Finite Group Databasebeta

← #10894942

Subgroups — #10894942

Order 768.

754 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
1#83172C211yesmaximal, characteristic
2#76105C211yesmaximal, characteristic, complemented
3C3×#2041C211yesmaximal, characteristic
4#71762C211yesmaximal, characteristic
5#70188C211yesmaximal, characteristic, complemented
6#70187C211yesmaximal, characteristic, complemented
7#70186C211yesmaximal, characteristic, complemented
8#70185C211yesmaximal, characteristic, complemented
9#70154C211yesmaximal, characteristic, complemented
10#70154C211yesmaximal, characteristic, complemented
11#70153C211yesmaximal, characteristic, complemented
12#70153C211yesmaximal, characteristic, complemented
13#69613C211yesmaximal, characteristic, complemented
14#68644C211yesmaximal, characteristic, complemented
15#68604C211yesmaximal, characteristic, complemented
16#29133—31nomaximal, Sylow
17C2×C3×#127C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×#102C2211yescharacteristic
22C2×C3×#102C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C3×#389C2211yescharacteristic
25#5221C2211yescharacteristic
26#5221C2211yescharacteristic
27#5220—21no—
28#5220—21no—
29#5220—21no—
30#5220—21no—
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#815C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38C2×C8×Dic3C2211yescharacteristic
39#4984C2211yescharacteristic
40#4983—21no—
41#4982—21no—
42#4977C2211yescharacteristic
43#4975C2211yescharacteristic
44#4972C2211yescharacteristic
45#4936C2211yescharacteristic
46#4932C2211yescharacteristic
47#4928C2211yescharacteristic
48#4659—21no—
49#4658—21no—
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2647—31no—
61#2328—31no—
62#2101—31no—
63#2101—31no—
64#2041—31no—
65#2041D311yescharacteristic
66#1936—31no—
67#1936—31no—
68#1934—31no—
69#1934—31no—
70#1804—31no—
71#1804—31no—
72#1802—31no—
73#1802—31no—
74#1768—31no—
75C22×C3×D4—21no—
76C22×C3×D4—21no—
77C22×C3×D4C2311yescharacteristic
78C22×C3×D4C2311yescharacteristic
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C3×#127—22no—
84C3×#127—22no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92C2×#217—21no—
93C2×#217—21no—
94C2×#217—21no—
95D4×Dic3—22no—
96D4×Dic3—22no—
97D4×Dic3—22no—
98D4×Dic3—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112C3×#102—22no—
113C3×#102—22no—
114C3×#102—41no—
115C3×#102—41no—
116C3×#95—21no—
117#832—22no—
118#832—22no—
119#832—41no—
120#832—41no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95D411yescharacteristic
129C3⋊#95D411yescharacteristic
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#815—41no—
133Dic3⋊C8—22no—
134Dic3⋊C8—22no—
135C8×Dic3—41no—
136C2×#127C2×D311yescharacteristic
137C2×C4×D4—31no—
138C2×C4×D4—31no—
139C2×C4⋊C8—31no—
140C2×C4⋊C8—31no—
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×#102C2×D311yescharacteristic
146C2×#102C2×D311yescharacteristic
147C2×#102C2×D311yescharacteristic
148C2×#102C2×D311yescharacteristic
149C2×#97—31no—
150C2×C4×C8—31no—
151#400—61no—
152#400—61no—
153#398—61no—
154#398—61no—
155#390—31no—
156#390—31no—
157#389—31no—
158#389—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#386—61no—
162#386—61no—
163#386—61no—
164#386—61no—
165#384—31no—
166#382—31no—
167#382—31no—
168#376—31no—
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#336C2×D311yescharacteristic
179C24×C3—21no—
180C24×C3—21no—
181C2×C3×D4—24no—
182C2×C3×D4—24no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4—21no—
186C2×C3×D4—21no—
