Finite Group Databasebeta

← #11233300

Subgroups — #11233300

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×#5567C211yesmaximal, characteristic, complemented
2C2×#5551C211yesmaximal, characteristic
3C3×#2536C211yesmaximal, characteristic, complemented
4#82759C212yesmaximal, complemented
5#82757C212yesmaximal, complemented
6#82455C211yesmaximal, characteristic, complemented
7#82348C211yesmaximal, characteristic, complemented
8#77756C214yesmaximal, complemented
9#77034C214yesmaximal, complemented
10#75674C214yesmaximal, complemented
11#75506C214yesmaximal, complemented
12#74190C211yesmaximal, characteristic, complemented
13#74152C211yesmaximal, characteristic, complemented
14#73894C212yesmaximal, complemented
15#73877C212yesmaximal, complemented
16#45335—31nomaximal, Sylow
17C22×#235C2212yes—
18C2×C3×C22⋊Q8C2212yes—
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×D4C2211yescharacteristic
21C2×#947C2211yescharacteristic
22C2×#944C2212yes—
23C2×#938C2212yes—
24C2×#924C2212yes—
25#5567—28no—
26#5551—28no—
27C2×D3×C4⋊C4C2212yes—
28C2×#889C2214yes—
29C2×#887C2212yes—
30C2×#887C2212yes—
31C2×#883C2214yes—
32C2×C4×D12C2211yescharacteristic
33C2×#870C2212yes—
34C2×#869C2211yescharacteristic
35C2×C4×Dic6C2211yescharacteristic
36C2×C4×Dic6C2212yes—
37C3×#398C2212yes—
38C3×#394C2212yes—
39C3×#388C2214yes—
40C3×#382C2211yescharacteristic
41C3×#380C2212yes—
42#5229C2211yescharacteristic
43#5222C2212yes—
44C4×#217C2212yes—
45C4×#217C2212yes—
46#4989C2214yes—
47#4989C2214yes—
48#4989C2214yes—
49#4988C2211yescharacteristic
50#4988C2212yes—
51#4985C2212yes—
52#4984C2212yes—
53#4983C2214yes—
54#4975C2211yescharacteristic
55#4973C2212yes—
56#4972C2214yes—
57#4959C2214yes—
58#4959C2214yes—
59#4957C2214yes—
60#4957C2214yes—
61#4953C2212yes—
62#4949C2214yes—
63#4948C2214yes—
64#4947C2212yes—
65#4945C2214yes—
66#4944C2214yes—
67#4940C2214yes—
68#4932C2211yescharacteristic
69#4677C2212yes—
70#4673C2212yes—
71#4668C2214yes—
72#4667C2214yes—
73#4666C2214yes—
74#4657C2214yes—
75#4654C2214yes—
76#4654C2214yes—
77#4652C2214yes—
78C2×#557—31no—
79C2×#547—31no—
80#2711—34no—
81#2694—34no—
82#2580—31no—
83#2536D311yescharacteristic, complemented
84#2473—34no—
85#2458—34no—
86#2432—32no—
87#2429—32no—
88#2423—32no—
89#2398—31no—
90#2341—32no—
91#2327—31no—
92#2326—31no—
93C22×C3×Q8C2311yescharacteristic
94C22×C3×D4C2311yescharacteristic
95C23×C3×C4C2312yes—
96C22×#67C2314yes—
97C2×#235—28no—
98C2×#235—28no—
99C2×#235—44no—
100C22×D12C2311yescharacteristic
101C22×C4×D3C2314yes—
102C22×C4×D3C2312yes—
103C22×Dic6C2311yescharacteristic
