Finite Group Databasebeta

← #11069332

Subgroups — #11069332

Order 768.

2958 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5718C211yesmaximal, characteristic
2C2×#5675C212yesmaximal, complemented
3C2×Q8×D12C211yesmaximal, characteristic, complemented
4C3×#2548C211yesmaximal, characteristic, complemented
5#83184C211yesmaximal, characteristic, complemented
6#82961C212yesmaximal, complemented
7C4×#944C211yesmaximal, characteristic, complemented
8#77139C218yesmaximal, complemented
9#76457C212yesmaximal, complemented
10#76445C212yesmaximal, complemented
11#76322C214yesmaximal, complemented
12#76316C214yesmaximal, complemented
13#73711C211yesmaximal, characteristic, complemented
14#73617C211yesmaximal, characteristic, complemented
15#39363—31nomaximal, Sylow
16C22×D3×Q8C2212yes—
17C22×#237C2212yes—
18C2×C3×#127C2211yescharacteristic
19C2×C3×#124C2212yes—
20C2×C3×C4×Q8C2211yescharacteristic
21C2×#944C2211yescharacteristic
22C2×#940C2212yes—
23C2×#939C2214yes—
24C2×#938C2212yes—
25C2×#936C2212yes—
26C2×#936C2212yes—
27C2×D4×Dic3C2212yes—
28#5718—28no—
29#5675—216no—
30Q8×D12—28no—
31C2×#897C2214yes—
32C2×#897C2212yes—
33C2×#883C2218yes—
34C2×#878C2218yes—
35C2×C4×D12C2212yes—
36C2×C4×D12C2211yescharacteristic
37C2×#872C2212yes—
38C2×#869C2212yes—
39C2×#869C2211yescharacteristic
40C2×#869C2212yes—
41C2×C4×Dic6C2211yescharacteristic
42C3×#394C2214yes—
43C3×#390C2214yes—
44C3×#390C2212yes—
45C3×C4×C4⋊C4C2211yescharacteristic
46#5229C2211yescharacteristic
47#5221C2214yes—
48#5221C2214yes—
49#4988C2212yes—
50#4984C2214yes—
51#4976C2214yes—
52#4975C2211yescharacteristic
53#4973C2211yescharacteristic
54#4972C2212yes—
55#4961C2218yes—
56#4959C2218yes—
57#4956C2218yes—
58#4950C2218yes—
59#4948C2218yes—
60#4947C2218yes—
61C4×(C2×D3)⋊C4C2212yes—
62C4×Dic3⋊C4C2214yes—
63C42×Dic3C2211yescharacteristic
64#4929C2212yes—
65#4669C2214yes—
66#4661C2214yes—
67C2×#550—31no—
68C2×D4×Q8—31no—
69C2×#536—32no—
70#2858—31no—
71#2681—38no—
72#2620—32no—
73#2548D311yescharacteristic, complemented
74#2546—31no—
75#2452—34no—
76#2445—34no—
77#2436—32no—
78#2432—32no—
79C4×C4⋊Q8—31no—
80#2293—31no—
81C22×C3×Q8C2311yescharacteristic
82C22×C3×D4C2312yes—
83C22×C3×D4C2312yes—
84C22×C3×D4C2312yes—
85C22×#67C2314yes—
86C22×#67C2314yes—
87C23×Dic3C2314yes—
88C2×D3×Q8—28no—
89C2×D3×Q8—28no—
90C2×D3×Q8—44no—
91C2×#237—28no—
92C2×#237—28no—
93C2×#237—44no—
94C22×D12C2311yescharacteristic
95C22×C4×D3C2314yes—
96C22×C4×D3C2312yes—
97C22×Dic6C2312yes—
98C22×Dic6C2314yes—
99C22×Dic6C2311yescharacteristic
100C3×#127—24no—
101C3×#124—28no—
