Finite Group Databasebeta

← #10845397

Subgroups — #10845397

Order 768.

926 conjugacy classes of subgroups.

The subgroup lattice has 726 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4870C211yesmaximal, characteristic, complemented
2C2×#4729C211yesmaximal, characteristic, complemented
3#73942C211yesmaximal, characteristic, complemented
4#73935C211yesmaximal, characteristic
5C3×#2077C211yesmaximal, characteristic, complemented
6#70578C211yesmaximal, characteristic, complemented
7#69934C211yesmaximal, characteristic, complemented
8#69517C211yesmaximal, characteristic, complemented
9#69513C211yesmaximal, characteristic, complemented
10#68885C211yesmaximal, characteristic, complemented
11#68833C211yesmaximal, characteristic, complemented
12#68822C211yesmaximal, characteristic, complemented
13#68803C211yesmaximal, characteristic, complemented
14#68795C211yesmaximal, characteristic, complemented
15#68780C211yesmaximal, characteristic, complemented
16#25346—31nomaximal, Sylow
17C2×#930C2211yescharacteristic
18C2×C4×#67C2211yescharacteristic
19C22×#208C2211yescharacteristic
20C22×ΓL(1,𝔽25)C2211yescharacteristic
21C2×#895C2211yescharacteristic
22C2×#893C2211yescharacteristic
23C2×#888C2211yescharacteristic
24C2×#884—21no—
25C2×C3×#106C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27C3×#380C2211yescharacteristic
28C3×#380C2211yescharacteristic
29#5212C2211yescharacteristic
30C2×#821C2211yescharacteristic
31C2×#821C2211yescharacteristic
32C2×(C2×D3)⋊C8C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×Dic3⋊C8C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36#4987C2211yescharacteristic
37#4987C2211yescharacteristic
38#4987C2211yescharacteristic
39#4975C2211yescharacteristic
40Dic3×C4⋊C4C2211yescharacteristic
41C2×#809C2211yescharacteristic
42C2×#809C2211yescharacteristic
43C2×#808C2211yescharacteristic
44C2×#807C2211yescharacteristic
45#4959—21no—
46#4957—21no—
47#4947—21no—
48#4870—24no—
49#4729—24no—
50#4673—21no—
51#4672—21no—
52#4667—21no—
53#4650—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57#4551C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#476—31no—
63C2×#436—31no—
64#2563—31no—
65#2537—31no—
66#2077—31no—
67#2077D311yescharacteristic, complemented
68#2033—31no—
69#2033—31no—
70#1990—31no—
71#1947—31no—
72#1895—31no—
73#1895—31no—
74#1809—31no—
75#1809—31no—
76C4×#102—31no—
77C23×C3×C4C2311yescharacteristic
78C22×#67—21no—
79C22×#67D411yescharacteristic
80C22×D12—21no—
81C22×D12C2311yescharacteristic
82C22×C4×D3—21no—
83C22×C4×D3C2311yescharacteristic
84C22×Dic6—21no—
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95#930—22no—
96#930—41no—
97C4×#67—22no—
98C4×#67—22no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4—21no—
104C2×(C2×D3)⋊C4—21no—
105C2×(C2×D3)⋊C4D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4D411yescharacteristic
113C2×C4×Dic3—21no—
114C2×C4×Dic3C2311yescharacteristic
115C2×#208—22no—
116C2×#208—22no—
117C2×#208—41no—
118C22×C3⋊C8C2311yescharacteristic
119C22×C3⋊C8C2311yescharacteristic
120C2×ΓL(1,𝔽25)—22no—
121C2×ΓL(1,𝔽25)—22no—
122C2×ΓL(1,𝔽25)—41no—
123#895—22no—
124#895—41no—
125#893—22no—
126#893—22no—
127#888—22no—
128#888—22no—
129#884—44no—
130C3×#106—22no—
131C3×C22⋊C8—22no—
