Finite Group Databasebeta

← #10827352

Subgroups — #10827352

Order 768.

1494 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5555C211yesmaximal, characteristic
2C3×#2409C211yesmaximal, characteristic, complemented
3C2×#5227C211yesmaximal, characteristic, complemented
4C22×#835C211yesmaximal, characteristic, complemented
5#71699C212yesmaximal, complemented
6#71692C212yesmaximal, complemented
7#71610C211yesmaximal, characteristic, complemented
8#69687C212yesmaximal, complemented
9#69668C212yesmaximal, complemented
10#69515C211yesmaximal, characteristic, complemented
11#69501C211yesmaximal, characteristic, complemented
12#23961—31nomaximal, Sylow
13C23×C3×Q8C2211yescharacteristic
14C22×#237—21no—
15C2×C3×C22⋊Q8C2212yes—
16C2×C3×C4×D4C2211yescharacteristic
17C22×#215C2212yes—
18C2×#930C2211yescharacteristic
19C2×#930C2211yescharacteristic
20C22×C4⋊Dic3C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C23×C3⋊C8C2211yescharacteristic
23#5555—24no—
24#5555—24no—
25C2×#897—21no—
26C2×#886—21no—
27C2×#886—21no—
28C2×C4×D12C2211yescharacteristic
29C2×#869C2211yescharacteristic
30C3×#395—21no—
31C3×#393—21no—
32C3×#391C2212yes—
33C3×#386—21no—
34C3×#385—21no—
35C3×#380C2211yescharacteristic
36#5227—24no—
37C2×#835—24no—
38C2×#835C2212yes—
39C2×#835C2212yes—
40C2×#835C2212yes—
41C2×#830C2211yescharacteristic
42C2×#830C2212yes—
43C2×#809C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#547—31no—
49C2×#448—31no—
50C22×#103—31no—
51#2409D311yescharacteristic, complemented
52#2024—32no—
53#2000—32no—
54#1893—31no—
55#1877—32no—
56#1872—32no—
57#1869—31no—
58#1823—31no—
59C22×C3×Q8—21no—
60C22×C3×Q8—24no—
61C22×C3×Q8—22no—
62C22×C3×Q8—22no—
63C22×C3×Q8—42no—
64C22×C3×Q8C2312yes—
65C22×C3×D4D411yescharacteristic
66C23×C3×C4—21no—
67C23×C3×C4C2311yescharacteristic
68C23×C3×C4C2311yescharacteristic
69C22×#67—21no—
70C22×#67—21no—
71C23×Dic3—21no—
72C23×Dic3—21no—
73C2×#237—28no—
74C2×#237—44no—
75C2×#237—44no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×Dic6—21no—
79C3×C22⋊Q8—24no—
80C3×C22⋊Q8—42no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C2×C3×C4⋊C4—22no—
84C2×C3×C4⋊C4D411yescharacteristic
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C22⋊C4—21no—
87C2×C3×C22⋊C4—22no—
88C2×C3×C22⋊C4D411yescharacteristic
89C2×C3×C22⋊C4D411yescharacteristic
90C2×C3×C42C2311yescharacteristic
91C2×#215—24no—
92C2×#215—24no—
93C2×#215—42no—
94#930—22no—
95#930—22no—
96#930—22no—
97#930D414yes—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—22no—
105C2×C4⋊Dic3—22no—
106C2×C4⋊Dic3—22no—
107C2×C4⋊Dic3D414yes—
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C22×C3⋊C8—22no—
112C22×C3⋊C8—42no—
113C22×C3⋊C8C2312yes—
114C22×C3⋊C8C2312yes—
115#897—28no—
116#886—28no—
117#886—28no—
118C4×D12—22no—
119C4×D12—22no—
120#869—22no—
121#869—22no—
122C3×#95—21no—
123C3×#95—22no—
124C3×#95—22no—
125#835—24no—
126#835—24no—
127#835—24no—
128#830—22no—
129#830—42no—
130#809—42no—
131C22×#41—61no—
132C23×Q8C2×D311yescharacteristic
133C22×SD16—32no—
134C23×C8—31no—
135#547—64no—
136#547—64no—
137C2×C4⋊Q8—31no—
138C2×#123—61no—
139C2×#123—61no—
140C2×C22⋊Q8—61no—
