Finite Group Databasebeta

← #10972906

Subgroups — #10972906

Order 768.

2654 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5716C211yesmaximal, characteristic
2C2×C4×D3×Q8C211yesmaximal, characteristic, complemented
3C3×#2548C211yesmaximal, characteristic, complemented
4#83179C211yesmaximal, characteristic, complemented
5#82961C212yesmaximal, complemented
6#82953C212yesmaximal, complemented
7#82478C211yesmaximal, characteristic, complemented
8#77107C214yesmaximal, complemented
9#76977C214yesmaximal, complemented
10#76475C214yesmaximal, complemented
11#76364C214yesmaximal, complemented
12#76335C212yesmaximal, complemented
13#76297C212yesmaximal, complemented
14#73752C211yesmaximal, characteristic, complemented
15#73721C211yesmaximal, characteristic, complemented
16#35249—31nomaximal, Sylow
17C22×D3×Q8C2211yescharacteristic
18C22×#237C2212yes—
19C2×C3×#127C2211yescharacteristic
20C2×C3×#124C2212yes—
21C2×C3×C4×Q8C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×#938C2212yes—
24C2×#938C2212yes—
25C2×#936C2212yes—
26C2×#936C2212yes—
27C2×D4×Dic3C2212yes—
28C2×D4×Dic3C2212yes—
29#5716—28no—
30C4×D3×Q8—28no—
31C2×D3×C4⋊C4C2211yescharacteristic
32C2×D3×C4⋊C4C2212yes—
33C2×#887C2211yescharacteristic
34C2×#887C2212yes—
35C2×#883C2218yes—
36C2×C42×D3C2212yes—
37C2×C42×D3C2211yescharacteristic
38C2×#869C2211yescharacteristic
39C2×C4×Dic6C2212yes—
40C2×C4×Dic6C2211yescharacteristic
41C3×#394C2214yes—
42C3×#390C2212yes—
43C3×#390C2214yes—
44C3×C4×C4⋊C4C2211yescharacteristic
45#5229C2212yes—
46#5222C2214yes—
47#5221C2212yes—
48#5221C2214yes—
49#4987C2212yes—
50#4984C2214yes—
51#4979C2214yes—
52#4977C2212yes—
53#4975C2212yes—
54#4956C2218yes—
55#4951C2218yes—
56#4950C2218yes—
57#4949C2218yes—
58#4947C2218yes—
59#4946C2218yes—
60#4945C2218yes—
61C4×(C2×D3)⋊C4C2212yes—
62#4935C2212yes—
63C4×C4⋊Dic3C2211yescharacteristic
64#4932C2212yes—
65#4929C2212yes—
66#4928C2214yes—
67#4664C2212yes—
68#4656C2214yes—
69#4651C2212yes—
70C2×#532—31no—
71C22×C4×Q8—31no—
72#2854—34no—
73#2632—34no—
74#2580—34no—
75#2567—31no—
76#2567—31no—
77#2548—31no—
78#2548D311yescharacteristic, complemented
79#2501—32no—
80#2436—32no—
81#2406—34no—
82#2326—32no—
83#2326—32no—
84#2283—31no—
85C22×C3×Q8C2311yescharacteristic
86C22×C3×D4C2312yes—
87C22×C3×D4C2312yes—
88C22×C3×D4C2312yes—
89C22×#67C2314yes—
90C23×Dic3C2314yes—
91C2×D3×Q8—24no—
92C2×D3×Q8—24no—
93C2×D3×Q8—24no—
94C2×#237—28no—
95C2×#237—48no—
96C22×C4×D3C2312yes—
97C22×C4×D3C2312yes—
98C22×C4×D3C2311yescharacteristic
99C22×C4×D3C2311yescharacteristic
100C22×C4×D3C2311yescharacteristic
101C22×Dic6C2312yes—
102C22×Dic6C2311yescharacteristic
103C22×Dic6C2312yes—
104C3×#127—24no—
105C3×#124—28no—
