Finite Group Databasebeta

← #10876105

Subgroups — #10876105

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2513C211yesmaximal, characteristic, complemented
2#73770C211yesmaximal, characteristic, complemented
3#69657C211yesmaximal, characteristic
4#69657C211yesmaximal, characteristic
5#69657C211yesmaximal, characteristic
6#69653C211yesmaximal, characteristic, complemented
7#69645C211yesmaximal, characteristic, complemented
8#69641C211yesmaximal, characteristic, complemented
9#69625C211yesmaximal, characteristic, complemented
10#69613C211yesmaximal, characteristic, complemented
11#69556C211yesmaximal, characteristic, complemented
12#69542C211yesmaximal, characteristic, complemented
13#69483C211yesmaximal, characteristic, complemented
14#69474C211yesmaximal, characteristic, complemented
15#69458C211yesmaximal, characteristic, complemented
16#27581—31nomaximal, Sylow
17C22×#213C2211yescharacteristic
18C2×C4×D12C2211yescharacteristic
19C2×C4×D12C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C3×#397—21no—
22C3×#397—21no—
23C3×#390C2211yescharacteristic
24C3×#390C2211yescharacteristic
25C3×#390C2211yescharacteristic
26C3×#389C2211yescharacteristic
27C3×#388—21no—
28C3×#388—21no—
29C3×#382C2211yescharacteristic
30C3×C4×C4⋊C4C2211yescharacteristic
31C3×C4×C4⋊C4C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35#4989—21no—
36#4989—21no—
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#809C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4936C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4667—21no—
47D3×#95—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2513D311yescharacteristic, complemented
61#2349—31no—
62#2101—31no—
63#2101—31no—
64#2073—31no—
65#2067—31no—
66#1936—31no—
67#1918—31no—
68#1917—31no—
69#1879—31no—
70#1879—31no—
71#1879—31no—
72#1862—31no—
73#1804—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C2×C3×C4⋊C4—21no—
81C2×C3×C4⋊C4D411yescharacteristic
82C2×C3×C4⋊C4D411yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—21no—
88C2×C3×C22⋊C4—21no—
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C42—21no—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#213—22no—
97C2×#213—22no—
98C2×#213—22no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C4×D12—22no—
112C4×D12—22no—
113C4×D12—22no—
114C4×D12—22no—
115C4×D12—41no—
116C4×D12—41no—
117C3×#95—21no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3×#95D411yescharacteristic
122C3×#95D411yescharacteristic
123#832—22no—
124#832—41no—
125#832—41no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130#809—22no—
131#809—22no—
132#809—22no—
133#807—22no—
134C4⋊(C3⋊C8)—41no—
135#803—41no—
136C22×D8—31no—
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×#107—31no—
141C2×C4⋊C8—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×#97—31no—
149#397—21no—
150#397—21no—
151#390C2×D311yescharacteristic
152#390C2×D311yescharacteristic
153#390C2×D311yescharacteristic
154#389—31no—
155#389—31no—
156#389C2×D311yescharacteristic
157#388—21no—
158#388—21no—
159#385—61no—
160#385—61no—
161#382—31no—
162#382C2×D311yescharacteristic
163#380—61no—
164C4×C4⋊C4C2×D311yescharacteristic
165C4×C4⋊C4C2×D311yescharacteristic
166C2×#95—61no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#336—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C23×D3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2411yescharacteristic
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C2×D12—21no—
201C2×D12—21no—
202C2×D12—21no—
203C2×D12#4112yes—
204C2×D12#4112yes—
205C2×D12#4112yes—
206C2×C4×D3—24no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C4×D3—81no—
214C2×C4×D3—81no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—41no—
222C3×C4⋊C4—41no—
223C3×C4⋊C4—81no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—81no—
227C3×C22⋊C4—81no—
228C3×C22⋊C4—81no—
229C3×C22⋊C4—81no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—81no—
237#213—44no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—42no—
240(C2×D3)⋊C4—42no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3#4112yes—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C2×C3⋊C8—21no—
261C22×D4—31no—
262C22×D4C22×D311yescharacteristic
263C23×C4—61no—
264C23×C4—61no—
265C23×C4—61no—
266C2×D8—62no—
267C2×D8—62no—
268C2×D8—62no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C22×C8—31no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—62no—
277C4×D4—121no—
278C4×D4—121no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4#6711yescharacteristic
286C2×C4⋊C4#6711yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C42—21no—
300C2×C42C22×D311yescharacteristic
301C2×C42C22×D311yescharacteristic
302C2×C42C22×D311yescharacteristic
303#107—62no—
304C4⋊C8—121no—
305#102—62no—
306#102—62no—
307#102—62no—
308#102—62no—
309#102—121no—
310#102—121no—
311#97—121no—
312#95—21no—
313#95—21no—
314#95—21no—
