43#4975C2211yescharacteristic
44#4975C2212yes—
45Dic3×C4⋊C4C2212yes—
46#4961C2214yes—
47#4960C2212yes—
48#4957C2214yes—
49#4957C2214yes—
50#4957C2214yes—
51#4957C2214yes—
52#4953C2212yes—
53#4952C2214yes—
54#4951C2214yes—
55#4940C2214yes—
56#4939C2212yes—
57#4939C2214yes—
58C4×(C2×D3)⋊C4C2212yes—
59#4936C2211yescharacteristic
60C4×C4⋊Dic3C2212yes—
61C4×Dic3⋊C4C2214yes—
62C42×Dic3C2211yescharacteristic
63#4674C2212yes—
64#4674C2214yes—
65#4672C2214yes—
66#4671C2212yes—
67#4667C2214yes—
68#4666C2212yes—
69#4666C2212yes—
70#4666C2212yes—
71#4665C2214yes—
72#4664C2212yes—
73#4650C2214yes—
74#4647C2214yes—
75#4647C2214yes—
76#2859—31no—
77#2856—32no—
78#2535—31no—
79#2535—31no—
80#2513—34no—
81#2490—32no—
82#2490—32no—
83#2420—34no—
84#2414—34no—
85#2363—34no—
86#2357—31no—
87#2357D311yescharacteristic, complemented
88C4×#127—31no—
89#2285—32no—
90C42×D4—31no—
91C22×C3×D4C2311yescharacteristic
92C23×C3×C4C2312yes—
93C22×#67C2314yes—
94C22×#67C2314yes—
95C22×D12C2311yescharacteristic
96C22×D12C2311yescharacteristic
97C22×D12C2314yes—
98C22×C4×D3C2312yes—
99C22×C4×D3C2312yes—
100C22×C4×D3C2312yes—
101C22×C4×D3C2312yes—
102C3×C4×D4—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C22⋊C4C2314yes—
107C2×C3×C22⋊C4C2312yes—
108C2×C3×C22⋊C4C2314yes—
109C2×C3×C42C2312yes—
110C2×C3×C42C2314yes—
111C2×C3×C42C2311yescharacteristic
112C2×#217C2314yes—
113C2×#217C2314yes—
114#940—24no—
115#938—44no—
116C4×#67—48no—
117C2×(C2×D3)⋊C4C2312yes—
118C2×(C2×D3)⋊C4C2312yes—
119C2×(C2×D3)⋊C4C2312yes—
120C2×(C2×D3)⋊C4C2314yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2314yes—
131C2×Dic3⋊C4C2314yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2312yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140C2×C4×Dic3C2314yes—
141C2×C4×Dic3C2311yescharacteristic
142#895—44no—
143#893—28no—
144#890—44no—
145#884—216no—
146#884—48no—
147C4×D12—24no—
148C4×D12—24no—
149#872—48no—
150C42×D3—28no—
151C3×#95C2312yes—
152C3×#95C2312yes—
153C3×#95C2314yes—
154C3×#95C2311yescharacteristic
155C3×#95C2311yescharacteristic
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C2×#127—31no—
164C2×#125—32no—
165C2×#121—32no—
166C2×#121—32no—
167C2×#121—34no—
168C2×#121—34no—
169C2×C4×D4—34no—
170C2×C4×D4—32no—
171C2×C4×D4—31no—
172C2×C4×D4—31no—
173C2×C4×D4C2×D311yescharacteristic
174C2×#117—34no—
175C22×C42—32no—
176#397—34no—
177#397—32no—
178#397—34no—
179#397—32no—
180#394—34no—
181#394—34no—
182#392—32no—
183#392—32no—
184#390—31no—
185#390—31no—
186#390—34no—
187#390—34no—
188#390—34no—
189#390—34no—
190#390—32no—
191#390—34no—
192#390C2×D312yes—
193#389—34no—
194#389—34no—
195#388—34no—
196#388—34no—
197#388—34no—
198#388C2×D314yes—
199#386—34no—
200#386—34no—
201#384—31no—
202#384—32no—
203#382—31no—
204#382—34no—
205#382C2×D311yescharacteristic
206#381—32no—
207#381C2×D312yes—
208#380—32no—
209#380—32no—
210#380—34no—
211C4×C4⋊C4—32no—
212C4×C4⋊C4—32no—
