Finite Group Databasebeta

← #10867965

Subgroups — #10867965

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76304C211yesmaximal, characteristic
2#74235C211yesmaximal, characteristic, complemented
3C3×#2073C211yesmaximal, characteristic, complemented
4#70207C211yesmaximal, characteristic
5#70206C211yesmaximal, characteristic
6#70170C211yesmaximal, characteristic, complemented
7Dic3×#102C211yesmaximal, characteristic, complemented
8#69653C211yesmaximal, characteristic, complemented
9#69540C211yesmaximal, characteristic, complemented
10#68876C211yesmaximal, characteristic, complemented
11#68859C211yesmaximal, characteristic, complemented
12#68813C211yesmaximal, characteristic, complemented
13#68789C211yesmaximal, characteristic, complemented
14#68623C211yesmaximal, characteristic, complemented
15#68615C211yesmaximal, characteristic, complemented
16#27038—31nomaximal, Sylow
17C2×#938C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#832C2211yescharacteristic
25C2×#832C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×(C2×D3)⋊C8C2211yescharacteristic
28C2×#817C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4987C2211yescharacteristic
31#4987C2211yescharacteristic
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#808C2211yescharacteristic
36C2×#807C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4958—21no—
39#4957—21no—
40#4945—21no—
41Dic3×C22⋊C4—21no—
42C4×(C2×D3)⋊C4C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4674—21no—
47#4669—21no—
48#4660—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#336C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2495—31no—
61#2363—31no—
62#2073D311yescharacteristic, complemented
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1920—31no—
71#1864—31no—
72#1854—31no—
73#1825—31no—
74C4×#102—31no—
75C22×C3×D4C2311yescharacteristic
76C22×#67—21no—
77C23×Dic3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4C2311yescharacteristic
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C22⋊C4—21no—
85C2×C3×C22⋊C4—21no—
86C2×C3×C42C2311yescharacteristic
87C2×#217—21no—
88C2×#217—21no—
89#938—22no—
90#938—22no—
91D4×Dic3—22no—
92D4×Dic3—41no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×(C2×D3)⋊C4D411yescharacteristic
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×Dic3⋊C4—21no—
104C2×Dic3⋊C4—21no—
105C2×C4×Dic3—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C2×C4×Dic3C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111#892—22no—
112#892—41no—
113C3×#102—22no—
114C3×#102—22no—
115C3×#95—21no—
116C3×#95—21no—
117#832—22no—
118#832—41no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128(C2×D3)⋊C8—22no—
129(C2×D3)⋊C8—41no—
130#817—22no—
131C8×Dic3—41no—
132#808—22no—
133#807—22no—
134#807—41no—
135C4⋊(C3⋊C8)—22no—
136C2×#121—31no—
137C2×C4×D4—31no—
138C2×#117—31no—
139C2×#107—31no—
140C2×#107—31no—
141C2×#106—31no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#102—31no—
145C2×#102—31no—
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×C22⋊C8—31no—
149C2×C22⋊C8—31no—
150C2×C4×C8—31no—
151#400—61no—
152#397—61no—
153#390—61no—
154#390C2×D311yescharacteristic
155#389—31no—
156#389—31no—
157#389—31no—
158#388—61no—
159#388—61no—
160#386—61no—
161#383—61no—
162#382C2×D311yescharacteristic
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4—31no—
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C24×C3—21no—
180C23×D3C2×D411yescharacteristic
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4#4112yes—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2411yescharacteristic
193C2×#67—24no—
194C2×#67—81no—
195C2×#67—81no—
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—81no—
203C22×Dic3C2×D411yescharacteristic
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—21no—
210C2×C4×D3—21no—
211C2×C4×D3—21no—
212C2×C3×C8—21no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C22⋊C4—42no—
225C3×C22⋊C4—81no—
226C3×C42—21no—
227C3×C42—21no—
228#217—24no—
229#217—42no—
230(C2×D3)⋊C4—22no—
231(C2×D3)⋊C4—41no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—81no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—81no—
242C4⋊Dic3#4112yes—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—21no—
247C4×Dic3—21no—
248C4×Dic3—42no—
249C4×Dic3—42no—
250C4×Dic3—81no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8#4112yes—
257C22×D4—61no—
258C22×D4C22×D311yescharacteristic
259C23×C4—31no—
260C23×C4—61no—
261C22×C8—31no—
262C22×C8—31no—
263C22×C8C22×D311yescharacteristic
264C22×C8C22×D311yescharacteristic
265#121—62no—
266#121—62no—
267C4×D4—62no—
268C4×D4—121no—
269#117—62no—
270#117—121no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—31no—
284C2×C22⋊C4—31no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—31no—
291C2×C42—61no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—121no—
297#106—62no—
298#106—62no—
299C4⋊C8—62no—
300#102—22no—
301#102—22no—
302#102—62no—
303#102—121no—
304C22⋊C8—62no—
305C22⋊C8—121no—
306C4×C8—121no—
307#95—21no—
308#95—21no—
309#95—31no—
310#95—31no—