315#95—21no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C22×D3—22no—
331C22×D3—22no—
332C22×D3—22no—
333C22×D3—41no—
334C22×D3—41no—
335C22×D3—41no—
336C22×D3—41no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4H(5,𝔽2)11yescharacteristic
373C2×C3×C42−1+411yescharacteristic
374C2×Dic3—22no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383D12—22no—
384D12—22no—
385D12—22no—
386D12#12614yes—
387C4×D3—84no—
388C4×D3—84no—
389C4×D3—84no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24—61no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—32no—
403C2×D4—32no—
404C2×D4—32no—
405C2×D4—61no—
406C2×D4—61no—
407C2×D4—61no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—64no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—122no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—241no—
420C22×C4—241no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C2×#6711yescharacteristic
426C22×C4C2×#6711yescharacteristic
427C22×C4C2×#6711yescharacteristic
428C22×C4C2×#6711yescharacteristic
429C22×C4C23×D311yescharacteristic
430D8—124no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—32no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—81no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—42no—
461C22⋊C4—42no—
462C22⋊C4—64no—
463C22⋊C4—81no—
464C22⋊C4—81no—
465C22⋊C4—81no—
466C22⋊C4—81no—
467C22⋊C4—122no—
468C22⋊C4—122no—
469C42—21no—
470C42—21no—
471C42—21no—
472C42—21no—
473C42—21no—
474C42—21no—
475C42—81no—
476C22×C3—81no—
477C22×C3—81no—
478C22×C3—81no—
479C22×C3—81no—
480C22×C3—81no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3#52511yescharacteristic
484C22×C3#54611yescharacteristic
485C22×C3#54811yescharacteristic
486C22×C3#57511yescharacteristic
487C22×C3#57511yescharacteristic
488C22×C3#57611yescharacteristic
489C22×C3#57611yescharacteristic
490C2×D3—24no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—42no—
495C2×D3—42no—
496C2×D3—42no—
497C3×C4—41no—
498C3×C4—41no—
499C3×C4—41no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—41no—
503C3×C4—81no—
504C3×C4—81no—
505C3×C4—81no—
506C3×C4C2×#12611yescharacteristic
507C3×C4#53911yescharacteristic
508C3×C4#55211yescharacteristic
509C3×C4#55311yescharacteristic
510Dic3—42no—
511Dic3—81no—
512Dic3—81no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—62no—
521C23—62no—
522C23—62no—
523C23—121no—
524C23—121no—
525C23—121no—
526C23—121no—
527C23C22×#6711yescharacteristic
528D4—34no—
529D4—41no—
530D4—41no—
531D4—41no—
532D4—41no—
533D4—62no—
534D4—62no—
535D4—62no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—62no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—121no—
570C2×C4—121no—
571C2×C4—121no—
572C2×C4—244no—
573C2×C4—244no—
574C2×C4—244no—
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4#100911yescharacteristic
580C2×C4#101011yescharacteristic
581C8—121no—
582C8—121no—
583C8—121no—
584C8—121no—
585C2×C3—161no—
586C2×C3#248111yescharacteristic
587C2×C3#311111yescharacteristic
588C2×C3#312211yescharacteristic
589C2×C3#331911yescharacteristic
590C2×C3#332211yescharacteristic
591C2×C3#337011yescharacteristic
592C2×C3#337011yescharacteristic
593D3—44no—
594C22—64no—
595C22—81no—
596C22—81no—
597C22—81no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22—122no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—122no—
607C22—122no—
608C22#582211yescharacteristic
609C22#582211yescharacteristic
610C22#582811yescharacteristic
611C22#582911yescharacteristic
612C22#583111yescharacteristic
613C22#583511yescharacteristic
614C22#583511yescharacteristic
615C4—41no—
616C4—41no—
617C4—41no—
618C4—41no—
619C4—41no—
620C4—41no—
621C4—81no—
622C4—81no—
623C4—81no—
624C4—122no—
625C4—241no—
626C4—241no—
627C4D3×#12611yescharacteristic
628C4#570711yescharacteristic
629C4#570811yescharacteristic
630C4#570911yescharacteristic
631C3#2758111yescharacteristic, complemented, Sylow
632C2—124no—
633C2—161no—
634C2#8308211yescharacteristic
635C2#8314211yescharacteristic
636C2#8343511yescharacteristic
637C2#8361711yescharacteristic
638C2#8365311yescharacteristic
639C2#8371411yescharacteristic
640C2#8371411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2513C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenter#2513D3
hypocenterC3#27581
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2513

Complements

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

NH ≅ G/Ntype
1. C3×#2513C2semidirect
2. #73770C2semidirect
6. #69653C2semidirect
7. #69645C2semidirect
8. #69641C2semidirect
9. #69625C2semidirect
10. #69613C2semidirect
11. #69556C2semidirect
12. #69542C2semidirect
13. #69483C2semidirect
14. #69474C2semidirect
15. #69458C2semidirect
60. #2513D3semidirect
631. C3#27581semidirect

Maximal subgroups

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

subgroupindex
C3×#25132
#737702
#696572
#696532
#696452
#696412
#696252
#696132
#695562
#695422
#694832
#694742
#694582
#275813