Finite Group Databasebeta

← #10877478

Subgroups — #10877478

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76002C211yesmaximal, characteristic, complemented
2#74168C211yesmaximal, characteristic
3C3×#2101C211yesmaximal, characteristic, complemented
4#69615C211yesmaximal, characteristic, complemented
5#69554C211yesmaximal, characteristic, complemented
6#68903C211yesmaximal, characteristic, complemented
7#68883C211yesmaximal, characteristic
8#68880C211yesmaximal, characteristic
9#68872C211yesmaximal, characteristic, complemented
10D3×#340C211yesmaximal, characteristic, complemented
11#68852C211yesmaximal, characteristic, complemented
12#68848C211yesmaximal, characteristic, complemented
13#68812C211yesmaximal, characteristic, complemented
14#68679C211yesmaximal, characteristic, complemented
15#68647C211yesmaximal, characteristic, complemented
16#27701—31nomaximal, Sylow
17C22×D3×C8C2211yescharacteristic
18C2×#895C2211yescharacteristic
19C2×D3×C4⋊C4C2211yescharacteristic
20C2×D3×C4⋊C4C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#382C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×(C2×D3)⋊C8C2211yescharacteristic
27C2×C8×Dic3C2211yescharacteristic
28#4989—21no—
29#4988C2211yescharacteristic
30#4987C2211yescharacteristic
31#4983—21no—
32#4977C2211yescharacteristic
33#4977C2211yescharacteristic
34#4972C2211yescharacteristic
35C2×#809C2211yescharacteristic
36C2×#809C2211yescharacteristic
37#4939C2211yescharacteristic
38C4×(C2×D3)⋊C4C2211yescharacteristic
39C2×C4×C3⋊C8C2211yescharacteristic
40#4675—21no—
41#4675—21no—
42#4667—21no—
43#4665—21no—
44#4665—21no—
45#4658—21no—
46C3×#340C2211yescharacteristic
47C3×#340C2211yescharacteristic
48C3×#340C2211yescharacteristic
49C3×#340C2211yescharacteristic
50C3×#336C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2504—31no—
61#2442—31no—
62#2101—31no—
63#2101D311yescharacteristic, complemented
64#2073—31no—
65#2025—31no—
66#1991—31no—
67#1934—31no—
68#1934—31no—
69#1893—31no—
70#1874—31no—
71#1835—31no—
72#1824—31no—
73C8×C22⋊C4—31no—
74C2×#340—31no—
75C22×D12—21no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
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)⋊C4D411yescharacteristic
95C2×(C2×D3)⋊C4D411yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×C4⋊Dic3C2311yescharacteristic
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4—21no—
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4—21no—
103C2×Dic3⋊C4—21no—
104C2×C4×Dic3C2311yescharacteristic
105C22×C3⋊C8C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C2×D3×C8—22no—
108C2×D3×C8—22no—
109C2×D3×C8—41no—
110#895—22no—
111#895—22no—
112D3×C4⋊C4—22no—
113D3×C4⋊C4—22no—
114D3×C4⋊C4—22no—
115D3×C4⋊C4—41no—
116#870—22no—
117#870—41no—
118C3×#95—21no—
119C3×#95—21no—
120C3×#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95D411yescharacteristic
128C3⋊#95D411yescharacteristic
129#821—22no—
130#821—41no—
131(C2×D3)⋊C8—22no—
132C8×Dic3—22no—
133#809—22no—
134#809—41no—
135C4×C3⋊C8—41no—
136C23×C8—31no—
137C2×#125—31no—
138C2×#121—31no—
139C22×C4⋊C4—31no—
140C22×C4⋊C4—31no—
141C2×#102—31no—
142C2×#102—31no—
143C2×#102—31no—
144C2×#102—31no—
145C2×C22⋊C8—31no—
146C2×C4×C8—31no—
147C2×C4×C8—31no—
148#400—61no—
149#398—61no—
150#397—61no—
151#397—61no—
152#390—31no—
153#390—31no—
154#389—31no—
155#389C2×D311yescharacteristic
156#386—61no—
157#386—61no—
158#385—61no—
159#382—31no—
160#382—31no—
161#382C2×D311yescharacteristic
162#380—31no—
163#380—61no—
164C4×C22⋊C4—31no—
165#340—31no—
166#340—31no—
167#340—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340C2×D311yescharacteristic
