Finite Group Databasebeta

← #10845523

Subgroups — #10845523

Order 768.

916 conjugacy classes of subgroups.

The subgroup lattice has 720 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83063C211yesmaximal, characteristic, complemented
2C2×#5066C211yesmaximal, characteristic, complemented
3C2×#4706C211yesmaximal, characteristic, complemented
4#74368C211yesmaximal, characteristic
5C3×#1992C211yesmaximal, characteristic, complemented
6#69812C211yesmaximal, characteristic, complemented
7#69568C211yesmaximal, characteristic, complemented
8#68891C211yesmaximal, characteristic, complemented
9#68889C211yesmaximal, characteristic, complemented
10#68885C211yesmaximal, characteristic, complemented
11#68883C211yesmaximal, characteristic, complemented
12#68870C211yesmaximal, characteristic, complemented
13#68868C211yesmaximal, characteristic, complemented
14#68823C211yesmaximal, characteristic, complemented
15#68823C211yesmaximal, characteristic, complemented
16#25355—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×ΓL(1,𝔽25)C2211yescharacteristic
19C22×D3×C8C2211yescharacteristic
20C2×#895—21no—
21C2×#895C2211yescharacteristic
22C2×#895C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C3×#97C2211yescharacteristic
26C3×#384C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2211yescharacteristic
30C2×(C2×D3)⋊C8C2211yescharacteristic
31#5066—24no—
32#4991—21no—
33#4991—21no—
34#4991—21no—
35#4990—21no—
36#4990—21no—
37#4987—21no—
38#4987C2211yescharacteristic
39#4987C2211yescharacteristic
40#4987C2211yescharacteristic
41#4987C2211yescharacteristic
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4986—21no—
45C2×#809C2211yescharacteristic
46C2×#809C2211yescharacteristic
47C2×#808C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51C4×C4⋊Dic3C2211yescharacteristic
52C2×C4⋊(C3⋊C8)C2211yescharacteristic
53#4706—24no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478—31no—
63C2×#435—31no—
64#2562—31no—
65#2537—31no—
66#2077—31no—
67#2075—31no—
68#2034—31no—
69#2034—31no—
70#1992D311yescharacteristic, complemented
71#1948—31no—
72#1895—31no—
73#1893—31no—
74#1809—31no—
75#1807—31no—
76#1762—31no—
77C22×D12—21no—
78C22×D12—21no—
79C22×D12D411yescharacteristic
80C22×D12C2311yescharacteristic
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C3×#125—22no—
87C3×#125—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
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)⋊C4—21no—
106C2×(C2×D3)⋊C4D411yescharacteristic
107C2×(C2×D3)⋊C4D411yescharacteristic
108C2×C4⋊Dic3D411yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4×Dic3—21no—
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C2×ΓL(1,𝔽25)—22no—
116C2×ΓL(1,𝔽25)—22no—
117C2×ΓL(1,𝔽25)—41no—
118C2×D3×C8—22no—
119C2×D3×C8—22no—
120C2×D3×C8—41no—
121#895—22no—
122#895—22no—
123#895—41no—
124#895—41no—
125#895—44no—
126C4×D12—22no—
127C4×D12—22no—
128C4×D12—22no—
129C4×D12—22no—
130C3×#97—22no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138#821—22no—
139#821—22no—
140(C2×D3)⋊C8—22no—
141(C2×D3)⋊C8—22no—
142#809—22no—
143#809—22no—
144#808—22no—
145#808—41no—
146#808—41no—
147#807—22no—
148C4⋊(C3⋊C8)D414yescomplemented
