Finite Group Databasebeta

← #11233310

Subgroups — #11233310

Order 768.

2218 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic
2C2×#5721C211yesmaximal, characteristic, complemented
3#84065C212yesmaximal, complemented
4Q8×#217C212yesmaximal, complemented
5#83247C214yesmaximal, complemented
6#83227C211yesmaximal, characteristic, complemented
7#82796C214yesmaximal, complemented
8#82768C214yesmaximal, complemented
9#82758C212yesmaximal, complemented
10#82753C212yesmaximal, complemented
11#82481C211yesmaximal, characteristic, complemented
12#82456C211yesmaximal, characteristic, complemented
13#82375C211yesmaximal, characteristic
14#82366C214yesmaximal
15#82348C211yesmaximal, characteristic
16#45336—31nomaximal, Sylow
17C23×C3×Q8C2212yes—
18C3×D4×Q8—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2212yes—
22C2×C3×C22⋊Q8C2214yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2212yes—
26C2×Q8×Dic3C2212yes—
27C2×Q8×Dic3C2212yes—
28C2×#944C2211yescharacteristic
29C2×#944C2212yes—
30C2×#924C2212yes—
31#5721—28no—
32C2×#890C2212yes—
33C2×#889C2214yes—
34C2×#887C2212yes—
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37#5230C2214yes—
38#5230C2214yes—
39#5229C2211yescharacteristic
40#5229C2212yes—
41#5229C2214yes—
42#5229C2214yes—
43#5229C2212yes—
44#5222C2212yes—
45#5222C2212yes—
46#5212C2212yes—
47#5212C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50C4×#217C2212yes—
51C4×#217C2212yes—
52C4×#217C2214yes—
53#4986C2214yes—
54#4986C2212yes—
55#4985C2212yes—
56#4983C2212yes—
57#4982C2214yes—
58#4980C2214yes—
59#4980C2214yes—
60#4979C2212yes—
61#4978C2214yes—
62#4978C2214yes—
63#4977C2211yescharacteristic
64#4976C2214yes—
65#4973C2211yescharacteristic
66#4972C2212yes—
67#4953C2214yes—
68#4953C2212yes—
69#4948C2214yes—
70#4947C2212yes—
71#4945C2214yes—
72#4944C2214yes—
73#4944C2214yes—
74#4936C2211yescharacteristic
75#4933C2214yes—
76#4933C2214yes—
77#4932C2211yescharacteristic
78C2×#557—31no—
79C2×D4×Q8D311yescharacteristic
80#2713—34no—
81#2690—34no—
82#2581—31no—
83#2536—31no—
84#2465—34no—
85#2460—34no—
86#2434—32no—
87#2429—32no—
88#2421—32no—
89#2398—31no—
90Q8×C22⋊C4—32no—
91#2327—31no—
92#2327—31no—
93C22×C3×Q8—28no—
94C22×C3×Q8—28no—
95C22×C3×Q8—44no—
96C22×C3×Q8C2314yes—
97C22×C3×Q8C2312yes—
98C22×C3×Q8C2311yescharacteristic
99C22×C3×D4C2311yescharacteristic
100C23×C3×C4C2312yes—
101C23×C3×C4C2314yes—
102C22×Dic6C2311yescharacteristic
103C22×Dic6C2311yescharacteristic
104C3×C4⋊Q8—28no—
105C3×C4⋊Q8D418yes—
106C3×C22⋊Q8—28no—
107C3×C22⋊Q8—216no—
108C3×C4×Q8—24no—
109C3×C4×D4—28no—
110C3×C4×D4—24no—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C4⋊C4C2312yes—
114C2×C3×C4⋊C4C2311yescharacteristic
115C2×C3×C4⋊C4C2312yes—
116C2×C3×C4⋊C4C2312yes—
117C2×C3×C22⋊C4C2312yes—
118C2×C3×C22⋊C4C2314yes—
119C2×C3×C42C2312yes—
120C2×C3×C42C2311yescharacteristic
121C2×#217C2312yes—
122C2×#217C2314yes—
123C2×#217C2312yes—
124C2×#217C2312yes—
125C2×#217C2312yes—
126C2×#217C2314yes—
127Q8×Dic3—28no—