102C3×C4×Q8—24no—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C4⋊C4C2311yescharacteristic
105C2×C3×C4⋊C4C2312yes—
106C2×C3×C22⋊C4C2318yes—
107C2×C3×C22⋊C4C2318yes—
108C2×C3×C42C2312yes—
109C2×C3×C42C2312yes—
110C2×C3×C42C2311yescharacteristic
111C2×#217C2314yes—
112C2×#217C2318yes—
113C2×#217C2318yes—
114#944D418yescomplemented
115#940—28no—
116#939—216no—
117#938—28no—
118#936—28no—
119#936—44no—
120D4×Dic3—28no—
121C2×(C2×D3)⋊C4C2314yes—
122C2×(C2×D3)⋊C4C2314yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×C4⋊Dic3C2312yes—
126C2×C4⋊Dic3C2318yes—
127C2×C4⋊Dic3C2311yescharacteristic
128C2×C4⋊Dic3C2314yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×Dic3⋊C4C2312yes—
132C2×Dic3⋊C4C2314yes—
133C2×Dic3⋊C4C2314yes—
134C2×Dic3⋊C4C2314yes—
135C2×C4×Dic3C2312yes—
136C2×C4×Dic3C2311yescharacteristic
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2311yescharacteristic
140#897—28no—
141#897—216no—
142#883—232no—
143#878—232no—
144C4×D12—28no—
145C4×D12D418yes—
146#872—28no—
147#869—28no—
148#869—24no—
149#869—28no—
150C4×Dic6—24no—
151C3×#95C2312yes—
152C3⋊#95C2318yes—
153C3⋊#95C2314yes—
154C3⋊#95C2314yes—
155C3⋊#95C2314yes—
156C22×#41—32no—
157C23×Q8—32no—
158#550—68no—
159D4×Q8—68no—
160#536—616no—
161C2×C4⋊Q8—31no—
162C2×C4⋊Q8—31no—
163C2×C4⋊Q8—32no—
164C2×C4⋊Q8—32no—
165C2×#127—32no—
166C2×#127C2×D311yescharacteristic
167C2×#124—32no—
168C2×#124—32no—
169C2×#124C2×D312yes—
170C2×#123—38no—
171C2×C22⋊Q8—32no—
172C2×C22⋊Q8—34no—
173C2×C22⋊Q8—38no—
174C2×#121—32no—
175C2×#121—34no—
176C2×C4×Q8—31no—
177C2×C4×Q8C2×D311yescharacteristic
178C2×C4×D4—32no—
179C2×C4×D4—31no—
180C2×C4×D4—32no—
181C2×#117—32no—
182#398—34no—
183#396—38no—
184#394—38no—
185#394—38no—
186#394C2×D314yes—
187#393—38no—
188#391—31no—
189#391—31no—
190#391—32no—
191#390—34no—
192#390—34no—
193#390—32no—
194#390C2×D312yes—
195#390C2×D314yes—
196#389—34no—
197#389—32no—
198#389—34no—
199#388—34no—
200#386—38no—
201#385—38no—
202#384—31no—
203C4×C4⋊C4—34no—
204C4×C4⋊C4C2×D311yescharacteristic
205C4×C22⋊C4—32no—
206C43—31no—
207C24×C3C2414yes—
208C23×D3C2412yes—
209C2×C3×Q8—22no—
210C2×C3×Q8—22no—
211C2×C3×Q8C2×D414yes—
212C2×C3×D4—24no—
213C2×C3×D4—24no—
214C2×C3×D4—24no—
215C2×C3×D4—24no—
216C2×C3×D4—24no—
217C2×C3×D4—28no—
218C2×C3×D4—44no—
219C22×C3×C4C2414yes—
220C22×C3×C4C2412yes—
221C22×C3×C4C2412yes—
222C22×C3×C4C2411yescharacteristic
223C22×C3×C4C2411yescharacteristic
224C22×C3×C4C2411yescharacteristic
225C2×#67—28no—
226C2×#67—28no—
227C2×#67—28no—
228C2×#67—44no—
229C2×#67—48no—
230C22×Dic3—28no—
231C22×Dic3—28no—
232C22×Dic3—44no—
