Finite Group Databasebeta

← #10955481

Subgroups — #10955481

Order 768.

3118 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5524C211yesmaximal, characteristic
2C2×#5523C211yesmaximal, characteristic, complemented
3C22×#892C211yesmaximal, characteristic, complemented
4C3×#2317C211yesmaximal, characteristic, complemented
5#82267C211yesmaximal, characteristic, complemented
6#82223C211yesmaximal, characteristic, complemented
7#82186C211yesmaximal, characteristic, complemented
8#75985C214yesmaximal, complemented
9#75984C214yesmaximal, complemented
10#74592C214yesmaximal, complemented
11#74380C214yesmaximal, complemented
12#74324C214yesmaximal, complemented
13#74229C214yesmaximal, complemented
14#34201—31nomaximal, Sylow
15C22×#235C2211yescharacteristic
16C23×C4×D3C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C22×C3×C4⋊C4C2211yescharacteristic
20C22×(C2×D3)⋊C4C2212yes—
21C2×#926C2211yescharacteristic
22C22×C4⋊Dic3C2211yescharacteristic
23C22×C4×Dic3C2212yes—
24#5524C4116yescomplemented
25#5523—28no—
26C2×#897C2212yes—
27C2×#895C2212yes—
28C2×#892—28no—
29C2×#892C2214yes—
30C2×#892C2214yes—
31C2×#892C2212yescomplemented
32C2×D3×C4⋊C4C2212yescomplemented
33C2×#886C2212yes—
34C2×#882C2212yes—
35C2×#67⋊C4C2214yes—
36C2×C4×D12C2214yes—
37C2×C4×D12C2212yescomplemented
38C2×C4×Dic6C2212yescomplemented
39C3×#389C2214yescomplemented
40C3×#385C2214yes—
41C3×#384C2212yescomplemented
42C3×#382C2212yescomplemented
43#5213C2214yes—
44#4990C2218yes—
45#4990C2218yescomplemented
46#4988C2218yes—
47#4988C2212yescomplemented
48#4987C2212yes—
49#4987C2212yes—
50#4987C2214yescomplemented
51Dic3×C4⋊C4C2214yescomplemented
52Dic3×C4⋊C4C2218yescomplemented
53#4973C2212yescomplemented
54#4955C2212yes—
55#4943C2212yes—
56Dic3×C22⋊C4C2214yes—
57C4×(C2×D3)⋊C4C2214yes—
58C4×C4⋊Dic3C2214yescomplemented
59#4669C2218yes—
60#4666C2218yescomplemented
61#4665C2218yes—
62#4654C2218yescomplemented
63C2×#525—31no—
64C2×#520—31no—
65C22×#117—31no—
66#2363—34no—
67#2362—34no—
68#2354—34no—
69#2352—34no—
70#2321—34no—
71#2317D311yescharacteristic, complemented
72#2311—31no—
73C4×#123—34no—
74#2280—31no—
75#2274—31no—
76C24×D3C2311yescharacteristic
77C23×C3×C4C2312yes—
78C23×C3×C4C2311yescharacteristic
79C22×#67C2312yes—
80C23×Dic3C2311yescharacteristic
81C23×Dic3C2312yes—
82C2×#235—24no—
83C2×#235—24no—
84C2×#235C2×C418yes—
85C22×D12C2311yescharacteristic
86C22×D12C2314yes—
87C22×C4×D3—24no—
88C22×C4×D3—24no—
89C22×C4×D3C2×C418yescomplemented
90C22×C4×D3C2312yes—
91C22×C4×D3C2312yes—
92C22×C4×D3C2312yes—
93C22×C4×D3C2314yes—
94C22×Dic6C2311yescharacteristic
95C3×#117—24no—
96C3×#117C2×C418yescomplemented
97C2×C3×C4⋊C4—24no—
98C2×C3×C4⋊C4C2312yes—
99C2×C3×C4⋊C4C2311yescharacteristic
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2312yes—
102C2×C3×C4⋊C4C2318yes—
