Finite Group Databasebeta

← #11233689

Subgroups — #11233689

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×C3×D4×Q8C211yesmaximal, characteristic, complemented
2C2×#5730C211yesmaximal, characteristic, complemented
3#84068C214yesmaximal, complemented
4#83241C212yesmaximal, complemented
5#83236C212yesmaximal, complemented
6#83228C211yesmaximal, characteristic, complemented
7#82817C214yesmaximal, complemented
8#82807C214yesmaximal, complemented
9#82797C214yesmaximal, complemented
10#82501C211yesmaximal, characteristic, complemented
11#82467C211yesmaximal, characteristic, complemented
12#82388C211yesmaximal, characteristic, complemented
13#82363C212yesmaximal, complemented
14#82343C212yesmaximal
15#82329C211yesmaximal, characteristic
16#45357—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×#940C2211yescharacteristic
29C2×C4×#67C2214yes—
30#5730—28no—
31C2×#897C2212yes—
32C2×#893C2212yes—
33C2×D3×C4⋊C4C2212yes—
34C2×#890C2212yes—
35C2×C4×D12C2211yescharacteristic
36C2×C4×Dic6C2211yescharacteristic
37#5231C2214yes—
38#5231C2212yes—
39#5230C2212yes—
40#5230C2214yes—
41#5229C2214yes—
42#5229C2214yes—
43#5221C2211yescharacteristic
44#5221C2212yes—
45#5213C2214yes—
46#5213C2214yes—
47#5212C2214yes—
48#5210C2214yes—
49#5210C2214yes—
50#4994C2214yes—
51#4993C2212yes—
52#4989C2214yes—
53#4989C2214yes—
54#4988C2211yescharacteristic
55#4984C2212yes—
56#4983C2212yes—
57#4981C2212yes—
58#4981C2214yes—
59#4980C2214yes—
60#4979C2212yes—
61#4977C2211yescharacteristic
62#4976C2214yes—
63Dic3×C4⋊C4C2212yes—
64#4972C2212yes—
65#4957C2214yes—
66#4956C2214yes—
67#4956C2214yes—
68#4949C2214yes—
69#4949C2212yes—
70#4948C2214yes—
71#4947C2212yes—
72#4939C2212yes—
73#4936C2211yescharacteristic
74#4932C2211yescharacteristic
75C4×Dic3⋊C4C2212yes—
76C4×Dic3⋊C4C2212yes—
77#4928C2212yes—
78C2×#554—31no—
79C2×D4×Q8D311yescharacteristic, complemented
80#2732—34no—
81#2692—34no—
82#2580—31no—
83#2580—31no—
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×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⋊C4C2314yes—
115C2×C3×C4⋊C4C2312yes—
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#940Q818yescomplemented
129C4×#67—48no—
130C2×(C2×D3)⋊C4C2314yes—
131C2×(C2×D3)⋊C4C2314yes—
132C2×(C2×D3)⋊C4C2312yes—
133C2×(C2×D3)⋊C4C2314yes—
134C2×(C2×D3)⋊C4C2314yes—
135C2×C4⋊Dic3C2311yescharacteristic
136C2×C4⋊Dic3C2312yes—
137C2×C4⋊Dic3C2311yescharacteristic
138C2×C4⋊Dic3C2314yes—
139C2×C4⋊Dic3C2314yes—
140C2×Dic3⋊C4C2312yes—
141C2×Dic3⋊C4C2312yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2312yes—
144C2×Dic3⋊C4C2314yes—
145C2×Dic3⋊C4C2312yes—
146C2×Dic3⋊C4C2312yes—
147C2×Dic3⋊C4C2314yes—
148C2×C4×Dic3C2311yescharacteristic
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2314yes—
151C2×C4×Dic3C2311yescharacteristic
152#897—28no—
153#893—28no—
154D3×C4⋊C4—28no—
155#890—28no—
156C4×D12—24no—
157C4×Dic6—24no—
