Finite Group Databasebeta

← #10868228

Subgroups — #10868228

Order 768.

736 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2363C211yesmaximal, characteristic
2#73513C211yesmaximal, characteristic, complemented
3#71679C211yesmaximal, characteristic
4#71679C211yesmaximal, characteristic
5#71609C211yesmaximal, characteristic, complemented
6#71606C211yesmaximal, characteristic, complemented
7#71576C211yesmaximal, characteristic, complemented
8#71575C211yesmaximal, characteristic, complemented
9#69652C211yesmaximal, characteristic, complemented
10#69640C211yesmaximal, characteristic, complemented
11#69620C211yesmaximal, characteristic, complemented
12#69600C211yesmaximal, characteristic, complemented
13#69466C211yesmaximal, characteristic, complemented
14#69461C211yesmaximal, characteristic, complemented
15#69452C211yesmaximal, characteristic, complemented
16#27056—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×#117C2211yescharacteristic
20C3×#390—21no—
21C3×#390C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#388—21no—
24C3×#386—21no—
25C3×C4×C4⋊C4C2211yescharacteristic
26C3×C4×C22⋊C4—21no—
27C3×C4×C22⋊C4C2211yescharacteristic
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#5212C2211yescharacteristic
33#5212C2211yescharacteristic
34C2×#830C2211yescharacteristic
35C2×#830C2211yescharacteristic
36C2×#808C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4951—21no—
41#4948—21no—
42#4935C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×#803C2211yescharacteristic
48#4660—21no—
49#4655—21no—
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2497—31no—
61#2363D311yescharacteristic
62#2071—31no—
63#2065—31no—
64#2033—31no—
65#2033—31no—
66#1947—31no—
67#1943—31no—
68#1924—31no—
69#1921—31no—
70#1918—31no—
71#1862—31no—
72#1854—31no—
73#1825—31no—
74#1762—31no—
75C22×C3×D4—21no—
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C23×C3×C4C2311yescharacteristic
79C3×#121—22no—
80C3×#121—22no—
81C3×C4×D4—22no—
82C3×C4×D4—41no—
83C3×#117—22no—
84C3×#117—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4D411yescharacteristic
95C2×C3×C22⋊C4D411yescharacteristic
96C2×C3×C42—21no—
97C2×C3×C42—21no—
98C2×C3×C42C2311yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×#217—21no—
101C2×#217—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8C2311yescharacteristic
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C3×#95—21no—
116C3×#95—21no—
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119#832—22no—
120#832—22no—
121#832—22no—
122#832—41no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128#830—22no—
129#830—41no—
130#808—41no—
131#807—22no—
132#807—22no—
133#807—22no—
134C4⋊(C3⋊C8)—22no—
135#803—41no—
136C2×#121C2×D311yescharacteristic
137C2×C4×D4C2×D311yescharacteristic
138C2×#117C2×D311yescharacteristic
139C2×#107—31no—
140C2×#107—31no—
141C2×#107—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102—31no—
147C2×#102—31no—
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×#97—31no—
151#400—61no—
152#396—61no—
153#390—21no—
154#390C2×D311yescharacteristic
155#389—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#388—21no—
159#388—61no—
160#387—61no—
161#386—21no—
162#384—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—21no—
168C4×C22⋊C4C2×D311yescharacteristic
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#336—31no—
178#336—31no—
179C24×C3—21no—
180C24×C3C2×D411yescharacteristic
181C2×C3×D4—24no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—81no—
188C2×C3×D4—81no—
189C2×C3×D4#4112yes—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—42no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2411yescharacteristic
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—81no—
217C3×C4⋊C4#4112yes—
218C3×C4⋊C4#4112yes—
219C3×C22⋊C4—24no—
220C3×C22⋊C4—22no—
221C3×C22⋊C4—41no—
222C3×C22⋊C4—42no—
223C3×C22⋊C4—42no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—81no—
226C3×C42—21no—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231C3×C42—81no—
232#217—81no—
233#217—81no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244Dic3⋊C4—42no—
245Dic3⋊C4—81no—
246C4×Dic3—81no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
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#4112yes—
259C22×D4—21no—
260C22×D4C22×D311yescharacteristic
261C23×C4—21no—
262C23×C4C22×D311yescharacteristic
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267#121—22no—
268#121—22no—
269C4×D4—22no—
270C4×D4—41no—
271#117—22no—
272#117—41no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—31no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4—61no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4C22×D311yescharacteristic
283C2×C4⋊C4C22×D311yescharacteristic
284C2×C4⋊C4C22×D311yescharacteristic
285C2×C22⋊C4—21no—
286C2×C22⋊C4—21no—
287C2×C22⋊C4—21no—
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4#6711yescharacteristic
293C2×C22⋊C4#6711yescharacteristic
294C2×C42—21no—
295C2×C42—21no—
296C2×C42—61no—
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—62no—
300#107—62no—
301#107—62no—
302#106—121no—
303C4⋊C8—62no—
