Finite Group Databasebeta

← #10973003

Subgroups — #10973003

Order 768.

3398 conjugacy classes of subgroups.

The subgroup lattice has 728 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5805C211yesmaximal, characteristic, complemented
2C2×#5805C211yesmaximal, characteristic, complemented
3C2×#5510C211yesmaximal, characteristic, complemented
4C2×C4×#235C211yesmaximal, characteristic, complemented
5C2×C3×#390C211yesmaximal, characteristic, complemented
6#82157C211yesmaximal, characteristic, complemented
7#82109C218yesmaximal, complemented
8#82070C211yesmaximal, characteristic, complemented
9#76476C212yesmaximal, complemented
10#76474C212yesmaximal, complemented
11#76457C212yesmaximal, complemented
12#76456C212yesmaximal, complemented
13#76444C214yesmaximal, complemented
14#76443C214yesmaximal, complemented
15#35256—31nomaximal, Sylow
16C23×C3×D4C2211yescharacteristic
17C22×#235C2211yescharacteristic
18C22×#235C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×C3×C22⋊C4C2214yes—
21C22×C3×C42C2211yescharacteristic
22C2×#953C2214yes—
23C2×#953C2214yes—
24#5805—28no—
25#5805—28no—
26C2×#940C2212yes—
27C2×#940C2212yes—
28C2×#938C2214yes—
29C2×#938C2212yes—
30C2×#938C2212yes—
31C2×#936C2212yes—
32C2×#936C2212yes—
33C2×#930C2214yes—
34C2×C4×#67C2214yes—
35C2×#926C2211yescharacteristic
36C2×#926C2211yescharacteristic
37#5510—28no—
38C2×#897C2212yescomplemented
39C2×#897C2212yescomplemented
40C2×#895C2212yescomplemented
41C2×#895C2212yescomplemented
42C2×#884C2218yescomplemented
43C4×#235—28no—
44C2×C4×D12C2212yescomplemented
45C2×#872C2212yescomplemented
46C2×C42×D3C2212yescomplemented
47C2×C4×Dic6C2212yescomplemented
48C3×#390C2214yes—
49C3×#390C2212yes—
50C3×#390C2212yes—
51#5213C2214yes—
52#5210C2214yes—
53#4977C2212yescomplemented
54#4975C2212yescomplemented
55#4972C2214yescomplemented
56#4960C2218yescomplemented
57#4959C2218yescomplemented
58#4959C2218yescomplemented
59#4957C2218yescomplemented
60#4956C2218yescomplemented
61#4951C2218yescomplemented
62#4948C2218yescomplemented
63#4939C2211yescharacteristic, complemented
64#4939C2211yescharacteristic, complemented
65C4×(C2×D3)⋊C4C2214yescomplemented
66#4929C2211yescharacteristic, complemented
67#4929C2211yescharacteristic, complemented
68C2×#537—31no—
69C2×#535—31no—
70C2×#535—31no—
71C2×C4×#41—31no—
72#2856—32no—
73#2850—32no—
74#2620—32no—
75#2619—32no—
76#2428—34no—
77#2426—34no—
78#2424—38no—
79#2317—31no—
80#2297—31no—
81C2×#390D311yescharacteristic, complemented
82C25×C3C2312yes—
83C22×C3×D4—24no—
84C22×C3×D4—24no—
85C22×C3×D4—24no—
86C22×C3×D4C2312yes—