106C3×C4×Q8—24no—
107C2×C3×C4⋊C4C2311yescharacteristic
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C22⋊C4C2318yes—
111C2×C3×C22⋊C4C2318yes—
112C2×C3×C42C2312yes—
113C2×C3×C42C2312yes—
114C2×C3×C42C2311yescharacteristic
115C2×#217C2314yes—
116C2×#217C2318yes—
117C2×#217C2314yes—
118C2×#217C2318yes—
119Q8×Dic3—24no—
120#938—28no—
121#938—28no—
122#936—28no—
123#936—28no—
124D4×Dic3—28no—
125D4×Dic3—28no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×C4⋊Dic3C2312yes—
128C2×C4⋊Dic3C2312yes—
129C2×C4⋊Dic3C2312yes—
130C2×C4⋊Dic3C2311yescharacteristic
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2318yes—
133C2×Dic3⋊C4C2312yes—
134C2×Dic3⋊C4C2312yes—
135C2×Dic3⋊C4C2312yes—
136C2×Dic3⋊C4C2318yes—
137C2×C4×Dic3C2312yes—
138C2×C4×Dic3C2312yes—
139C2×C4×Dic3C2312yes—
140C2×C4×Dic3C2312yes—
141C2×C4×Dic3C2311yescharacteristic
142C2×C4×Dic3C2311yescharacteristic
143C2×C4×Dic3C2311yescharacteristic
144D3×C4⋊C4—28no—
145D3×C4⋊C4—24no—
146#887—24no—
147#887—28no—
148#883—416no—
149C42×D3—28no—
150C42×D3D418yes—
151#869—24no—
152C4×Dic6—24no—
153C4×Dic6—28no—
154C3×#95C2312yes—
155C3⋊#95C2312yes—
156C3⋊#95C2314yes—
157C3⋊#95C2314yes—
158C3⋊#95C2318yes—
159C3⋊#95C2314yes—
160C3⋊#95C2318yes—
161C22×#41—32no—
162C23×Q8—31no—
163#532—68no—
164C2×C4⋊Q8—31no—
165C2×#127C2×D311yescharacteristic
166C2×#124—32no—
167C2×#124—32no—
168C2×#124C2×D312yes—
169C2×#123—38no—
170C2×#121—32no—
171C2×#121—32no—
172C2×C4×Q8—31no—
173C2×C4×Q8—31no—
174C2×C4×Q8—32no—
175C2×C4×Q8—31no—
176C2×C4×Q8—32no—
177C2×C4×Q8—68no—
178C2×C4×Q8C2×D311yescharacteristic
179C2×C4×D4—32no—
180C2×C4×D4—32no—
181C22×C4⋊C4—32no—
182C22×C4⋊C4—31no—
183C22×C42—31no—
184C22×C42—32no—
185#399—38no—
186#395—34no—
187#395—34no—
188#394—34no—
189#394C2×D314yes—
190#393—38no—
191#391—32no—
192#391—32no—
193#390—34no—
194#390—32no—
195#390C2×D314yes—
196#390C2×D312yes—
197#389—34no—
198#389—34no—
199#388—38no—
200#388—38no—
201#386—38no—
202#385—38no—
203#385—38no—
204#384—32no—
205#384—32no—
206#383—32no—
207#382—32no—
208#381—32no—
209#380—32no—
210C4×C4⋊C4—31no—
211C4×C4⋊C4C2×D311yescharacteristic
212C4×C22⋊C4—32no—
213#376—32no—
214C24×C3C2414yes—
215C23×D3C2411yescharacteristic
216C2×C3×Q8—22no—
217C2×C3×Q8—22no—
218C2×C3×Q8#4114yescomplemented
219C2×C3×D4—24no—
220C2×C3×D4—24no—
221C2×C3×D4—24no—
222C2×C3×D4—24no—
223C2×C3×D4—24no—
224C2×C3×D4—28no—
225C2×C3×D4—44no—
226C22×C3×C4C2414yes—
227C22×C3×C4C2412yes—
228C22×C3×C4C2411yescharacteristic
229C22×C3×C4C2412yes—
230C22×C3×C4C2411yescharacteristic
231C22×C3×C4C2411yescharacteristic
232C2×#67—28no—
233C2×#67—216no—
234C22×Dic3—28no—
235C22×Dic3—48no—
236C22×Dic3C2414yes—
237C22×Dic3C2414yes—