158C3⋊#95C2314yes—
159C3⋊#95C2312yes—
160C3⋊#95C2312yes—
161C3⋊#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2312yes—
164C3⋊#95C2312yes—
165C3⋊#95C2312yes—
166C3⋊#95C2312yes—
167C3⋊#95C2314yes—
168C3⋊#95C2314yes—
169C3⋊#95C2314yes—
170C3⋊#95C2314yes—
171C23×Q8C2×D312yes—
172#554—68no—
173D4×Q8—28no—
174C2×C4⋊Q8C2×D312yes—
175C2×C4⋊Q8C2×D311yescharacteristic
176C2×#127—31no—
177C2×#125—32no—
178C2×C22⋊Q8—34no—
179C2×C22⋊Q8—34no—
180C2×C22⋊Q8—32no—
181C2×C22⋊Q8C2×D314yes—
182C2×C22⋊Q8C2×D312yes—
183C2×C4×Q8—31no—
184C2×C4×Q8C2×D311yescharacteristic
185C2×C4×D4—34no—
186C2×C4×D4—32no—
187C2×C4×D4—31no—
188C2×C4×D4C2×D311yescharacteristic
189C2×C4×D4C2×D312yes—
190C22×C4⋊C4—32no—
191#399—32no—
192#399—32no—
193#399—34no—
194#398—32no—
195#398—32no—
196#398—34no—
197#396—34no—
198#396—34no—
199#395—32no—
200#395—34no—
201#395—32no—
202#393—32no—
203#393—34no—
204#393—32no—
205#391—34no—
206#391—34no—
207#390—32no—
208#390—31no—
209#390—31no—
210#390—34no—
211#390—32no—
212#389—32no—
213#389—32no—
214#389—34no—
215#389—32no—
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#385—34no—
225#382—31no—
226#382—31no—
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—28no—
236C2×C3×Q8—24no—
237C2×C3×Q8—28no—
238C2×C3×Q8—44no—
239C2×C3×Q8—44no—
240C2×C3×Q8—432no—
241C2×C3×Q8C2×D418yes—
242C2×C3×D4—22no—
243C2×C3×D4—22no—
244C2×C3×D4C2×Q814yes—
245C22×C3×C4—28no—
246C22×C3×C4—24no—
247C22×C3×C4—28no—
248C22×C3×C4—44no—
249C22×C3×C4—44no—
250C22×C3×C4C2412yes—
251C22×C3×C4C2412yes—
252C22×C3×C4C2411yescharacteristic
253C22×C3×C4C2411yescharacteristic
254C22×C3×C4C2414yes—
255C22×C3×C4C2412yes—
256C22×C3×C4C2411yescharacteristic
257C2×#67—44no—
258C2×#67—44no—
259C2×#67C2×Q8116yescomplemented
260C22×Dic3C2414yes—
261C22×Dic3C2412yes—
262C22×Dic3C2412yes—
263C22×Dic3C2412yes—
264C22×Dic3C2412yes—
265C22×Dic3C2412yes—
266C2×D12—22no—
267C2×D12—22no—
268C2×D12C2×Q814yescomplemented
269C2×C4×D3—24no—
270C2×C4×D3—28no—
271C2×C4×D3—44no—
272C2×C4×D3—44no—
273C2×C4×D3—44no—
274C2×Dic6—22no—
275C2×Dic6—42no—
276C3×C4⋊C4—22no—
277C3×C4⋊C4—24no—
278C3×C4⋊C4—22no—
279C3×C4⋊C4—28no—
280C3×C4⋊C4C2×D418yes—
281C3×C4⋊C4C2×D418yes—
282C3×C22⋊C4—28no—
283C3×C22⋊C4—24no—
284C3×C42—24no—
285C3×C42C2×D414yes—
286#217—44no—
287#217—44no—
288#217—44no—
289(C2×D3)⋊C4—24no—
290(C2×D3)⋊C4—28no—
291(C2×D3)⋊C4—28no—
292(C2×D3)⋊C4—44no—
293(C2×D3)⋊C4—44no—
294C4⋊Dic3—22no—
295C4⋊Dic3—22no—
296C4⋊Dic3—28no—
297C4⋊Dic3—24no—
298C4⋊Dic3—44no—
299Dic3⋊C4—24no—
300Dic3⋊C4—28no—
301Dic3⋊C4—24no—
302Dic3⋊C4—24no—
303Dic3⋊C4—42no—
304Dic3⋊C4—42no—
305Dic3⋊C4—44no—
306Dic3⋊C4—44no—
307C4×Dic3—24no—
308C4×Dic3—22no—
309C4×Dic3—44no—
310C4×Dic3C2×Q814yescomplemented
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—34no—
321C23×C4—32no—
322C23×C4C22×D314yes—
323C23×C4C22×D312yes—
324C4⋊Q8—28no—
325C4⋊Q8#6718yescomplemented
326#127—38no—
327#125—68no—