104C22×Dic6C2312yes—
105C3×C22⋊Q8—44no—
106C3×C4×Q8—24no—
107C3×C4×D4—24no—
108C2×C3×C4⋊C4C2312yes—
109C2×C3×C4⋊C4C2311yescharacteristic
110C2×C3×C4⋊C4C2311yescharacteristic
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2314yes—
113C2×C3×C22⋊C4C2312yes—
114C2×C3×C22⋊C4C2314yes—
115C2×C3×C22⋊C4C2314yes—
116C2×C3×C42C2312yes—
117C2×C3×C42C2311yescharacteristic
118C2×#217C2312yes—
119C2×#217C2314yes—
120C2×#217C2312yes—
121C2×#217C2312yes—
122C2×#217C2312yes—
123#947—24no—
124#944—28no—
125#938—28no—
126C2×(C2×D3)⋊C4C2314yes—
127C2×(C2×D3)⋊C4C2312yes—
128C2×(C2×D3)⋊C4C2314yes—
129C2×(C2×D3)⋊C4C2314yes—
130C2×(C2×D3)⋊C4C2314yes—
131C2×C4⋊Dic3C2311yescharacteristic
132C2×C4⋊Dic3C2311yescharacteristic
133C2×C4⋊Dic3C2314yes—
134C2×C4⋊Dic3C2312yes—
135C2×C4⋊Dic3C2312yes—
136#924—28no—
137C2×Dic3⋊C4C2314yes—
138C2×Dic3⋊C4C2314yes—
139C2×Dic3⋊C4C2312yes—
140C2×Dic3⋊C4C2314yes—
141C2×Dic3⋊C4C2314yes—
142C2×Dic3⋊C4C2314yes—
143C2×Dic3⋊C4C2314yes—
144C2×C4×Dic3C2314yes—
145C2×C4×Dic3C2311yescharacteristic
146C2×C4×Dic3C2311yescharacteristic
147C2×C4×Dic3C2312yes—
148C2×C4×Dic3C2314yes—
149D3×C4⋊C4—28no—
150#889—216no—
151#887—28no—
152#887—44no—
153#883—216no—
154C4×D12—24no—
155#870—28no—
156#869D418yes—
157C4×Dic6—24no—
158C4×Dic6—28no—
159C3×#95C2314yes—
160C3×#95C2312yes—
161C3×#95C2312yes—
162C3⋊#95C2312yes—
163C3⋊#95C2314yes—
164C3⋊#95C2314yes—
165C3⋊#95C2314yes—
166C3⋊#95C2312yes—
167C3⋊#95C2312yes—
168C3⋊#95C2312yes—
169C3⋊#95C2312yes—
170C3⋊#95C2312yes—
171C3⋊#95C2314yes—
172C22×#41—32no—
173#557—68no—
174#547—68no—
175C2×C4⋊Q8—32no—
176C2×C4⋊Q8—31no—
177C2×#125—34no—
178C2×#125—32no—
179C2×#124—31no—
180C2×#123—34no—
181C2×C22⋊Q8—32no—
182C2×C22⋊Q8C2×D312yes—
183C2×#121—32no—
184C2×C4×Q8—32no—
185C2×C4×Q8—32no—
186C2×C4×Q8—31no—
187C2×C4×Q8—32no—
188C2×C4×Q8C2×D311yescharacteristic
189C2×C4×D4—31no—
190C2×C4×D4C2×D311yescharacteristic
191C22×C4⋊C4—32no—
192#400—32no—
193#399—34no—
194#399—32no—
195#398—34no—
196#398—34no—
197#398C2×D312yes—
198#397—32no—
199#396—34no—
200#394—32no—
201#394—34no—
202#394—34no—
203#394C2×D312yes—
204#393—32no—
205#393—32no—
206#391—31no—
207#391—32no—
208#390—32no—
209#390—31no—
210#390—34no—
211#390—34no—
212#389—32no—
213#389—34no—
214#388—34no—
215#388—34no—
216#388—34no—
217#388C2×D314yes—
218#387—34no—
219#387—34no—
220#387—34no—
221#385—34no—
222#385—34no—
223#385—34no—
224#385—34no—
225#384—31no—
226#382C2×D311yescharacteristic
227#380—34no—
228#380C2×D312yes—
229C4×C22⋊C4—32no—
230C4×C22⋊C4—32no—
231C4×C22⋊C4—34no—
232#376—31no—
233C24×C3C2412yes—
234C23×D3C2412yes—