141C2×C22⋊Q8C2×D312yes—
142C2×#121—31no—
143C2×#121—31no—
144C2×C4×D4—31no—
145C2×C4×D4C2×D311yescharacteristic
146C22×C4⋊C4—31no—
147C22×C4⋊C4—31no—
148#448—64no—
149C2×#103—32no—
150C2×#103—32no—
151C2×#103—32no—
152C2×#103—64no—
153C2×#102—32no—
154C2×C22⋊C8—31no—
155C2×C22⋊C8—32no—
156#395—21no—
157#393—21no—
158#391C2×D312yes—
159#386—21no—
160#385—21no—
161#380C2×D311yescharacteristic
162#340—32no—
163#340—32no—
164#340—32no—
165#336—32no—
166C24×C3C2×D411yescharacteristic
167C24×C3C2×D411yescharacteristic
168C23×D3—21no—
169C2×C3×Q8—22no—
170C2×C3×Q8—28no—
171C2×C3×Q8—22no—
172C2×C3×Q8—22no—
173C2×C3×Q8—22no—
174C2×C3×Q8—22no—
175C2×C3×Q8—44no—
176C2×C3×Q8—48no—
177C2×C3×Q8—44no—
178C2×C3×Q8—42no—
179C2×C3×Q8#4114yescomplemented
180C2×C3×D4—22no—
181C2×C3×D4—22no—
182C2×C3×D4SD1614yescomplemented
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—22no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—22no—
189C22×C3×C4—22no—
190C22×C3×C4—22no—
191C22×C3×C4—44no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D412yes—
194C22×C3×C4C2×D412yes—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C2×#67—24no—
201C2×#67—24no—
202C2×#67—42no—
203C2×#67—42no—
204C2×#67—42no—
205C2×#67—42no—
206C22×Dic3—24no—
207C22×Dic3—24no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—42no—
212C22×Dic3—42no—
213C22×Dic3—42no—
214C22×Dic3—42no—
215#237—432no—
216C2×D12—22no—
217C2×D12—22no—
218C2×D12—21no—
219C2×D12C2×D412yes—
220C2×C4×D3—24no—
221C2×C4×D3—42no—
222C2×C4×D3—42no—
223C2×Dic6—24no—
224C2×Dic6—42no—
225C2×Dic6—42no—
226C3×C4⋊C4—22no—
227C3×C4⋊C4—22no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—44no—
230C3×C22⋊C4—22no—
231C3×C22⋊C4—22no—
232C3×C22⋊C4—44no—
233C3×C22⋊C4—42no—
234C3×C42—21no—
235C3×C42—21no—
236#215—48no—
237(C2×D3)⋊C4—24no—
238(C2×D3)⋊C4—24no—
239(C2×D3)⋊C4—24no—
240C4⋊Dic3—24no—
241C4⋊Dic3—24no—
242C4⋊Dic3—24no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3C2×D412yes—
247C4⋊Dic3C2×D412yes—
248C4⋊Dic3C2×D412yes—
249C2×C3⋊C8—22no—
250C2×C3⋊C8—22no—
251C2×C3⋊C8—22no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—22no—
254C2×C3⋊C8—22no—
255C2×C3⋊C8—44no—
256C2×#41—68no—
257C2×#41—124no—
258C2×#41—124no—
259C22×Q8—21no—
260C22×Q8—24no—
261C22×Q8—22no—
262C22×Q8—22no—
263C22×Q8—42no—
264C22×Q8—61no—
265C22×Q8C22×D312yes—
266C22×D4—31no—
267C22×D4—61no—
268C22×D4—61no—
269C22×D4#6711yescharacteristic
270C23×C4—21no—
271C23×C4—61no—
272C23×C4—61no—
273C23×C4—61no—
274C23×C4C22×D311yescharacteristic
275C23×C4C22×D311yescharacteristic
276C2×SD16—64no—
277C2×SD16—64no—
278C2×SD16—122no—
279C22×C8—32no—
280C22×C8—32no—
281C22×C8—62no—
282C22×C8—122no—
283C4⋊Q8—62no—
284C4⋊Q8—62no—
285#123—68no—
286#123—68no—
287C22⋊Q8—24no—
288C22⋊Q8—42no—
289C22⋊Q8—68no—
290#121—34no—
291#121—62no—
292#121—62no—
293#121—62no—
294C4×D4—22no—
295C4×D4—22no—
296C4×D4—62no—
297C4×D4—62no—
298C2×C4⋊C4—22no—
299C2×C4⋊C4—34no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4—31no—
303C2×C4⋊C4—62no—
304C2×C4⋊C4—62no—
305C2×C4⋊C4—62no—
306C2×C4⋊C4—61no—
307C2×C4⋊C4—61no—
308C2×C4⋊C4—61no—
309C2×C4⋊C4#6711yescharacteristic