311#95—61no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318C23×C3—22no—
319C23×C3—41no—
320C23×C3—41no—
321C23×C3—41no—
322C23×C3—41no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3C22×D411yescharacteristic
326C22×D3—21no—
327C22×D3—21no—
328C22×D3—21no—
329C22×D3—21no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—21no—
333C3×D4—22no—
334C3×D4—22no—
335C3×D4—41no—
336C3×D4—41no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4H(5,𝔽2)11yescharacteristic
357C2×C3×C42−1+411yescharacteristic
358#67—84no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—22no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—84no—
380C2×Dic3#13512yes—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—41no—
385C3×C8—41no—
386C3×C8—41no—
387C3⋊C8—22no—
388C3⋊C8—41no—
389C24—21no—
390C24—31no—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—64no—
397C2×D4—241no—
398C2×D4—241no—
399C2×D4#23712yes—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—31no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—122no—
417C22×C4—122no—
418C22×C4—241no—
419C22×C4D3×D411yescharacteristic
420C22×C4D3×D411yescharacteristic
421C22×C4C23×D311yescharacteristic
422C2×C8—21no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—32no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—32no—
441C4⋊C4—32no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—241no—
449C4⋊C4—241no—
450C4⋊C4—241no—
451C22⋊C4—42no—
452C22⋊C4—62no—
453C22⋊C4—64no—
454C22⋊C4—81no—
455C22⋊C4—121no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—122no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—61no—
463C42—61no—
464C42—122no—
465C42—122no—
466C42—241no—
467C22×C3—42no—
468C22×C3—42no—
469C22×C3—42no—
470C22×C3—81no—
471C22×C3—81no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3#52511yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3#57511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C2×D3—41no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—41no—
486C2×D3—41no—
487C2×D3—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4#52911yescharacteristic
494C3×C4#53911yescharacteristic
495C3×C4#55311yescharacteristic
496C3×C4#55611yescharacteristic
497Dic3—22no—
498Dic3—42no—
499Dic3—41no—
500Dic3—42no—
501Dic3—41no—
502Dic3—81no—
503Dic3—81no—
504C23—22no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—61no—
512C23—61no—
513C23—61no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23C2×D3×D411yescharacteristic
519D4—22no—
520D4—22no—
521D4—41no—
522D4—41no—
523D4—244no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—32no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—62no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—62no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4—121no—
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—244no—
565C2×C4C2×#23711yescharacteristic
566C2×C4D3×#4111yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4#101011yescharacteristic
571C8—41no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3—82no—
576C2×C3#246611yescharacteristic
577C2×C3#310211yescharacteristic
578C2×C3#311811yescharacteristic
579C2×C3#329611yescharacteristic
580C2×C3#330911yescharacteristic
581C2×C3#334111yescharacteristic
582C2×C3#335111yescharacteristic
583D3—81no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—81no—
588C22—81no—
589C22—81no—
590C22—81no—
591C22—121no—
592C22—121no—
593C22—121no—
594C22—121no—
595C22—121no—
596C22—121no—
597C22—121no—
598C22#560611yescharacteristic
599C22#565511yescharacteristic
600C22#567611yescharacteristic
601C22D3×#13511yescharacteristic
602C22#577011yescharacteristic
603C22#577311yescharacteristic
604C22#577411yescharacteristic
605C4—41no—
606C4—41no—
607C4—62no—
608C4—81no—
609C4—81no—
610C4—81no—
611C4—121no—
612C4—122no—
613C4—121no—
614C4—122no—
615C4—241no—
616C4—241no—
617C4#564911yescharacteristic
618C4#566511yescharacteristic
619C4#570211yescharacteristic
620C4#570911yescharacteristic
621C3#2703811yescharacteristic, complemented, Sylow
622C2—82no—
623C2—241no—
624C2#7747111yescharacteristic
625C2#8050411yescharacteristic
626C2#8050911yescharacteristic
627C2#8138911yescharacteristic
628C2#8139611yescharacteristic
629C2#8165211yescharacteristic
630C2#8172411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2073C2
socleC23×C3C22×D4
prime radicalC2×#938C22
hypercenter#2073D3
hypocenterC3#27038
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2073

Complements

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

NH ≅ G/Ntype
2. #74235C2semidirect
3. C3×#2073C2semidirect
6. #70170C2semidirect
7. Dic3×#102C2semidirect
8. #69653C2semidirect
9. #69540C2semidirect
10. #68876C2semidirect
11. #68859C2semidirect
12. #68813C2semidirect
13. #68789C2semidirect
14. #68623C2semidirect
15. #68615C2semidirect
62. #2073D3semidirect
621. C3#27038semidirect

Maximal subgroups

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

subgroupindex
#763042
#742352
C3×#20732
#702072
#702062
#701702
Dic3×#1022
#696532
#695402
#688762
#688592
#688132
#687892
#686232
#686152
#270383