Finite Group Databasebeta

← #11233301

Subgroups — #11233301

Order 768.

2282 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5726C211yesmaximal, characteristic
2C2×#5721C211yesmaximal, characteristic, complemented
3C3×#2327C211yesmaximal, characteristic, complemented
4#82496C212yesmaximal, complemented
5#82457C212yesmaximal, complemented
6#76623C212yesmaximal, complemented
7#76498C212yesmaximal, complemented
8#75325C214yesmaximal, complemented
9#75155C214yesmaximal, complemented
10#75151C214yesmaximal, complemented
11#75009C214yesmaximal, complemented
12#74240C211yesmaximal, characteristic, complemented
13#74190C211yesmaximal, characteristic, complemented
14#73419C211yesmaximal, characteristic, complemented
15#73387C211yesmaximal, characteristic, complemented
16#45335—31nomaximal, Sylow
17C22×#239C2212yes—
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×C4×Q8C2212yes—
20C2×#946C2212yes—
21C2×Q8×Dic3C2212yes—
22C2×#944C2212yes—
23#5726—28no—
24#5721—28no—
25C2×#897C2212yes—
26C2×#895C2212yes—
27C2×#894C2214yes—
28C2×D3×C4⋊C4C2212yes—
29C2×#890C2212yes—
30C2×#889C2214yes—
31C2×#887C2212yes—
32C2×#875C2211yescharacteristic
33C2×C4×D12C2211yescharacteristic
34C2×C4×D12C2211yescharacteristic
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37C3×#391C2212yes—
38C3×#387C2214yes—
39C3×#387C2214yes—
40C3×#382C2211yescharacteristic
41C3×#376C2211yescharacteristic
42#4994C2214yes—
43#4988C2214yes—
44#4988C2212yes—
45#4987C2212yes—
46#4987C2214yes—
47#4984C2212yes—
48#4981C2214yes—
49#4940C2214yes—
50#4939C2211yescharacteristic
51#4939C2214yes—
52C4×(C2×D3)⋊C4C2214yes—
53C4×(C2×D3)⋊C4C2212yes—
54C4×(C2×D3)⋊C4C2212yes—
55#4935C2211yescharacteristic
56#4929C2211yescharacteristic
57#4928C2214yes—
58#4677C2214yes—
59#4677C2212yes—
60#4674C2212yes—
61#4673C2212yes—
62#4673C2212yes—
63#4672C2212yes—
64#4672C2214yes—
65#4672C2214yes—
66#4672C2212yes—
67#4670C2214yes—
68#4670C2214yes—
69#4670C2214yes—
70#4668C2214yes—
71#4668C2214yes—
72#4668C2214yes—
73#4668C2214yes—
74#4661C2212yes—
75#4661C2214yes—
76#4659C2212yes—
77#4653C2214yes—
78C2×#557—31no—
79C2×#547—31no—
80#2711—34no—
81#2694—34no—
82#2580—31no—
83#2536—31no—
84#2473—34no—
85#2458—34no—
86#2432—32no—
87#2429—32no—
88#2423—32no—
89#2398—31no—
90#2341—32no—
91#2327D311yescharacteristic, complemented
92#2326—31no—
93C22×C3×Q8C2312yes—
94C2×#239—28no—
95C2×#239—28no—
96C2×#239—44no—
97C22×D12C2314yes—
98C22×D12C2311yescharacteristic
99C22×D12C2311yescharacteristic
100C22×C4×D3C2312yes—
101C22×C4×D3C2314yes—
102C22×C4×D3C2312yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2311yescharacteristic
105C3×C4⋊Q8D418yes—
106C3×C4×Q8—44no—
107C2×C3×C4⋊C4C2312yes—
108C2×C3×C4⋊C4C2314yes—
109C2×C3×C4⋊C4C2312yes—
110C2×C3×C4⋊C4C2312yes—
111C2×C3×C42C2311yescharacteristic
112C2×C3×C42C2311yescharacteristic
113C2×C3×C42C2314yes—
114C2×C3×C42C2311yescharacteristic
115#946—28no—
116Q8×Dic3—28no—
117#944—28no—
118C2×(C2×D3)⋊C4C2314yes—
119C2×(C2×D3)⋊C4C2314yes—
120C2×(C2×D3)⋊C4C2312yes—
121C2×(C2×D3)⋊C4C2312yes—
122C2×(C2×D3)⋊C4C2312yes—
123C2×(C2×D3)⋊C4C2314yes—
124C2×(C2×D3)⋊C4C2312yes—
125C2×(C2×D3)⋊C4C2314yes—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2314yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2312yes—
