Finite Group Databasebeta

← #11233694

Subgroups — #11233694

Order 768.

2406 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5579C211yesmaximal, characteristic, complemented
2C2×D4×Dic6C211yesmaximal, characteristic, complemented
3C3×#2580C211yesmaximal, characteristic, complemented
4#82467C211yesmaximal, characteristic, complemented
5#82329C211yesmaximal, characteristic
6#78104C214yesmaximal, complemented
7#76824C214yesmaximal, complemented
8#76324C212yesmaximal, complemented
9#76306C212yesmaximal
10#76062C212yesmaximal, complemented
11#76010C212yesmaximal, complemented
12#74541C214yesmaximal, complemented
13#74405C214yesmaximal, complemented
14#74190C211yesmaximal, characteristic, complemented
15#74154C211yesmaximal, characteristic, complemented
16#45357—31nomaximal, Sylow
17C23×Dic6C2212yes—
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#946C2212yes—
21C2×#940C2211yescharacteristic
22C2×D4×Dic3C2212yes—
23C2×#924C2212yes—
24#5579—28no—
25D4×Dic6—28no—
26C2×#896C2214yes—
27C2×#896C2214yes—
28C2×D3×C4⋊C4C2212yes—
29C2×#888C2212yes—
30C2×#67⋊C4C2214yes—
31C2×#878C2214yes—
32C2×C4×D12C2212yes—
33C2×C4×D12C2211yescharacteristic
34C2×#870C2212yes—
35C2×#869C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37C3×#399C2212yes—
38C3×#393C2212yes—
39C3×#390C2212yes—
40C3×#389C2212yes—
41C3×#385C2214yes—
42C3×#382C2211yescharacteristic
43#5230C2212yes—
44#5221C2211yescharacteristic
45#5213C2214yes—
46#4989C2214yes—
47#4988C2212yes—
48#4988C2211yescharacteristic
49#4984C2212yes—
50#4983C2212yes—
51#4981C2212yes—
52#4979C2214yes—
53#4979C2212yes—
54#4977C2211yescharacteristic
55Dic3×C4⋊C4C2212yes—
56#4972C2214yes—
57#4972C2212yes—
58#4958C2214yes—
59#4957C2214yes—
60#4956C2214yes—
61#4956C2214yes—
62#4949C2214yes—
63#4948C2214yes—
64#4946C2214yes—
65#4942C2214yes—
66#4932C2211yescharacteristic
67C4×Dic3⋊C4C2212yes—
68C4×Dic3⋊C4C2212yes—
69#4677C2212yes—
70#4676C2214yes—
71#4673C2214yes—
72#4673C2212yes—
73#4668C2214yes—
74#4657C2214yes—
75#4653C2214yes—
76#4646C2214yes—
77#4646C2214yes—
78C2×#554—31no—
79C2×D4×Q8—31no—
80#2732—34no—
81#2692—34no—
82#2580—31no—
83#2580D311yescharacteristic, complemented
84#2550—32no—
85#2454—32no—
86#2452—32no—
87#2449—34no—
88#2409—34no—
89#2399—31no—
90#2366—32no—
91#2331—31no—
92#2328—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C23×Dic3C2314yes—
98C22×D12C2311yescharacteristic
99C22×C4×D3C2312yes—
100C22×C4×D3C2314yes—
101C22×Dic6—28no—
102C22×Dic6—48no—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2314yes—
105C22×Dic6C2312yes—
106C3×C4×Q8—24no—
107C3×C4×D4—24no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C4⋊C4C2311yescharacteristic
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42C2312yes—
117C2×C3×C42C2311yescharacteristic
118C2×#217C2314yes—
119C2×#217C2314yes—
120#946—28no—
121#940Q818yescomplemented
122D4×Dic3—28no—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2314yes—
125C2×(C2×D3)⋊C4C2312yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×C4⋊Dic3C2311yescharacteristic
129C2×C4⋊Dic3C2311yescharacteristic
130C2×C4⋊Dic3C2314yes—
131C2×C4⋊Dic3C2312yes—
132C2×C4⋊Dic3C2312yes—
133#924—28no—
134C2×Dic3⋊C4C2314yes—
135C2×Dic3⋊C4C2314yes—
136C2×Dic3⋊C4C2312yes—
