Finite Group Databasebeta

← #10867971

Subgroups — #10867971

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#76236C211yesmaximal, characteristic, complemented
2#73649C211yesmaximal, characteristic
3C3×#1920C211yesmaximal, characteristic, complemented
4#70584C211yesmaximal, characteristic, complemented
5#70580C211yesmaximal, characteristic, complemented
6#69630C211yesmaximal, characteristic, complemented
7#69468C211yesmaximal, characteristic, complemented
8#68873C211yesmaximal, characteristic, complemented
9#68859C211yesmaximal, characteristic, complemented
10#68823C211yesmaximal, characteristic
11#68822C211yesmaximal, characteristic
12#68819C211yesmaximal, characteristic, complemented
13#68780C211yesmaximal, characteristic, complemented
14#68637C211yesmaximal, characteristic, complemented
15#68615C211yesmaximal, characteristic, complemented
16#27038—31nomaximal, Sylow
17C2×#895C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#892C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C3×#389C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#821C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×(C2×D3)⋊C8C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4990—21no—
29#4987C2211yescharacteristic
30#4987C2211yescharacteristic
31#4985—21no—
32#4984C2211yescharacteristic
33Dic3×C4⋊C4C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4939—21no—
41#4939C2211yescharacteristic
42C4×(C2×D3)⋊C4—21no—
43C4×(C2×D3)⋊C4C2211yescharacteristic
44#4936C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4675—21no—
47#4670—21no—
48#4665—21no—
49#4651—21no—
50C3×#340C2211yescharacteristic
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#336C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2495—31no—
61#2363—31no—
62#2073—31no—
63#2067—31no—
64#2034—31no—
65#2033—31no—
66#1949—31no—
67#1942—31no—
68#1923—31no—
69#1920—31no—
70#1920D311yescharacteristic, complemented
71#1864—31no—
72#1854—31no—
73#1825—31no—
74C4×#102—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3C2311yescharacteristic
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C2×C3×C4⋊C4—21no—
82C2×C3×C4⋊C4C2311yescharacteristic
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C42—21no—
87C2×C3×C42C2311yescharacteristic
88C2×(C2×D3)⋊C4—21no—
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×(C2×D3)⋊C4D411yescharacteristic
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3C2311yescharacteristic
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107#895—22no—
108#895—22no—
109#893—22no—
110#893—41no—
111#892—22no—
112#892—41no—
113C3×#106—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3⋊#95—21no—
117C3⋊#95—21no—
118C3⋊#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95D411yescharacteristic
124C3⋊#95D411yescharacteristic
125#821—22no—
126#821—22no—
127(C2×D3)⋊C8—22no—
128(C2×D3)⋊C8—41no—
129C8×Dic3—41no—
130#809—22no—
131#809—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×#106C2×D311yescharacteristic
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×C4×C8—31no—
151#400—61no—
152#397—61no—
153#390—31no—
154#390—61no—
155#389—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#388—61no—
159#388—61no—
160#386—61no—
161#383—61no—
162#382—31no—
163#380—31no—
164#380—31no—
165C4×C4⋊C4—31no—
166C4×C4⋊C4C2×D311yescharacteristic
167C4×C22⋊C4—31no—
168C4×C22⋊C4—61no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340C2×D311yescharacteristic
175#340C2×D311yescharacteristic
176#340C2×D311yescharacteristic
177#336—31no—
178#336C2×D311yescharacteristic
179C23×D3—21no—
180C23×D3C2×D411yescharacteristic
181C22×C3×C4—21no—
182C22×C3×C4—21no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C22×C3×C4C2×D411yescharacteristic
186C22×C3×C4C2×D411yescharacteristic
187C22×C3×C4C2411yescharacteristic
188C22×Dic3—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3C2×D411yescharacteristic
192C22×Dic3C2×D411yescharacteristic
193C22×Dic3C2×D411yescharacteristic
194C2×D12—21no—
195C2×D12—21no—
196C2×D12—21no—
197C2×D12—21no—
198C2×D12—21no—
199C2×D12—24no—
200C2×D12—81no—
201C2×D12—81no—
202C2×D12#4112yes—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—21no—
208C2×C4×D3—21no—
209C2×C4×D3—42no—
210C2×C4×D3—42no—
211C2×C4×D3—81no—
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×C4⋊C4—81no—
225C3×C4⋊C4#4112yes—
226C3×C4⋊C4#4112yes—
227C3×C42—21no—
228C3×C42—21no—
229C3×C42—81no—
230(C2×D3)⋊C4—24no—
231(C2×D3)⋊C4—22no—
232(C2×D3)⋊C4—41no—
233(C2×D3)⋊C4—42no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—42no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—81no—
246Dic3⋊C4—81no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—42no—
250C4×Dic3—42no—
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—31no—
258C22×D4—61no—
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—21no—
272C2×C4⋊C4—31no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4—31no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4C22×D311yescharacteristic
278C2×C4⋊C4C22×D311yescharacteristic
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—31no—
282C2×C22⋊C4—31no—
283C2×C22⋊C4—61no—
284C2×C22⋊C4—61no—