128Q8×Dic3—44no—
129#944—28no—
130#944—24no—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2314yes—
136#924—44no—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×C4×Dic3C2311yescharacteristic
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2312yes—
147C2×C4×Dic3C2314yes—
148C2×C4×Dic3C2312yes—
149C2×C4×Dic3C2314yes—
150#890—28no—
151#889—216no—
152#887—28no—
153C4×Dic6—24no—
154C4×Dic6—24no—
155C3⋊#95C2312yes—
156C3⋊#95C2312yes—
157C3⋊#95C2312yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2314yes—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2314yes—
164C3⋊#95C2312yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C23×Q8C2×D312yes—
173#557—68no—
174D4×Q8—28no—
175C2×C4⋊Q8—31no—
176C2×C4⋊Q8—32no—
177C2×C4⋊Q8C2×D312yes—
178C2×C4⋊Q8C2×D311yescharacteristic
179C2×#125—34no—
180C2×#125—32no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8C2×D312yes—
183C2×C22⋊Q8C2×D314yes—
184C2×C4×Q8—32no—
185C2×C4×Q8—32no—
186C2×C4×Q8—32no—
187C2×C4×Q8—31no—
188C2×C4×Q8—31no—
189C2×C4×Q8C2×D311yescharacteristic
190C2×C4×D4C2×D311yescharacteristic
191C2×C4×D4C2×D312yes—
192#400—32no—
193#400—32no—
194#400—34no—
195#398—34no—
196#398—32no—
197#397—32no—
198#397—34no—
199#396—34no—
200#395—34no—
201#395—34no—
202#395—32no—
203#394—32no—
204#394—32no—
205#393—32no—
206#391—32no—
207#391—31no—
208#391—32no—
209#391—34no—
210#391—34no—
211#391—31no—
212#389—32no—
213#389—34no—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387—34no—
219#387—34no—
220#387—34no—
221#387—34no—
222#387—34no—
223#385—34no—
224#385—34no—
225#382—31no—
226#382—31no—
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C2×C3×Q8—22no—
235C2×C3×Q8—28no—
236C2×C3×Q8—24no—
237C2×C3×Q8—28no—
238C2×C3×Q8—24no—
239C2×C3×Q8—22no—
240C2×C3×Q8—44no—
241C2×C3×Q8—432no—
242C2×C3×Q8C2×D418yes—
243C2×C3×Q8#4114yes—
244C2×C3×D4—24no—
245C2×C3×D4—22no—
246C22×C3×C4—28no—
247C22×C3×C4—24no—
248C22×C3×C4—28no—
249C22×C3×C4—24no—
250C22×C3×C4—24no—
251C22×C3×C4—44no—
252C22×C3×C4C2412yes—
253C22×C3×C4C2412yes—
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2411yescharacteristic
256C22×C3×C4C2414yes—
257C22×C3×C4C2412yes—
258C22×C3×C4C2411yescharacteristic
259C22×Dic3C2412yes—
260C22×Dic3C2412yes—
261C22×Dic3C2414yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C2×Dic6—22no—
267C2×Dic6—24no—
268C2×Dic6—22no—
269C2×Dic6—22no—
270C2×Dic6—22no—
271C3×C4⋊C4—22no—
272C3×C4⋊C4—24no—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—28no—
275C3×C4⋊C4C2×D418yes—
276C3×C4⋊C4C2×D418yes—
277C3×C22⋊C4—28no—
278C3×C22⋊C4—24no—
279C3×C42—24no—
280C3×C42C2×D414yes—
281#217—28no—
282#217—24no—
283#217—24no—
284#217—24no—
285#217—24no—
286#217—44no—
287C4⋊Dic3—22no—
288C4⋊Dic3—22no—
289C4⋊Dic3—28no—
290C4⋊Dic3—24no—
291C4⋊Dic3—44no—
292Dic3⋊C4—24no—
293Dic3⋊C4—24no—
294Dic3⋊C4—28no—
295Dic3⋊C4—28no—
296Dic3⋊C4—28no—
297Dic3⋊C4—28no—
298Dic3⋊C4—44no—
299C4×Dic3—24no—
300C4×Dic3—24no—
301C4×Dic3—28no—
302C4×Dic3—22no—
303C4×Dic3—22no—
