Finite Group Databasebeta

← #10844939

Subgroups — #10844939

Order 768.

788 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4845C211yesmaximal, characteristic
2C2×#4828C211yesmaximal, characteristic, complemented
3#74343C211yesmaximal, characteristic, complemented
4#73735C211yesmaximal, characteristic
5C3×#1948C211yesmaximal, characteristic, complemented
6#70589C211yesmaximal, characteristic, complemented
7#70585C211yesmaximal, characteristic
8#70575C211yesmaximal, characteristic, complemented
9#70574C211yesmaximal, characteristic
10#69581C211yesmaximal, characteristic
11#69475C211yesmaximal, characteristic, complemented
12#68864C211yesmaximal, characteristic, complemented
13#68863C211yesmaximal, characteristic
14#68857C211yesmaximal, characteristic
15#68851C211yesmaximal, characteristic, complemented
16#25311—31nomaximal, Sylow
17C22×Dic12C2211yescharacteristic
18C22×#204C2211yescharacteristic
19C2×#875C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C4×Dic6C2211yescharacteristic
22C2×C3×#106C2211yescharacteristic
23C2×C3×#106C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C3×#384C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27C2×#821C2211yescharacteristic
28C2×#816C2211yescharacteristic
29C2×#815C2211yescharacteristic
30C2×Dic3⋊C8C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#4988C2211yescharacteristic
34#4988C2211yescharacteristic
35#4984—21no—
36#4984C2211yescharacteristic
37#4984C2211yescharacteristic
38#4981—21no—
39#4981—21no—
40#4973—21no—
41#4973C2211yescharacteristic
42#4973C2211yescharacteristic
43#4973C2211yescharacteristic
44#4973C2211yescharacteristic
45C2×#810C2211yescharacteristic
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#809C2211yescharacteristic
50C2×#809C2211yescharacteristic
51#4940—21no—
52#4940—21no—
53C2×#803C2211yescharacteristic
54#4845—24no—
55#4828—24no—
56#4674—21no—
57#4673—21no—
58C3×#340C2211yescharacteristic
59C3×#340C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#464—31no—
63C2×#446—31no—
64#2592—31no—
65#2582—31no—
66#2066—31no—
67#2065—31no—
68#2056—31no—
69#2055—31no—
70#1971—31no—
71#1968—31no—
72#1959—31no—
73#1948D311yescharacteristic, complemented
74#1936—31no—
75#1933—31no—
76#1763—31no—
77C22×D12C2311yescharacteristic
78C22×Dic6D411yescharacteristic
79C22×Dic6C2311yescharacteristic
80C22×Dic6C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C42—21no—
90C2×C3×C42C2311yescharacteristic
91C2×(C2×D3)⋊C4—21no—
92C2×(C2×D3)⋊C4—21no—
93C2×(C2×D3)⋊C4—21no—
94C2×(C2×D3)⋊C4—21no—
95C2×(C2×D3)⋊C4—21no—
96C2×(C2×D3)⋊C4—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
99C2×C4⋊Dic3D411yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×Dic3⋊C4—21no—
102C2×Dic3⋊C4D411yescharacteristic
103C2×Dic3⋊C4D411yescharacteristic
104C2×C4×Dic3—21no—
105C2×C4×Dic3C2311yescharacteristic
106C2×C4×Dic3C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108C22×C3⋊C8C2311yescharacteristic
109C2×Dic12—22no—
110C2×Dic12—22no—
111C2×Dic12—41no—
112C2×#204—22no—
113C2×#204—22no—
114C2×#204—41no—
115#875—22no—
116#875—22no—
117C4×Dic6—22no—