187C2×C3×D4—21no—
188C2×C3×D4—21no—
189C2×C3×D4—21no—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—81no—
194C2×C3×D4—81no—
195C2×C3×D4—81no—
196C2×C3×D4—81no—
197C2×C3×D4#4112yes—
198C2×C3×D4#4112yes—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2411yescharacteristic
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—42no—
209C22×Dic3—42no—
210C22×Dic3—81no—
211C22×Dic3—81no—
212C22×Dic3—81no—
213C22×Dic3—81no—
214C22×Dic3C2×D411yescharacteristic
215C22×Dic3C2×D411yescharacteristic
216C22×Dic3C2×D411yescharacteristic
217C22×Dic3C2×D411yescharacteristic
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—81no—
229C3×C42—21no—
230C3×C42—21no—
231#217—42no—
232#217—42no—
233#217—81no—
234#217—81no—
235#217—81no—
236#217—81no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—81no—
246Dic3⋊C4—81no—
247Dic3⋊C4—81no—
248Dic3⋊C4—81no—
249Dic3⋊C4—81no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3#4112yes—
253C4×Dic3#4112yes—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×D4—21no—
261C22×D4—21no—
262C22×D4C22×D311yescharacteristic
263C22×D4C22×D311yescharacteristic
264C23×C4—61no—
265C23×C4—61no—
266C22×C8—31no—
267C22×C8—31no—
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270#127—22no—
271#127—22no—
272C4×D4—62no—
273C4×D4—62no—
274C4×D4—62no—
275C4×D4—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C22⋊C4—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C22⋊C4—61no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42C22×D311yescharacteristic
297C4⋊C8—62no—
298C4⋊C8—62no—
299#102—22no—
300#102—22no—
301#102—41no—
302#102—41no—
303#102—62no—
304#102—62no—
305#102—121no—
306#102—121no—
307#97—121no—
308C4×C8—121no—
309#95—21no—
310#95—31no—
311#95—31no—
312#95—31no—
313#95—31no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321C23×C3—22no—
322C23×C3—22no—
323C23×C3—41no—
324C23×C3—41no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C22×D411yescharacteristic
336C3×D4—22no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—22no—
340C3×D4—41no—
341C3×D4—41no—
342C3×D4—41no—
343C3×D4—41no—
344C3×D4—84no—
345C3×D4—84no—
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—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4C2×#4111yescharacteristic
361C2×C3×C4C2×#4111yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—41no—
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—84no—
387C2×Dic3—84no—
388C2×Dic3#12412yes—
389C2×Dic3#12412yes—
390C3×C8—41no—
391C3×C8—41no—
392C3⋊C8—41no—
393C3⋊C8—41no—
394C24—21no—
395C24—21no—
396C2×D4—24no—
397C2×D4—24no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—21no—
408C2×D4—81no—
409C2×D4—81no—
410C2×D4—81no—
411C2×D4—81no—
412C2×D4#23712yes—
413C2×D4#23712yes—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—31no—
417C22×C4—31no—
418C22×C4—31no—
419C22×C4—31no—
420C22×C4—61no—
421C22×C4—61no—
422C22×C4—61no—
423C22×C4—61no—
424C22×C4—122no—
425C22×C4—122no—
426C22×C4—241no—
427C22×C4—241no—
428C22×C4—241no—
429C22×C4—241no—
430C22×C4D3×D411yescharacteristic
431C22×C4D3×D411yescharacteristic
432C22×C4C23×D311yescharacteristic
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—61no—
457C4⋊C4—81no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C4⋊C4—241no—
461C4⋊C4—241no—
462C4⋊C4—241no—
463C22⋊C4—122no—
464C22⋊C4—122no—
465C22⋊C4—241no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C42—21no—
470C42—21no—
471C42—32no—
472C42—32no—
473C42—61no—