149C22×M16—31no—
150C23×C8—31no—
151C2×#125C2×D311yescharacteristic
152C2×#121—31no—
153C2×#121—31no—
154C2×#121—61no—
155C2×C4×D4—31no—
156C2×C4×D4—31no—
157#478—64no—
158#435—64no—
159C2×#107—31no—
160C2×#106—31no—
161C2×#106—31no—
162C2×#106—31no—
163C2×C4⋊C8—31no—
164C2×#102—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×C22⋊C8—31no—
169C2×C22⋊C8—31no—
170C2×#97C2×D311yescharacteristic
171#400—61no—
172#394—61no—
173#394—61no—
174#394—61no—
175#388—61no—
176#388—61no—
177#384C2×D311yescharacteristic
178#380—31no—
179#380—31no—
180#380—31no—
181#380—31no—
182#380—31no—
183#380—31no—
184#380—61no—
185C4×C4⋊C4—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#340C2×D311yescharacteristic
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193#340C2×D311yescharacteristic
194C23×D3—21no—
195C23×D3C2×D411yescharacteristic
196C23×D3C2×D411yescharacteristic
197C22×C3×C4—21no—
198C22×C3×C4—21no—
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×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—21no—
216C2×D12—41no—
217C2×D12—42no—
218C2×D12—42no—
219C2×D12—42no—
220C2×D12—42no—
221C2×D12—81no—
222C2×D12—81no—
223C2×D12#4112yes—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—21no—
229C2×C4×D3—21no—
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×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C2×C3×C8—21no—
243C2×C3×C8—21no—
244C2×C3×C8#4112yes—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4—42no—
255C3×C4⋊C4—42no—
256C3×C4⋊C4—81no—
257C3×C42C2×D412yes—
258C3×C42#4112yes—
259(C2×D3)⋊C4—22no—
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—42no—
266(C2×D3)⋊C4—42no—
267(C2×D3)⋊C4—42no—
268(C2×D3)⋊C4—81no—
269(C2×D3)⋊C4—81no—
270C4⋊Dic3—22no—
271C4⋊Dic3—21no—
272C4⋊Dic3—21no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4×Dic3—81no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8C2×D412yes—
283C2×C3⋊C8C2×D412yes—
284ΓL(1,𝔽25)—44no—
285D3×C8—44no—
286C22×D4—31no—
287C22×D4—31no—
288C22×D4—61no—
289C22×D4—61no—
290C23×C4—31no—
291C23×C4—31no—
292C23×C4—61no—
293C2×M16—62no—
294C2×M16—62no—
295C2×M16—121no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8—62no—
299C22×C8—62no—
300C22×C8—121no—
301C22×C8C22×D311yescharacteristic
302C22×C8C22×D311yescharacteristic
303#125—22no—
304#125—22no—
305#121—62no—
306#121—62no—
307#121—121no—
308#121—121no—
309#121—124no—
310C4×D4—62no—
311C4×D4—62no—
312C4×D4—62no—
313C4×D4—62no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—31no—
320C2×C4⋊C4—31no—
321C2×C4⋊C4—31no—
322C2×C4⋊C4C22×D311yescharacteristic
323C2×C4⋊C4C22×D311yescharacteristic
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D311yescharacteristic
326C2×C22⋊C4—31no—
327C2×C22⋊C4—31no—
328C2×C22⋊C4—61no—
329C2×C22⋊C4—61no—
330C2×C22⋊C4—61no—
331C2×C22⋊C4—61no—
332C2×C22⋊C4—61no—
333C2×C22⋊C4—61no—
334C2×C22⋊C4—61no—
335C2×C22⋊C4—61no—
336C2×C22⋊C4—61no—
337C2×C42—61no—
338C2×C42C22×D311yescharacteristic
339#107—62no—
340#106—62no—
341#106—121no—
342#106—121no—
343C4⋊C8—34no—
344#102—62no—
345#102—62no—
346#102—62no—
347#102—62no—
348C22⋊C8—62no—
349C22⋊C8—62no—
350#97—22no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356#95—61no—