328C22⋊Q8—28no—
329C22⋊Q8—216no—
330C22⋊Q8—616no—
331C22⋊Q8—68no—
332C22⋊Q8—128no—
333C4×Q8—24no—
334C4×Q8—64no—
335C4×D4—28no—
336C4×D4—24no—
337C4×D4—68no—
338C4×D4—64no—
339C4×D4—128no—
340C2×C4⋊C4—31no—
341C2×C4⋊C4—34no—
342C2×C4⋊C4—32no—
343C2×C4⋊C4—31no—
344C2×C4⋊C4—32no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—32no—
347C2×C4⋊C4—32no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—32no—
351C2×C4⋊C4—32no—
352C2×C4⋊C4—34no—
353C2×C4⋊C4—68no—
354C2×C4⋊C4C22×D311yescharacteristic
355C2×C4⋊C4C22×D312yes—
356C2×C4⋊C4C22×D311yescharacteristic
357C2×C4⋊C4C22×D314yes—
358C2×C4⋊C4C22×D312yes—
359C2×C4⋊C4C22×D312yes—
360C2×C22⋊C4—32no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—34no—
363C2×C22⋊C4—34no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4—34no—
367C2×C22⋊C4—34no—
368C2×C22⋊C4C22×D312yes—
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—32no—
385#95—32no—
386#95—34no—
387#95—34no—
388#95—34no—
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—28no—
401C3×Q8—48no—
402C3×Q8—416no—
403C3×D4—28no—
404C2×C3×C4—22no—
405C2×C3×C4—24no—
406C2×C3×C4—24no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—22no—
412C2×C3×C4—24no—
413C2×C3×C4—22no—
414C2×C3×C4—22no—
415C2×C3×C4—416no—
416C2×C3×C4—48no—
417C2×C3×C4C22×D412yes—
418C2×C3×C4C22×D418yes—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C22×Q812yes—
422C2×C3×C4H(5,𝔽2)14yes—
423C2×C3×C42−1+412yes—
424#67—416no—
425C2×Dic3—22no—
426C2×Dic3—22no—
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—24no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—22no—
435C2×Dic3—24no—
436C2×Dic3—22no—
437C2×Dic3—24no—
438C2×Dic3—22no—
439C2×Dic3—24no—
440C2×Dic3—22no—
441C2×Dic3C22×Q814yes—
442D12—28no—
443C4×D3—48no—
444C4×D3—88no—
445Dic6—44no—
446C24—32no—
447C24C23×D312yes—
448C2×Q8—24no—
449C2×Q8—28no—
450C2×Q8—22no—
451C2×Q8—28no—
452C2×Q8—44no—
453C2×Q8—44no—
454C2×Q8—432no—
455C2×Q8—62no—
456C2×Q8—122no—
457C2×Q8C2×#6718yes—
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—28no—
468C22×C4—28no—
469C22×C4—24no—
470C22×C4—32no—
471C22×C4—34no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—32no—
475C22×C4—32no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—64no—
479C22×C4—68no—
480C22×C4—124no—
481C22×C4—124no—
482C22×C4—124no—
483C22×C4C23×D312yes—
484C22×C4C23×D312yes—
485C22×C4C23×D312yes—
486C22×C4C23×D311yescharacteristic
487C22×C4C23×D311yescharacteristic
488C22×C4C23×D314yes—
489C22×C4C23×D311yescharacteristic
490C4⋊C4—22no—
491C4⋊C4—24no—
492C4⋊C4—22no—
493C4⋊C4—28no—
494C4⋊C4—64no—
495C4⋊C4—64no—
496C4⋊C4—64no—
497C4⋊C4—62no—
498C4⋊C4—68no—
499C4⋊C4—62no—
500C4⋊C4—64no—
501C4⋊C4—68no—
502C4⋊C4—124no—
503C4⋊C4—124no—
504C4⋊C4—122no—
505C4⋊C4—122no—
506C4⋊C4—124no—
507C4⋊C4C2×#6718yes—
508C4⋊C4C2×#6718yes—
509C22⋊C4—28no—
510C22⋊C4—24no—
511C22⋊C4—68no—
512C22⋊C4—68no—
513C22⋊C4—64no—