118C4×Dic6—22no—
119C4×Dic6—22no—
120C4×Dic6—22no—
121C3×#106—41no—
122C3×#106—41no—
123C3×C4⋊C8D414yes—
124C3×#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135#821—22no—
136#816—22no—
137#815—22no—
138Dic3⋊C8—22no—
139Dic3⋊C8—22no—
140C8×Dic3—22no—
141#810—22no—
142#810—22no—
143#810—22no—
144#810—41no—
145#809—22no—
146#809—41no—
147#803—22no—
148C22×Q16—31no—
149C22×SD16—31no—
150C2×#124—31no—
151C2×C4×Q8—31no—
152C2×C4×Q8—31no—
153#464—64no—
154#446—64no—
155C2×#106C2×D311yescharacteristic
156C2×#106C2×D311yescharacteristic
157C2×C4⋊C8—31no—
158C2×C4⋊C8—31no—
159C2×C4⋊C8C2×D311yescharacteristic
160C2×#103—31no—
161C2×#103—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103—31no—
165C2×#102—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#97—31no—
169C2×#97—31no—
170C2×C4×C8—31no—
171#398—61no—
172#398—61no—
173#397—61no—
174#393—61no—
175#391—31no—
176#391—31no—
177#391—31no—
178#391—31no—
179#391—61no—
180#390—31no—
181#390—31no—
182#389—31no—
183#389—31no—
184#389—61no—
185#388—61no—
186#388—61no—
187#384C2×D311yescharacteristic
188C4×C4⋊C4C2×D311yescharacteristic
189#340—31no—
190#340—31no—
191#340C2×D311yescharacteristic
192#340C2×D311yescharacteristic
193C23×D3—21no—
194C22×C3×C4—21no—
195C22×C3×C4—21no—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2411yescharacteristic
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3C2×D411yescharacteristic
205C22×Dic3C2×D411yescharacteristic
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C2×D12—21no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C2×Dic6—21no—
217C2×Dic6—21no—
218C2×Dic6—22no—
219C2×Dic6—21no—
220C2×Dic6—22no—
221C2×Dic6—21no—
222C2×Dic6—21no—
223C2×Dic6—21no—
224C2×Dic6—21no—
225C2×Dic6—21no—
226C2×Dic6—21no—
227C2×Dic6—41no—
228C2×Dic6#4112yes—
229C2×C3×C8—21no—
230C2×C3×C8—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8C2×D412yes—
234C2×C3×C8C2×D412yes—
235C3×C4⋊C4—21no—
236C3×C4⋊C4—21no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—21no—
239C3×C4⋊C4—21no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—42no—
244C3×C4⋊C4—81no—
245C3×C42—81no—
246C3×C42C2×D412yes—
247C3×C42#4112yes—
248(C2×D3)⋊C4—42no—
249(C2×D3)⋊C4—42no—
250(C2×D3)⋊C4—42no—
251(C2×D3)⋊C4—42no—
252(C2×D3)⋊C4—81no—
253(C2×D3)⋊C4—81no—
254C4⋊Dic3—22no—
255C4⋊Dic3—21no—
256C4⋊Dic3—21no—
257C4⋊Dic3—81no—
258C4⋊Dic3—81no—
259Dic3⋊C4—22no—
260Dic3⋊C4—22no—
261Dic3⋊C4—42no—
262C4×Dic3—21no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3—42no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8—21no—
271C2×C3⋊C8—21no—
272C2×C3⋊C8#4112yes—
273Dic12—44no—
274#204—44no—
275C22×Q8—31no—
276C22×Q8—31no—
277C22×Q8—31no—
278C22×D4—31no—
279C2×Q16—62no—
280C2×Q16—62no—
281C2×Q16—121no—
282C2×SD16—62no—
283C2×SD16—62no—
284C2×SD16—121no—
285C22×C8—31no—
286C22×C8—31no—
287C22×C8C22×D311yescharacteristic
288C22×C8C22×D311yescharacteristic
289#124—62no—
290#124—62no—
291C4×Q8—62no—
292C4×Q8—62no—
293C4×Q8—62no—
294C4×Q8—62no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—31no—