130C2×(C2×D3)⋊C4C2312yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2314yes—
133C2×C4⋊Dic3C2311yescharacteristic
134C2×C4⋊Dic3C2311yescharacteristic
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2314yes—
137C2×C4⋊Dic3C2314yes—
138C2×C4⋊Dic3C2314yes—
139C2×Dic3⋊C4C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2312yes—
146C2×C4×Dic3C2312yes—
147#897—44no—
148#895—28no—
149#894—216no—
150D3×C4⋊C4—28no—
151#890—28no—
152#889—216no—
153#887—28no—
154#875—24no—
155C4×D12—24no—
156C4×D12—24no—
157C4×Dic6—24no—
158C4×Dic6—24no—
159C3×#95C2314yes—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3×#95C2314yes—
163C3×#95C2312yes—
164C3×#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2314yes—
167C3⋊#95C2314yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C22×#41—32no—
173#557—68no—
174#547—68no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8C2×D311yescharacteristic
177C2×#125—34no—
178C2×#125—32no—
179C2×#124—31no—
180C2×#123—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8—32no—
183C2×#121—32no—
184C2×C4×Q8—32no—
185C2×C4×Q8—32no—
186C2×C4×Q8—31no—
187C2×C4×Q8—31no—
188C2×C4×Q8C2×D312yes—
189C2×C4×D4—31no—
190C2×C4×D4—31no—
191C22×C4⋊C4—32no—
192#400—32no—
193#399—32no—
194#399—34no—
195#398—34no—
196#398—34no—
197#398—32no—
198#397—32no—
199#396—34no—
200#394—32no—
201#394—34no—
202#394—34no—
203#394—32no—
204#393—32no—
205#393—32no—
206#391—31no—
207#391C2×D312yes—
208#390—32no—
209#390—34no—
210#390—31no—
211#390—34no—
212#389—32no—
213#389—34no—
214#388—34no—
215#388—34no—
216#388—34no—
217#388—34no—
218#387—34no—
219#387C2×D314yes—
220#387C2×D314yes—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382C2×D311yescharacteristic
227#380—32no—
228#380—34no—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—34no—
231C4×C22⋊C4—32no—
232#376C2×D311yescharacteristic
233C23×D3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—24no—
236C2×C3×Q8—24no—
237C2×C3×Q8C2×D418yes—
238C22×C3×C4C2412yes—
239C22×C3×C4C2412yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2411yescharacteristic
242C22×C3×C4C2411yescharacteristic
243C22×C3×C4C2414yes—
244C22×C3×C4C2411yescharacteristic
245C22×Dic3C2412yes—
246C22×Dic3C2412yes—
247C22×Dic3C2412yes—
248C22×Dic3C2412yes—
249C22×Dic3C2412yes—
250C22×Dic3C2412yes—
251#239—432no—
252C2×D12—22no—
253C2×D12—22no—
254C2×D12—28no—
255C2×D12—22no—
256C2×D12—28no—
257C2×D12—42no—
258C2×D12—44no—
259C2×D12#4114yes—
260C2×C4×D3—28no—
261C2×C4×D3—24no—
262C2×C4×D3—24no—
263C2×C4×D3—28no—
264C2×C4×D3—24no—
265C2×C4×D3—24no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×Dic6—22no—
269C2×Dic6—24no—
270C2×Dic6—22no—
271C2×Dic6—22no—
272C2×Dic6—22no—
273C3×C4⋊C4—24no—
274C3×C4⋊C4—44no—
275C3×C4⋊C4C2×D418yes—
276C3×C4⋊C4C2×D418yes—
277C3×C42—22no—
278C3×C42—22no—
279C3×C42—44no—
280C3×C42C2×D414yes—
281(C2×D3)⋊C4—28no—
282(C2×D3)⋊C4—24no—
283(C2×D3)⋊C4—24no—
284(C2×D3)⋊C4—28no—
285(C2×D3)⋊C4—28no—
286(C2×D3)⋊C4—24no—
287(C2×D3)⋊C4—24no—
288(C2×D3)⋊C4—44no—
289(C2×D3)⋊C4—42no—
290(C2×D3)⋊C4—44no—
291(C2×D3)⋊C4—44no—
292(C2×D3)⋊C4—42no—
293(C2×D3)⋊C4—44no—
294C4⋊Dic3—22no—
295C4⋊Dic3—22no—
296C4⋊Dic3—22no—
297C4⋊Dic3—22no—
298C4⋊Dic3—28no—
299C4⋊Dic3—44no—
300C4⋊Dic3—44no—
301C4⋊Dic3—44no—
302Dic3⋊C4—24no—
303Dic3⋊C4—24no—