238C22×Dic3C2412yes—
239C22×Dic3C2411yescharacteristic
240C22×Dic3C2411yescharacteristic
241C22×Dic3C2411yescharacteristic
242C22×Dic3C2412yes—
243D3×Q8—232no—
244#237—432no—
245C2×C4×D3—24no—
246C2×C4×D3—24no—
247C2×C4×D3—24no—
248C2×C4×D3—22no—
249C2×C4×D3—22no—
250C2×C4×D3—24no—
251C2×C4×D3—24no—
252C2×C4×D3—24no—
253C2×C4×D3—28no—
254C2×C4×D3C2×D414yes—
255C2×C4×D3C2×D414yes—
256C2×C4×D3C2×D414yes—
257C2×Dic6—22no—
258C2×Dic6—24no—
259C2×Dic6—22no—
260C2×Dic6—24no—
261C2×Dic6—24no—
262C2×Dic6—24no—
263C2×Dic6—44no—
264C2×Dic6#4114yescomplemented
265C3×C4⋊C4—24no—
266C3×C4⋊C4—22no—
267C3×C4⋊C4—24no—
268C3×C22⋊C4—216no—
269C3×C22⋊C4—48no—
270C3×C42—24no—
271C3×C42—24no—
272C3×C42C2×D414yes—
273#217—28no—
274#217—28no—
275#217—216no—
276#217—48no—
277(C2×D3)⋊C4—28no—
278C4⋊Dic3—22no—
279C4⋊Dic3—24no—
280C4⋊Dic3—24no—
281C4⋊Dic3—24no—
282C4⋊Dic3—24no—
283C4⋊Dic3—48no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—24no—
287Dic3⋊C4—48no—
288C4×Dic3—24no—
289C4×Dic3—24no—
290C4×Dic3—24no—
291C4×Dic3—24no—
292C4×Dic3C2×D414yes—
293C4×Dic3C2×D414yes—
294C4×Dic3C2×D414yes—
295C2×#41—68no—
296C2×#41—128no—
297C22×Q8—31no—
298C22×Q8—32no—
299C22×Q8—32no—
300C22×Q8—64no—
301C22×Q8—64no—
302C22×Q8—64no—
303C22×Q8C22×D311yescharacteristic
304C22×D4—34no—
305C22×D4C22×D312yes—
306C22×D4C22×D312yes—
307C22×D4C22×D312yes—
308C23×C4—31no—
309C23×C4—31no—
310C23×C4—31no—
311C23×C4—34no—
312C23×C4—32no—
313C23×C4—32no—
314C4⋊Q8—64no—
315#127—24no—
316#124—28no—
317#124—68no—
318#124—68no—
319#123—1216no—
320#121—68no—
321#121—68no—
322C4×Q8—24no—
323C4×Q8—68no—
324C4×Q8—64no—
325C4×Q8—68no—
326C4×Q8—64no—
327C4×Q8—64no—
328C4×D4—68no—
329C4×D4—68no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—32no—
332C2×C4⋊C4—32no—
333C2×C4⋊C4—32no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—32no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—38no—
339C2×C4⋊C4—38no—
340C2×C4⋊C4—64no—
341C2×C4⋊C4—68no—
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C4⋊C4C22×D312yes—
344C2×C4⋊C4C22×D312yes—
345C2×C22⋊C4—34no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—38no—
348C2×C22⋊C4—34no—
349C2×C22⋊C4—38no—
350C2×C22⋊C4C22×D318yes—
351C2×C22⋊C4C22×D318yes—
352C2×C42—31no—
353C2×C42—31no—
354C2×C42—31no—
355C2×C42—38no—
356C2×C42—32no—
357C2×C42—32no—
358C2×C42—32no—
359C2×C42—32no—
360C2×C42—68no—
361C2×C42C22×D312yes—
362C2×C42C22×D312yes—
363C2×C42C22×D311yescharacteristic
364#95—34no—
365#95—34no—
366#95—34no—
367#95—32no—
368#95—38no—
369#95—38no—
370#95C22×D312yes—
371C23×C3—24no—
372C23×C3—24no—
373C23×C3—24no—
374C23×C3—28no—
375C23×C3—28no—
376C23×C3C2511yescharacteristic
377C22×D3—21no—
378C22×D3—21no—
379C22×D3—22no—
