Finite Group Databasebeta

← #11233333

Subgroups — #11233333

Order 768.

2410 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#5730C211yesmaximal, characteristic, complemented
3#84068C214yesmaximal, complemented
4#83258C212yesmaximal, complemented
5#83233C212yesmaximal, complemented
6#83228C211yesmaximal, characteristic, complemented
7#82831C214yesmaximal, complemented
8#82818C214yesmaximal, complemented
9#82796C214yesmaximal, complemented
10#82492C211yesmaximal, characteristic, complemented
11#82480C211yesmaximal, characteristic, complemented
12#82416C211yesmaximal, characteristic
13#82408C212yesmaximal, complemented
14#82399C212yesmaximal, complemented
15#82328C211yesmaximal, characteristic, complemented
16#45337—31nomaximal, Sylow
17C23×C3×Q8C2212yes—
18C3×D4×Q8—28no—
19C2×C3×C4⋊Q8C2212yes—
20C2×C3×C4⋊Q8C2211yescharacteristic
21C2×C3×C22⋊Q8C2214yes—
22C2×C3×C22⋊Q8C2212yes—
23C2×C3×C4×Q8C2211yescharacteristic
24C2×C3×C4×D4C2211yescharacteristic
25C2×C3×C4×D4C2212yes—
26C2×#946C2214yes—
27C2×#946C2214yes—
28C2×#938C2212yes—
29C2×#936C2211yescharacteristic
30C2×C4×#67C2214yes—
31#5730—28no—
32C2×#897C2212yes—
33C2×#893C2212yes—
34C2×D3×C4⋊C4C2212yes—
35C2×#890C2212yes—
36C2×C4×D12C2211yescharacteristic
37C2×C4×Dic6C2211yescharacteristic
38#5231C2214yes—
39#5231C2212yes—
40#5230C2214yes—
41#5230C2212yes—
42#5229C2214yes—
43#5229C2214yes—
44#5221C2211yescharacteristic
45#5213C2214yes—
46#5213C2214yes—
47#5212C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50#4992C2212yes—
51#4991C2212yes—
52#4990C2214yes—
53#4989C2214yes—
54#4987C2212yes—
55#4986C2214yes—
56#4986C2212yes—
57#4985C2212yes—
58#4984C2212yes—
59#4982C2214yes—
60#4980C2214yes—
61#4978C2214yes—
62#4975C2211yescharacteristic
63Dic3×C4⋊C4C2212yes—
64#4973C2211yescharacteristic
65#4961C2214yes—
66#4960C2212yes—
67#4956C2214yes—
68#4953C2212yes—
69#4953C2214yes—
70#4945C2214yes—
71#4944C2214yes—
72C4×(C2×D3)⋊C4C2212yes—
73C4×(C2×D3)⋊C4C2212yes—
74#4936C2211yescharacteristic
75#4932C2211yescharacteristic
76#4929C2212yes—
77#4928C2212yes—
78C2×#554—31no—
79C2×D4×Q8D311yescharacteristic, complemented
80#2728—34no—
81#2690—34no—
82#2581—31no—
83#2535—31no—
84#2492—32no—
85#2463—34no—
86#2440—32no—
87#2427—32no—
88#2409—34no—
89#2398—31no—
90#2361—32no—
91#2331—31no—
92#2326—31no—
93C22×C3×Q8—28no—
94C22×C3×Q8—48no—
95C22×C3×Q8C2311yescharacteristic
96C22×C3×Q8C2314yes—
97C22×C3×Q8C2312yes—
98C22×C3×D4C2311yescharacteristic
99C23×C3×C4C2314yes—
100C23×C3×C4C2312yes—
101C22×#67C2314yes—
102C22×D12C2311yescharacteristic
103C22×C4×D3C2312yes—
104C22×C4×D3C2314yes—
105C22×Dic6C2311yescharacteristic
106C3×C4⋊Q8—28no—
107C3×C4⋊Q8D418yescomplemented
108C3×C22⋊Q8—28no—
109C3×C22⋊Q8—216no—
110C3×C4×Q8—24no—
111C3×C4×D4—28no—
112C3×C4×D4—24no—
113C2×C3×C4⋊C4C2311yescharacteristic
114C2×C3×C4⋊C4C2312yes—
115C2×C3×C4⋊C4C2314yes—
116C2×C3×C4⋊C4C2311yescharacteristic
117C2×C3×C4⋊C4C2312yes—
118C2×C3×C4⋊C4C2312yes—
119C2×C3×C22⋊C4C2312yes—
120C2×C3×C22⋊C4C2314yes—
121C2×C3×C42C2312yes—
122C2×C3×C42C2311yescharacteristic
123C2×#217C2314yes—
124C2×#217C2314yes—
125C2×#217C2314yes—
126#946—216no—
127#946—48no—
128#938—28no—
129#936—24no—
130C4×#67—48no—
131C2×(C2×D3)⋊C4C2312yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2312yes—