298C2×C4⋊C4—31no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—61no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C4⋊C4C22×D311yescharacteristic
308C2×C22⋊C4—61no—
309C2×C22⋊C4—61no—
310C2×C22⋊C4—61no—
311C2×C22⋊C4—61no—
312C2×C22⋊C4—61no—
313C2×C22⋊C4—61no—
314C2×C42—21no—
315C2×C42—31no—
316C2×C42—31no—
317C2×C42—61no—
318C2×C42C22×D311yescharacteristic
319#106—41no—
320#106—41no—
321C4⋊C8—62no—
322C4⋊C8—62no—
323C4⋊C8D1214yes—
324#103—62no—
325#103—62no—
326#103—62no—
327#103—62no—
328#103—121no—
329#102—62no—
330#102—62no—
331#102—121no—
332#97—62no—
333#97—62no—
334C4×C8—62no—
335#95—21no—
336#95—61no—
337#95—61no—
338#95—61no—
339#95—61no—
340#95—61no—
341#95—61no—
342#95—61no—
343#95—61no—
344#95—61no—
345#95—61no—
346C23×C3C22×D411yescharacteristic
347C22×D3—41no—
348C22×D3—41no—
349C22×D3—41no—
350C22×D3—41no—
351C22×D3—41no—
352C22×D3—41no—
353C22×D3—41no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4—41no—
370C2×C3×C4#12112yes—
371C2×C3×C4#12112yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4C2×#4111yescharacteristic
375C2×C3×C4C2×#4111yescharacteristic
376C2×C3×C4H(5,𝔽2)11yescharacteristic
377C2×C3×C42−1+411yescharacteristic
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—21no—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—22no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—21no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3#13712yes—
399D12—41no—
400D12—41no—
401D12—41no—
402D12—41no—
403Dic6—22no—
404Dic6—22no—
405Dic6—44no—
406Dic6—41no—
407Dic6—41no—
408Dic6—41no—
409Dic6—41no—
410Dic6—41no—
411Dic6—41no—
412C3×C8—22no—
413C3×C8—22no—
414C3⋊C8—22no—
415C3⋊C8—41no—
416C24—61no—
417C2×Q8—32no—
418C2×Q8—62no—
419C2×Q8—62no—
420C2×Q8—61no—
421C2×Q8—61no—
422C2×Q8—61no—
423C2×Q8—61no—
424C2×Q8—61no—
425C2×Q8—61no—
426C2×Q8—61no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—121no—
432C2×D4—61no—
433C2×D4—61no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—31no—
443C22×C4—31no—
444C22×C4—31no—
445C22×C4—31no—
446C22×C4—61no—
447C22×C4—61no—
448C22×C4—61no—
449C22×C4D3×D411yescharacteristic
450C22×C4D3×D411yescharacteristic
451C22×C4C23×D311yescharacteristic
452Q16—124no—
453SD16—124no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—32no—
459C2×C8—61no—
460C2×C8—61no—
461C2×C8—61no—
462C2×C8—61no—
463C2×C8—61no—
464C2×C8C2×D1212yes—
465C2×C8C2×D1212yes—
466C4⋊C4—21no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—42no—
475C4⋊C4—62no—
476C4⋊C4—62no—
477C4⋊C4—62no—
478C4⋊C4—61no—
479C4⋊C4—61no—
480C4⋊C4—81no—
481C4⋊C4—122no—
482C4⋊C4—241no—
483C4⋊C4—241no—
484C22⋊C4—122no—
485C22⋊C4—122no—
486C22⋊C4—122no—
487C22⋊C4—122no—
488C22⋊C4—241no—
489C22⋊C4—241no—
490C42—61no—
491C42—61no—
492C42—61no—
493C42—61no—
494C42—81no—
495C42—122no—
496C42C2×D1212yes—
497C42#23912yes—
498C22×C3C2×#12111yescharacteristic
499C22×C3#54811yescharacteristic
500C22×C3#55011yescharacteristic
501C22×C3C2×#13711yescharacteristic
502C22×C3#57511yescharacteristic
503C22×C3#57711yescharacteristic
504C22×C3#57711yescharacteristic
505C2×D3—81no—
506C2×D3—81no—
507C2×D3—81no—
508C2×D3—81no—
509C2×D3—81no—
510C2×D3—81no—
511C2×D3—81no—