87C22×C3×D4C2312yes—
88C23×C3×C4C2312yes—
89C23×C3×C4C2312yes—
90C23×C3×C4C2311yescharacteristic
91C22×#67C2312yes—
92C22×#67C2312yes—
93C22×#67C2314yes—
94C22×#67C2318yes—
95C22×#67C2314yes—
96C2×#235—24no—
97C2×#235—24no—
98C2×#235—24no—
99C2×#235—24no—
100C2×#235—24no—
101C2×#235D418yescomplemented
102C22×D12C2311yescharacteristic
103C22×D12C2311yescharacteristic
104C22×D12C2314yes—
105C22×C4×D3C2312yes—
106C22×C4×D3C2314yes—
107C22×C4×D3C2312yes—
108C22×Dic6C2311yescharacteristic
109C22×Dic6C2311yescharacteristic
110C2×C3×C4⋊C4—24no—
111C2×C3×C4⋊C4C2312yes—
112C2×C3×C4⋊C4C2312yes—
113C2×C3×C22⋊C4—48no—
114C2×C3×C22⋊C4C2318yes—
115C2×C3×C22⋊C4C2318yes—
116C2×C3×C42—24no—
117C2×C3×C42C2311yescharacteristic
118C2×C3×C42C2311yescharacteristic
119C2×C3×C42C2312yes—
120#953—216no—
121#953—216no—
122C2×#217C2314yes—
123C2×#217C2318yes—
124C2×#217C2314yes—
125C2×#217C2312yes—
126C2×#217C2312yes—
127#940—28no—
128#940—28no—
129#938—28no—
130#938—28no—
131#938—216no—
132#936—28no—
133#936—28no—
134#930—216no—
135C4×#67—216no—
136C2×(C2×D3)⋊C4C2314yes—
137C2×(C2×D3)⋊C4C2314yes—
138C2×(C2×D3)⋊C4C2314yes—
139C2×(C2×D3)⋊C4C2314yes—
140C2×(C2×D3)⋊C4C2314yes—
141#926—24no—
142#926D418yescomplemented
143C2×C4⋊Dic3C2311yescharacteristic
144C2×C4⋊Dic3C2311yescharacteristic
145C2×C4⋊Dic3C2314yes—
146C2×C4⋊Dic3C2312yes—
147C2×Dic3⋊C4C2314yes—
148C2×Dic3⋊C4C2318yes—
149C2×C4×Dic3C2312yes—
150C2×C4×Dic3C2314yes—
151C2×C4×Dic3C2312yes—
152#897—28no—
153#897—28no—
154#895—28no—
155#895—28no—
156#884—416no—
157C4×D12—28no—
158#872—28no—
159C42×D3—28no—
160C4×Dic6—28no—
161C3⋊#95C2314yes—
162C3⋊#95C2314yes—
163C3⋊#95C2318yes—
164C22×#41—31no—
165C22×#41—31no—
166C23×D4C2×D311yescharacteristic
167#537—68no—
168#535—68no—
169#535—68no—
170C2×#127—32no—
171C2×#127—32no—
172C2×#124—32no—
173C2×#124—32no—
174C2×C22⋊Q8—32no—
175C2×C22⋊Q8—32no—
176C2×#121—32no—
177C2×#121—32no—
178C2×#121—34no—
179C2×#121—34no—
180C2×#121—32no—
181C2×#121—32no—
182C2×#121—38no—
183C2×#120—34no—
184C2×#120—34no—
185C4×#41—68no—
186C2×C4×Q8—32no—
187C2×C4×D4—34no—
188C2×C4×D4—32no—
189C2×#117—31no—
190C2×#117—31no—
191C2×#117—32no—
192C22×C4⋊C4C2×D311yescharacteristic
193C22×C22⋊C4C2×D314yes—
194C22×C42—32no—
195C22×C42C2×D311yescharacteristic
196#396—38no—
197#394—38no—
198#394—38no—
199#392—38no—
200#391—31no—
201#391—31no—
202#390—38no—
203#390—31no—
204#390—31no—
205#390C2×D312yes—
206#390C2×D312yes—
207#390C2×D314yes—
208#389—34no—
209#388—38no—
210#386—34no—
211#386—38no—
212#385—34no—
213#384—32no—
214#382—32no—
215C4×C22⋊C4—34no—
216C24×C3—24no—
217C24×C3—24no—
