Finite Group Databasebeta

← #10945137

Subgroups — #10945137

Order 768.

984 conjugacy classes of subgroups.

The subgroup lattice has 692 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic, complemented
2#83057C211yesmaximal, characteristic, complemented
3C4×D4×Dic3C211yesmaximal, characteristic, complemented
4C2×#5016C211yesmaximal, characteristic
5#71738C211yesmaximal, characteristic, complemented
6#70624C211yesmaximal, characteristic, complemented
7#70623C211yesmaximal, characteristic, complemented
8#70570C211yesmaximal, characteristic, complemented
9#70569C211yesmaximal, characteristic, complemented
10#70458C211yesmaximal, characteristic, complemented
11#70443C211yesmaximal, characteristic, complemented
12#70172C211yesmaximal, characteristic
13#70172C211yesmaximal, characteristic
14#70171C211yesmaximal, characteristic, complemented
15#70171C211yesmaximal, characteristic, complemented
16#33028—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×D4×Dic3C2211yescharacteristic
21C22×#214C2211yescharacteristic
22C22×#213C2211yescharacteristic
23C22×C4×Dic3—21no—
24C2×#875C2211yescharacteristic
25C3×#478—24no—
26C2×C3×#107C2211yescharacteristic
27C2×C3×#106C2211yescharacteristic
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#102C2211yescharacteristic
31C4×#217—21no—
32C2×#821C2211yescharacteristic
33C2×#821C2211yescharacteristic
34C2×#816C2211yescharacteristic
35C2×#816C2211yescharacteristic
36#5016—24no—
37#4990—21no—
38#4990—21no—
39#4990—21no—
40#4990—21no—
41#4988C2211yescharacteristic
42#4988C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44Dic3×C4⋊C4C2211yescharacteristic
45#4973C2211yescharacteristic
46#4973C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#809C2211yescharacteristic
49Dic3×C22⋊C4—21no—
50Dic3×C22⋊C4—21no—
51C4×C4⋊Dic3C2211yescharacteristic
52C42×Dic3C2211yescharacteristic
53C2×C4⋊(C3⋊C8)C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#478D311yescharacteristic, complemented
63C2×#463—31no—
64#2861—31no—
65C42×D4—31no—
66#1989—31no—
67#1983—31no—
68#1928—31no—
69#1927—31no—
70#1926—31no—
71#1925—31no—
72#1879—31no—
73#1879—31no—
74#1877—31no—
75#1877—31no—
76#1831—31no—
77C22×C3×D4C2311yescharacteristic
78C23×C3×C4—21no—
79C23×Dic3—21no—
80C23×Dic3—21no—
81C22×D12C2311yescharacteristic
82C22×Dic6C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×#125—22no—
86C3×#125—22no—
87C3×C4×D4—22no—
88C3×C4×D4D414yes—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C22⋊C4—21no—
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98D4×Dic3—22no—
99D4×Dic3—22no—
100D4×Dic3—22no—
101D4×Dic3—22no—
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×#213—22no—
106C2×#213—22no—
107C2×#213—22no—
108C2×(C2×D3)⋊C4—21no—
109C2×(C2×D3)⋊C4—21no—
110C2×(C2×D3)⋊C4—21no—
111C2×(C2×D3)⋊C4—21no—
112C2×(C2×D3)⋊C4—21no—
113C2×(C2×D3)⋊C4—21no—
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118C2×C4×Dic3—21no—
119C2×C4×Dic3—44no—
120C2×C4×Dic3D411yescharacteristic
121C2×C4×Dic3D411yescharacteristic
122C2×C4×Dic3D411yescharacteristic
123C2×C4×Dic3D411yescharacteristic
124C2×C4×Dic3C2311yescharacteristic
125C2×C4×Dic3C2311yescharacteristic
126C22×C3⋊C8C2311yescharacteristic
127C22×C3⋊C8C2311yescharacteristic
128#875—22no—
129#875—22no—
130C3×#107—22no—
131C3×#106—22no—
132C3×C4⋊C8—22no—
133C3×#102—22no—
134C3×#102—22no—
135C3⋊#95—21no—
136C3⋊#95—21no—
137C3⋊#95—21no—
138C3⋊#95—21no—
139C3⋊#95—21no—
140C3⋊#95—21no—
141C3⋊#95—21no—
142#821—22no—
143#821—22no—
144#816—22no—
145#816—22no—