134C2×(C2×D3)⋊C4C2312yes—
135C2×(C2×D3)⋊C4C2312yes—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×C4⋊Dic3C2311yescharacteristic
139C2×C4⋊Dic3C2312yes—
140C2×C4⋊Dic3C2311yescharacteristic
141C2×C4⋊Dic3C2314yes—
142C2×C4⋊Dic3C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2314yes—
147C2×C4×Dic3C2311yescharacteristic
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2314yes—
151#897—28no—
152#893—28no—
153D3×C4⋊C4—28no—
154#890—28no—
155C4×D12—24no—
156C4×Dic6—24no—
157C3⋊#95C2314yes—
158C3⋊#95C2314yes—
159C3⋊#95C2314yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C3⋊#95C2314yes—
172C23×Q8C2×D312yes—
173#554—68no—
174D4×Q8—28no—
175C2×C4⋊Q8C2×D312yes—
176C2×C4⋊Q8C2×D311yescharacteristic
177C2×#125—32no—
178C2×#124—31no—
179C2×C22⋊Q8—34no—
180C2×C22⋊Q8—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8C2×D314yes—
183C2×C22⋊Q8C2×D312yes—
184C2×#121—32no—
185C2×C4×Q8—31no—
186C2×C4×Q8C2×D311yescharacteristic
187C2×C4×D4—34no—
188C2×C4×D4—32no—
189C2×C4×D4—31no—
190C2×C4×D4C2×D311yescharacteristic
191C2×C4×D4C2×D312yes—
192C22×C4⋊C4—32no—
193#400—32no—
194#400—32no—
195#400—34no—
196#398—34no—
197#397—32no—
198#397—34no—
199#397—32no—
200#395—32no—
201#395—34no—
202#394—32no—
203#394—34no—
204#393—32no—
205#393—34no—
206#392—32no—
207#391—34no—
208#391—34no—
209#391—31no—
210#391—32no—
211#390—31no—
212#389—32no—
213#389—32no—
214#388—34no—
215#388—34no—
216#388—34no—
217#387—34no—
218#387—34no—
219#386—34no—
220#386—34no—
221#386—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382—31no—
227#380—34no—
228#380—32no—
229C4×C4⋊C4—32no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—32no—
232#376—31no—
233C24×C3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—28no—
237C2×C3×Q8—24no—
238C2×C3×Q8—28no—
239C2×C3×Q8—44no—
240C2×C3×Q8—44no—
241C2×C3×Q8—432no—
242C2×C3×Q8C2×D418yes—
243C2×C3×D4—22no—
244C2×C3×D4—22no—
245C2×C3×D4#4114yes—
246C22×C3×C4—28no—
247C22×C3×C4—24no—
248C22×C3×C4—28no—
249C22×C3×C4—44no—
250C22×C3×C4—44no—
251C22×C3×C4C2412yes—
252C22×C3×C4C2412yes—
253C22×C3×C4C2411yescharacteristic
254C22×C3×C4C2411yescharacteristic
255C22×C3×C4C2414yes—
256C22×C3×C4C2412yes—
257C22×C3×C4C2411yescharacteristic
258C2×#67—28no—
259C2×#67—28no—
260C2×#67—44no—
261C22×Dic3C2412yes—
262C22×Dic3C2414yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C22×Dic3C2412yes—
267C2×D12—24no—
268C2×D12—22no—
269C2×C4×D3—24no—
270C2×C4×D3—24no—
271C2×C4×D3—24no—
272C2×C4×D3—28no—
273C2×C4×D3—44no—
274C2×C4×D3—44no—
275C2×Dic6—22no—
276C2×Dic6—22no—
277C2×Dic6—22no—
278C3×C4⋊C4—22no—
279C3×C4⋊C4—24no—
280C3×C4⋊C4—22no—
281C3×C4⋊C4—28no—
282C3×C4⋊C4C2×D418yes—
283C3×C4⋊C4C2×D418yes—
284C3×C22⋊C4—28no—
285C3×C22⋊C4—24no—
286C3×C42—24no—
287C3×C42C2×D414yes—
288#217—28no—
289#217—28no—
290#217—44no—
291(C2×D3)⋊C4—24no—
292(C2×D3)⋊C4—24no—
293(C2×D3)⋊C4—24no—
294(C2×D3)⋊C4—28no—
295(C2×D3)⋊C4—24no—
296(C2×D3)⋊C4—24no—
297(C2×D3)⋊C4—44no—
298C4⋊Dic3—22no—
299C4⋊Dic3—22no—
300C4⋊Dic3—28no—
301C4⋊Dic3—24no—
302C4⋊Dic3—44no—
303Dic3⋊C4—24no—
304Dic3⋊C4—28no—
305Dic3⋊C4—28no—
306Dic3⋊C4—44no—
307C4×Dic3—24no—
308C4×Dic3—22no—
309C4×Dic3—22no—
310C4×Dic3—44no—
311C22×Q8—28no—
312C22×Q8—31no—
313C22×Q8—48no—
314C22×Q8C22×D311yescharacteristic
315C22×Q8C22×D314yes—