514C22⋊C4—124no—
515C22⋊C4—124no—
516C22⋊C4—124no—
517C22⋊C4—124no—
518C22⋊C4—124no—
519C42—24no—
520C42—34no—
521C42—62no—
522C42—64no—
523C42—124no—
524C42C2×#6714yes—
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—28no—
541C3×C4—22no—
542C3×C4—24no—
543C3×C4—22no—
544C3×C4—42no—
545C3×C4—42no—
546C3×C4D4×Q814yes—
547Dic3—24no—
548Dic3—42no—
549Dic3—42no—
550Dic3—42no—
551Dic3—44no—
552Dic3—42no—
553C23—24no—
554C23—22no—
555C23—24no—
556C23—22no—
557C23—34no—
558C23—64no—
559C23—62no—
560C23—64no—
561C23—62no—
562C23C2×D3×Q814yes—
563C23C24×D311yescharacteristic
564Q8—28no—
565Q8—48no—
566Q8—416no—
567Q8—124no—
568D4—28no—
569D4—68no—
570D4—1216no—
571C2×C4—24no—
572C2×C4—24no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—24no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—34no—
583C2×C4—416no—
584C2×C4—48no—
585C2×C4—62no—
586C2×C4—62no—
587C2×C4—64no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—62no—
592C2×C4—64no—
593C2×C4—64no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—64no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—62no—
601C2×C4—128no—
602C2×C4—248no—
603C2×C4#100414yes—
604C2×C4C2×D3×Q812yes—
605C2×C4#100712yes—
606C2×C4C22×#6712yes—
607C2×C4C22×#6712yes—
608C2×C4C22×#6718yes—
609C2×C4C22×#6712yes—
610C2×C3—44no—
611C2×C3C2×D4×Q811yescharacteristic
612C2×C3#348411yescharacteristic
613C2×C3#348711yescharacteristic
614C2×C3#350112yes—
615C2×C3#355412yes—
616D3—44no—
617C22—24no—
618C22—28no—
619C22—42no—
620C22—44no—
621C22—42no—
622C22—64no—
623C22—68no—
624C22—122no—
625C22—124no—
626C22—122no—
627C22C2×#100412yes—
628C22C22×D3×Q811yescharacteristic
629C22C2×#100711yescharacteristic
630C22#596812yes—
631C22C23×#6711yescharacteristic
632C4—24no—
633C4—22no—
634C4—28no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—64no—
639C4—122no—
640C4—124no—
641C4—122no—
642C4—122no—
643C4—122no—
644C4Q8×#6714yes—
645C3#4535711yescharacteristic, complemented, Sylow
646C2—44no—
647C2—124no—
648C2#8676112yes—
649C2#8676512yes—
650C2#8723911yescharacteristic
651C2C2×Q8×#6711yescharacteristic
652C2#8729011yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×D4×Q8C2
prime radicalC2×#940C22
hypercenterC2×D4×Q8D3
hypocenterC3#45357
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. #83241C2semidirect
5. #83236C2semidirect
6. #83228C2semidirect
7. #82817C2semidirect
8. #82807C2semidirect
9. #82797C2semidirect
10. #82501C2semidirect
11. #82467C2semidirect
12. #82388C2semidirect
13. #82363C2semidirect
79. C2×D4×Q8D3semidirect
107. C3×C4⋊Q8D4semidirect
128. #940Q8semidirect
259. C2×#67C2×Q8semidirect
268. C2×D12C2×Q8semidirect
310. C4×Dic3C2×Q8semidirect
325. C4⋊Q8#67semidirect
645. C3#45357semidirect

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
#832412
#832362
#832282
#828172
#828072
#827972
#825012
#824672
#823882
#823632
#823432
#823292
#453573