380C22×D3C22×D412yes—
381C22×D3H(5,𝔽2)14yes—
382C3×Q8—28no—
383C3×D4—48no—
384C3×D4—48no—
385C3×D4—48no—
386C2×C3×C4—22no—
387C2×C3×C4—22no—
388C2×C3×C4—22no—
389C2×C3×C4—22no—
390C2×C3×C4—24no—
391C2×C3×C4—24no—
392C2×C3×C4—24no—
393C2×C3×C4—24no—
394C2×C3×C4C22×D412yes—
395C2×C3×C4C22×D412yes—
396C2×C3×C4C22×D412yes—
397C2×C3×C4C2×#4114yes—
398C2×C3×C4C2×#4112yes—
399C2×C3×C4C2×#4112yes—
400C2×C3×C42−1+414yes—
401#67—416no—
402C2×Dic3—22no—
403C2×Dic3—21no—
404C2×Dic3—22no—
405C2×Dic3—21no—
406C2×Dic3—22no—
407C2×Dic3—22no—
408C2×Dic3—22no—
409C2×Dic3—24no—
410C2×Dic3—24no—
411C2×Dic3—24no—
412C2×Dic3—28no—
413C2×Dic3—24no—
414C2×Dic3—24no—
415C2×Dic3—416no—
416C2×Dic3C22×D412yes—
417C2×Dic3C22×D412yes—
418C2×Dic3C22×D412yes—
419C2×Dic3C2×#4114yes—
420C4×D3—216no—
421C4×D3—28no—
422C4×D3—28no—
423C4×D3—28no—
424C4×D3—48no—
425Dic6—216no—
426Dic6—28no—
427Dic6—48no—
428C24—31no—
429C24C23×D314yes—
430#41—1232no—
431C2×Q8—22no—
432C2×Q8—22no—
433C2×Q8—34no—
434C2×Q8—64no—
435C2×Q8—64no—
436C2×Q8—632no—
437C2×Q8—62no—
438C2×Q8—64no—
439C2×Q8—64no—
440C2×Q8—62no—
441C2×Q8—124no—
442C2×Q8#23714yescomplemented
443C2×D4—24no—
444C2×D4—24no—
445C2×D4—24no—
446C2×D4—24no—
447C2×D4—24no—
448C2×D4—28no—
449C2×D4—44no—
450C2×D4—68no—
451C2×D4—616no—
452C22×C4—31no—
453C22×C4—31no—
454C22×C4—32no—
455C22×C4—31no—
456C22×C4—34no—
457C22×C4—34no—
458C22×C4—34no—
459C22×C4—34no—
460C22×C4—34no—
461C22×C4—32no—
462C22×C4—64no—
463C22×C4—64no—
464C22×C4—64no—
465C22×C4—64no—
466C22×C4—68no—
467C22×C4—68no—
468C22×C4—64no—
469C22×C4—62no—
470C22×C4—62no—
471C22×C4—64no—
472C22×C4—128no—
473C22×C4C23×D314yes—
474C22×C4C23×D312yes—
475C22×C4C23×D311yescharacteristic
476C22×C4C23×D312yes—
477C22×C4C23×D311yescharacteristic
478C22×C4C23×D311yescharacteristic
479C4⋊C4—24no—
480C4⋊C4—22no—
481C4⋊C4—24no—
482C4⋊C4—64no—
483C4⋊C4—64no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—62no—
487C4⋊C4—64no—
488C4⋊C4—64no—
489C4⋊C4—64no—
490C4⋊C4—128no—
491C4⋊C4—128no—
492C22⋊C4—216no—
493C22⋊C4—48no—
494C22⋊C4—68no—
495C22⋊C4—68no—
496C22⋊C4—68no—
497C22⋊C4—616no—
498C22⋊C4—128no—
499C42—24no—
500C42—24no—
501C42—34no—
502C42—34no—
503C42—34no—
504C42—64no—
505C42—64no—
506C42—64no—
507C42—64no—
508C42D3×D414yes—
509C22×C3—44no—
510C22×C3—44no—
511C22×C3—44no—
512C22×C3—48no—
513C22×C3—48no—
514C22×C3C23×D411yescharacteristic
515C22×C3C22×#4111yescharacteristic
516C22×C3C22×#4111yescharacteristic
517C22×C3C2×H(5,𝔽2)12yes—
518C22×C3C2×2−1+412yes—
519C2×D3—22no—
520C2×D3—22no—
521C2×D3—24no—
522C2×D3—24no—
523C2×D3#53514yes—
524C3×C4—24no—
525C3×C4—22no—
526C3×C4—42no—
527C3×C4—44no—
528C3×C4#52514yes—
529C3×C4#53614yes—
530Dic3—22no—