213C4×C4⋊C4—34no—
214C4×C4⋊C4C2×D311yescharacteristic
215C4×C4⋊C4C2×D312yes—
216C4×C22⋊C4—32no—
217C4×C22⋊C4—32no—
218C4×C22⋊C4—32no—
219C4×C22⋊C4—32no—
220C4×C22⋊C4C2×D312yes—
221C43—31no—
222C24×C3C2412yes—
223C23×D3C2412yes—
224C23×D3C2412yes—
225C2×C3×D4—22no—
226C2×C3×D4—22no—
227C2×C3×D4—22no—
228C22×C3×C4—24no—
229C22×C3×C4—44no—
230C22×C3×C4C2412yes—
231C22×C3×C4C2412yes—
232C22×C3×C4C2414yes—
233C22×C3×C4C2411yescharacteristic
234C22×C3×C4C2411yescharacteristic
235C22×C3×C4C2412yes—
236C22×C3×C4C2411yescharacteristic
237C2×#67—28no—
238C2×#67—28no—
239C2×#67—28no—
240C2×#67—28no—
241C2×#67—44no—
242C2×#67—44no—
243C22×Dic3C2412yes—
244C22×Dic3C2412yes—
245C22×Dic3C2414yes—
246C22×Dic3C2412yes—
247C22×Dic3C2412yes—
248C2×D12—24no—
249C2×D12—22no—
250C2×D12—22no—
251C2×D12—22no—
252C2×D12—22no—
253C2×D12—28no—
254C2×D12—44no—
255C2×D12—44no—
256C2×C4×D3—24no—
257C2×C4×D3—24no—
258C2×C4×D3—24no—
259C2×C4×D3—24no—
260C2×C4×D3—24no—
261C2×C4×D3—24no—
262C2×C4×D3—24no—
263C2×C4×D3—24no—
264C2×C4×D3—24no—
265C2×C4×D3—44no—
266C2×C4×D3#4118yes—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—22no—
269C3×C4⋊C4—24no—
270C3×C22⋊C4—28no—
271C3×C22⋊C4—24no—
272C3×C22⋊C4—44no—
273C3×C42—24no—
274C3×C42—44no—
275C3×C42#4114yes—
276#217—44no—
277#217—44no—
278(C2×D3)⋊C4—24no—
279(C2×D3)⋊C4—24no—
280(C2×D3)⋊C4—24no—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—24no—
283(C2×D3)⋊C4—44no—
284(C2×D3)⋊C4—44no—
285(C2×D3)⋊C4—44no—
286(C2×D3)⋊C4—44no—
287(C2×D3)⋊C4—44no—
288C4⋊Dic3—22no—
289C4⋊Dic3—22no—
290C4⋊Dic3—24no—
291C4⋊Dic3—44no—
292Dic3⋊C4—28no—
293Dic3⋊C4—28no—
294Dic3⋊C4—28no—
295Dic3⋊C4—44no—
296C4×Dic3—22no—
297C4×Dic3—24no—
298C4×Dic3—24no—
299C4×Dic3—24no—
300C4×Dic3—24no—
301C4×Dic3—44no—
302C4×Dic3#4114yes—
303C22×D4—34no—
304C22×D4—34no—
305C22×D4—31no—
306C22×D4—31no—
307C22×D4—34no—
308C22×D4C22×D311yescharacteristic
309C23×C4—32no—
310C23×C4—32no—
311C23×C4—32no—
312C23×C4—32no—
313C23×C4C22×D312yes—
314#127—64no—
315#125—124no—
316#121—616no—
317#121—128no—
318#121—124no—
319#121—124no—
320C4×D4—24no—
321C4×D4—68no—
322C4×D4—64no—
323C4×D4—64no—
324C4×D4—128no—
325#117—128no—
326C2×C4⋊C4—34no—
327C2×C4⋊C4—34no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—32no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—34no—
333C2×C4⋊C4—34no—
334C2×C4⋊C4C22×D312yes—
335C2×C4⋊C4C22×D311yescharacteristic
336C2×C4⋊C4C22×D312yes—
337C2×C22⋊C4—32no—
338C2×C22⋊C4—32no—
339C2×C22⋊C4—32no—
340C2×C22⋊C4—32no—
341C2×C22⋊C4—34no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—34no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4C22×D312yes—
350C2×C22⋊C4C22×D314yes—
351C2×C22⋊C4C22×D314yes—
352C2×C42—31no—
353C2×C42—32no—
354C2×C42—32no—
355C2×C42—31no—
356C2×C42—32no—
357C2×C42—32no—
358C2×C42—34no—
359C2×C42—68no—
360C2×C42C22×D312yes—
361C2×C42C22×D314yes—
362C2×C42C22×D311yescharacteristic
363#95—32no—
364#95—32no—
365#95—32no—