285C2×C22⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C22⋊C4—61no—
288C2×C22⋊C4—61no—
289C2×C22⋊C4—61no—
290C2×C42—21no—
291C2×C42—31no—
292C2×C42—61no—
293C2×C42—61no—
294C2×C42C22×D311yescharacteristic
295#107—62no—
296#107—121no—
297#106—22no—
298#106—62no—
299C4⋊C8—62no—
300#102—62no—
301#102—62no—
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×C3C22×D411yescharacteristic
319C22×D3—22no—
320C22×D3—21no—
321C22×D3—21no—
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—21no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C2×C3×C4—22no—
334C2×C3×C4—22no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
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×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4H(5,𝔽2)11yescharacteristic
356C2×C3×C42−1+411yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3#13512yes—
375D12—22no—
376D12—22no—
377D12—41no—
378D12—41no—
379D12—84no—
380C4×D3—41no—
381C4×D3—41no—
382C4×D3—41no—
383C4×D3—41no—
384C4×D3—84no—
385C3×C8—41no—
386C3×C8—41no—
387C3⋊C8—22no—
388C3⋊C8—41no—
389C24—31no—
390C24—61no—
391C2×D4—32no—
392C2×D4—64no—
393C2×D4—61no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—241no—
399C2×D4—241no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—31no—
405C22×C4—31no—
406C22×C4—31no—
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—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—81no—
447C4⋊C4—241no—
448C4⋊C4—241no—
449C4⋊C4#23512yes—
450C4⋊C4#23912yes—
451C22⋊C4—62no—
452C22⋊C4—64no—
453C22⋊C4—121no—
454C22⋊C4—122no—
455C22⋊C4—122no—
456C22⋊C4—122no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—21no—
461C42—21no—
462C42—61no—
463C42—61no—
464C42—81no—
465C42—122no—
466C42—122no—
467C22×C3#52511yescharacteristic
468C22×C3#54611yescharacteristic
469C22×C3#54711yescharacteristic
470C22×C3C2×#13511yescharacteristic
471C22×C3#57511yescharacteristic
472C22×C3#57611yescharacteristic
473C22×C3#57711yescharacteristic
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—41no—
477C2×D3—42no—
478C2×D3—41no—
479C2×D3—42no—
480C2×D3—41no—
481C2×D3—42no—
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—81no—
485C2×D3—81no—
486C2×D3—81no—
487C2×D3—81no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—42no—
491C3×C4—42no—
492C3×C4—81no—
493C3×C4—81no—
494C3×C4#52911yescharacteristic
495C3×C4#53911yescharacteristic
496C3×C4#55311yescharacteristic
497C3×C4#55611yescharacteristic
498Dic3—22no—
499Dic3—41no—
500Dic3—41no—
501Dic3—81no—
502Dic3—81no—
503Dic3—81no—
504C23—61no—
505C23—61no—
506C23—61no—
507C23—61no—
508C23—61no—
509C23—61no—
510C23—62no—
511C23—61no—
512C23—121no—
513C23—121no—
514C23—121no—
515C23—121no—
516C23—121no—
517C23—121no—
518C23C2×D3×D411yescharacteristic
519D4—62no—
520D4—62no—
521D4—121no—
522D4—121no—
523D4—244no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—32no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—61no—
544C2×C4—61no—
545C2×C4—61no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
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×#23511yescharacteristic
566C2×C4C2×#23911yescharacteristic
567C2×C4D3×#4111yescharacteristic
568C2×C4D3×#4111yescharacteristic
569C2×C4#100911yescharacteristic
570C2×C4#101011yescharacteristic
571C8—41no—
572C8—41no—
573C8—62no—
574C8—121no—
575C2×C3#246611yescharacteristic
576C2×C3#310211yescharacteristic
577C2×C3#311811yescharacteristic
578C2×C3#329611yescharacteristic
579C2×C3#330911yescharacteristic
580C2×C3#334111yescharacteristic
581C2×C3#335111yescharacteristic
582D3—81no—
583D3—82no—
584C22—121no—
585C22—121no—
586C22—122no—
587C22—121no—
588C22—121no—
589C22—122no—
590C22—121no—
591C22—121no—
592C22—121no—
593C22—122no—
594C22—241no—
595C22—241no—
596C22—241no—
597C22—241no—
598C22#555011yescharacteristic
599C22#562711yescharacteristic
600C22#565511yescharacteristic
601C22#576211yescharacteristic
602C22D3×#13511yescharacteristic
603C22#577411yescharacteristic
604C22#577711yescharacteristic
605C4—41no—
606C4—42no—
607C4—42no—
608C4—41no—
609C4—62no—
610C4—81no—
611C4—81no—
612C4—121no—
613C4—121no—
614C4—241no—
615C4—241no—
616C4—241no—
617C4#552811yescharacteristic
618C4#558411yescharacteristic
619C4#569511yescharacteristic
620C4#570711yescharacteristic
621C3#2703811yescharacteristic, complemented, Sylow
622C2—241no—
623C2—242no—
624C2#7500011yescharacteristic
625C2#7967711yescharacteristic
626C2#7979911yescharacteristic
627C2#8104811yescharacteristic
628C2#8120411yescharacteristic
629C2#8138911yescharacteristic
630C2#8144011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#1920D3
hypocenterC3#27038
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
1. #76236C2semidirect
3. C3×#1920C2semidirect
4. #70584C2semidirect
5. #70580C2semidirect
6. #69630C2semidirect
7. #69468C2semidirect
8. #68873C2semidirect
9. #68859C2semidirect
12. #68819C2semidirect
13. #68780C2semidirect
14. #68637C2semidirect
15. #68615C2semidirect
70. #1920D3semidirect
621. C3#27038semidirect

Maximal subgroups

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

subgroupindex
#762362
#736492
C3×#19202
#705842
#705802
#696302
#694682
#688732
#688592
#688232
#688222
#688192
#687802
#686372
#686152
#270383