233C22×Dic3C2412yes—
234C22×Dic3C2412yes—
235C22×Dic3C2414yes—
236C22×Dic3C2412yes—
237C22×Dic3C2412yes—
238C22×Dic3C2412yes—
239D3×Q8—432no—
240#237—432no—
241C2×D12—24no—
242C2×D12C2×D414yes—
243C2×C4×D3—24no—
244C2×C4×D3—24no—
245C2×C4×D3—28no—
246C2×C4×D3—28no—
247C2×C4×D3—44no—
248C2×C4×D3C2×D418yes—
249C2×Dic6—22no—
250C2×Dic6—22no—
251C2×Dic6—24no—
252C2×Dic6—24no—
253C2×Dic6—28no—
254C2×Dic6—24no—
255C2×Dic6—28no—
256C2×Dic6—44no—
257C2×Dic6C2×D414yes—
258C3×C4⋊C4—24no—
259C3×C4⋊C4—22no—
260C3×C4⋊C4—24no—
261C3×C22⋊C4—216no—
262C3×C22⋊C4—48no—
263C3×C42—24no—
264C3×C42—24no—
265C3×C42C2×D414yes—
266#217—216no—
267#217—28no—
268#217—48no—
269(C2×D3)⋊C4—28no—
270(C2×D3)⋊C4—28no—
271(C2×D3)⋊C4—44no—
272(C2×D3)⋊C4C2×D418yes—
273C4⋊Dic3—22no—
274C4⋊Dic3—24no—
275C4⋊Dic3—24no—
276C4⋊Dic3—216no—
277C4⋊Dic3—28no—
278C4⋊Dic3C2×D414yes—
279Dic3⋊C4—24no—
280Dic3⋊C4—28no—
281Dic3⋊C4—44no—
282Dic3⋊C4C2×D4116yes—
283C4×Dic3—24no—
284C4×Dic3—22no—
285C4×Dic3—24no—
286C4×Dic3—42no—
287C4×Dic3C2×D414yes—
288C2×#41—68no—
289C2×#41—68no—
290C2×#41—124no—
291C22×Q8—31no—
292C22×Q8—32no—
293C22×Q8—34no—
294C22×Q8—68no—
295C22×Q8—68no—
296C22×Q8—124no—
297C22×Q8C22×D311yescharacteristic
298C22×D4—34no—
299C22×D4—34no—
300C22×D4—31no—
301C22×D4C22×D312yes—
302C22×D4C22×D312yes—
303C22×D4C22×D312yes—
304C23×C4—32no—
305C23×C4—34no—
306C23×C4—34no—
307C4⋊Q8—38no—
308C4⋊Q8—68no—
309C4⋊Q8—68no—
310C4⋊Q8—64no—
311#127—24no—
312#127—68no—
313#124—28no—
314#124—68no—
315#124—124no—
316#123—632no—
317C22⋊Q8—632no—
318C22⋊Q8—616no—
319C22⋊Q8—68no—
320#121—616no—
321#121—68no—
322C4×Q8—24no—
323C4×Q8—64no—
324C4×D4—38no—
325C4×D4—68no—
326C4×D4—68no—
327#117—68no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—34no—
330C2×C4⋊C4—34no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—34no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—38no—
337C2×C4⋊C4—34no—
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D312yes—
340C2×C4⋊C4C22×D312yes—
341C2×C22⋊C4—32no—
342C2×C22⋊C4—34no—
343C2×C22⋊C4—38no—
344C2×C22⋊C4—34no—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—38no—
347C2×C22⋊C4—34no—
348C2×C22⋊C4C22×D318yes—
349C2×C22⋊C4C22×D318yes—
350C2×C42—31no—
351C2×C42—32no—
352C2×C42—31no—
353C2×C42—32no—
354C2×C42—32no—
355C2×C42C22×D312yes—
356C2×C42C22×D312yes—
357C2×C42C22×D311yescharacteristic
358#95—34no—
359#95—34no—
360#95—34no—
361#95—38no—
362#95C22×D312yes—
363C23×C3—24no—
364C23×C3—24no—
365C23×C3—24no—
366C23×C3—28no—
367C23×C3—28no—
368C23×C3C2511yescharacteristic
369C22×D3—22no—
370C22×D3—22no—
371C22×D3—24no—