357#95—61no—
358C23×C3C22×D411yescharacteristic
359C22×D3—21no—
360C22×D3—21no—
361C22×D3—21no—
362C22×D3—21no—
363C22×D3—21no—
364C22×D3—21no—
365C22×D3—21no—
366C22×D3—22no—
367C22×D3—21no—
368C22×D3—21no—
369C22×D3—21no—
370C22×D3—21no—
371C22×D3—21no—
372C22×D3—21no—
373C22×D3—41no—
374C22×D3—41no—
375C22×D3—41no—
376C22×D3—41no—
377C22×D3—41no—
378C22×D3—41no—
379C22×D3AGL(1,ℤ/8)12yes—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—41no—
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#12112yes—
397C2×C3×C4#12112yes—
398C2×C3×C4C22×D411yescharacteristic
399C2×C3×C4C2×#4111yescharacteristic
400C2×C3×C4C2×#4111yescharacteristic
401C2×C3×C4C2×#4111yescharacteristic
402C2×C3×C4H(5,𝔽2)11yescharacteristic
403C2×C3×C42−1+411yescharacteristic
404C2×Dic3—21no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—21no—
410C2×Dic3—41no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3—41no—
419D12—22no—
420D12—22no—
421D12—44no—
422D12—41no—
423D12—41no—
424D12—84no—
425D12—84no—
426C4×D3—41no—
427C4×D3—41no—
428C4×D3—41no—
429C4×D3—41no—
430C4×D3—41no—
431C4×D3—41no—
432C4×D3—41no—
433C4×D3—41no—
434C4×D3—84no—
435C3×C8—22no—
436C3×C8—41no—
437C3⋊C8—22no—
438C3⋊C8—22no—
439C24—31no—
440C24—31no—
441C24—61no—
442C2×D4—32no—
443C2×D4—62no—
444C2×D4—62no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4—61no—
450C2×D4—121no—
451C2×D4—122no—
452C2×D4—122no—
453C2×D4—122no—
454C2×D4—122no—
455C2×D4—241no—
456C2×D4—241no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—31no—
462C22×C4—31no—
463C22×C4—61no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—61no—
468C22×C4—61no—
469C22×C4—61no—
470C22×C4—61no—
471C22×C4—61no—
472C22×C4—61no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—122no—
479C22×C4—122no—
480C22×C4—122no—
481C22×C4D3×D411yescharacteristic
482C22×C4D3×D411yescharacteristic
483C22×C4C23×D311yescharacteristic
484M16—124no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—32no—
491C2×C8—32no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—124no—
497C2×C8#23912yes—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—62no—
510C4⋊C4—61no—
511C4⋊C4—61no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—81no—
517C22⋊C4—62no—
518C22⋊C4—62no—
519C22⋊C4—122no—
520C22⋊C4—122no—
521C22⋊C4—122no—
522C22⋊C4—122no—
523C22⋊C4—122no—
524C22⋊C4—122no—
525C22⋊C4—122no—
526C22⋊C4—241no—
527C22⋊C4—241no—
528C42—241no—
529C42#23512yes—
530C42D3×D412yes—
531C22×C3C2×#12111yescharacteristic
532C22×C3#54611yescharacteristic
533C22×C3#54711yescharacteristic
534C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
535C22×C3#57511yescharacteristic
536C22×C3#57711yescharacteristic
537C22×C3#57711yescharacteristic
538C2×D3—22no—
539C2×D3—22no—
540C2×D3—22no—
541C2×D3—41no—
542C2×D3—41no—
543C2×D3—41no—
544C2×D3—41no—
545C2×D3—41no—
546C2×D3—41no—
547C2×D3—41no—
548C2×D3—42no—
549C2×D3—41no—
550C2×D3—42no—
551C2×D3—41no—
552C2×D3—42no—
553C2×D3—41no—
554C2×D3—41no—
555C2×D3—81no—
556C2×D3—81no—
557C2×D3—81no—
558C2×D3—81no—
559C3×C4—22no—
560C3×C4—22no—
561C3×C4—81no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4#52911yescharacteristic
566C3×C4#54211yescharacteristic