235C2×C3×Q8—22no—
236C2×C3×Q8—22no—
237C2×C3×Q8—22no—
238C2×C3×D4—22no—
239C2×C3×D4—42no—
240C22×C3×C4—24no—
241C22×C3×C4—24no—
242C22×C3×C4—24no—
243C22×C3×C4C2412yes—
244C22×C3×C4C2414yes—
245C22×C3×C4C2412yes—
246C22×C3×C4C2412yes—
247C22×C3×C4C2411yescharacteristic
248C22×C3×C4C2411yescharacteristic
249C22×C3×C4C2411yescharacteristic
250C2×#67—28no—
251C2×#67—28no—
252C2×#67—44no—
253C22×Dic3C2412yes—
254C22×Dic3C2412yes—
255C22×Dic3C2412yes—
256C22×Dic3C2412yes—
257C22×Dic3C2414yes—
258C22×Dic3C2412yes—
259#235—432no—
260C2×D12—22no—
261C2×D12—22no—
262C2×D12#4114yes—
263C2×C4×D3—28no—
264C2×C4×D3—24no—
265C2×C4×D3—28no—
266C2×C4×D3—44no—
267C2×C4×D3—44no—
268C2×Dic6—22no—
269C2×Dic6—24no—
270C2×Dic6—24no—
271C2×Dic6—24no—
272C2×Dic6C2×D418yes—
273C3×C4⋊C4—22no—
274C3×C4⋊C4—24no—
275C3×C4⋊C4—22no—
276C3×C4⋊C4—24no—
277C3×C4⋊C4—44no—
278C3×C22⋊C4—24no—
279C3×C22⋊C4—44no—
280C3×C22⋊C4—44no—
281C3×C42—24no—
282C3×C42C2×D414yes—
283#217—28no—
284#217—24no—
285#217—24no—
286#217—42no—
287#217—42no—
288(C2×D3)⋊C4—28no—
289(C2×D3)⋊C4—24no—
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—28no—
303Dic3⋊C4—44no—
304Dic3⋊C4—44no—
305C4×Dic3—24no—
306C4×Dic3—22no—
307C4×Dic3—22no—
308C4×Dic3—28no—
309C4×Dic3—44no—
310C2×#41—68no—
311C2×#41—68no—
312C2×#41—124no—
313C22×Q8—32no—
314C22×Q8—31no—
315C22×Q8C22×D311yescharacteristic
316C22×D4—34no—
317C22×D4—31no—
318C22×D4C22×D311yescharacteristic
319C23×C4—34no—
320C23×C4—32no—
321C23×C4C22×D312yes—
322C4⋊Q8—38no—
323C4⋊Q8—68no—
324#125—68no—
325#125—616no—
326#124—64no—
327#123—616no—
328C22⋊Q8—44no—
329C22⋊Q8—68no—
330#121—68no—
331C4×Q8—24no—
332C4×Q8—68no—
333C4×Q8—68no—
334C4×Q8—64no—
335C4×Q8—124no—
336C4×D4—24no—
337C4×D4—64no—
338C2×C4⋊C4—32no—
339C2×C4⋊C4—32no—
340C2×C4⋊C4—34no—
341C2×C4⋊C4—31no—
342C2×C4⋊C4—34no—
343C2×C4⋊C4—32no—
344C2×C4⋊C4—31no—
345C2×C4⋊C4—34no—
346C2×C4⋊C4—34no—
347C2×C4⋊C4—34no—
348C2×C4⋊C4—34no—
349C2×C4⋊C4—34no—
350C2×C4⋊C4—68no—
351C2×C4⋊C4C22×D311yescharacteristic
352C2×C4⋊C4C22×D311yescharacteristic
353C2×C4⋊C4C22×D312yes—
354C2×C4⋊C4C22×D312yes—
355C2×C4⋊C4C22×D314yes—
356C2×C22⋊C4—34no—
357C2×C22⋊C4—32no—
358C2×C22⋊C4—32no—
359C2×C22⋊C4—32no—
360C2×C22⋊C4—34no—
361C2×C22⋊C4—34no—
362C2×C22⋊C4—32no—
363C2×C22⋊C4—32no—
364C2×C22⋊C4—34no—
365C2×C22⋊C4—34no—
366C2×C22⋊C4C22×D312yes—
367C2×C22⋊C4C22×D314yes—
368C2×C22⋊C4C22×D314yes—
369C2×C42—34no—
370C2×C42—32no—
371C2×C42—31no—
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—32no—
381#95—32no—
382#95—34no—
383#95—34no—
384#95—34no—
385#95—34no—
386#95C22×D314yes—