146#810—22no—
147#809—22no—
148C4⋊(C3⋊C8)—22no—
149C22×SD16—31no—
150C22×D8—31no—
151C2×#125C2×D311yescharacteristic
152C2×#124—31no—
153C2×C4×D4—31no—
154C2×C4×D4—31no—
155C2×C4×D4C2×D311yescharacteristic
156C22×C42—61no—
157#478—24no—
158#463—64no—
159C2×#107C2×D311yescharacteristic
160C2×#106C2×D311yescharacteristic
161C2×C4⋊C8—31no—
162C2×C4⋊C8C2×D311yescharacteristic
163C2×#103—31no—
164C2×#103—31no—
165C2×#103—31no—
166C2×#102—31no—
167C2×#102—31no—
168C2×#102—31no—
169C2×#102C2×D311yescharacteristic
170C2×#102C2×D311yescharacteristic
171#391—31no—
172#391—31no—
173#390—31no—
174#390—31no—
175#388—61no—
176#388—61no—
177#388—61no—
178#388—61no—
179C4×C4⋊C4—31no—
180C4×C4⋊C4—31no—
181C4×C4⋊C4—31no—
182C4×C22⋊C4—61no—
183C4×C22⋊C4—61no—
184C4×C22⋊C4—61no—
185C43—31no—
186#340—31no—
187#340—31no—
188#340—31no—
189#340—31no—
190#336—31no—
191#336—31no—
192#336—31no—
193#336—31no—
194C24×C3—21no—
195C23×D3—21no—
196C2×C3×D4—21no—
197C2×C3×D4—21no—
198C2×C3×D4—21no—
199C2×C3×D4C2×D412yes—
200C2×C3×D4#4112yes—
201C2×C3×D4#4112yes—
202C22×C3×C4—21no—
203C22×C3×C4—24no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—42no—
207C22×C3×C4—42no—
208C22×C3×C4C2×D411yescharacteristic
209C22×C3×C4C2×D411yescharacteristic
210C22×C3×C4C2411yescharacteristic
211C22×Dic3—24no—
212C22×Dic3—24no—
213C22×Dic3—21no—
214C22×Dic3—21no—
215C22×Dic3—21no—
216C22×Dic3—42no—
217C22×Dic3—42no—
218C22×Dic3—42no—
219C22×Dic3—42no—
220C22×Dic3C2×D411yescharacteristic
221C22×Dic3C2×D411yescharacteristic
222C22×Dic3C2×D411yescharacteristic
223C22×Dic3C2×D411yescharacteristic
224C2×D12—21no—
225C2×D12—21no—
226C2×D12—21no—
227C2×D12—21no—
228C2×D12—21no—
229C2×D12—21no—
230C2×Dic6—21no—
231C2×Dic6—21no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×C3×C8—21no—
237C2×C3×C8—21no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—21no—
241C2×C3×C8—21no—
242C3×C4⋊C4—21no—
243C3×C4⋊C4—21no—
244C3×C4⋊C4—21no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—42no—
248C3×C4⋊C4—42no—
249C3×C4⋊C4C2×D412yes—
250C3×C22⋊C4—24no—
251C3×C42C2×D412yes—
252C3×C42#4112yes—
253#217—24no—
254#217—24no—
255#214—44no—
256#213—44no—
257(C2×D3)⋊C4—42no—
258(C2×D3)⋊C4—42no—
259(C2×D3)⋊C4—42no—
260(C2×D3)⋊C4—42no—
261(C2×D3)⋊C4—42no—
262(C2×D3)⋊C4—42no—
263C4⋊Dic3—21no—
264C4⋊Dic3—21no—
265C4⋊Dic3#4112yes—
266C4⋊Dic3#4112yes—
267C4×Dic3—21no—
268C4×Dic3—21no—
269C4×Dic3—21no—
270C4×Dic3—21no—
271C4×Dic3—22no—
272C4×Dic3—22no—
273C4×Dic3—41no—
274C4×Dic3—41no—
275C4×Dic3—42no—
276C4×Dic3—42no—
277C4×Dic3—42no—
278C2×C3⋊C8—21no—
279C2×C3⋊C8—21no—
280C2×C3⋊C8—21no—
281C2×C3⋊C8—21no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C22×Q8—31no—
285C22×D4—31no—
286C22×D4C22×D311yescharacteristic
287C23×C4—21no—
288C23×C4—61no—
289C23×C4—61no—
290C2×SD16—62no—
291C2×SD16—62no—
292C2×SD16—62no—
293C2×D8—62no—
294C2×D8—62no—
295C2×D8—62no—
296C22×C8—31no—
297C22×C8—31no—
298C22×C8C22×D311yescharacteristic
299C22×C8C22×D311yescharacteristic
300#125—22no—
301#125—22no—
302#124—62no—
303#124—62no—
304C4×D4—22no—
305C4×D4—62no—
306C4×D4—62no—
307C4×D4—62no—
308C4×D4—62no—
309C4×D4#6714yes—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—31no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4C22×D311yescharacteristic
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D311yescharacteristic
317C2×C22⋊C4—21no—
318C2×C22⋊C4—61no—
319C2×C22⋊C4—61no—
320C2×C22⋊C4—61no—
321C2×C22⋊C4—61no—
322C2×C22⋊C4—61no—
323C2×C22⋊C4—61no—
324C2×C22⋊C4—61no—
325C2×C22⋊C4—61no—