304Dic3⋊C4—28no—
305Dic3⋊C4—28no—
306Dic3⋊C4—28no—
307C4×Dic3—24no—
308C4×Dic3—24no—
309C4×Dic3—28no—
310C2×#41—68no—
311C2×#41—68no—
312C2×#41—124no—
313C22×Q8—31no—
314C22×Q8—31no—
315C22×Q8C22×D312yes—
316C22×D4—31no—
317C22×D4—34no—
318C22×D4—31no—
319C23×C4—32no—
320C23×C4—34no—
321C23×C4—32no—
322C4⋊Q8—68no—
323C4⋊Q8D1218yes—
324#125—616no—
325#125—68no—
326#124—64no—
327#123—616no—
328C22⋊Q8—68no—
329C22⋊Q8—124no—
330#121—68no—
331C4×Q8—44no—
332C4×Q8—68no—
333C4×Q8—68no—
334C4×Q8—64no—
335C4×Q8—64no—
336C4×D4—64no—
337C4×D4—64no—
338C2×C4⋊C4—31no—
339C2×C4⋊C4—34no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—31no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—34no—
351C2×C4⋊C4—68no—
352C2×C4⋊C4C22×D312yes—
353C2×C4⋊C4C22×D314yes—
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D312yes—
356C2×C22⋊C4—32no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—34no—
359C2×C22⋊C4—34no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—32no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—32no—
366C2×C22⋊C4—32no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4—34no—
369C2×C42—34no—
370C2×C42—32no—
371C2×C42—32no—
372C2×C42C22×D311yescharacteristic
373C2×C42C22×D311yescharacteristic
374C2×C42C22×D314yes—
375C2×C42C22×D311yescharacteristic
376#95—32no—
377#95—34no—
378#95—32no—
379#95—32no—
380#95—32no—
381#95—34no—
382#95—34no—
383#95C22×D312yes—
384#95C22×D312yes—
385#95C22×D312yes—
386#95C22×D312yes—
387#95C22×D314yes—
388#95C22×D314yes—
389C23×C3C2511yescharacteristic
390C22×D3—22no—
391C22×D3—22no—
392C22×D3—24no—
393C22×D3—24no—
394C22×D3—22no—
395C22×D3—22no—
396C22×D3—22no—
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—22no—
400C22×D3—22no—
401C22×D3C2×#4114yes—
402C3×Q8—216no—
403C2×C3×C4—24no—
404C2×C3×C4—24no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—24no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—22no—
416C2×C3×C4C22×D412yes—
417C2×C3×C4C22×D418yes—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C2×#4112yes—
421C2×C3×C4H(5,𝔽2)12yes—
422C2×Dic3—22no—
423C2×Dic3—22no—
424C2×Dic3—22no—
425C2×Dic3—22no—
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—24no—
435C2×Dic3—24no—
436C2×Dic3—22no—
437C2×Dic32−1+414yes—
438D12—28no—
439D12—416no—
440D12—44no—
441C4×D3—416no—
442C4×D3—48no—
443C4×D3—48no—
444Dic6—28no—
445Dic6—44no—
446C24—32no—
447C24—32no—
448#41—1232no—
449C2×Q8—24no—
450C2×Q8—24no—
451C2×Q8—64no—
452C2×Q8—62no—
453C2×Q8—62no—
454C2×Q8—62no—
455C2×Q8—62no—
456C2×Q8C2×D1218yes—
457C2×D4—34no—
458C2×D4—68no—
459C2×D4—62no—
460C2×D4—62no—
461C2×D4—68no—
462C2×D4—62no—
463C2×D4—124no—
464C2×D4—122no—
465C22×C4—32no—
466C22×C4—32no—
467C22×C4—32no—
468C22×C4—32no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—68no—
472C22×C4—64no—
473C22×C4—68no—
474C22×C4—64no—
475C22×C4—64no—
476C22×C4—64no—
477C22×C4—124no—
478C22×C4—124no—
479C22×C4C23×D312yes—
480C22×C4C23×D312yes—
481C22×C4C23×D314yes—
482C22×C4C23×D311yescharacteristic
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D314yes—
485C22×C4C23×D311yescharacteristic
486C4⋊C4—24no—
487C4⋊C4—44no—
488C4⋊C4—62no—
489C4⋊C4—68no—
490C4⋊C4—62no—
491C4⋊C4—62no—
492C4⋊C4—64no—
493C4⋊C4—64no—
494C4⋊C4—68no—
495C4⋊C4—62no—
496C4⋊C4—68no—
497C4⋊C4—68no—