366#95—34no—
367#95—34no—
368#95—34no—
369#95—32no—
370#95C22×D311yescharacteristic
371#95C22×D311yescharacteristic
372#95C22×D312yes—
373#95C22×D314yes—
374#95C22×D312yes—
375C23×C3—24no—
376C23×C3—22no—
377C23×C3—22no—
378C23×C3—22no—
379C23×C3—24no—
380C23×C3C2511yescharacteristic
381C22×D3—22no—
382C22×D3—22no—
383C22×D3—22no—
384C22×D3—22no—
385C22×D3—22no—
386C22×D3—22no—
387C22×D3—22no—
388C22×D3—24no—
389C22×D3—22no—
390C22×D3—24no—
391C22×D3—22no—
392C22×D3C2×#4114yes—
393C3×D4—44no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—22no—
398C2×C3×C4—22no—
399C2×C3×C4—24no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—24no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—84no—
407C2×C3×C4C2×#4112yes—
408C2×C3×C4C2×#4112yes—
409C2×C3×C4C2×#4112yes—
410C2×C3×C4C2×#4114yes—
411C2×C3×C4C2×#4112yes—
412C2×C3×C4H(5,𝔽2)12yes—
413#67—416no—
414#67—88no—
415C2×Dic3—22no—
416C2×Dic3—22no—
417C2×Dic3—24no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—24no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3C2×#4114yes—
427C2×Dic3C2×#4114yes—
428C2×Dic3H(5,𝔽2)14yes—
429D12—28no—
430D12—44no—
431D12—88no—
432C4×D3—216no—
433C4×D3—48no—
434C4×D3—48no—
435C4×D3—84no—
436C24—32no—
437C24—32no—
438C24C23×D312yes—
439C2×D4—22no—
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—64no—
443C2×D4—68no—
444C2×D4—68no—
445C2×D4—68no—
446C2×D4—68no—
447C2×D4—68no—
448C2×D4—62no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—124no—
453C2×D4—124no—
454C2×D4—124no—
455C2×D4—124no—
456C22×C4—24no—
457C22×C4—32no—
458C22×C4—38no—
459C22×C4—32no—
460C22×C4—32no—
461C22×C4—34no—
462C22×C4—32no—
463C22×C4—44no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—64no—
467C22×C4—64no—
468C22×C4—64no—
469C22×C4—64no—
470C22×C4—64no—
471C22×C4—64no—
472C22×C4—64no—
473C22×C4—124no—
474C22×C4C23×D312yes—
475C22×C4C23×D312yes—
476C22×C4C23×D314yes—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C22×C4C23×D312yes—
480C22×C4C23×D311yescharacteristic
481C4⋊C4—22no—
482C4⋊C4—24no—
483C4⋊C4—24no—
484C4⋊C4—68no—
485C4⋊C4—68no—
486C4⋊C4—68no—
487C4⋊C4—62no—
488C4⋊C4—62no—
489C4⋊C4—64no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C22⋊C4—24no—
493C22⋊C4—28no—
494C22⋊C4—44no—
495C22⋊C4—64no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—64no—
499C22⋊C4—68no—
500C22⋊C4—124no—
501C22⋊C4—124no—
502C22⋊C4—124no—
503C22⋊C4—124no—
504C22⋊C4—124no—
505C22⋊C4—124no—
506C22⋊C4—124no—
507C42—24no—
508C42—34no—
509C42—44no—
510C42—64no—
511C42—64no—
512C42—64no—
513C42—64no—
514C42—62no—
515C42—124no—
516C42#23514yes—
517C22×C3—42no—
518C22×C3—44no—
519C22×C3—42no—
520C22×C3—44no—
521C22×C3—42no—
522C22×C3C22×#4111yescharacteristic
523C22×C3C22×#4112yes—
524C22×C3C22×#4111yescharacteristic
525C22×C3C2×H(5,𝔽2)11yescharacteristic
526C22×C3C2×H(5,𝔽2)12yes—
527C2×D3—24no—
528C2×D3—24no—