474C42—61no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—42no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—81no—
482C22×C3—81no—
483C22×C3—81no—
484C22×C3—81no—
485C22×C3—81no—
486C22×C3—81no—
487C22×C3—81no—
488C22×C3—81no—
489C22×C3#52511yescharacteristic
490C22×C3C2×#12411yescharacteristic
491C22×C3#53711yescharacteristic
492C22×C3#57611yescharacteristic
493C22×C3#57611yescharacteristic
494C22×C3#57711yescharacteristic
495C22×C3#57711yescharacteristic
496C3×C4—41no—
497C3×C4—41no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4#55211yescharacteristic
501C3×C4#55211yescharacteristic
502C3×C4#55311yescharacteristic
503C3×C4#55311yescharacteristic
504Dic3—22no—
505Dic3—22no—
506Dic3—41no—
507Dic3—41no—
508Dic3—81no—
509Dic3—81no—
510Dic3—81no—
511Dic3—81no—
512C23—22no—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23—41no—
525C23—41no—
526C23C2×D3×D411yescharacteristic
527D4—22no—
528D4—22no—
529D4—22no—
530D4—22no—
531D4—41no—
532D4—41no—
533D4—41no—
534D4—41no—
535D4—84no—
536D4—84no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—32no—
544C2×C4—32no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—61no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
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—121no—
573C2×C4—244no—
574C2×C4—244no—
575C2×C4C2×#23711yescharacteristic
576C2×C4C2×#23711yescharacteristic
577C2×C4#100411yescharacteristic
578C2×C4D3×#4111yescharacteristic
579C2×C4D3×#4111yescharacteristic
580C2×C4#101011yescharacteristic
581C8—41no—
582C8—41no—
583C8—121no—
584C8—121no—
585C2×C3—82no—
586C2×C3—82no—
587C2×C3#256211yescharacteristic
588C2×C3#312211yescharacteristic
589C2×C3#312311yescharacteristic
590C2×C3#313611yescharacteristic
591C2×C3#313611yescharacteristic
592C2×C3#325711yescharacteristic
593C2×C3#325711yescharacteristic
594C22—42no—
595C22—42no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—81no—
601C22—81no—
602C22—81no—
603C22—81no—
604C22—81no—
605C22—81no—
606C22—81no—
607C22—81no—
608C22#558611yescharacteristic
609C22#560811yescharacteristic
610C22D3×#12411yescharacteristic
611C22#575711yescharacteristic
612C22#576211yescharacteristic
613C22#577411yescharacteristic
614C22#577511yescharacteristic
615C4—41no—
616C4—41no—
617C4—62no—
618C4—62no—
619C4—81no—
620C4—81no—
621C4—121no—
622C4—121no—
623C4—241no—
624C4—241no—
625C4—241no—
626C4—241no—
627C4#560111yescharacteristic
628C4#560111yescharacteristic
629C4#565011yescharacteristic
630C4#565011yescharacteristic
631C3#2913311yescharacteristic, complemented, Sylow
632C2—82no—
633C2—82no—
634C2#7695511yescharacteristic
635C2#7999311yescharacteristic
636C2#8013711yescharacteristic
637C2#8034611yescharacteristic
638C2#8040311yescharacteristic
639C2#8156311yescharacteristic
640C2#8171411yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2041C2
socleC23×C3C22×D4
prime radicalC2×C3×#127C22
hypercenter#2041D3
hypocenterC3#29133
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2041

Complements

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

NH ≅ G/Ntype
2. #76105C2semidirect
5. #70188C2semidirect
6. #70187C2semidirect
7. #70186C2semidirect
8. #70185C2semidirect
9. #70154C2semidirect
10. #70154C2semidirect
11. #70153C2semidirect
12. #70153C2semidirect
13. #69613C2semidirect
14. #68644C2semidirect
15. #68604C2semidirect
631. C3#29133semidirect

Maximal subgroups

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

subgroupindex
#831722
#761052
C3×#20412
#717622
#701882
#701872
#701862
#701852
#701542
#701532
#696132
#686442
#686042
#291333