387#95C22×D312yes—
388#95C22×D312yes—
389C23×C3—22no—
390C23×C3—22no—
391C23×C3—22no—
392C23×C3—22no—
393C23×C3—22no—
394C23×C3—22no—
395C23×C3—22no—
396C23×C3C2511yescharacteristic
397C22×D3—22no—
398C22×D3—22no—
399C22×D3—24no—
400C22×D3—24no—
401C22×D3C2×#4114yes—
402C3×Q8—44no—
403C3×D4—44no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—22no—
407C2×C3×C4—22no—
408C2×C3×C4—22no—
409C2×C3×C4—22no—
410C2×C3×C4—22no—
411C2×C3×C4—24no—
412C2×C3×C4—22no—
413C2×C3×C4—22no—
414C2×C3×C4—24no—
415C2×C3×C4—24no—
416C2×C3×C4—24no—
417C2×C3×C4—48no—
418C2×C3×C4C22×D412yes—
419C2×C3×C4C22×D412yes—
420C2×C3×C4C22×D412yes—
421C2×C3×C4C2×#4112yes—
422C2×C3×C4H(5,𝔽2)12yes—
423#67—416no—
424C2×Dic3—22no—
425C2×Dic3—24no—
426C2×Dic3—24no—
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—22no—
437C2×Dic3—22no—
438C2×Dic3—22no—
439C2×Dic3C22×D418yes—
440C2×Dic32−1+414yes—
441D12—28no—
442C4×D3—416no—
443C4×D3—48no—
444Dic6—28no—
445Dic6—216no—
446C24—32no—
447C24C23×D312yes—
448#41—1232no—
449C2×Q8—22no—
450C2×Q8—22no—
451C2×Q8—22no—
452C2×Q8—38no—
453C2×Q8—64no—
454C2×Q8—62no—
455C2×Q8—64no—
456C2×Q8—64no—
457C2×D4—22no—
458C2×D4—34no—
459C2×D4—42no—
460C2×D4—68no—
461C2×D4—68no—
462C2×D4—62no—
463C2×D4—62no—
464C2×D4—124no—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—24no—
468C22×C4—32no—
469C22×C4—34no—
470C22×C4—32no—
471C22×C4—32no—
472C22×C4—32no—
473C22×C4—32no—
474C22×C4—68no—
475C22×C4—68no—
476C22×C4—64no—
477C22×C4—124no—
478C22×C4—124no—
479C22×C4C23×D312yes—
480C22×C4C23×D314yes—
481C22×C4C23×D312yes—
482C22×C4C23×D312yes—
483C22×C4C23×D311yescharacteristic
484C22×C4C23×D311yescharacteristic
485C22×C4C23×D311yescharacteristic
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—24no—
489C4⋊C4—24no—
490C4⋊C4—38no—
491C4⋊C4—38no—
492C4⋊C4—44no—
493C4⋊C4—68no—
494C4⋊C4—68no—
495C4⋊C4—68no—
496C4⋊C4—68no—
497C4⋊C4—62no—
498C4⋊C4—64no—
499C4⋊C4—62no—
500C4⋊C4—124no—
501C4⋊C4—124no—
502C4⋊C4—124no—
503C22⋊C4—24no—
504C22⋊C4—44no—
505C22⋊C4—44no—
506C22⋊C4—68no—
507C22⋊C4—68no—
508C22⋊C4—68no—
509C22⋊C4—64no—
510C22⋊C4—64no—
511C22⋊C4—64no—
512C22⋊C4—124no—
513C22⋊C4—124no—
514C22⋊C4—122no—
515C22⋊C4—122no—
516C42—24no—
517C42—68no—
518C42—64no—
519C42—62no—
520C42—62no—
521C42—124no—
522C42D3×D414yes—
523C22×C3—42no—
524C22×C3—42no—
525C22×C3—42no—
526C22×C3—42no—
527C22×C3—42no—
528C22×C3—42no—
529C22×C3—42no—
530C22×C3C23×D411yescharacteristic
531C22×C3C22×#4111yescharacteristic
532C22×C3C2×H(5,𝔽2)11yescharacteristic
533C22×C3C2×2−1+412yes—
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×C4#54614yes—