372C22×D3—24no—
373C22×D3C22×D414yes—
374C3×Q8—28no—
375C3×D4—48no—
376C3×D4—48no—
377C3×D4—84no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—24no—
382C2×C3×C4—22no—
383C2×C3×C4—24no—
384C2×C3×C4—24no—
385C2×C3×C4—24no—
386C2×C3×C4C22×D412yes—
387C2×C3×C4C22×D414yes—
388C2×C3×C4C22×D412yes—
389C2×C3×C4C22×D412yes—
390C2×C3×C4C22×D412yes—
391C2×C3×C4H(5,𝔽2)12yes—
392C2×C3×C42−1+414yes—
393#67—416no—
394#67—88no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—22no—
399C2×Dic3—22no—
400C2×Dic3—22no—
401C2×Dic3—22no—
402C2×Dic3—24no—
403C2×Dic3—24no—
404C2×Dic3—22no—
405C2×Dic3—24no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—24no—
409C2×Dic3—416no—
410C2×Dic3C22×D414yes—
411C2×Dic3C22×D414yes—
412C2×Dic3C22×D414yes—
413D12—28no—
414C4×D3—216no—
415C4×D3—416no—
416Dic6—28no—
417Dic6—416no—
418Dic6—48no—
419C24—32no—
420C24C23×D314yes—
421#41—1232no—
422C2×Q8—22no—
423C2×Q8—22no—
424C2×Q8—34no—
425C2×Q8—68no—
426C2×Q8—62no—
427C2×Q8—68no—
428C2×Q8—64no—
429C2×Q8—64no—
430C2×Q8—62no—
431C2×Q8—64no—
432C2×Q8—124no—
433C2×Q8—1232no—
434C2×Q8D3×D414yes—
435C2×D4—24no—
436C2×D4—24no—
437C2×D4—24no—
438C2×D4—24no—
439C2×D4—24no—
440C2×D4—28no—
441C2×D4—34no—
442C2×D4—44no—
443C2×D4—68no—
444C2×D4—68no—
445C2×D4—68no—
446C2×D4—64no—
447C2×D4—124no—
448C2×D4—128no—
449C22×C4—32no—
450C22×C4—32no—
451C22×C4—32no—
452C22×C4—38no—
453C22×C4—32no—
454C22×C4—32no—
455C22×C4—34no—
456C22×C4—68no—
457C22×C4—68no—
458C22×C4—68no—
459C22×C4—64no—
460C22×C4—64no—
461C22×C4—68no—
462C22×C4—124no—
463C22×C4—124no—
464C22×C4C23×D314yes—
465C22×C4C23×D312yes—
466C22×C4C23×D312yes—
467C22×C4C23×D311yescharacteristic
468C22×C4C23×D311yescharacteristic
469C22×C4C23×D311yescharacteristic
470C4⋊C4—24no—
471C4⋊C4—22no—
472C4⋊C4—24no—
473C4⋊C4—34no—
474C4⋊C4—316no—
475C4⋊C4—68no—
476C4⋊C4—68no—
477C4⋊C4—616no—
478C4⋊C4—64no—
479C4⋊C4—64no—
480C4⋊C4—62no—
481C4⋊C4—64no—
482C4⋊C4—124no—
483C22⋊C4—216no—
484C22⋊C4—38no—
485C22⋊C4—48no—
486C22⋊C4—68no—
487C22⋊C4—68no—
488C22⋊C4—68no—
489C22⋊C4—616no—
490C22⋊C4—124no—
491C22⋊C4—128no—
492C42—24no—
493C42—24no—
494C42—34no—
495C42—64no—
496C42—62no—
497C42—64no—
498C42—122no—
499C42D3×D414yes—
500C22×C3—44no—
501C22×C3—44no—
502C22×C3—48no—
503C22×C3—44no—
504C22×C3—48no—
505C22×C3C23×D411yescharacteristic
506C22×C3C23×D411yescharacteristic
507C22×C3C2×H(5,𝔽2)11yescharacteristic
508C22×C3C2×2−1+412yes—
509C22×C3#58512yes—
510C2×D3—24no—
511C2×D3—24no—
512C2×D3—24no—
513C2×D3—44no—
514C2×D3—44no—
515C3×C4—24no—
516C3×C4—22no—
517C3×C4—44no—
518C3×C4—42no—
519C3×C4#53614yescomplemented