103C2×C3×C4⋊C4C2312yes—
104C2×C3×C22⋊C4C2312yes—
105C2×C3×C22⋊C4C2312yes—
106C2×C3×C42C2312yes—
107C2×C3×C42C2312yes—
108C2×#217C2312yes—
109C2×(C2×D3)⋊C4—28no—
110C2×(C2×D3)⋊C4C2318yes—
111C2×(C2×D3)⋊C4C2314yes—
112C2×(C2×D3)⋊C4C2314yes—
113C2×(C2×D3)⋊C4C2318yes—
114C2×(C2×D3)⋊C4C2314yes—
115C2×(C2×D3)⋊C4C2314yes—
116C2×(C2×D3)⋊C4C2314yes—
117#926—24no—
118C2×C4⋊Dic3D418yescomplemented
119C2×C4⋊Dic3C2312yes—
120C2×C4⋊Dic3C2314yes—
121C2×C4⋊Dic3C2312yes—
122C2×C4⋊Dic3C2312yes—
123C2×Dic3⋊C4C2314yes—
124C2×C4×Dic3—28no—
125C2×C4×Dic3C2318yes—
126C2×C4×Dic3C2312yes—
127C2×C4×Dic3C2314yes—
128C2×C4×Dic3C2318yes—
129#897C2×C4116yescomplemented
130#895C2×C4116yescomplemented
131#892—216no—
132#892—216no—
133#892—28no—
134D3×C4⋊C4—28no—
135#886C2×C4116yescomplemented
136#882C2×C4116yescomplemented
137#67⋊C4—216no—
138C4×D12—28no—
139C4×D12C2×C4132yescomplemented
140C4×Dic6—28no—
141C3×#95C2318yes—
142C3⋊#95C2314yes—
143C3⋊#95C2318yes—
144C3⋊#95C2318yes—
145C3⋊#95C2314yes—
146C22×#41—31no—
147C24×C4—31no—
148#525—316no—
149C2×#123—32no—
150C2×C22⋊Q8—32no—
151C2×#121—32no—
152C2×#120—32no—
153#520—68no—
154C2×C4×Q8—32no—
155C2×C4×D4—34no—
156C2×C4×D4—34no—
157C2×C4×D4—32no—
158C2×#117—31no—
159C2×#117—34no—
160C2×#117—32no—
161C2×#117—34no—
162C2×#117—68no—
163C2×#117C2×D311yescharacteristic
164C2×#117C2×D311yescharacteristic
165C22×C4⋊C4—31no—
166C22×C4⋊C4—32no—
167C22×C4⋊C4C2×D311yescharacteristic
168C22×C22⋊C4—32no—
169C22×C42—32no—
170#391—32no—
171#390—32no—
172#390—38no—
173#389C2×D314yescomplemented
174#388—38no—
175#388—38no—
176#388—38no—
177#387—38no—
178#386—34no—
179#386—38no—
180#385C2×D314yes—
181#384C2×D312yescomplemented
182#382C2×D312yescomplemented
183#381—32no—
184#380—34no—
185#380—32no—
186#380—32no—
187#379—32no—
188C4×C4⋊C4—38no—
189C4×C4⋊C4—34no—
190C4×C4⋊C4—34no—
191C4×C22⋊C4—34no—
192C4×C22⋊C4—34no—
193C4×C22⋊C4—38no—
194C24×C3C2411yescharacteristic
195C23×D3—22no—
196C23×D3—22no—
197C23×D3—42no—
198C23×D3—42no—
199C23×D3C22×C414yes—
200C23×D3C2412yes—
201C23×D3C2412yes—
202C22×C3×C4—24no—
203C22×C3×C4—44no—
204C22×C3×C4C22×C414yes—
205C22×C3×C4C2×D414yes—
206C22×C3×C4#4114yes—
207C22×C3×C4C2412yes—
208C22×C3×C4C2414yes—
209C22×C3×C4C2412yes—
210C22×C3×C4C2414yes—
211C22×C3×C4C2411yescharacteristic
212C22×C3×C4C2411yescharacteristic
213C2×#67—24no—
214C2×#67—24no—
215C2×#67C22×C418yes—
216C22×Dic3—22no—
217C22×Dic3—22no—
218C22×Dic3—44no—
219C22×Dic3C22×C414yes—
220C22×Dic3C2×D418yes—
221C22×Dic3C2414yes—
222C22×Dic3C2412yes—
223C22×Dic3C2412yes—
224C22×Dic3C2414yes—
225#235—232no—
226C2×D12—22no—
227C2×D12—22no—
228C2×D12—44no—
229C2×D12—44no—
230C2×D12C22×C414yes—
231C2×D12C22×C4116yes—
232C2×C4×D3—232no—
233C2×C4×D3—24no—
234C2×C4×D3—24no—
235C2×C4×D3—24no—
236C2×C4×D3—24no—
237C2×C4×D3—24no—
238C2×C4×D3—24no—