137C2×Dic3⋊C4C2312yes—
138C2×Dic3⋊C4C2312yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2312yes—
143C2×C4×Dic3C2311yescharacteristic
144C2×C4×Dic3C2312yes—
145C2×C4×Dic3C2314yes—
146C2×C4×Dic3C2311yescharacteristic
147#896—216no—
148#896—48no—
149D3×C4⋊C4—28no—
150#888—28no—
151#67⋊C4—48no—
152#878—216no—
153C4×D12—24no—
154C4×D12—28no—
155#870—28no—
156#869D418yescomplemented
157C4×Dic6—24no—
158C3×#95C2312yes—
159C3×#95C2314yes—
160C3×#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2314yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C23×Q8—32no—
172#554—68no—
173D4×Q8—68no—
174C2×C4⋊Q8—32no—
175C2×C4⋊Q8—31no—
176C2×#127—31no—
177C2×#125—32no—
178C2×C22⋊Q8—32no—
179C2×C22⋊Q8—32no—
180C2×C22⋊Q8—34no—
181C2×C22⋊Q8—34no—
182C2×C22⋊Q8—34no—
183C2×C4×Q8—31no—
184C2×C4×Q8C2×D311yescharacteristic
185C2×C4×D4—32no—
186C2×C4×D4—34no—
187C2×C4×D4—31no—
188C2×C4×D4—32no—
189C2×C4×D4C2×D311yescharacteristic
190C22×C4⋊C4—32no—
191#399—34no—
192#399—32no—
193#399C2×D312yes—
194#398—32no—
195#398—32no—
196#398—34no—
197#396—34no—
198#396—34no—
199#395—32no—
200#395—32no—
201#395—34no—
202#393—32no—
203#393—34no—
204#393C2×D312yes—
205#391—34no—
206#391—34no—
207#390—31no—
208#390—32no—
209#390—31no—
210#390—34no—
211#390C2×D312yes—
212#389—32no—
213#389—32no—
214#389—34no—
215#389C2×D312yes—
216#387—34no—
217#386—34no—
218#386—34no—
219#386—34no—
220#386—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385C2×D314yes—
225#382—31no—
226#382C2×D311yescharacteristic
227#380—34no—
228C4×C4⋊C4—32no—
229C4×C4⋊C4—32no—
230C4×C4⋊C4—32no—
231#376—31no—
232C24×C3C2412yes—
233C23×D3C2412yes—
234C2×C3×Q8—22no—
235C2×C3×Q8—42no—
236C2×C3×D4—22no—
237C2×C3×D4—22no—
238C2×C3×D4C2×Q814yes—
239C22×C3×C4—24no—
240C22×C3×C4—44no—
241C22×C3×C4C2414yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2412yes—
244C22×C3×C4C2412yes—
245C22×C3×C4C2411yescharacteristic
246C22×C3×C4C2411yescharacteristic
247C22×C3×C4C2411yescharacteristic
248C2×#67—44no—
249C2×#67—44no—
250C2×#67C2×Q8116yescomplemented
251C22×Dic3—28no—
252C22×Dic3—28no—
253C22×Dic3—44no—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2414yes—
258C22×Dic3C2412yes—
259C22×Dic3C2412yes—
260C2×D12—22no—
261C2×D12—22no—
262C2×D12C2×Q814yescomplemented
263C2×C4×D3—24no—
264C2×C4×D3—28no—
265C2×C4×D3—44no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×Dic6—22no—
269C2×Dic6—24no—
270C2×Dic6—28no—
271C2×Dic6—28no—
272C2×Dic6—432no—
273C2×Dic6—44no—
274C2×Dic6—44no—
275C2×Dic6C2×D418yes—
276C3×C4⋊C4—22no—
277C3×C4⋊C4—22no—
278C3×C4⋊C4—24no—
279C3×C4⋊C4—24no—
280C3×C4⋊C4—44no—
281C3×C22⋊C4—24no—
282C3×C22⋊C4—44no—
283C3×C22⋊C4—44no—
284C3×C42—24no—
285C3×C42C2×D414yes—
286#217—28no—
287#217—44no—
288(C2×D3)⋊C4—24no—
289(C2×D3)⋊C4—28no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—44no—
293C4⋊Dic3—22no—
294C4⋊Dic3—22no—
295C4⋊Dic3—44no—
296C4⋊Dic3C2×D418yes—
297C4⋊Dic3C2×D418yes—
298Dic3⋊C4—24no—
299Dic3⋊C4—28no—
300Dic3⋊C4—28no—
301Dic3⋊C4—28no—
302Dic3⋊C4—24no—
303Dic3⋊C4—24no—
304Dic3⋊C4—44no—
305Dic3⋊C4—42no—
306Dic3⋊C4—42no—
307C4×Dic3—24no—
308C4×Dic3—22no—
309C4×Dic3—44no—
310C4×Dic3C2×Q814yescomplemented
311C22×Q8—32no—
312C22×Q8—31no—
313C22×Q8—34no—
314C22×Q8—68no—
315C22×Q8—128no—