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—21no—
200C2×D12—42no—
201C2×D12—42no—
202C2×D12—42no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3—21no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×C4×D3—81no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C4×D3#4112yes—
214C2×C4×D3#4112yes—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—81no—
230C3×C42—21no—
231C3×C42—21no—
232(C2×D3)⋊C4—22no—
233(C2×D3)⋊C4—22no—
234(C2×D3)⋊C4—42no—
235(C2×D3)⋊C4—42no—
236(C2×D3)⋊C4—81no—
237(C2×D3)⋊C4—81no—
238(C2×D3)⋊C4—81no—
239(C2×D3)⋊C4—81no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—42no—
247Dic3⋊C4—42no—
248Dic3⋊C4—42no—
249Dic3⋊C4—42no—
250Dic3⋊C4—81no—
251C4×Dic3—21no—
252C4×Dic3—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—
259D3×C8—44no—
260C22×D4—31no—
261C22×D4—61no—
262C23×C4—31no—
263C23×C4—61no—
264C23×C4—61no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—62no—
268C22×C8—62no—
269C22×C8—121no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#125—62no—
273#125—121no—
274#121—62no—
275#121—62no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—62no—
282C2×C4⋊C4—62no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—121no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—31no—
294C2×C22⋊C4—31no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4—61no—
298C2×C22⋊C4—61no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C42—31no—
302C2×C42C22×D311yescharacteristic
303#102—62no—
304#102—62no—
305#102—121no—
306#102—121no—
307C22⋊C8—62no—
308C4×C8—62no—
309C4×C8—121no—
310#95—21no—
311#95—21no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321C23×C3C22×D411yescharacteristic
322C22×D3—21no—
323C22×D3—21no—
324C22×D3—21no—
325C22×D3—21no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C22×D3—41no—
331C22×D3—41no—
332C22×D3—41no—
333C22×D3#12412yes—
334C22×D3#13512yes—
335C22×D3#13512yes—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
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×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C2×#4111yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C42−1+411yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—22no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
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#12412yes—
378D12—41no—
379D12—41no—
380D12—41no—
381D12—41no—
382D12—84no—
383C4×D3—22no—
384C4×D3—22no—
385C4×D3—22no—
386C4×D3—22no—
387C4×D3—84no—
388C4×D3—84no—
389C3×C8—41no—
390C3×C8—41no—
391C3⋊C8—22no—
392C3⋊C8—41no—
393C24—31no—
394C24—61no—
395C2×D4—61no—
396C2×D4—61no—
397C2×D4—61no—
398C2×D4—61no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—122no—
402C2×D4—122no—
403C2×D4—122no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—31no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—31no—
412C22×C4—31no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—61no—
420C22×C4—122no—
421C22×C4—122no—
422C22×C4—122no—
423C22×C4—241no—
424C22×C4—241no—
425C22×C4—241no—
426C22×C4D3×D411yescharacteristic
427C22×C4D3×D411yescharacteristic
428C22×C4C23×D311yescharacteristic
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—32no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—61no—
441C2×C8—124no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—32no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—61no—
456C4⋊C4—81no—
457C4⋊C4—122no—
458C4⋊C4—122no—
459C4⋊C4—122no—
460C4⋊C4—122no—
461C4⋊C4—241no—
462C22⋊C4—62no—
463C22⋊C4—62no—
464C22⋊C4—122no—
465C22⋊C4—122no—
466C22⋊C4—241no—
467C22⋊C4—241no—
468C22⋊C4—241no—
469C22⋊C4—241no—
470C42—21no—
471C42—21no—
472C42—61no—
473C42—61no—
474C22×C3C2×#12411yescharacteristic