520C3×C4D4214yes—
521Dic3—24no—
522Dic3—24no—
523Dic3—24no—
524Dic3—44no—
525Dic3—42no—
526Dic3—42no—
527C23—24no—
528C23—24no—
529C23—24no—
530C23—28no—
531C23—28no—
532C23—34no—
533C23—62no—
534C23—62no—
535C23—64no—
536C23—64no—
537C23C24×D311yescharacteristic
538Q8—28no—
539Q8—68no—
540Q8—1216no—
541Q8—128no—
542D4—48no—
543D4—48no—
544D4—68no—
545D4—84no—
546D4—1216no—
547D4—248no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—34no—
557C2×C4—34no—
558C2×C4—34no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—62no—
562C2×C4—62no—
563C2×C4—62no—
564C2×C4—62no—
565C2×C4—616no—
566C2×C4—62no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—62no—
570C2×C4—64no—
571C2×C4—62no—
572C2×C4—62no—
573C2×C4—64no—
574C2×C4—1216no—
575C2×C4—1216no—
576C2×C4C2×D3×D412yes—
577C2×C4C2×D3×D414yes—
578C2×C4C2×D3×D412yes—
579C2×C4C2×D3×D412yes—
580C2×C4C2×D3×D412yes—
581C2×C4#100912yes—
582C2×C4#101014yes—
583C2×C3—84no—
584C2×C3C2×#53611yescharacteristic
585C2×C3C2×D4211yescharacteristic
586C2×C3#349712yes—
587C2×C3#350011yescharacteristic
588C2×C3#355412yes—
589D3—44no—
590C22—48no—
591C22—48no—
592C22—44no—
593C22—44no—
594C22—44no—
595C22—64no—
596C22—64no—
597C22—64no—
598C22—124no—
599C22—124no—
600C22C22×D3×D411yescharacteristic
601C22C22×D3×D411yescharacteristic
602C22C2×#100911yescharacteristic
603C22C2×#101012yes—
604C22#596612yes—
605C4—22no—
606C4—24no—
607C4—44no—
608C4—42no—
609C4—64no—
610C4—64no—
611C4—64no—
612C4—122no—
613C4—122no—
614C4—124no—
615C4#560914yes—
616C4#567514yescomplemented
617C3#3936311yescharacteristic, complemented, Sylow
618C2—84no—
619C2—124no—
620C2C2×#560911yescharacteristic
621C2C2×#567511yescharacteristic
622C2#8631711yescharacteristic
623C2#8678412yes—
624C2#8706712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2548C2
prime radicalC2×#5718C2
hypercenter#2548D3
hypocenterC3#39363
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. C2×#5675C2semidirect
3. C2×Q8×D12C2semidirect
4. C3×#2548C2semidirect
5. #83184C2semidirect
6. #82961C2semidirect
7. C4×#944C2semidirect
8. #77139C2semidirect
9. #76457C2semidirect
10. #76445C2semidirect
11. #76322C2semidirect
12. #76316C2semidirect
13. #73711C2semidirect
14. #73617C2semidirect
73. #2548D3semidirect
114. #944D4semidirect
519. C3×C4#536semidirect
616. C4#5675semidirect
617. C3#39363semidirect

Maximal subgroups

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

subgroupindex
C2×#57182
C2×#56752
C2×Q8×D122
C3×#25482
#831842
#829612
C4×#9442
#771392
#764572
#764452
#763222
#763162
#737112
#736172
#393633