304C4×Dic3—44no—
305C22×Q8—28no—
306C22×Q8—28no—
307C22×Q8—31no—
308C22×Q8—31no—
309C22×Q8—44no—
310C22×Q8C22×D314yes—
311C22×Q8C22×D311yescharacteristic
312C22×Q8C22×D312yes—
313C22×D4C22×D311yescharacteristic
314C23×C4C22×D312yes—
315C23×C4C22×D314yes—
316C4⋊Q8—28no—
317C4⋊Q8—64no—
318C4⋊Q8—68no—
319C4⋊Q8#6718yes—
320#125—616no—
321#125—68no—
322C22⋊Q8—28no—
323C22⋊Q8—216no—
324C22⋊Q8—124no—
325C4×Q8—24no—
326C4×Q8—68no—
327C4×Q8—68no—
328C4×Q8—64no—
329C4×Q8—64no—
330C4×Q8—124no—
331C4×D4—28no—
332C4×D4—24no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—32no—
335C2×C4⋊C4—34no—
336C2×C4⋊C4—32no—
337C2×C4⋊C4—32no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—34no—
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D312yes—
348C2×C4⋊C4C22×D314yes—
349C2×C4⋊C4C22×D312yes—
350C2×C4⋊C4C22×D312yes—
351C2×C22⋊C4—32no—
352C2×C22⋊C4—32no—
353C2×C22⋊C4—34no—
354C2×C22⋊C4—32no—
355C2×C22⋊C4—32no—
356C2×C22⋊C4—34no—
357C2×C22⋊C4C22×D312yes—
358C2×C22⋊C4C22×D314yes—
359C2×C42—34no—
360C2×C42—32no—
361C2×C42—32no—
362C2×C42—31no—
363C2×C42—31no—
364C2×C42—34no—
365C2×C42C22×D312yes—
366C2×C42C22×D311yescharacteristic
367#95—32no—
368#95—32no—
369#95—34no—
370#95—34no—
371#95—32no—
372#95—32no—
373#95—32no—
374#95—32no—
375#95—34no—
376#95—34no—
377#95—32no—
378#95—34no—
379#95—32no—
380#95—34no—
381#95—34no—
382#95—34no—
383#95—34no—
384C23×C3—22no—
385C23×C3—22no—
386C23×C3—22no—
387C23×C3—22no—
388C23×C3—22no—
389C23×C3—22no—
390C23×C32−1+414yes—
391C23×C3C2511yescharacteristic
392C3×Q8—28no—
393C3×Q8—216no—
394C3×Q8—416no—
395C3×D4—28no—
396C2×C3×C4—22no—
397C2×C3×C4—24no—
398C2×C3×C4—24no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—24no—
407C2×C3×C4—416no—
408C2×C3×C4—48no—
409C2×C3×C4C22×D412yes—
410C2×C3×C4C22×D418yes—
411C2×C3×C4C22×D412yes—
412C2×C3×C4C22×D412yes—
413C2×C3×C4C2×#4114yes—
414C2×C3×C4C2×#4112yes—
415C2×C3×C4H(5,𝔽2)12yes—
416C2×Dic3—22no—
417C2×Dic3—22no—
418C2×Dic3—22no—
419C2×Dic3—22no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—22no—
426C2×Dic3—24no—
427C2×Dic3—24no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic32−1+414yes—
436Dic6—28no—
437Dic6—44no—
438C24C23×D312yes—
439C2×Q8—24no—
440C2×Q8—28no—
441C2×Q8—24no—
442C2×Q8—22no—
443C2×Q8—22no—
444C2×Q8—28no—
445C2×Q8—44no—
446C2×Q8—432no—
447C2×Q8—64no—
448C2×Q8—62no—
449C2×Q8—62no—
450C2×Q8—62no—
451C2×Q8—62no—
452C2×Q8#23714yes—
453C2×Q8C2×#6718yes—
454C2×D4—24no—
455C2×D4—22no—
456C22×C4—28no—
457C22×C4—24no—
458C22×C4—24no—
459C22×C4—28no—
460C22×C4—24no—
461C22×C4—32no—
462C22×C4—32no—
463C22×C4—32no—
464C22×C4—34no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—44no—
469C22×C4C23×D312yes—
470C22×C4C23×D312yes—
471C22×C4C23×D312yes—
472C22×C4C23×D311yescharacteristic
473C22×C4C23×D311yescharacteristic
474C22×C4C23×D314yes—
475C22×C4C23×D311yescharacteristic
476C4⋊C4—22no—
477C4⋊C4—24no—
478C4⋊C4—22no—
479C4⋊C4—28no—
480C4⋊C4—68no—
481C4⋊C4—68no—
482C4⋊C4—64no—
483C4⋊C4—62no—
484C4⋊C4—64no—