529C2×D3—24no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—42no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—44no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—44no—
539C3×C4—22no—
540C3×C4—24no—
541C3×C4—22no—
542C3×C4—42no—
543C3×C4—44no—
544C3×C4—42no—
545C3×C4#55514yes—
546Dic3—42no—
547Dic3—42no—
548Dic3—42no—
549Dic3—44no—
550Dic3#55318yes—
551C23—22no—
552C23—24no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—34no—
557C23—62no—
558C23—62no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—64no—
565C23—62no—
566C23—62no—
567C23—64no—
568C23C24×D311yescharacteristic
569D4—44no—
570D4—68no—
571D4—1216no—
572D4—124no—
573D4—248no—
574D4—248no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—24no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—22no—
587C2×C4—34no—
588C2×C4—34no—
589C2×C4—34no—
590C2×C4—616no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—64no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—64no—
602C2×C4—84no—
603C2×C4—128no—
604C2×C4—128no—
605C2×C4—244no—
606C2×C4C2×#23512yes—
607C2×C4C2×#23512yes—
608C2×C4C2×#23512yes—
609C2×C4D3×#4114yes—
610C2×C4D3×#4112yes—
611C2×C4#100912yes—
612C2×C3—82no—
613C2×C3#344411yescharacteristic
614C2×C3C2×#55312yes—
615C2×C3C2×#55511yescharacteristic
616C2×C3#350812yes—
617C2×C3#350811yescharacteristic
618D3—44no—
619D3—82no—
620C22—44no—
621C22—42no—
622C22—42no—
623C22—44no—
624C22—42no—
625C22—64no—
626C22—64no—
627C22—64no—
628C22—122no—
629C22—122no—
630C22—122no—
631C22—122no—
632C22—124no—
633C22—122no—
634C22—122no—
635C22—124no—
636C22—122no—
637C22C22×#23511yescharacteristic
638C22C2×D3×#4112yes—
639C22C2×D3×#4111yescharacteristic
640C22C2×#100911yescharacteristic
641C22D3×H(5,𝔽2)12yes—
642C4—22no—
643C4—22no—
644C4—24no—
645C4—38no—
646C4—44no—
647C4—42no—
648C4—42no—
649C4—124no—
650C4—122no—
651C4—122no—
652C4—122no—
653C4#553214yes—
654C3#3621111yescharacteristic, complemented, Sylow
655C2—82no—
656C2—124no—
657C2—242no—
658C2C2×#553211yescharacteristic
659C2#8583611yescharacteristic
660C2#8586211yescharacteristic
661C2D3×#55312yes—
662C2#8705112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2357C2
hypercenter#2357D3
hypocenterC3#36211
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2357

Complements

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

NH ≅ G/Ntype
1. C3×#2357C2semidirect
2. #82416C2semidirect
3. C4×#940C2semidirect
4. #76476C2semidirect
5. #76449C2semidirect
6. #75553C2semidirect
7. #75495C2semidirect
8. #74239C2semidirect
9. #74220C2semidirect
10. #74066C2semidirect
11. #73787C2semidirect
12. #73753C2semidirect
13. #73645C2semidirect
14. C4×#893C2semidirect
15. #73470C2semidirect
87. #2357D3semidirect
654. C3#36211semidirect

Maximal subgroups

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

subgroupindex
C3×#23572
#824162
C4×#9402
#764762
#764492
#755532
#754952
#742392
#742202
#740662
#737872
#737532
#736452
C4×#8932
#734702
#362113