512C3×C4—22no—
513C3×C4—22no—
514C3×C4—81no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—81no—
518C3×C4#52911yescharacteristic
519C3×C4#54211yescharacteristic
520C3×C4#54611yescharacteristic
521C3×C4#54811yescharacteristic
522Dic3—22no—
523Dic3—42no—
524Dic3—41no—
525Dic3—41no—
526Dic3—41no—
527Dic3—81no—
528Dic3—81no—
529C23—121no—
530C23—121no—
531C23—121no—
532C23—121no—
533C23—121no—
534C23—121no—
535C23—121no—
536C23C2×D3×D411yescharacteristic
537Q8—62no—
538Q8—62no—
539Q8—124no—
540Q8—121no—
541Q8—121no—
542Q8—121no—
543Q8—121no—
544Q8—121no—
545Q8—121no—
546D4—121no—
547D4—121no—
548D4—121no—
549D4—121no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—32no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—61no—
568C2×C4—61no—
569C2×C4—62no—
570C2×C4—61no—
571C2×C4—61no—
572C2×C4—61no—
573C2×C4—61no—
574C2×C4—61no—
575C2×C4—61no—
576C2×C4—61no—
577C2×C4—61no—
578C2×C4—61no—
579C2×C4—121no—
580C2×C4—121no—
581C2×C4—121no—
582C2×C4—121no—
583C2×C4—121no—
584C2×C4—121no—
585C2×C4—121no—
586C2×C4—121no—
587C2×C4#89512yes—
588C2×C4#89512yes—
589C2×C4C22×D1211yescharacteristic
590C2×C4#100411yescharacteristic
591C2×C4C2×#23911yescharacteristic
592C2×C4#100711yescharacteristic
593C2×C4D3×#4111yescharacteristic
594C2×C4D3×#4111yescharacteristic
595C8—22no—
596C8—22no—
597C8—62no—
598C8—121no—
599C2×C3#242611yescharacteristic
600C2×C3#305611yescharacteristic
601C2×C3#306011yescharacteristic
602C2×C3#336911yescharacteristic
603C2×C3#337111yescharacteristic
604C2×C3#340011yescharacteristic
605C2×C3#340111yescharacteristic
606D3—161no—
607C22—241no—
608C22—241no—
609C22—241no—
610C22—241no—
611C22—241no—
612C22—241no—
613C22—241no—
614C22C2×#89511yescharacteristic
615C22#559911yescharacteristic
616C22#563311yescharacteristic
617C22#576211yescharacteristic
618C22#576211yescharacteristic
619C22#577311yescharacteristic
620C22D3×#13711yescharacteristic
621C4—22no—
622C4—22no—
623C4—62no—
624C4—81no—
625C4—81no—
626C4—81no—
627C4—81no—
628C4—121no—
629C4—121no—
630C4—121no—
631C4—122no—
632C4—241no—
633C4—241no—
634C4#555411yescharacteristic
635C4#557611yescharacteristic
636C4#569111yescharacteristic
637C4#573311yescharacteristic
638C3#2531111yescharacteristic, complemented, Sylow
639C2—481no—
640C2#7621811yescharacteristic
641C2#7912011yescharacteristic
642C2#7921211yescharacteristic
643C2#8037711yescharacteristic
644C2#8042311yescharacteristic
645C2#8100411yescharacteristic
646C2#8105111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1948C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#1948D3
hypocenterC3#25311
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. C2×#4828C2semidirect
3. #74343C2semidirect
5. C3×#1948C2semidirect
6. #70589C2semidirect
8. #70575C2semidirect
11. #69475C2semidirect
12. #68864C2semidirect
15. #68851C2semidirect
73. #1948D3semidirect
638. C3#25311semidirect

Maximal subgroups

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

subgroupindex
C2×#48452
C2×#48282
#743432
#737352
C3×#19482
#705892
#705852
#705752
#705742
#695812
#694752
#688642
#688632
#688572
#688512
#253113