316C22×Q8C22×D312yes—
317C22×D4—34no—
318C22×D4—31no—
319C22×D4C22×D311yescharacteristic
320C23×C4—32no—
321C23×C4—34no—
322C23×C4C22×D314yes—
323C23×C4C22×D312yes—
324C4⋊Q8—28no—
325C4⋊Q8#6718yescomplemented
326#125—68no—
327#124—64no—
328C22⋊Q8—28no—
329C22⋊Q8—216no—
330C22⋊Q8—616no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333#121—68no—
334C4×Q8—24no—
335C4×Q8—64no—
336C4×D4—28no—
337C4×D4—24no—
338C4×D4—68no—
339C4×D4—64no—
340C4×D4—128no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—34no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—68no—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D312yes—
353C2×C4⋊C4C22×D311yescharacteristic
354C2×C4⋊C4C22×D314yes—
355C2×C4⋊C4C22×D312yes—
356C2×C4⋊C4C22×D312yes—
357C2×C22⋊C4—34no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—32no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4C22×D314yes—
368C2×C22⋊C4C22×D312yes—
369C2×C42—32no—
370C2×C42—31no—
371C2×C42—31no—
372C2×C42—34no—
373C2×C42C22×D312yes—
374C2×C42C22×D311yescharacteristic
375#95—32no—
376#95—32no—
377#95—32no—
378#95—32no—
379#95—32no—
380#95—32no—
381#95—34no—
382#95—34no—
383#95—34no—
384#95—32no—
385#95—32no—
386#95—34no—
387#95—34no—
388#95—34no—
389#95—34no—
390C23×C3—22no—
391C23×C3—24no—
392C23×C3—24no—
393C23×C3—22no—
394C23×C3C2×#4114yes—
395C23×C3C2511yescharacteristic
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—22no—
401C22×D3—22no—
402C22×D32−1+414yes—
403C3×Q8—28no—
404C3×Q8—48no—
405C3×Q8—416no—
406C3×D4—28no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—24no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—24no—
416C2×C3×C4—22no—
417C2×C3×C4—22no—
418C2×C3×C4—416no—
419C2×C3×C4—48no—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×D412yes—
422C2×C3×C4C22×D418yes—
423C2×C3×C4C22×D412yes—
424C2×C3×C4C2×#4112yes—
425C2×C3×C4H(5,𝔽2)14yes—
426C2×C3×C42−1+412yes—
427#67—416no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—24no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—22no—
436C2×Dic3—24no—
437C2×Dic3—24no—
438C2×Dic3—22no—
439C2×Dic3—22no—
440C2×Dic3—22no—
441C2×Dic3—22no—
442C2×Dic3—22no—
443C2×Dic3—22no—
444C2×Dic3—24no—
445D12—28no—
446C4×D3—48no—
447C4×D3—88no—
448Dic6—44no—
449C24—32no—
450C24C23×D312yes—
451C2×Q8—24no—
452C2×Q8—28no—
453C2×Q8—22no—
454C2×Q8—28no—
455C2×Q8—44no—
456C2×Q8—44no—
457C2×Q8—432no—
458C2×Q8—62no—
459C2×Q8—62no—
460C2×Q8—62no—
461C2×Q8C2×#6718yes—
462C2×D4—22no—
463C2×D4—22no—
464C2×D4—64no—
465C2×D4—68no—
466C2×D4—68no—
467C2×D4—62no—
468C2×D4—124no—
469C2×D4#23914yes—
470C22×C4—28no—
471C22×C4—28no—
472C22×C4—24no—
473C22×C4—32no—
474C22×C4—34no—
475C22×C4—32no—
476C22×C4—32no—
477C22×C4—32no—
478C22×C4—32no—
479C22×C4—44no—
480C22×C4—44no—
481C22×C4—64no—
482C22×C4—64no—
483C22×C4—64no—
484C22×C4—68no—
485C22×C4—124no—
486C22×C4—124no—
487C22×C4C23×D312yes—
488C22×C4C23×D312yes—
489C22×C4C23×D312yes—
490C22×C4C23×D311yescharacteristic
491C22×C4C23×D311yescharacteristic
492C22×C4C23×D314yes—
493C22×C4C23×D311yescharacteristic
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—28no—
498C4⋊C4—68no—
499C4⋊C4—64no—
500C4⋊C4—62no—
501C4⋊C4—68no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—68no—
505C4⋊C4—124no—
506C4⋊C4—124no—
507C4⋊C4C2×#6718yes—