316C22×Q8C22×D311yescharacteristic
317C22×D4—34no—
318C22×D4—31no—
319C22×D4C22×D311yescharacteristic
320C23×C4—34no—
321C23×C4—34no—
322C23×C4—32no—
323C23×C4C22×D312yes—
324C4⋊Q8—38no—
325C4⋊Q8—68no—
326#127—38no—
327#125—68no—
328C22⋊Q8—616no—
329C22⋊Q8—616no—
330C22⋊Q8—68no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333C4×Q8—24no—
334C4×Q8—64no—
335C4×D4—24no—
336C4×D4—68no—
337C4×D4—68no—
338C4×D4—64no—
339C4×D4—128no—
340C2×C4⋊C4—32no—
341C2×C4⋊C4—32no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—32no—
346C2×C4⋊C4—31no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—32no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—32no—
353C2×C4⋊C4—34no—
354C2×C4⋊C4—68no—
355C2×C4⋊C4C22×D311yescharacteristic
356C2×C4⋊C4C22×D311yescharacteristic
357C2×C4⋊C4C22×D312yes—
358C2×C4⋊C4C22×D312yes—
359C2×C4⋊C4C22×D314yes—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4C22×D312yes—
368C2×C22⋊C4C22×D314yes—
369C2×C22⋊C4C22×D314yes—
370C2×C42—31no—
371C2×C42—32no—
372C2×C42—31no—
373C2×C42—34no—
374C2×C42C22×D312yes—
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—32no—
378#95—32no—
379#95—32no—
380#95—34no—
381#95—32no—
382#95—32no—
383#95—34no—
384#95—34no—
385#95—34no—
386#95C22×D312yes—
387#95C22×D312yes—
388#95C22×D314yes—
389C23×C3—22no—
390C23×C3—24no—
391C23×C3—22no—
392C23×C3—24no—
393C23×C3C22×Q814yes—
394C23×C3C2511yescharacteristic
395C22×D3—22no—
396C22×D3—24no—
397C22×D3—22no—
398C22×D3—24no—
399C22×D3C22×Q814yes—
400C3×Q8—44no—
401C3×D4—28no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—24no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—24no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—24no—
414C2×C3×C4—22no—
415C2×C3×C4—48no—
416C2×C3×C4C22×D412yes—
417C2×C3×C4C22×D412yes—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×Q812yes—
420C2×C3×C42−1+412yes—
421#67—416no—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—24no—
425C2×Dic3—24no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—416no—
437C2×Dic3C22×D418yes—
438C2×Dic3C22×Q814yes—
439C2×Dic3H(5,𝔽2)14yes—
440D12—28no—
441C4×D3—48no—
442C4×D3—88no—
443Dic6—28no—
444Dic6—48no—
445Dic6—416no—
446C24—32no—
447C24C23×D312yes—
448C2×Q8—22no—
449C2×Q8—38no—
450C2×Q8—42no—
451C2×Q8—64no—
452C2×Q8—68no—
453C2×Q8—68no—
454C2×Q8—62no—
455C2×Q8—124no—
456C2×Q8—124no—
457C2×Q8—1232no—
458C2×D4—22no—
459C2×D4—22no—
460C2×D4—316no—
461C2×D4—34no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—124no—
465C2×D4—124no—
466C2×D4D3×Q814yes—
467C22×C4—24no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—34no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—44no—
475C22×C4—68no—
476C22×C4—68no—
477C22×C4—68no—
478C22×C4—64no—
479C22×C4—124no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4C23×D314yes—
484C22×C4C23×D312yes—
485C22×C4C23×D312yes—
486C22×C4C23×D312yes—
487C22×C4C23×D311yescharacteristic
488C22×C4C23×D311yescharacteristic
489C22×C4C23×D311yescharacteristic
490C4⋊C4—22no—
491C4⋊C4—22no—
492C4⋊C4—24no—
493C4⋊C4—24no—
494C4⋊C4—38no—
495C4⋊C4—38no—
496C4⋊C4—44no—
497C4⋊C4—68no—
498C4⋊C4—64no—
499C4⋊C4—68no—
500C4⋊C4—68no—
501C4⋊C4—64no—
502C4⋊C4—62no—
503C4⋊C4—64no—
504C4⋊C4—62no—
505C4⋊C4—124no—
506C4⋊C4—122no—