218C24×C3—24no—
219C24×C3—44no—
220C24×C3—44no—
221C24×C3C2414yes—
222C24×C3C2411yescharacteristic
223C23×D3C2412yes—
224C23×D3C2412yes—
225C2×C3×D4—24no—
226C2×C3×D4—24no—
227C2×C3×D4—24no—
228C2×C3×D4—24no—
229C2×C3×D4—24no—
230C2×C3×D4—24no—
231C2×C3×D4—416no—
232C22×C3×C4—24no—
233C22×C3×C4—24no—
234C22×C3×C4—24no—
235C22×C3×C4—42no—
236C22×C3×C4—44no—
237C22×C3×C4C2×D414yes—
238C22×C3×C4C2×D414yes—
239C22×C3×C4#4114yes—
240C22×C3×C4C2414yes—
241C22×C3×C4C2412yes—
242C22×C3×C4C2412yes—
243C22×C3×C4C2411yescharacteristic
244C22×C3×C4C2411yescharacteristic
245C2×#67—24no—
246C2×#67—24no—
247C2×#67—24no—
248C2×#67—24no—
249C2×#67—24no—
250C2×#67—28no—
251C2×#67—28no—
252C2×#67—216no—
253C2×#67—48no—
254C2×#67—48no—
255C2×#67—48no—
256C2×#67—48no—
257C2×#67C2×D418yes—
258C22×Dic3C2412yes—
259C22×Dic3C2412yes—
260C22×Dic3C2412yes—
261C22×Dic3C2414yes—
262C22×Dic3C2412yes—
263#235—232no—
264#235—416no—
265C2×D12—22no—
266C2×D12—22no—
267C2×D12—22no—
268C2×D12—22no—
269C2×D12—22no—
270C2×D12—28no—
271C2×D12—48no—
272C2×D12C2×D414yes—
273C2×C4×D3—24no—
274C2×C4×D3—24no—
275C2×C4×D3—24no—
276C2×C4×D3—24no—
277C2×C4×D3—24no—
278C2×C4×D3—28no—
279C2×C4×D3—28no—
280C2×C4×D3—44no—
281C2×C4×D3C2×D418yes—
282C2×Dic6—22no—
283C2×Dic6—22no—
284C2×Dic6—22no—
285C2×Dic6—22no—
286C2×Dic6—22no—
287C2×Dic6C2×D414yes—
288C3×C4⋊C4—24no—
289C3×C4⋊C4—24no—
290C3×C22⋊C4—216no—
291C3×C22⋊C4—48no—
292C3×C42—24no—
293C3×C42—22no—
294C3×C42—22no—
295#217—24no—
296#217—28no—
297#217—28no—
298#217—216no—
299#217C2×D418yes—
300(C2×D3)⋊C4—28no—
301(C2×D3)⋊C4—28no—
302(C2×D3)⋊C4—28no—
303(C2×D3)⋊C4—28no—
304(C2×D3)⋊C4—44no—
305C4⋊Dic3—22no—
306C4⋊Dic3—22no—
307C4⋊Dic3—28no—
308C4⋊Dic3C2×D418yes—
309Dic3⋊C4—28no—
310Dic3⋊C4—48no—
311C4×Dic3—24no—
312C4×Dic3—28no—
313C4×Dic3C2×D418yes—
314C25C22×D312yes—
315C2×#41—38no—
316C2×#41—64no—
317C2×#41—64no—
318C2×#41—64no—
319C2×#41—64no—
320C2×#41—64no—
321C22×Q8—31no—
322C22×Q8—31no—
323C22×D4—24no—
324C22×D4—24no—
325C22×D4—24no—
326C22×D4—32no—
327C22×D4—32no—
328C22×D4—34no—
329C22×D4—38no—
330C22×D4—34no—
331C22×D4—31no—
332C22×D4—31no—
333C22×D4—34no—
334C22×D4C22×D312yes—
335C22×D4C22×D312yes—
336C23×C4—32no—
337C23×C4—32no—
338C23×C4—34no—
339C23×C4C22×D312yes—
340C23×C4C22×D312yes—
341C23×C4C22×D311yescharacteristic
342#127—68no—
343#127—68no—
344#124—68no—
345#124—68no—
346C22⋊Q8—68no—
347C22⋊Q8—68no—
348#121—68no—
349#121—68no—
350#121—616no—
351#121—616no—
352#121—68no—
353#121—68no—
354#121—1216no—
355#120—616no—
356#120—616no—
357C4×Q8—68no—
358C4×D4—68no—
359C4×D4—616no—
360#117—38no—
361#117—68no—