310C2×C4⋊C4C22×D312yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—21no—
313C2×C22⋊C4—61no—
314C2×C22⋊C4—61no—
315C2×C22⋊C4—61no—
316C2×C22⋊C4#6711yescharacteristic
317C2×C22⋊C4#6711yescharacteristic
318C2×C42C22×D311yescharacteristic
319#103—64no—
320#103—64no—
321#103—64no—
322#102—122no—
323C22⋊C8—62no—
324C22⋊C8—122no—
325#95—21no—
326#95—22no—
327#95—22no—
328C23×C3—22no—
329C23×C3—22no—
330C23×C3—21no—
331C23×C3—22no—
332C23×C3—21no—
333C23×C3—22no—
334C23×C3#12012yes—
335C23×C3C2×SD1612yes—
336C23×C3C2×SD1612yes—
337C23×C3AGL(1,ℤ/8)12yes—
338C23×C3C22×D411yescharacteristic
339C22×D3—22no—
340C22×D3—42no—
341C22×D3—41no—
342C22×D3—41no—
343C22×D3—42no—
344C3×Q8—24no—
345C3×Q8—24no—
346C3×Q8—48no—
347C3×Q8—42no—
348C3×Q8—42no—
349C3×D4—28no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—24no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—24no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—48no—
362C2×C3×C4—42no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—42no—
367C2×C3×C4—42no—
368C2×C3×C4—42no—
369C2×C3×C4—42no—
370C2×C3×C4#12012yes—
371C2×C3×C4C2×SD1612yes—
372C2×C3×C4C22×D411yescharacteristic
373C2×C3×C4C22×D411yescharacteristic
374C2×C3×C4C2×#4112yes—
375C2×C3×C4H(5,𝔽2)12yes—
376#67—48no—
377#67—48no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—48no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387C2×Dic3—41no—
388C2×Dic3—48no—
389D12—24no—
390D12—41no—
391D12—41no—
392C4×D3—48no—
393Dic6—48no—
394C3⋊C8—42no—
395C3⋊C8—42no—
396C24—61no—
397C24C2×#6711yescharacteristic
398C24C2×#6711yescharacteristic
399#41—1232no—
400C2×Q8—22no—
401C2×Q8—28no—
402C2×Q8—22no—
403C2×Q8—22no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—44no—
407C2×Q8—48no—
408C2×Q8—44no—
409C2×Q8—42no—
410C2×Q8—64no—
411C2×Q8—122no—
412C2×Q8—122no—
413C2×Q8#23714yescomplemented
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—32no—
417C2×D4—64no—
418C2×D4—64no—
419C2×D4—61no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—122no—
423C2×D4—122no—
424C2×D4—122no—
425C2×D4—122no—
426C2×D4#21514yescomplemented
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—44no—
436C22×C4—42no—
437C22×C4—64no—
438C22×C4—64no—
439C22×C4—64no—
440C22×C4—61no—
441C22×C4—61no—
442C22×C4—61no—
443C22×C4—122no—
444C22×C4—122no—
445C22×C4—122no—
446C22×C4—122no—
447C22×C4—122no—
448C22×C4—122no—
449C22×C4D3×D412yes—
450C22×C4D3×D412yes—
451C22×C4C2×#6711yescharacteristic
452C22×C4C2×#6711yescharacteristic
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C22×C4C23×D311yescharacteristic
456SD16—128no—
457C2×C8—62no—
458C2×C8—62no—
459C2×C8—62no—
460C2×C8—62no—
461C2×C8—62no—
462C2×C8—62no—
463C2×C8—124no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—22no—
467C4⋊C4—32no—
468C4⋊C4—32no—
469C4⋊C4—32no—
470C4⋊C4—44no—
471C4⋊C4—64no—
472C4⋊C4—64no—
473C4⋊C4—64no—
474C4⋊C4—61no—
475C4⋊C4—61no—
476C4⋊C4—61no—
477C22⋊C4—22no—
478C22⋊C4—22no—
479C22⋊C4—44no—
480C22⋊C4—42no—
481C22⋊C4—64no—
482C22⋊C4—64no—
483C22⋊C4—64no—
484C42—21no—
485C42—21no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3—24no—
490C22×C3—22no—