475C22×C3#54611yescharacteristic
476C22×C3#54711yescharacteristic
477C22×C3C2×#13511yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C2×D3—22no—
482C2×D3—22no—
483C2×D3—22no—
484C2×D3—22no—
485C2×D3—22no—
486C2×D3—22no—
487C2×D3—81no—
488C2×D3—81no—
489C2×D3—81no—
490C2×D3—81no—
491C2×D3—81no—
492C2×D3—81no—
493C2×D3—81no—
494C2×D3#48814yes—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—81no—
498C3×C4—81no—
499C3×C4—81no—
500C3×C4—81no—
501C3×C4#52911yescharacteristic
502C3×C4#53911yescharacteristic
503C3×C4#55111yescharacteristic
504C3×C4#55211yescharacteristic
505Dic3—22no—
506Dic3—41no—
507Dic3—41no—
508Dic3—42no—
509Dic3—81no—
510Dic3—81no—
511C23—32no—
512C23—32no—
513C23—32no—
514C23—61no—
515C23—61no—
516C23—61no—
517C23—61no—
518C23—121no—
519C23—121no—
520C23—121no—
521C23—121no—
522C23—121no—
523C23—121no—
524C23—121no—
525C23C2×D3×D411yescharacteristic
526D4—121no—
527D4—121no—
528D4—121no—
529D4—121no—
530D4—244no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—32no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—62no—
553C2×C4—62no—
554C2×C4—62no—
555C2×C4—62no—
556C2×C4—62no—
557C2×C4—61no—
558C2×C4—61no—
559C2×C4—61no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—61no—
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—244no—
572C2×C4—244no—
573C2×C4#100711yescharacteristic
574C2×C4D3×#4111yescharacteristic
575C2×C4D3×#4111yescharacteristic
576C2×C4D3×#4111yescharacteristic
577C2×C4D3×#4111yescharacteristic
578C2×C4#100911yescharacteristic
579C8—41no—
580C8—41no—
581C8—62no—
582C8—121no—
583C2×C3#248811yescharacteristic
584C2×C3C2×#48811yescharacteristic
585C2×C3#313611yescharacteristic
586C2×C3#329711yescharacteristic
587C2×C3#330111yescharacteristic
588C2×C3#333711yescharacteristic
589C2×C3#333911yescharacteristic
590D3—24no—
591D3—161no—
592C22—34no—
593C22—62no—
594C22—62no—
595C22—62no—
596C22—62no—
597C22—62no—
598C22—62no—
599C22—241no—
600C22—241no—
601C22—241no—
602C22—241no—
603C22—241no—
604C22—241no—
605C22—241no—
606C22#563111yescharacteristic
607C22#565411yescharacteristic
608C22D3×#12411yescharacteristic
609C22D3×#13511yescharacteristic
610C22D3×#13511yescharacteristic
611C22#577411yescharacteristic
612C22#577811yescharacteristic
613C4—41no—
614C4—41no—
615C4—62no—
616C4—81no—
617C4—81no—
618C4—81no—
619C4—81no—
620C4—121no—
621C4—121no—
622C4—122no—
623C4—241no—
624C4—241no—
625C4#569211yescharacteristic
626C4#569311yescharacteristic
627C4#570511yescharacteristic
628C4#570611yescharacteristic
629C3#2770111yescharacteristic, complemented, Sylow
630C2—64no—
631C2—481no—
632C2#7834711yescharacteristic
633C2#8119511yescharacteristic
634C2#8125411yescharacteristic
635C2#8138211yescharacteristic
636C2#8152511yescharacteristic
637C2D3×#48811yescharacteristic
638C2#8168011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2101C2
socleC23×C3C22×D4
prime radicalC2×#895C22
hypercenter#2101D3
hypocenterC3#27701
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2101

Complements

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

NH ≅ G/Ntype
1. #76002C2semidirect
3. C3×#2101C2semidirect
4. #69615C2semidirect
5. #69554C2semidirect
6. #68903C2semidirect
9. #68872C2semidirect
10. D3×#340C2semidirect
11. #68852C2semidirect
12. #68848C2semidirect
13. #68812C2semidirect
14. #68679C2semidirect
15. #68647C2semidirect
63. #2101D3semidirect
629. C3#27701semidirect

Maximal subgroups

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

subgroupindex
#760022
#741682
C3×#21012
#696152
#695542
#689032
#688832
#688802
#688722
D3×#3402
#688522
#688482
#688122
#686792
#686472
#277013