547Dic3—28no—
548Dic3—24no—
549Dic3—42no—
550Dic3—42no—
551Dic3—42no—
552Dic3—42no—
553C23—22no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—22no—
560C23—34no—
561C23—62no—
562C23—62no—
563C23—64no—
564C23—64no—
565C23C24×D311yescharacteristic
566Q8—44no—
567Q8—68no—
568Q8—616no—
569D4—44no—
570D4—68no—
571D4—1216no—
572C2×C4—22no—
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—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—38no—
586C2×C4—34no—
587C2×C4—48no—
588C2×C4—62no—
589C2×C4—64no—
590C2×C4—64no—
591C2×C4—62no—
592C2×C4—62no—
593C2×C4—62no—
594C2×C4—62no—
595C2×C4—62no—
596C2×C4—62no—
597C2×C4—62no—
598C2×C4—62no—
599C2×C4—62no—
600C2×C4—64no—
601C2×C4—62no—
602C2×C4—62no—
603C2×C4—128no—
604C2×C4—1216no—
605C2×C4C2×D3×D412yes—
606C2×C4C2×D3×D412yes—
607C2×C4C2×D3×D412yes—
608C2×C4#100412yes—
609C2×C4D3×#4112yes—
610C2×C3—82no—
611C2×C3C2×#54611yescharacteristic
612C2×C3#348611yescharacteristic
613C2×C3#348711yescharacteristic
614C2×C3#351512yes—
615C2×C3#355412yes—
616D3—44no—
617C22—42no—
618C22—42no—
619C22—42no—
620C22—42no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—68no—
625C22—64no—
626C22—122no—
627C22—122no—
628C22—124no—
629C22C22×D3×D411yescharacteristic
630C22C2×#100411yescharacteristic
631C22C2×D3×#4111yescharacteristic
632C22#596412yes—
633C22D3×2−1+412yes—
634C4—22no—
635C4—22no—
636C4—42no—
637C4—42no—
638C4—42no—
639C4—44no—
640C4—64no—
641C4—68no—
642C4—122no—
643C4—122no—
644C4—122no—
645C4—122no—
646C4#561214yes—
647C3#4533511yescharacteristic, complemented, Sylow
648C2—82no—
649C2—124no—
650C2C2×#561211yescharacteristic
651C2#8602311yescharacteristic
652C2#8635511yescharacteristic
653C2#8672912yes—
654C2#8683712yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC3×#2536C2
prime radicalC2×#5551C2
hypercenter#2536D3
hypocenterC3#45335
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2536

Complements

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

NH ≅ G/Ntype
1. C2×#5567C2semidirect
3. C3×#2536C2semidirect
4. #82759C2semidirect
5. #82757C2semidirect
6. #82455C2semidirect
7. #82348C2semidirect
8. #77756C2semidirect
9. #77034C2semidirect
10. #75674C2semidirect
11. #75506C2semidirect
12. #74190C2semidirect
13. #74152C2semidirect
14. #73894C2semidirect
15. #73877C2semidirect
83. #2536D3semidirect
647. C3#45335semidirect

Maximal subgroups

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

subgroupindex
C2×#55672
C2×#55512
C3×#25362
#827592
#827572
#824552
#823482
#777562
#770342
#756742
#755062
#741902
#741522
#738942
#738772
#453353