508C4⋊C4C2×#6718yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—64no—
512C22⋊C4—64no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—64no—
516C22⋊C4—64no—
517C22⋊C4—68no—
518C22⋊C4—64no—
519C22⋊C4—124no—
520C22⋊C4—124no—
521C42—24no—
522C42—62no—
523C42—62no—
524C42—64no—
525C42—124no—
526C42C2×#6714yes—
527C22×C3—24no—
528C22×C3—28no—
529C22×C3—42no—
530C22×C3—42no—
531C22×C3—44no—
532C22×C3C23×D411yescharacteristic
533C22×C3C22×#4111yescharacteristic
534C22×C3C2×H(5,𝔽2)12yes—
535C22×C3C2×2−1+411yescharacteristic
536C22×C3C2×2−1+412yes—
537C2×D3—24no—
538C2×D3—24no—
539C2×D3—24no—
540C2×D3—42no—
541C2×D3—42no—
542C2×D3—42no—
543C2×D3—42no—
544C3×C4—28no—
545C3×C4—22no—
546C3×C4—24no—
547C3×C4—22no—
548C3×C4—42no—
549C3×C4—42no—
550C3×C4#54814yes—
551Dic3—42no—
552Dic3—42no—
553Dic3—42no—
554Dic3—42no—
555Dic3—44no—
556Dic3—42no—
557C23—24no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—34no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23—62no—
566C23—62no—
567C23—62no—
568C23C2×#23914yes—
569C23C24×D311yescharacteristic
570Q8—28no—
571Q8—48no—
572Q8—416no—
573Q8—124no—
574D4—28no—
575D4—68no—
576D4—1216no—
577C2×C4—24no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—416no—
589C2×C4—48no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—62no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—64no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—64no—
604C2×C4—62no—
605C2×C4—62no—
606C2×C4—62no—
607C2×C4—128no—
608C2×C4—248no—
609C2×C4#100414yes—
610C2×C4C2×#23912yes—
611C2×C4#100712yes—
612C2×C4C22×#6712yes—
613C2×C4C22×#6712yes—
614C2×C4C22×#6718yes—
615C2×C4C22×#6712yes—
616C2×C3—44no—
617C2×C3C2×#54811yescharacteristic
618C2×C3#348411yescharacteristic
619C2×C3#348611yescharacteristic
620C2×C3#352012yes—
621C2×C3#355212yes—
622D3—44no—
623C22—24no—
624C22—28no—
625C22—42no—
626C22—44no—
627C22—42no—
628C22—64no—
629C22—64no—
630C22—64no—
631C22—122no—
632C22—122no—
633C22—122no—
634C22—122no—
635C22C2×#100412yes—
636C22C22×#23911yescharacteristic
637C22C2×#100711yescharacteristic
638C22D3×2−1+412yes—
639C22C23×#6711yescharacteristic
640C4—24no—
641C4—22no—
642C4—28no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—122no—
647C4—122no—
648C4—122no—
649C4—124no—
650C4—122no—
651C4—122no—
652C4#581714yes—
653C3#4533711yescharacteristic, complemented, Sylow
654C2—44no—
655C2—124no—
656C2#8675112yes—
657C2#8676412yes—
658C2#8723911yescharacteristic
659C2C2×#581711yescharacteristic
660C2#8729411yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×D4×Q8C2
prime radical#82416C2
hypercenterC2×D4×Q8D3
hypocenterC3#45337
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D4×Q8C2semidirect
2. C2×#5730C2semidirect
3. #84068C2semidirect
4. #83258C2semidirect
5. #83233C2semidirect
6. #83228C2semidirect
7. #82831C2semidirect
8. #82818C2semidirect
9. #82796C2semidirect
10. #82492C2semidirect
11. #82480C2semidirect
13. #82408C2semidirect
14. #82399C2semidirect
15. #82328C2semidirect
79. C2×D4×Q8D3semidirect
107. C3×C4⋊Q8D4semidirect
325. C4⋊Q8#67semidirect
653. C3#45337semidirect

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×#57302
#840682
#832582
#832332
#832282
#828312
#828182
#827962
#824922
#824802
#824162
#824082
#823992
#823282
#453373