362#117—64no—
363C2×C4⋊C4—24no—
364C2×C4⋊C4—32no—
365C2×C4⋊C4—34no—
366C2×C4⋊C4—31no—
367C2×C4⋊C4—31no—
368C2×C4⋊C4—34no—
369C2×C4⋊C4—38no—
370C2×C4⋊C4C22×D312yes—
371C2×C4⋊C4C22×D312yes—
372C2×C22⋊C4—32no—
373C2×C22⋊C4—32no—
374C2×C22⋊C4—38no—
375C2×C22⋊C4—34no—
376C2×C22⋊C4—34no—
377C2×C22⋊C4—34no—
378C2×C22⋊C4—34no—
379C2×C22⋊C4—34no—
380C2×C22⋊C4—34no—
381C2×C22⋊C4—34no—
382C2×C22⋊C4—48no—
383C2×C22⋊C4C22×D318yes—
384C2×C22⋊C4C22×D318yes—
385C2×C42—24no—
386C2×C42—32no—
387C2×C42—32no—
388C2×C42—34no—
389C2×C42—68no—
390C2×C42C22×D311yescharacteristic
391C2×C42C22×D312yes—
392C2×C42C22×D311yescharacteristic
393#95—34no—
394#95—34no—
395#95—38no—
396C23×C3—22no—
397C23×C3—22no—
398C23×C3—24no—
399C23×C3—24no—
400C23×C3—24no—
401C23×C3—28no—
402C23×C3—28no—
403C23×C3—416no—
404C23×C3—48no—
405C23×C3—88no—
406C23×C3—84no—
407C23×C3—84no—
408C23×C3C22×D412yes—
409C23×C3C22×D412yes—
410C23×C3C2×#4112yes—
411C23×C3C2511yescharacteristic
412C22×D3—22no—
413C22×D3—22no—
414C22×D3—24no—
415C22×D3—24no—
416C22×D3—22no—
417C22×D3—22no—
418C22×D3—22no—
419C22×D3—24no—
420C22×D3—24no—
421C22×D3C22×D414yes—
422C3×D4—48no—
423C3×D4—48no—
424C2×C3×C4—22no—
425C2×C3×C4—22no—
426C2×C3×C4—22no—
427C2×C3×C4—24no—
428C2×C3×C4—28no—
429C2×C3×C4—22no—
430C2×C3×C4—22no—
431C2×C3×C4—48no—
432C2×C3×C4—84no—
433C2×C3×C4#121116yescomplemented
434C2×C3×C4C22×D412yes—
435C2×C3×C4C22×D412yes—
436C2×C3×C4C22×D412yes—
437C2×C3×C4C22×D412yes—
438C2×C3×C4C2×#4112yes—
439C2×C3×C4C2×#4112yes—
440C2×C3×C4H(5,𝔽2)14yes—
441#67—216no—
442#67—48no—
443#67—816no—
444#67—88no—
445#67—88no—
446C2×Dic3—22no—
447C2×Dic3—22no—
448C2×Dic3—24no—
449C2×Dic3—24no—
450C2×Dic3—22no—
451C2×Dic3—22no—
452C2×Dic3—22no—
453C2×Dic3—22no—
454C2×Dic3—22no—
455C2×Dic3—24no—
456C2×Dic3—24no—
457C2×Dic3C22×D418yes—
458C2×Dic3C22×D414yes—
459D12—28no—
460D12—44no—
461D12—88no—
462C4×D3—216no—
463C4×D3—416no—
464C4×D3—48no—
465Dic6—28no—
466Dic6—44no—
467C24—24no—
468C24—24no—
469C24—24no—
470C24—32no—
471C24—32no—
472C24—44no—
473C24—44no—
474C24C23×D314yes—
475C24C23×D311yescharacteristic
476#41—632no—
477#41—1216no—
478C2×Q8—34no—
479C2×Q8—62no—
480C2×Q8—62no—
481C2×Q8—62no—
482C2×Q8—62no—
483C2×Q8—62no—
484C2×D4—24no—
485C2×D4—24no—
486C2×D4—24no—
487C2×D4—24no—
488C2×D4—24no—
489C2×D4—24no—
490C2×D4—38no—
491C2×D4—34no—
492C2×D4—416no—
493C2×D4—68no—
494C2×D4—616no—
495C2×D4—64no—
496C2×D4—64no—
497C2×D4—64no—
498C2×D4—64no—
499C2×D4—64no—
500C2×D4—68no—
501C2×D4—68no—
502C2×D4—62no—
503C2×D4—62no—
504C2×D4—62no—
505C2×D4—62no—
506C2×D4—62no—
507C2×D4—128no—
508C2×D4—128no—