132C3×#95—21no—
133C3×#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139#821—22no—
140#821—22no—
141(C2×D3)⋊C8—22no—
142#818—41no—
143Dic3⋊C8D414yescomplemented
144C8×Dic3—22no—
145#809—22no—
146#809—22no—
147#808—22no—
148#807—41no—
149C22×M16—31no—
150C22×M16—31no—
151C2×C4⋊Q8—31no—
152C2×#121—31no—
153C2×#121—31no—
154C2×#121—61no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#476—64no—
158#436—64no—
159C2×#107—31no—
160C2×#107—31no—
161C2×#106—31no—
162C2×#106C2×D311yescharacteristic
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8C2×D311yescharacteristic
170C2×C4×C8—31no—
171#394—61no—
172#394—61no—
173#393—61no—
174#393—61no—
175#390—61no—
176#384—31no—
177#382—61no—
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—61no—
183#380C2×D311yescharacteristic
184#380C2×D311yescharacteristic
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C24×C3C2×D411yescharacteristic
195C23×D3—21no—
196C23×D3C2×D411yescharacteristic
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C2×#67—22no—
210C2×#67—22no—
211C2×#67—22no—
212C2×#67—42no—
213C2×#67—42no—
214C2×#67—81no—
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×D12—21no—
222C2×D12—21no—
223C2×D12—21no—
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—42no—
227C2×D12—42no—
228C2×D12—81no—
229C2×D12#4112yes—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—21no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×Dic6—42no—
240C2×Dic6—81no—
241C2×Dic6—81no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8—21no—
245C2×C3×C8—21no—
246C2×C3×C8—21no—
247C2×C3×C8C2×D412yes—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—21no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C22⋊C4—42no—
257C3×C22⋊C4—81no—
258#217—22no—
259#217—81no—
260(C2×D3)⋊C4—22no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263(C2×D3)⋊C4—42no—
264(C2×D3)⋊C4—42no—
265(C2×D3)⋊C4—81no—
266(C2×D3)⋊C4—81no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273Dic3⋊C4—22no—
274Dic3⋊C4—42no—
275Dic3⋊C4—42no—
276Dic3⋊C4—81no—
277C4×Dic3—42no—
278C4×Dic3C2×D412yes—
279C4×Dic3#4112yes—
280#208—44no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8C2×D412yes—
286C2×C3⋊C8#4112yes—
287ΓL(1,𝔽25)—44no—
288C22×Q8—61no—
289C22×D4—31no—
290C22×D4—31no—
291C22×D4—61no—
292C22×D4—61no—
293C23×C4—31no—
294C23×C4—61no—
295C23×C4C22×D311yescharacteristic
296C2×M16—62no—
297C2×M16—62no—
298C2×M16—62no—
299C2×M16—62no—
300C2×M16—121no—
301C2×M16—121no—
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308#121—62no—
309#121—62no—
310#121—124no—
311#121—121no—
312#121—121no—
313C4×D4—62no—
314C4×D4—62no—
315C4×D4—62no—
316C4×D4—62no—
317C2×C4⋊C4—31no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—61no—
322C2×C4⋊C4—61no—
323C2×C4⋊C4—61no—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C4⋊C4C22×D311yescharacteristic
327C2×C4⋊C4C22×D311yescharacteristic
328C2×C22⋊C4—21no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—31no—
331C2×C22⋊C4—31no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C22⋊C4—61no—
338C2×C22⋊C4—61no—
339C2×C42—31no—
340C2×C42—61no—
341#107—121no—
342#107—121no—
343#106—22no—
344#106—62no—
345C4⋊C8—34no—
346#102—62no—
347#102—62no—
348#102—62no—
349#102—62no—
350C22⋊C8—22no—