304#102—62no—
305#102—62no—
306#102—62no—
307#102—121no—
308C22⋊C8—62no—
309C22⋊C8—121no—
310#97—121no—
311#95—21no—
312#95—21no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95#6711yescharacteristic
319#95#6711yescharacteristic
320C23×C3—22no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—21no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3C22×D411yescharacteristic
335C3×D4—22no—
336C3×D4—22no—
337C3×D4—41no—
338C3×D4—41no—
339C3×D4—84no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—22no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—22no—
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—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
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—84no—
368C2×C3×C4#13712yes—
369C2×C3×C4C2×#4111yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C42−1+411yescharacteristic
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C3⋊C8—22no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24—21no—
392C24C2×#6711yescharacteristic
393C2×D4—24no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—81no—
400C2×D4—81no—
401C2×D4#23512yes—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—81no—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C23×D311yescharacteristic
424C2×C8—32no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—81no—
449C4⋊C4—122no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4#23512yes—
454C4⋊C4#23712yes—
455C22⋊C4—24no—
456C22⋊C4—22no—
457C22⋊C4—41no—
458C22⋊C4—42no—
459C22⋊C4—42no—
460C22⋊C4—42no—
461C22⋊C4—81no—
462C22⋊C4—241no—
463C22⋊C4—241no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—42no—
469C42—81no—
470C42—241no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3—42no—
476C22×C3—41no—
477C22×C3—42no—
478C22×C3—41no—
479C22×C3—42no—
480C22×C3—41no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3#52511yescharacteristic
486C22×C3#54611yescharacteristic
487C22×C3#54711yescharacteristic
488C22×C3C2×#13711yescharacteristic
489C22×C3#57511yescharacteristic
490C22×C3#57611yescharacteristic
491C22×C3#57611yescharacteristic
492C3×C4—22no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—42no—
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—42no—
499C3×C4—81no—
500C3×C4#52911yescharacteristic
501C3×C4#53911yescharacteristic
502C3×C4#55311yescharacteristic
503C3×C4#55611yescharacteristic
504Dic3—81no—
505Dic3—81no—
506Dic3—81no—
507Dic3—81no—
508C23—22no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23C22×#6711yescharacteristic
523D4—22no—
524D4—22no—
525D4—41no—
526D4—41no—
527D4—84no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—22no—
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—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—84no—
556C2×C4—121no—
557C2×C4—121no—
558C2×C4—121no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4#95112yes—
569C2×C4C2×#23511yescharacteristic
570C2×C4C2×#23511yescharacteristic
571C2×C4C2×#23711yescharacteristic
572C2×C4#100411yescharacteristic
573C2×C4D3×#4111yescharacteristic
574C2×C4#101011yescharacteristic
575C8—62no—
576C8—121no—
577C8—121no—
578C8—121no—
579C2×C3—81no—
580C2×C3—82no—
581C2×C3#246611yescharacteristic
582C2×C3#311611yescharacteristic
583C2×C3#312211yescharacteristic
584C2×C3#331911yescharacteristic
585C2×C3#333311yescharacteristic
586C2×C3#334611yescharacteristic
587C2×C3#334711yescharacteristic
588C22—41no—
589C22—41no—
590C22—42no—
591C22—41no—
592C22—42no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22—42no—
598C22—81no—
599C22—81no—
600C22—81no—
601C22—81no—
602C22#579111yescharacteristic
603C22#580611yescharacteristic
604C22#582211yescharacteristic
605C22C2×#95111yescharacteristic
606C22#582411yescharacteristic
607C22#583511yescharacteristic
608C22#583511yescharacteristic
609C4—22no—
610C4—41no—
611C4—42no—
612C4—42no—
613C4—41no—
614C4—41no—
615C4—41no—
616C4—81no—
617C4—241no—
618C4—241no—
619C4—241no—
620C4—241no—
621C4#553811yescharacteristic
622C4#554211yescharacteristic
623C4#555811yescharacteristic
624C4#556411yescharacteristic
625C3#2705611yescharacteristic, complemented, Sylow
626C2—81no—
627C2—82no—
628C2#8236911yescharacteristic
629C2#8253511yescharacteristic
630C2#8254211yescharacteristic
631C2#8349911yescharacteristic
632C2#8357511yescharacteristic
633C2#8360911yescharacteristic
634C2#8364811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2363C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#2363D3
hypocenterC3#27056
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2363

Complements

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

NH ≅ G/Ntype
2. #73513C2semidirect
5. #71609C2semidirect
6. #71606C2semidirect
7. #71576C2semidirect
8. #71575C2semidirect
9. #69652C2semidirect
10. #69640C2semidirect
11. #69620C2semidirect
12. #69600C2semidirect
13. #69466C2semidirect
14. #69461C2semidirect
15. #69452C2semidirect
625. C3#27056semidirect

Maximal subgroups

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

subgroupindex
C3×#23632
#735132
#716792
#716092
#716062
#715762
#715752
#696522
#696402
#696202
#696002
#694662
#694612
#694522
#270563