509C2×D4—128no—
510C2×D4—128no—
511C2×D4—128no—
512C22×C4—24no—
513C22×C4—24no—
514C22×C4—24no—
515C22×C4—34no—
516C22×C4—32no—
517C22×C4—38no—
518C22×C4—32no—
519C22×C4—32no—
520C22×C4—32no—
521C22×C4—42no—
522C22×C4—44no—
523C22×C4—68no—
524C22×C4—64no—
525C22×C4—64no—
526C22×C4—64no—
527C22×C4—64no—
528C22×C4—64no—
529C22×C4—68no—
530C22×C4—124no—
531C22×C4#23514yes—
532C22×C4D3×D414yes—
533C22×C4D3×D414yes—
534C22×C4C23×D312yes—
535C22×C4C23×D314yes—
536C22×C4C23×D312yes—
537C22×C4C23×D311yescharacteristic
538C22×C4C23×D311yescharacteristic
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—38no—
542C4⋊C4—68no—
543C4⋊C4—68no—
544C4⋊C4—62no—
545C4⋊C4—62no—
546C4⋊C4—128no—
547C22⋊C4—216no—
548C22⋊C4—38no—
549C22⋊C4—48no—
550C22⋊C4—68no—
551C22⋊C4—68no—
552C22⋊C4—68no—
553C22⋊C4—616no—
554C22⋊C4—68no—
555C22⋊C4—68no—
556C22⋊C4—64no—
557C22⋊C4—68no—
558C22⋊C4—124no—
559C42—22no—
560C42—22no—
561C42—24no—
562C42—38no—
563C42—68no—
564C42—64no—
565C22×C3—24no—
566C22×C3—24no—
567C22×C3—22no—
568C22×C3—22no—
569C22×C3—44no—
570C22×C3—48no—
571C22×C3—48no—
572C22×C3—44no—
573C22×C3—44no—
574C22×C3—816no—
575C22×C3C2×#12114yes—
576C22×C3C23×D411yescharacteristic
577C22×C3C23×D411yescharacteristic
578C22×C3C22×#4111yescharacteristic
579C22×C3C2×H(5,𝔽2)12yes—
580C22×C3#58512yes—
581C2×D3—24no—
582C2×D3—24no—
583C2×D3—24no—
584C2×D3—44no—
585C2×D3—44no—
586C2×D3—42no—
587C2×D3—44no—
588C2×D3—44no—
589C2×D3—42no—
590C2×D3—42no—
591C3×C4—24no—
592C3×C4—42no—
593C3×C4—44no—
594C3×C4C2×#12114yes—
595C3×C4C2×#12114yes—
596Dic3—24no—
597Dic3—28no—
598Dic3—42no—
599Dic3—42no—
600Dic3—42no—
601C23—22no—
602C23—22no—
603C23—24no—
604C23—24no—
605C23—24no—
606C23—28no—
607C23—28no—
608C23—34no—
609C23—416no—
610C23—48no—
611C23—62no—
612C23—62no—
613C23—64no—
614C23—64no—
615C23—62no—
616C23—62no—
617C23—62no—
618C23—64no—
619C23—64no—
620C23—84no—
621C23—84no—
622C23—88no—
623C23C2×#23512yes—
624C23C2×D3×D412yes—
625C23C2×D3×D412yes—
626C23C24×D311yescharacteristic
627Q8—68no—
628Q8—124no—
629D4—48no—
630D4—48no—
631D4—616no—
632D4—68no—
633D4—128no—
634D4—124no—
635D4—248no—
636D4—2416no—
637D4—248no—
638D4—248no—
639C2×C4—22no—
640C2×C4—22no—
641C2×C4—22no—
642C2×C4—24no—
643C2×C4—28no—
644C2×C4—22no—
645C2×C4—22no—
646C2×C4—38no—
647C2×C4—34no—
648C2×C4—48no—
649C2×C4—62no—
650C2×C4—62no—
651C2×C4—64no—
652C2×C4—64no—
653C2×C4—616no—
654C2×C4—62no—
655C2×C4—62no—
656C2×C4—62no—
657C2×C4—62no—
658C2×C4—62no—
659C2×C4—64no—
660C2×C4—64no—
661C2×C4—84no—
662C2×C4—1216no—
663C2×C4—128no—
664C2×C4#884116yescomplemented
665C2×C4C2×#23512yes—