507C4⋊C4—122no—
508C4⋊C4—124no—
509C22⋊C4—24no—
510C22⋊C4—44no—
511C22⋊C4—44no—
512C22⋊C4—68no—
513C22⋊C4—68no—
514C22⋊C4—68no—
515C22⋊C4—64no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C42—24no—
520C42—34no—
521C42—64no—
522C42—62no—
523C42—124no—
524C42D3×D414yes—
525C22×C3—24no—
526C22×C3—28no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—44no—
530C22×C3C23×D411yescharacteristic
531C22×C3C23×Q811yescharacteristic
532C22×C3C2×H(5,𝔽2)12yes—
533C22×C3C2×2−1+411yescharacteristic
534C22×C3#58512yes—
535C2×D3—24no—
536C2×D3—28no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—44no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—42no—
543C3×C4—42no—
544C3×C4—42no—
545C3×C4—44no—
546C3×C4D4×Q814yes—
547Dic3—24no—
548Dic3—28no—
549Dic3—24no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—34no—
558C23—62no—
559C23—62no—
560C23—64no—
561C23—64no—
562C23C2×D3×Q814yes—
563C23C24×D311yescharacteristic
564Q8—44no—
565Q8—68no—
566Q8—128no—
567Q8—1216no—
568D4—28no—
569D4—68no—
570D4—1216no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—24no—
583C2×C4—22no—
584C2×C4—38no—
585C2×C4—34no—
586C2×C4—34no—
587C2×C4—48no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—64no—
592C2×C4—64no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—64no—
600C2×C4—62no—
601C2×C4—62no—
602C2×C4—128no—
603C2×C4—1216no—
604C2×C4—248no—
605C2×C4C2×D3×D412yes—
606C2×C4C2×D3×D412yes—
607C2×C4C2×D3×D412yes—
608C2×C4C2×D3×Q812yes—
609C2×C4#101012yes—
610C2×C3—44no—
611C2×C3C2×D4×Q811yescharacteristic
612C2×C3#348411yescharacteristic
613C2×C3#348711yescharacteristic
614C2×C3#350112yes—
615C2×C3#355412yes—
616D3—44no—
617C22—28no—
618C22—24no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—68no—
623C22—64no—
624C22—122no—
625C22—124no—
626C22—122no—
627C22C22×D3×D411yescharacteristic
628C22C22×D3×Q811yescharacteristic
629C22C2×#101011yescharacteristic
630C22#596412yes—
631C22D3×H(5,𝔽2)12yes—
632C4—22no—
633C4—22no—
634C4—42no—
635C4—44no—
636C4—42no—
637C4—42no—
638C4—64no—
639C4—68no—
640C4—64no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4#563214yes—
645C3#4535711yescharacteristic, complemented, Sylow
646C2—44no—
647C2—124no—
648C2C2×#563211yescharacteristic
649C2#8602311yescharacteristic
650C2#8633011yescharacteristic
651C2#8694912yes—
652C2#8706812yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2580C2
prime radicalC2×#940C22
hypercenter#2580D3
hypocenterC3#45357
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2580

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C2×D4×Dic6C2semidirect
3. C3×#2580C2semidirect
4. #82467C2semidirect
6. #78104C2semidirect
7. #76824C2semidirect
8. #76324C2semidirect
10. #76062C2semidirect
11. #76010C2semidirect
12. #74541C2semidirect
13. #74405C2semidirect
14. #74190C2semidirect
15. #74154C2semidirect
83. #2580D3semidirect
121. #940Q8semidirect
156. #869D4semidirect
250. C2×#67C2×Q8semidirect
262. C2×D12C2×Q8semidirect
310. C4×Dic3C2×Q8semidirect
645. C3#45357semidirect

Maximal subgroups

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

subgroupindex
C2×#55792
C2×D4×Dic62
C3×#25802
#824672
#823292
#781042
#768242
#763242
#763062
#760622
#760102
#745412
#744052
#741902
#741542
#453573