326C2×C42—31no—
327C2×C42—31no—
328C2×C42—31no—
329C2×C42—31no—
330C2×C42—31no—
331C2×C42—31no—
332C2×C42—61no—
333C2×C42—61no—
334C2×C42—61no—
335C2×C42—124no—
336C2×C42C22×D311yescharacteristic
337#107—22no—
338#106—22no—
339C4⋊C8—22no—
340C4⋊C8—62no—
341#103—62no—
342#103—62no—
343#103—62no—
344#102—22no—
345#102—22no—
346#102—62no—
347#102—62no—
348#102—62no—
349#95—61no—
350#95—61no—
351#95—61no—
352#95—61no—
353#95—61no—
354#95—61no—
355#95—61no—
356C23×C3—22no—
357C23×C3—22no—
358C23×C3—22no—
359C23×C3—41no—
360C23×C3—41no—
361C23×C3—41no—
362C23×C3—41no—
363C23×C3C22×D411yescharacteristic
364C22×D3—41no—
365C22×D3—41no—
366C22×D3—41no—
367C22×D3—41no—
368C22×D3—41no—
369C22×D3—41no—
370C22×D3—41no—
371C3×D4—22no—
372C3×D4—22no—
373C3×D4—22no—
374C3×D4#12414yes—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—22no—
380C2×C3×C4—48no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—41no—
384C2×C3×C4—41no—
385C2×C3×C4—41no—
386C2×C3×C4—41no—
387C2×C3×C4—41no—
388C2×C3×C4—41no—
389C2×C3×C4#12112yes—
390C2×C3×C4#12112yes—
391C2×C3×C4C22×D411yescharacteristic
392C2×C3×C4C2×#4111yescharacteristic
393C2×C3×C4C2×#4111yescharacteristic
394C2×C3×C4C2×#4111yescharacteristic
395C2×C3×C4C2×#4111yescharacteristic
396C2×C3×C4C2×#4111yescharacteristic
397C2×Dic3—21no—
398C2×Dic3—21no—
399C2×Dic3—21no—
400C2×Dic3—21no—
401C2×Dic3—21no—
402C2×Dic3—21no—
403C2×Dic3—22no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—21no—
407C2×Dic3—21no—
408C2×Dic3—21no—
409C2×Dic3—48no—
410C2×Dic3—48no—
411C2×Dic3—41no—
412C2×Dic3—41no—
413C2×Dic3—41no—
414C2×Dic3—41no—
415C2×Dic3—41no—
416C2×Dic3—41no—
417C2×Dic3—41no—
418C2×Dic3#13512yes—
419C2×Dic3AGL(1,ℤ/8)12yes—
420D12—41no—
421D12—41no—
422D12—41no—
423D12—41no—
424Dic6—41no—
425Dic6—41no—
426Dic6—41no—
427Dic6—41no—
428C3×C8—41no—
429C3×C8—41no—
430C3⋊C8—41no—
431C3⋊C8—41no—
432C24—21no—
433C24—61no—
434C2×Q8—61no—
435C2×Q8—61no—
436C2×Q8—61no—
437C2×Q8—61no—
438C2×Q8—61no—
439C2×Q8—61no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—61no—
444C2×D4—61no—
445C2×D4—61no—
446C2×D4—61no—
447C2×D4—61no—
448C2×D4—61no—
449C2×D4#23912yes—
450C2×D4#23912yes—
451C2×D4C2×#6712yes—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—31no—
457C22×C4—31no—
458C22×C4—31no—
459C22×C4—31no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—61no—
465C22×C4—61no—
466C22×C4—61no—
467C22×C4—122no—
468C22×C4—122no—
469C22×C4—122no—
470C22×C4—122no—
471C22×C4D3×D411yescharacteristic
472C22×C4D3×D411yescharacteristic
473C22×C4C23×D311yescharacteristic
474SD16—124no—
475D8—124no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—61no—
483C2×C8—61no—
484C2×C8—61no—
485C2×C8—61no—
486C2×C8—61no—
487C2×C8—61no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—32no—
494C4⋊C4—32no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—61no—
498C4⋊C4—61no—
499C4⋊C4C2×#6712yes—
500C22⋊C4—24no—
501C22⋊C4—64no—
502C22⋊C4—64no—
503C22⋊C4—122no—
504C22⋊C4—122no—
505C22⋊C4—122no—
506C22⋊C4—122no—
507C22⋊C4—122no—
508C22⋊C4—122no—
509C42—62no—
510C42—62no—
511C42—61no—
512C42—61no—
513C42—61no—
514C42—61no—
515C42—121no—
516C42—121no—
517C42—122no—
518C42—122no—
519C42—122no—
520C42#23712yes—
521C42C2×#6712yes—
522C22×C3—24no—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3C2×#12111yescharacteristic
530C22×C3#52511yescharacteristic
531C22×C3#52511yescharacteristic
532C22×C3C2×#13511yescharacteristic
533C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
534C22×C3#57511yescharacteristic