666C2×C4C2×#23512yes—
667C2×C4C2×D3×D412yes—
668C2×C4C2×D3×D412yes—
669C2×C4C2×D3×D412yes—
670C2×C4C2×D3×D412yes—
671C2×C4#100414yes—
672C2×C3—24no—
673C2×C3—84no—
674C2×C3C22×#12111yescharacteristic
675C2×C3#346012yes—
676C2×C3#348812yes—
677C2×C3#349811yescharacteristic
678C2×C3#350011yescharacteristic
679D3—44no—
680D3—82no—
681C22—24no—
682C22—22no—
683C22—22no—
684C22—24no—
685C22—48no—
686C22—44no—
687C22—44no—
688C22—44no—
689C22—48no—
690C22—64no—
691C22—64no—
692C22—64no—
693C22—816no—
694C22—124no—
695C22—124no—
696C22—124no—
697C22—122no—
698C22—122no—
699C22—122no—
700C22—124no—
701C22C2×#88414yes—
702C22C22×#23511yescharacteristic
703C22C22×D3×D411yescharacteristic
704C22C22×D3×D411yescharacteristic
705C22C2×#100412yes—
706C22#596412yes—
707C4—24no—
708C4—44no—
709C4—42no—
710C4—64no—
711C4—68no—
712C4—122no—
713C4—122no—
714C4—122no—
715C4C2×#88414yes—
716C4C2×#88414yes—
717C3#3525611yescharacteristic, complemented, Sylow
718C2—24no—
719C2—84no—
720C2—124no—
721C2—242no—
722C2C22×#88411yescharacteristic
723C2#8577712yes—
724C2#8594912yes—
725C2#8607611yescharacteristic
726C2#8619511yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, cosocleC23×C3C25
FittingC2×C3×#390C2
hypercenterC2×#390D3
hypocenterC3#35256
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
1. C2×#5805C2semidirect
2. C2×#5805C2semidirect
3. C2×#5510C2semidirect
4. C2×C4×#235C2semidirect
5. C2×C3×#390C2semidirect
6. #82157C2semidirect
7. #82109C2semidirect
8. #82070C2semidirect
9. #76476C2semidirect
10. #76474C2semidirect
11. #76457C2semidirect
12. #76456C2semidirect
13. #76444C2semidirect
14. #76443C2semidirect
38. C2×#897C22semidirect
39. C2×#897C22semidirect
40. C2×#895C22semidirect
41. C2×#895C22semidirect
42. C2×#884C22semidirect
44. C2×C4×D12C22semidirect
45. C2×#872C22semidirect
46. C2×C42×D3C22semidirect
47. C2×C4×Dic6C22semidirect
53. #4977C22semidirect
54. #4975C22semidirect
55. #4972C22semidirect
56. #4960C22semidirect
57. #4959C22semidirect
58. #4959C22semidirect
59. #4957C22semidirect
60. #4956C22semidirect
61. #4951C22semidirect
62. #4948C22semidirect
63. #4939C22semidirect
64. #4939C22semidirect
65. C4×(C2×D3)⋊C4C22semidirect
66. #4929C22semidirect
67. #4929C22semidirect
81. C2×#390D3semidirect
101. C2×#235D4semidirect
142. #926D4semidirect
433. C2×C3×C4#121semidirect
664. C2×C4#884semidirect
717. C3#35256semidirect

Maximal subgroups

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

subgroupindex
C2×#58052
C2×#55102
C2×C4×#2352
C2×C3×#3902
#821572
#821092
#820702
#764762
#764742
#764572
#764562
#764442
#764432
#352563