239C2×C4×D3—28no—
240C2×C4×D3—48no—
241C2×C4×D3C22×C418yes—
242C2×C4×D3C22×C418yes—
243C2×C4×D3C22×C418yes—
244C2×Dic6—22no—
245C2×Dic6—22no—
246C2×Dic6C22×C414yes—
247C3×C4⋊C4—22no—
248C3×C4⋊C4—24no—
249C3×C4⋊C4—24no—
250C3×C4⋊C4—22no—
251C3×C4⋊C4—48no—
252C3×C4⋊C4C22×C418yes—
253C3×C22⋊C4—24no—
254C3×C22⋊C4C22×C418yes—
255C3×C42—24no—
256C3×C42C22×C418yes—
257#217—24no—
258(C2×D3)⋊C4—28no—
259(C2×D3)⋊C4—216no—
260(C2×D3)⋊C4—44no—
261(C2×D3)⋊C4—44no—
262(C2×D3)⋊C4—48no—
263(C2×D3)⋊C4C22×C4116yes—
264(C2×D3)⋊C4C22×C4116yes—
265C4⋊Dic3—24no—
266C4⋊Dic3C22×C4116yes—
267C4⋊Dic3C2×D418yes—
268C4⋊Dic3C2×D418yes—
269Dic3⋊C4—28no—
270C4×Dic3—24no—
271C4×Dic3—28no—
272C4×Dic3—216no—
273C4×Dic3—48no—
274C25—31no—
275C2×#41—38no—
276C2×#41—64no—
277C2×#41—64no—
278C22×Q8—31no—
279C22×D4—32no—
280C22×D4—31no—
281C22×D4—34no—
282C23×C4—31no—
283C23×C4—32no—
284C23×C4—38no—
285C23×C4—32no—
286C23×C4—32no—
287C23×C4—32no—
288C23×C4—34no—
289C23×C4—64no—
290C23×C4—64no—
291C23×C4C22×D312yes—
292C23×C4C22×D311yescharacteristic
293#123—316no—
294C22⋊Q8—316no—
295#121—316no—
296#120—316no—
297C4×Q8—68no—
298C4×D4—332no—
299C4×D4—68no—
300C4×D4—616no—
301#117—24no—
302#117—616no—
303#117—616no—
304#117—68no—
305#117—64no—
306#117C4×D318yescomplemented
307C2×C4⋊C4—24no—
308C2×C4⋊C4—38no—
309C2×C4⋊C4—32no—
310C2×C4⋊C4—32no—
311C2×C4⋊C4—34no—
312C2×C4⋊C4—34no—
313C2×C4⋊C4—32no—
314C2×C4⋊C4—68no—
315C2×C4⋊C4C22×D311yescharacteristic
316C2×C4⋊C4C22×D312yes—
317C2×C4⋊C4C22×D312yes—
318C2×C4⋊C4C22×D311yescharacteristic
319C2×C4⋊C4C22×D318yes—
320C2×C4⋊C4C22×D312yes—
321C2×C22⋊C4—34no—
322C2×C22⋊C4—34no—
323C2×C22⋊C4—32no—
324C2×C22⋊C4—38no—
325C2×C22⋊C4—34no—
326C2×C22⋊C4—34no—
327C2×C22⋊C4—34no—
328C2×C22⋊C4—38no—
329C2×C22⋊C4—68no—
330C2×C22⋊C4C22×D312yes—
331C2×C22⋊C4C22×D312yes—
332C2×C42—38no—
333C2×C42—34no—
334C2×C42—32no—
335C2×C42—38no—
336C2×C42—68no—
337C2×C42C22×D312yes—
338C2×C42C22×D312yes—
339#95—34no—
340#95—34no—
341#95—38no—
342#95—38no—
343#95C22×D318yes—
344C23×C3—22no—
345C23×C3—22no—
346C23×C3C23×C412yes—
347C23×C3C22×D412yes—
348C23×C3C2×#4112yes—
349C23×C3C2511yescharacteristic
350C22×D3—28no—
351C22×D3—22no—
352C22×D3—22no—
353C22×D3—22no—
354C22×D3—22no—
355C22×D3—24no—
356C22×D3—22no—
357C22×D3—22no—
358C22×D3—24no—
359C22×D3—22no—
360C22×D3—22no—
361C22×D3—48no—
362C22×D3—44no—
363C22×D3—44no—
364C22×D3—48no—
365C22×D3C23×C414yes—
366C22×D3C23×C414yes—
367C2×C3×C4—24no—
368C2×C3×C4—24no—
369C2×C3×C4—22no—
370C2×C3×C4—24no—
371C2×C3×C4—22no—
372C2×C3×C4—24no—
373C2×C3×C4—28no—
374C2×C3×C4—24no—
375C2×C3×C4—48no—
376C2×C3×C4C4×D4116yes—
377C2×C3×C4C23×C418yes—
378C2×C3×C4C23×C412yes—
379C2×C3×C4C23×C412yes—
380C2×C3×C4C22×D412yes—
381C2×C3×C4C22×D412yes—
382C2×C3×C4C2×#4112yes—
383C2×C3×C4C2×#4112yes—
384#67—216no—