498C4⋊C4—124no—
499C4⋊C4—124no—
500C4⋊C4—124no—
501C4⋊C4C2×D1218yes—
502C4⋊C4C2×D1218yes—
503C22⋊C4—68no—
504C22⋊C4—64no—
505C22⋊C4—64no—
506C22⋊C4—68no—
507C22⋊C4—64no—
508C22⋊C4—64no—
509C22⋊C4—68no—
510C22⋊C4—124no—
511C22⋊C4—124no—
512C22⋊C4—122no—
513C22⋊C4—124no—
514C22⋊C4—124no—
515C22⋊C4—122no—
516C42—22no—
517C42—22no—
518C42—44no—
519C42—68no—
520C42—64no—
521C42—64no—
522C42C2×D1214yes—
523C22×C3C23×D411yescharacteristic
524C22×C3C22×#4111yescharacteristic
525C22×C3C2×H(5,𝔽2)11yescharacteristic
526C22×C3C2×2−1+412yes—
527C22×C3#58512yes—
528C2×D3—24no—
529C2×D3—28no—
530C2×D3—42no—
531C2×D3—42no—
532C2×D3—44no—
533C2×D3—42no—
534C2×D3—42no—
535C2×D3—42no—
536C2×D3—42no—
537C2×D3—42no—
538C2×D3—42no—
539C2×D3—42no—
540C3×C4—28no—
541C3×C4—22no—
542C3×C4—22no—
543C3×C4—42no—
544C3×C4—44no—
545C3×C4—42no—
546C3×C4#54614yes—
547Dic3—24no—
548Dic3—42no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553C23—34no—
554C23—62no—
555C23—62no—
556C23—64no—
557C23—64no—
558C23—62no—
559C23—62no—
560C23—62no—
561C23—62no—
562C23—62no—
563C23—62no—
564C23—62no—
565C23C24×D311yescharacteristic
566Q8—216no—
567Q8—68no—
568Q8—124no—
569D4—68no—
570D4—1216no—
571D4—124no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—22no—
585C2×C4—34no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—64no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—1216no—
602C2×C4—128no—
603C2×C4—128no—
604C2×C4C22×D1212yes—
605C2×C4C22×D1218yes—
606C2×C4C22×D1212yes—
607C2×C4C22×D1212yes—
608C2×C4#100412yes—
609C2×C4D3×#4112yes—
610C2×C3C2×#54611yescharacteristic
611C2×C3#348611yescharacteristic
612C2×C3#348711yescharacteristic
613C2×C3#351512yes—
614C2×C3#355412yes—
615D3—44no—
616D3—82no—
617C22—64no—
618C22—68no—
619C22—124no—
620C22—122no—
621C22—122no—
622C22—122no—
623C22—122no—
624C22—122no—
625C22—122no—
626C22—122no—
627C22—122no—
628C22—122no—
629C22C23×D1211yescharacteristic
630C22C2×#100411yescharacteristic
631C22C2×D3×#4111yescharacteristic
632C22D3×2−1+412yes—
633C22#596812yes—
634C4—28no—
635C4—22no—
636C4—22no—
637C4—42no—
638C4—44no—
639C4—42no—
640C4—64no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4—122no—
645C4—122no—
646C4#555414yes—
647C3#4533511yescharacteristic, complemented, Sylow
648C2—124no—
649C2—242no—
650C2C2×#555411yescharacteristic
651C2#8587411yescharacteristic
652C2#8588111yescharacteristic
653C2#8635112yes—
654C2#8651112yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2327C2
prime radicalC2×#5726C2
hypercenter#2327D3
hypocenterC3#45335
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2327

Complements

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

NH ≅ G/Ntype
2. C2×#5721C2semidirect
3. C3×#2327C2semidirect
4. #82496C2semidirect
5. #82457C2semidirect
6. #76623C2semidirect
7. #76498C2semidirect
8. #75325C2semidirect
9. #75155C2semidirect
10. #75151C2semidirect
11. #75009C2semidirect
12. #74240C2semidirect
13. #74190C2semidirect
14. #73419C2semidirect
15. #73387C2semidirect
91. #2327D3semidirect
647. C3#45335semidirect

Maximal subgroups

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

subgroupindex
C2×#57262
C2×#57212
C3×#23272
#824962
#824572
#766232
#764982
#753252
#751552
#751512
#750092
#742402
#741902
#734192
#733872
#453353