485C4⋊C4—64no—
486C4⋊C4—68no—
487C4⋊C4—62no—
488C4⋊C4—68no—
489C4⋊C4—68no—
490C4⋊C4—124no—
491C4⋊C4—124no—
492C4⋊C4C2×#6718yes—
493C4⋊C4C2×#6718yes—
494C22⋊C4—28no—
495C22⋊C4—24no—
496C22⋊C4—64no—
497C22⋊C4—64no—
498C22⋊C4—68no—
499C22⋊C4—64no—
500C22⋊C4—64no—
501C22⋊C4—124no—
502C42—24no—
503C42—62no—
504C42—62no—
505C42—68no—
506C42—64no—
507C42—64no—
508C42—124no—
509C42C2×#6714yes—
510C22×C3—24no—
511C22×C3—24no—
512C22×C3—24no—
513C22×C3—42no—
514C22×C3—42no—
515C22×C3—42no—
516C22×C3—42no—
517C22×C3C23×D411yescharacteristic
518C22×C3C22×#4111yescharacteristic
519C22×C3C2×H(5,𝔽2)11yescharacteristic
520C22×C3C2×2−1+412yes—
521C22×C3C2×2−1+412yes—
522C3×C4—24no—
523C3×C4—28no—
524C3×C4—22no—
525C3×C4—22no—
526C3×C4—42no—
527C3×C4—42no—
528C3×C4#54614yes—
529Dic3—24no—
530Dic3—42no—
531Dic3—42no—
532Dic3—42no—
533Dic3—42no—
534Dic3—42no—
535Dic3—44no—
536C23—22no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23#100714yes—
543C23C24×D311yescharacteristic
544Q8—28no—
545Q8—216no—
546Q8—416no—
547Q8—68no—
548Q8—124no—
549D4—28no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—34no—
562C2×C4—416no—
563C2×C4—48no—
564C2×C4—62no—
565C2×C4—62no—
566C2×C4—62no—
567C2×C4—62no—
568C2×C4—62no—
569C2×C4—64no—
570C2×C4—62no—
571C2×C4—64no—
572C2×C4—62no—
573C2×C4—62no—
574C2×C4—62no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—62no—
578C2×C4—62no—
579C2×C4—64no—
580C2×C4—62no—
581C2×C4—62no—
582C2×C4—62no—
583C2×C4C2×#23714yes—
584C2×C4C2×#23712yes—
585C2×C4#100412yes—
586C2×C4C22×#6712yes—
587C2×C4C22×#6712yes—
588C2×C4C22×#6718yes—
589C2×C4C22×#6712yes—
590C2×C3—44no—
591C2×C3C2×#54611yescharacteristic
592C2×C3#348611yescharacteristic
593C2×C3#348611yescharacteristic
594C2×C3#352312yes—
595C2×C3#355212yes—
596C22—24no—
597C22—24no—
598C22—24no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22C22×#23711yescharacteristic
604C22C2×#100411yescharacteristic
605C22C2×#100712yes—
606C22D3×2−1+412yes—
607C22C23×#6711yescharacteristic
608C4—24no—
609C4—28no—
610C4—22no—
611C4—22no—
612C4—42no—
613C4—42no—
614C4—64no—
615C4—122no—
616C4—122no—
617C4—122no—
618C4—122no—
619C4—122no—
620C4—124no—
621C4#580614yes—
622C3#4533611yescharacteristic, complemented, Sylow
623C2—44no—
624C2#8675212yes—
625C2#8675612yes—
626C2C2×#580611yescharacteristic
627C2#8725111yescharacteristic
628C2#8729411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×D4×Q8C2
prime radicalC22×C3×D4C23
hypercenterC2×D4×Q8D3
hypocenterC3#45336
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#5721C2semidirect
3. #84065C2semidirect
4. Q8×#217C2semidirect
5. #83247C2semidirect
6. #83227C2semidirect
7. #82796C2semidirect
8. #82768C2semidirect
9. #82758C2semidirect
10. #82753C2semidirect
11. #82481C2semidirect
12. #82456C2semidirect
622. C3#45336semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D4×Q82
C2×#57212
#840652
Q8×#2172
#832472
#832272
#827962
#827682
#827582
#827532
#824812
#824562
#823752
#823662
#823482
#453363