491C22×C3—24no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3#45014yes—
496C22×C3C2×#12011yescharacteristic
497C22×C3#54612yes—
498C22×C3C22×SD1611yescharacteristic
499C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
500C22×C3#57812yes—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—82no—
505C2×D3—82no—
506C3×C4—22no—
507C3×C4—22no—
508C3×C4—41no—
509C3×C4—44no—
510C3×C4—41no—
511C3×C4—82no—
512C3×C4#52511yescharacteristic
513C3×C4#53511yescharacteristic
514C3×C4#54612yes—
515Dic3—42no—
516Dic3—42no—
517Dic3—42no—
518C23—22no—
519C23—22no—
520C23—21no—
521C23—22no—
522C23—21no—
523C23—22no—
524C23—62no—
525C23—122no—
526C23—121no—
527C23—121no—
528C23—122no—
529C23#93712yes—
530C23C2×#21512yes—
531C23C2×#21512yes—
532C23#94912yes—
533C23C22×#6711yescharacteristic
534Q8—24no—
535Q8—24no—
536Q8—48no—
537Q8—42no—
538Q8—42no—
539Q8—128no—
540D4—28no—
541D4—64no—
542D4—128no—
543D4—128no—
544D4—121no—
545D4—121no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—48no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—42no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—128no—
570C2×C4—128no—
571C2×C4—121no—
572C2×C4—121no—
573C2×C4—121no—
574C2×C4—128no—
575C2×C4—121no—
576C2×C4—121no—
577C2×C4—121no—
578C2×C4#93712yes—
579C2×C4C2×#21512yes—
580C2×C4C2×D3×D411yescharacteristic
581C2×C4C2×D3×D411yescharacteristic
582C2×C4C2×#23712yes—
583C2×C4#100412yes—
584C8—122no—
585C8—122no—
586C2×C3—24no—
587C2×C3—42no—
588C2×C3#240411yescharacteristic
589C2×C3C2×#45011yescharacteristic
590C2×C3#301111yescharacteristic
591C2×C3#329812yes—
592C2×C3#330812yes—
593D3—82no—
594C22—24no—
595C22—24no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—24no—
600C22—42no—
601C22—41no—
602C22—41no—
603C22—122no—
604C22—122no—
605C22—122no—
606C22—242no—
607C22—242no—
608C22#504614yes—
609C22C2×#93711yescharacteristic
610C22#580612yes—
611C22C22×#21511yescharacteristic
612C22C2×#94911yescharacteristic
613C22#583812yes—
614C4—22no—
615C4—22no—
616C4—41no—
617C4—44no—
618C4—41no—
619C4—82no—
620C4—122no—
621C4—122no—
622C4—122no—
623C4#558611yescharacteristic
624C4#559112yes—
625C4#559211yescharacteristic
626C3#2396111yescharacteristic, complemented, Sylow
627C2—24no—
628C2—42no—
629C2—242no—
630C2#8258111yescharacteristic
631C2C2×#504611yescharacteristic
632C2#8341111yescharacteristic
633C2#8353612yes—
634C2#8355012yes—
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2409C2
socleC23×C3C22×D4
prime radicalC2×#5555C2
hypercenter#2409D3
hypocenterC3#23961
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. C3×#2409C2semidirect
3. C2×#5227C2semidirect
4. C22×#835C2semidirect
5. #71699C2semidirect
6. #71692C2semidirect
7. #71610C2semidirect
8. #69687C2semidirect
9. #69668C2semidirect
10. #69515C2semidirect
11. #69501C2semidirect
51. #2409D3semidirect
179. C2×C3×Q8#41semidirect
182. C2×C3×D4SD16semidirect
413. C2×Q8#237semidirect
426. C2×D4#215semidirect
626. C3#23961semidirect

Maximal subgroups

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

subgroupindex
C2×#55552
C3×#24092
C2×#52272
C22×#8352
#716992
#716922
#716102
#696872
#696682
#695152
#695012
#239613