351C22⋊C8—62no—
352C4×C8—62no—
353#95—21no—
354#95—21no—
355#95—61no—
356#95—61no—
357#95—61no—
358#95—61no—
359#95—61no—
360C23×C3—21no—
361C23×C3—21no—
362C23×C3—21no—
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3C22×D411yescharacteristic
368C22×D3—21no—
369C22×D3—22no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—21no—
374C22×D3—21no—
375C22×D3—21no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3—41no—
380C22×D3—41no—
381C22×D3—41no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4—41no—
390C2×C3×C4—41no—
391C2×C3×C4—41no—
392C2×C3×C4—41no—
393C2×C3×C4—41no—
394C2×C3×C4—41no—
395C2×C3×C4—41no—
396C2×C3×C4—41no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4C22×D411yescharacteristic
402C2×C3×C4C2×#4111yescharacteristic
403C2×C3×C4C2×#4111yescharacteristic
404C2×C3×C4C2×#4111yescharacteristic
405C2×C3×C4H(5,𝔽2)11yescharacteristic
406C2×C3×C42−1+411yescharacteristic
407#67—44no—
408#67—84no—
409C2×Dic3—21no—
410C2×Dic3—21no—
411C2×Dic3—21no—
412C2×Dic3—21no—
413C2×Dic3—21no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419C2×Dic3—41no—
420C2×Dic3—41no—
421C2×Dic3—41no—
422C2×Dic3—41no—
423C2×Dic3#12112yes—
424C2×Dic3#12112yes—
425C2×Dic3AGL(1,ℤ/8)12yes—
426C2×Dic3#13712yes—
427D12—22no—
428D12—22no—
429D12—41no—
430D12—41no—
431D12—84no—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—41no—
435C4×D3—41no—
436C4×D3—84no—
437Dic6—84no—
438C3×C8—22no—
439C3×C8—41no—
440C3⋊C8—22no—
441C3⋊C8—22no—
442C24—31no—
443C24—61no—
444C24D3×D411yescharacteristic
445C2×Q8—122no—
446C2×Q8—241no—
447C2×Q8—241no—
448C2×D4—32no—
449C2×D4—62no—
450C2×D4—62no—
451C2×D4—62no—
452C2×D4—61no—
453C2×D4—61no—
454C2×D4—61no—
455C2×D4—61no—
456C2×D4—61no—
457C2×D4—122no—
458C2×D4—122no—
459C2×D4—122no—
460C2×D4—122no—
461C2×D4—241no—
462C2×D4—241no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—31no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—122no—
486C22×C4—122no—
487C22×C4—122no—
488C22×C4D3×D411yescharacteristic
489C22×C4C23×D311yescharacteristic
490M16—124no—
491M16—124no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—32no—
498C2×C8—32no—
499C2×C8—61no—
500C2×C8—61no—
501C2×C8—61no—
502C2×C8—61no—
503C2×C8D3×D412yes—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—62no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4—241no—
522C22⋊C4—42no—
523C22⋊C4—62no—
524C22⋊C4—62no—
525C22⋊C4—81no—
526C22⋊C4—122no—
527C22⋊C4—122no—
528C22⋊C4—122no—
529C22⋊C4—122no—
530C22⋊C4—241no—
531C22⋊C4—241no—
532C22⋊C4—241no—
533C42—32no—
534C42—32no—
535C42—122no—
536C22×C3—41no—
537C22×C3—41no—
538C22×C3—41no—
539C22×C3—41no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3C2×#12111yescharacteristic
544C22×C3#54611yescharacteristic
545C22×C3#54711yescharacteristic
546C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
547C22×C3C2×#13711yescharacteristic
548C22×C3#57611yescharacteristic
549C22×C3#57711yescharacteristic
550C2×D3—41no—
551C2×D3—42no—
552C2×D3—41no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—42no—
556C2×D3—41no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—42no—
560C2×D3—81no—
561C2×D3—81no—
562C2×D3—81no—
563C2×D3—81no—
564C3×C4—41no—
565C3×C4—81no—
566C3×C4—81no—
567C3×C4—81no—
568C3×C4—81no—