567C3×C4#54611yescharacteristic
568C3×C4#54811yescharacteristic
569Dic3—41no—
570Dic3—41no—
571Dic3—81no—
572Dic3—81no—
573Dic3—81no—
574C23—32no—
575C23—61no—
576C23—61no—
577C23—61no—
578C23—61no—
579C23—61no—
580C23—61no—
581C23—62no—
582C23—61no—
583C23—61no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—121no—
590C23—121no—
591C23—121no—
592C23—121no—
593C23—121no—
594C23—121no—
595C23C2×D3×D411yescharacteristic
596D4—62no—
597D4—62no—
598D4—124no—
599D4—121no—
600D4—121no—
601D4—244no—
602D4—244no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—61no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—61no—
623C2×C4—61no—
624C2×C4—61no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—244no—
643C2×C4#88412yes—
644C2×C4#88412yes—
645C2×C4C2×#23511yescharacteristic
646C2×C4C2×D3×D411yescharacteristic
647C2×C4C2×#23911yescharacteristic
648C2×C4#100711yescharacteristic
649C2×C4D3×#4111yescharacteristic
650C2×C4#100911yescharacteristic
651C8—22no—
652C8—41no—
653C8—62no—
654C8—62no—
655C2×C3#242711yescharacteristic
656C2×C3#305511yescharacteristic
657C2×C3#306011yescharacteristic
658C2×C3#330711yescharacteristic
659C2×C3#331011yescharacteristic
660C2×C3#334511yescharacteristic
661C2×C3#334811yescharacteristic
662D3—42no—
663D3—81no—
664D3—82no—
665C22—62no—
666C22—62no—
667C22—62no—
668C22—121no—
669C22—121no—
670C22—121no—
671C22—121no—
672C22—121no—
673C22—121no—
674C22—121no—
675C22—122no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—122no—
680C22—121no—
681C22—122no—
682C22—241no—
683C22—241no—
684C22—241no—
685C22—241no—
686C22C2×#88411yescharacteristic
687C22#567611yescharacteristic
688C22#572611yescharacteristic
689C22D3×AGL(1,ℤ/8)11yescharacteristic
690C22#577511yescharacteristic
691C22#577711yescharacteristic
692C22#577811yescharacteristic
693C4—22no—
694C4—22no—
695C4—81no—
696C4—81no—
697C4—81no—
698C4—81no—
699C4—121no—
700C4—121no—
701C4—241no—
702C4—241no—
703C4—241no—
704C4#558011yescharacteristic
705C4#558211yescharacteristic
706C4#563011yescharacteristic
707C4#563311yescharacteristic
708C3#2535511yescharacteristic, complemented, Sylow
709C2—122no—
710C2—241no—
711C2—242no—
712C2#7662411yescharacteristic
713C2#7888811yescharacteristic
714C2#7889011yescharacteristic
715C2#8172511yescharacteristic
716C2#8174811yescharacteristic
717C2#8199211yescharacteristic
718C2#8199511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1992C2
socleC23×C3C22×D4
prime radical#74368C2
hypercenter#1992D3
hypocenterC3#25355
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
1. #83063C2semidirect
2. C2×#5066C2semidirect
3. C2×#4706C2semidirect
5. C3×#1992C2semidirect
6. #69812C2semidirect
7. #69568C2semidirect
8. #68891C2semidirect
9. #68889C2semidirect
10. #68885C2semidirect
11. #68883C2semidirect
12. #68870C2semidirect
13. #68868C2semidirect
14. #68823C2semidirect
15. #68823C2semidirect
70. #1992D3semidirect
148. C4⋊(C3⋊C8)D4semidirect
708. C3#25355semidirect

Maximal subgroups

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

subgroupindex
#830632
C2×#50662
C2×#47062
#743682
C3×#19922
#698122
#695682
#688912
#688892
#688852
#688832
#688702
#688682
#688232
#253553