385C2×Dic3—28no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—28no—
391C2×Dic3—24no—
392C2×Dic3—24no—
393C2×Dic3—24no—
394C2×Dic3—48no—
395C2×Dic3C23×C414yes—
396C2×Dic3C23×C414yes—
397C2×Dic3C22×D418yes—
398D12—28no—
399D12—416no—
400C4×D3—216no—
401C4×D3—216no—
402C4×D3—216no—
403C4×D3—88no—
404Dic6—28no—
405C24—32no—
406C24—32no—
407C24—34no—
408C24—62no—
409C24—62no—
410C24—122no—
411C24—122no—
412C24C23×D311yescharacteristic
413#41—632no—
414C2×Q8—34no—
415C2×Q8—62no—
416C2×Q8—62no—
417C2×D4—38no—
418C2×D4—34no—
419C2×D4—316no—
420C2×D4—64no—
421C2×D4—64no—
422C2×D4—62no—
423C2×D4—62no—
424C2×D4—124no—
425C2×D4—124no—
426C22×C4—24no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—32no—
430C22×C4—34no—
431C22×C4—38no—
432C22×C4—38no—
433C22×C4—38no—
434C22×C4—38no—
435C22×C4—34no—
436C22×C4—44no—
437C22×C4—68no—
438C22×C4—62no—
439C22×C4—62no—
440C22×C4—632no—
441C22×C4—64no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—64no—
446C22×C4—64no—
447C22×C4—128no—
448C22×C4—124no—
449C22×C4C2×C4×D314yes—
450C22×C4D3×D414yes—
451C22×C4#23914yes—
452C22×C4C23×D312yes—
453C22×C4C23×D314yes—
454C22×C4C23×D312yes—
455C22×C4C23×D314yes—
456C22×C4C23×D311yescharacteristic
457C22×C4C23×D311yescharacteristic
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—22no—
462C4⋊C4—316no—
463C4⋊C4—38no—
464C4⋊C4—38no—
465C4⋊C4—48no—
466C4⋊C4—68no—
467C4⋊C4—64no—
468C4⋊C4C2×C4×D318yes—
469C22⋊C4—24no—
470C22⋊C4—316no—
471C22⋊C4—316no—
472C22⋊C4—616no—
473C22⋊C4—68no—
474C22⋊C4—64no—
475C22⋊C4—124no—
476C22⋊C4—128no—
477C22⋊C4—124no—
478C22⋊C4C2×C4×D318yes—
479C42—24no—
480C42—616no—
481C42—68no—
482C42—64no—
483C42—128no—
484C42C2×C4×D318yes—
485C22×C3—24no—
486C22×C3—22no—
487C22×C3—22no—
488C22×C3—24no—
489C22×C3C2×C4×D414yes—
490C22×C3C24×C411yescharacteristic
491C22×C3C23×D411yescharacteristic
492C22×C3C22×#4111yescharacteristic
493C22×C3#58512yes—
494C22×C3#58512yes—
495C2×D3—24no—
496C2×D3—24no—
497C2×D3—24no—
498C2×D3—28no—
499C2×D3—24no—
500C2×D3—416no—
501C2×D3—42no—
502C2×D3—42no—
503C2×D3—42no—
504C2×D3—42no—
505C2×D3—44no—
506C2×D3—42no—
507C2×D3—42no—
508C3×C4—28no—
509C3×C4—42no—
510C3×C4—42no—
511C3×C4—44no—
512C3×C4C2×C4×D414yes—
513C3×C4C2×C4×D414yes—
514Dic3—24no—
515Dic3—24no—
516Dic3—28no—
517Dic3—44no—
518C23—22no—
519C23—22no—
520C23—34no—
521C23—34no—
522C23—62no—
523C23—62no—
524C23—62no—
525C23—62no—
526C23—64no—
527C23—62no—
528C23—62no—
529C23—62no—
530C23—64no—
531C23—62no—
532C23—68no—
533C23—128no—
534C23—124no—
535C23—128no—
536C23—124no—
537C23C22×C4×D312yes—
538C23C2×D3×D412yes—
539C23C2×#23912yes—
540C23C24×D311yescharacteristic
541Q8—68no—
542D4—616no—
543D4—68no—
544D4—1216no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—28no—
552C2×C4—24no—
553C2×C4—34no—
554C2×C4—34no—
555C2×C4—38no—
556C2×C4—48no—