535C22×C3#57511yescharacteristic
536C2×D3—81no—
537C2×D3—81no—
538C2×D3—81no—
539C2×D3—81no—
540C2×D3—81no—
541C2×D3—81no—
542C2×D3—81no—
543C3×C4—22no—
544C3×C4—22no—
545C3×C4—42no—
546C3×C4—81no—
547C3×C4—81no—
548C3×C4C2×#12411yescharacteristic
549C3×C4#52911yescharacteristic
550C3×C4#52911yescharacteristic
551C3×C4#53211yescharacteristic
552Dic3—22no—
553Dic3—22no—
554Dic3—41no—
555Dic3—42no—
556Dic3—41no—
557Dic3—42no—
558Dic3—81no—
559C23—22no—
560C23—22no—
561C23—22no—
562C23—41no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—121no—
567C23—121no—
568C23—121no—
569C23—121no—
570C23—121no—
571C23—121no—
572C23—121no—
573C23C2×D3×D411yescharacteristic
574Q8—121no—
575Q8—121no—
576Q8—121no—
577Q8—121no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—121no—
582D4—121no—
583D4—121no—
584D4—121no—
585D4#94714yes—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—22no—
591C2×C4—32no—
592C2×C4—32no—
593C2×C4—48no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—61no—
603C2×C4—61no—
604C2×C4—61no—
605C2×C4—61no—
606C2×C4—61no—
607C2×C4—61no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—62no—
613C2×C4—62no—
614C2×C4—128no—
615C2×C4—128no—
616C2×C4—121no—
617C2×C4—121no—
618C2×C4—121no—
619C2×C4—121no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4#93912yes—
624C2×C4#93912yes—
625C2×C4C2×#23711yescharacteristic
626C2×C4C2×#23911yescharacteristic
627C2×C4C2×#23911yescharacteristic
628C2×C4D3×#4111yescharacteristic
629C2×C4D3×#4111yescharacteristic
630C2×C4C22×#6711yescharacteristic
631C8—41no—
632C8—41no—
633C8—121no—
634C8—121no—
635C2×C3—44no—
636C2×C3#285011yescharacteristic
637C2×C3#303511yescharacteristic
638C2×C3#303611yescharacteristic
639C2×C3#310211yescharacteristic
640C2×C3#310511yescharacteristic
641C2×C3#311111yescharacteristic
642C2×C3#311411yescharacteristic
643D3—161no—
644C22—24no—
645C22—42no—
646C22—42no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—241no—
652C22—241no—
653C22—241no—
654C22—241no—
655C22—241no—
656C22—241no—
657C22—241no—
658C22#562711yescharacteristic
659C22#562711yescharacteristic
660C22D3×#13511yescharacteristic
661C22#576811yescharacteristic
662C22D3×AGL(1,ℤ/8)11yescharacteristic
663C22#577311yescharacteristic
664C22C2×#93911yescharacteristic
665C4—22no—
666C4—22no—
667C4—42no—
668C4—62no—
669C4—62no—
670C4—81no—
671C4—81no—
672C4—121no—
673C4—122no—
674C4—121no—
675C4—122no—
676C4—241no—
677C4#562411yescharacteristic
678C4#562411yescharacteristic
679C4C2×#94711yescharacteristic
680C4#583211yescharacteristic
681C3#3302811yescharacteristic, complemented, Sylow
682C2—44no—
683C2—481no—
684C2#8114011yescharacteristic
685C2#8114111yescharacteristic
686C2#8114211yescharacteristic
687C2#8114311yescharacteristic
688C2#8280911yescharacteristic
689C2#8390011yescharacteristic
690C2#8393111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#478C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#478D3
hypocenterC3#33028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. C2×C3×#478C2semidirect
2. #83057C2semidirect
3. C4×D4×Dic3C2semidirect
5. #71738C2semidirect
6. #70624C2semidirect
7. #70623C2semidirect
8. #70570C2semidirect
9. #70569C2semidirect
10. #70458C2semidirect
11. #70443C2semidirect
14. #70171C2semidirect
15. #70171C2semidirect
62. C2×#478D3semidirect
681. C3#33028semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4782
#830572
C4×D4×Dic32
C2×#50162
#717382
#706242
#706232
#705702
#705692
#704582
#704432
#701722
#701712
#330283