531Dic3—24no—
532Dic3—22no—
533Dic3—22no—
534Dic3—44no—
535Dic3—42no—
536Dic3—44no—
537C23—24no—
538C23—24no—
539C23—24no—
540C23—28no—
541C23—28no—
542C23—32no—
543C23—34no—
544C23—61no—
545C23—61no—
546C23—62no—
547C23C24×D311yescharacteristic
548Q8—28no—
549Q8—616no—
550Q8—68no—
551Q8—128no—
552D4—48no—
553D4—48no—
554D4—48no—
555D4—1216no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—32no—
565C2×C4—32no—
566C2×C4—32no—
567C2×C4—34no—
568C2×C4—68no—
569C2×C4—616no—
570C2×C4—62no—
571C2×C4—61no—
572C2×C4—62no—
573C2×C4—61no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—68no—
577C2×C4—68no—
578C2×C4—62no—
579C2×C4—64no—
580C2×C4—64no—
581C2×C4—64no—
582C2×C4—64no—
583C2×C4—64no—
584C2×C4—68no—
585C2×C4—128no—
586C2×C4—1216no—
587C2×C4C2×D3×D412yes—
588C2×C4C2×D3×D412yes—
589C2×C4C2×D3×D412yes—
590C2×C4C2×#23714yes—
591C2×C4C2×#23712yes—
592C2×C4D3×#4112yes—
593C2×C4#101014yes—
594C2×C3—84no—
595C2×C3C2×#52511yescharacteristic
596C2×C3C2×#53511yescharacteristic
597C2×C3C2×#53611yescharacteristic
598C2×C3#352212yes—
599C2×C3#352212yes—
600D3—24no—
601C22—34no—
602C22—48no—
603C22—44no—
604C22—44no—
605C22—44no—
606C22—48no—
607C22—64no—
608C22—62no—
609C22—62no—
610C22—64no—
611C22C22×D3×D411yescharacteristic
612C22C22×#23711yescharacteristic
613C22C2×D3×#4111yescharacteristic
614C22C2×#101012yes—
615C22D3×H(5,𝔽2)12yes—
616C4—22no—
617C4—24no—
618C4—42no—
619C4—44no—
620C4—62no—
621C4—62no—
622C4—62no—
623C4—64no—
624C4—124no—
625C4—124no—
626C4—122no—
627C4#560614yes—
628C4#567514yes—
629C3#3524911yescharacteristic, complemented, Sylow
630C2—64no—
631C2—84no—
632C2C2×#560611yescharacteristic
633C2C2×#567511yescharacteristic
634C2D3×#53511yescharacteristic
635C2#8677612yes—
636C2#8705512yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2548C2
prime radicalC2×#5716C2
hypercenter#2548D3
hypocenterC3#35249
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2548

Complements

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

NH ≅ G/Ntype
2. C2×C4×D3×Q8C2semidirect
3. C3×#2548C2semidirect
4. #83179C2semidirect
5. #82961C2semidirect
6. #82953C2semidirect
7. #82478C2semidirect
8. #77107C2semidirect
9. #76977C2semidirect
10. #76475C2semidirect
11. #76364C2semidirect
12. #76335C2semidirect
13. #76297C2semidirect
14. #73752C2semidirect
15. #73721C2semidirect
78. #2548D3semidirect
218. C2×C3×Q8#41semidirect
264. C2×Dic6#41semidirect
442. C2×Q8#237semidirect
629. C3#35249semidirect

Maximal subgroups

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

subgroupindex
C2×#57162
C2×C4×D3×Q82
C3×#25482
#831792
#829612
#829532
#824782
#771072
#769772
#764752
#763642
#763352
#762972
#737522
#737212
#352493