557C2×C4—62no—
558C2×C4—62no—
559C2×C4—62no—
560C2×C4—62no—
561C2×C4—68no—
562C2×C4—68no—
563C2×C4—616no—
564C2×C4—616no—
565C2×C4—616no—
566C2×C4—64no—
567C2×C4—64no—
568C2×C4—64no—
569C2×C4—128no—
570C2×C4—248no—
571C2×C4#893116yes—
572C2×C4C22×C4×D318yes—
573C2×C4C22×C4×D312yes—
574C2×C4C22×C4×D312yes—
575C2×C4C2×D3×D412yes—
576C2×C4C2×D3×D412yes—
577C2×C4C2×#23912yes—
578C2×C4C2×#23912yes—
579C2×C3—24no—
580C2×C3C22×C4×D411yescharacteristic
581C2×C3#342812yes—
582C2×C3#348511yescharacteristic
583C2×C3#348711yescharacteristic
584C2×C3#349312yes—
585D3—44no—
586D3—44no—
587C22—22no—
588C22—24no—
589C22—22no—
590C22—24no—
591C22—64no—
592C22—68no—
593C22—64no—
594C22—64no—
595C22—64no—
596C22—1216no—
597C22—122no—
598C22—122no—
599C22—122no—
600C22—122no—
601C22—124no—
602C22—122no—
603C22—122no—
604C22C2×#89314yes—
605C22C23×C4×D311yescharacteristic
606C22C22×D3×D411yescharacteristic
607C22C22×#23911yescharacteristic
608C22#596612yes—
609C22#596812yes—
610C4—28no—
611C4—44no—
612C4—42no—
613C4—42no—
614C4—68no—
615C4—64no—
616C4—64no—
617C4—124no—
618C4C2×#89314yes—
619C4C2×#89314yes—
620C3#3420111yescharacteristic, complemented, Sylow
621C2—24no—
622C2—124no—
623C2—124no—
624C2C22×#89311yescharacteristic
625C2#8588512yes—
626C2#8631511yescharacteristic
627C2#8635411yescharacteristic
628C2#8658412yes—
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×#2317C2
socle, cosocleC23×C3C25
prime radicalC2×#5524C2
hypercenter#2317D3
hypocenterC3#34201
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#5523C2semidirect
3. C22×#892C2semidirect
4. C3×#2317C2semidirect
5. #82267C2semidirect
6. #82223C2semidirect
7. #82186C2semidirect
8. #75985C2semidirect
9. #75984C2semidirect
10. #74592C2semidirect
11. #74380C2semidirect
12. #74324C2semidirect
13. #74229C2semidirect
24. #5524C4semidirect
31. C2×#892C22semidirect
32. C2×D3×C4⋊C4C22semidirect
37. C2×C4×D12C22semidirect
38. C2×C4×Dic6C22semidirect
39. C3×#389C22semidirect
41. C3×#384C22semidirect
42. C3×#382C22semidirect
45. #4990C22semidirect
47. #4988C22semidirect
50. #4987C22semidirect
51. Dic3×C4⋊C4C22semidirect
52. Dic3×C4⋊C4C22semidirect
53. #4973C22semidirect
58. C4×C4⋊Dic3C22semidirect
60. #4666C22semidirect
62. #4654C22semidirect
71. #2317D3semidirect
89. C22×C4×D3C2×C4semidirect
96. C3×#117C2×C4semidirect
118. C2×C4⋊Dic3D4semidirect
129. #897C2×C4semidirect
130. #895C2×C4semidirect
135. #886C2×C4semidirect
136. #882C2×C4semidirect
139. C4×D12C2×C4semidirect
173. #389C2×D3semidirect
181. #384C2×D3semidirect
182. #382C2×D3semidirect
306. #117C4×D3semidirect
620. C3#34201semidirect

Maximal subgroups

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

subgroupindex
C2×#55242
C2×#55232
C22×#8922
C3×#23172
#822672
#822232
#821862
#759852
#759842
#745922
#743802
#743242
#742292
#342013