569C3×C4#52911yescharacteristic
570C3×C4#54211yescharacteristic
571C3×C4#54611yescharacteristic
572C3×C4#54811yescharacteristic
573Dic3—22no—
574Dic3—41no—
575Dic3—81no—
576Dic3—81no—
577Dic3—81no—
578Dic3#46814yes—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—62no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—121no—
595C23—121no—
596C23—121no—
597C23—121no—
598C23—121no—
599C23—121no—
600C23C2×D3×D411yescharacteristic
601Q8—244no—
602D4—62no—
603D4—62no—
604D4—124no—
605D4—121no—
606D4—121no—
607D4—244no—
608D4—244no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—32no—
613C2×C4—32no—
614C2×C4—32no—
615C2×C4—32no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—41no—
631C2×C4—41no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—121no—
650C2×C4—244no—
651C2×C4C2×D3×D411yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C2×C4D3×#4111yescharacteristic
654C2×C4D3×#4111yescharacteristic
655C2×C4#100911yescharacteristic
656C2×C4#101011yescharacteristic
657C8—22no—
658C8—41no—
659C8—62no—
660C8—62no—
661C2×C3—81no—
662C2×C3#242711yescharacteristic
663C2×C3C2×#46811yescharacteristic
664C2×C3#306211yescharacteristic
665C2×C3#332511yescharacteristic
666C2×C3#333111yescharacteristic
667C2×C3#335411yescharacteristic
668C2×C3#335711yescharacteristic
669D3—81no—
670D3—82no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—41no—
678C22—121no—
679C22—121no—
680C22—122no—
681C22—121no—
682C22—121no—
683C22—122no—
684C22—121no—
685C22—121no—
686C22—121no—
687C22—122no—
688C22—241no—
689C22—241no—
690C22—241no—
691C22—241no—
692C22D3×#12111yescharacteristic
693C22#565411yescharacteristic
694C22#565511yescharacteristic
695C22#576211yescharacteristic
696C22#576911yescharacteristic
697C22D3×AGL(1,ℤ/8)11yescharacteristic
698C22D3×#13711yescharacteristic
699C4—34no—
700C4—41no—
701C4—62no—
702C4—81no—
703C4—81no—
704C4—81no—
705C4—81no—
706C4—121no—
707C4—241no—
708C4—241no—
709C4—241no—
710C4#565411yescharacteristic
711C4#567711yescharacteristic
712C4#570611yescharacteristic
713C4#571011yescharacteristic
714C3#2534611yescharacteristic, complemented, Sylow
715C2—81no—
716C2—241no—
717C2—242no—
718C2#7746611yescharacteristic
719C2#8066111yescharacteristic
720C2D3×#46811yescharacteristic
721C2#8133811yescharacteristic
722C2#8140411yescharacteristic
723C2#8144711yescharacteristic
724C2#8151311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2077C2
socleC23×C3C22×D4
prime radical#73935C2
hypercenter#2077D3
hypocenterC3#25346
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2077

Complements

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

NH ≅ G/Ntype
1. C2×#4870C2semidirect
2. C2×#4729C2semidirect
3. #73942C2semidirect
5. C3×#2077C2semidirect
6. #70578C2semidirect
7. #69934C2semidirect
8. #69517C2semidirect
9. #69513C2semidirect
10. #68885C2semidirect
11. #68833C2semidirect
12. #68822C2semidirect
13. #68803C2semidirect
14. #68795C2semidirect
15. #68780C2semidirect
67. #2077D3semidirect
143. Dic3⋊C8D4semidirect
714. C3#25346semidirect

Maximal subgroups

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

subgroupindex
C2×#48702
C2×#47292
#739422
#739352
C3×#20772
#705782
#699342
#695172
#695132
#688852
#688332
#688222
#688032
#687952
#687802
#253463