Finite Group Databasebeta

← #10828497

Subgroups — #10828497

Order 768.

1072 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5622C211yesmaximal, characteristic
2C2×#5045C211yesmaximal, characteristic
3#82227C211yesmaximal, characteristic
4C2×#4723C211yesmaximal, characteristic
5C3×#1873C211yesmaximal, characteristic
6#71585C211yesmaximal, characteristic
7#70246C211yesmaximal, characteristic, complemented
8#70246C211yesmaximal, characteristic, complemented
9#70238C211yesmaximal, characteristic, complemented
10#70236C211yesmaximal, characteristic, complemented
11#69910C211yesmaximal, characteristic
12#68907C211yesmaximal, characteristic, complemented
13#68906C211yesmaximal, characteristic, complemented
14#68827C211yesmaximal, characteristic, complemented
15#68826C211yesmaximal, characteristic, complemented
16#24065—31nomaximal, Sylow
17C2×C3×C22⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C22×C4×Dic3C2211yescharacteristic
24#5622—24no—
25#5622—24no—
26C2×#889—21no—
27C2×#888C2211yescharacteristic
28C2×#887—21no—
29C2×#887C2211yescharacteristic
30C2×#878—21no—
31C2×#877—21no—
32C2×C3×#103C2211yescharacteristic
33C2×C3×#103C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×#835C2211yescharacteristic
36C2×#835C2211yescharacteristic
37C2×#830C2211yescharacteristic
38C2×#818C2211yescharacteristic
39C2×#817C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#5045—24no—
43#4984C2211yescharacteristic
44#4984C2211yescharacteristic
45#4984C2211yescharacteristic
46#4984C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#808C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4950—21no—
52#4950—21no—
53C4×C4⋊Dic3—21no—
54C4×C4⋊Dic3—21no—
55#4723—24no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#533—31no—
65C2×#485—31no—
66C2×#485—31no—
67#2296—31no—
68#2058—31no—
69#2057—31no—
70#2043—31no—
71#2043—31no—
72#1873—31no—
73#1873D311yescharacteristic
74#1865—31no—
75#1865—31no—
76#1863—31no—
77#1863—31no—
78#1817—31no—
79C22×C3×Q8C2311yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×Dic3D411yescharacteristic
82C23×Dic3D411yescharacteristic
83C22×Dic6—21no—
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×C22⋊Q8—22no—
88C3×C22⋊Q8D414yes—
89C3×#117—22no—
90C3×#117—22no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C42—21no—
98C2×#217—21no—
99C2×#217—21no—
100Q8×Dic3—22no—
101Q8×Dic3—22no—
102#944—22no—
103#944—22no—
104#926—22no—
105#926—22no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3—21no—
108C2×C4⋊Dic3—21no—
109C2×C4⋊Dic3—21no—
110C2×C4⋊Dic3—21no—
111C2×C4⋊Dic3C2311yescharacteristic
112C2×C4⋊Dic3C2311yescharacteristic
113C2×C4⋊Dic3C2311yescharacteristic
114#924—22no—
115#924D414yes—
116C2×Dic3⋊C4—21no—
117C2×Dic3⋊C4—21no—
118C2×Dic3⋊C4—21no—
119C2×Dic3⋊C4—21no—
120C2×Dic3⋊C4—21no—
121C2×C4×Dic3—22no—
122C2×C4×Dic3—22no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3—21no—
126C2×C4×Dic3D411yescharacteristic
127C2×C4×Dic3D411yescharacteristic
128C2×C4×Dic3C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132#889—28no—
133#888—22no—
134#888—22no—
135#887—28no—
136#887—22no—
137#887—22no—
138#878—28no—
139#877—28no—
140C3×#103—22no—
141C3×#103—22no—
142C3×C22⋊C8—22no—
143#835—22no—
144#835—22no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148C3⋊#95—21no—
149#830—22no—
150#818—22no—
151#817—22no—
152#816—22no—
153#816—22no—
154#810—22no—
155#810—22no—
156#808—22no—
157#807—22no—
158#533—64no—
159#533—64no—
160C2×C4⋊Q8—31no—
161C2×C4⋊Q8—31no—
162C2×#125—61no—
163C2×C22⋊Q8—31no—
164C2×C22⋊Q8—61no—
165C2×C22⋊Q8C2×D311yescharacteristic
166C2×C4×Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—61no—
169C2×#117—31no—
170C2×#117—61no—
171C2×#117C2×D311yescharacteristic
172C22×C42—31no—
173#485—64no—
174#485—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#106—31no—
178C2×#106—31no—
179C2×#103—31no—
180C2×#103—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×#103C2×D311yescharacteristic
186C2×#103C2×D311yescharacteristic
187C2×C22⋊C8—31no—
188C2×C22⋊C8C2×D311yescharacteristic
189#389—31no—
190#389—31no—
191#389—31no—
192#389—31no—
193#385—61no—
194#385—61no—
195C4×C4⋊C4—61no—
196C4×C4⋊C4—61no—
197#340—31no—
198#340—31no—
199#340C2×D311yescharacteristic
200#340C2×D311yescharacteristic
201#336—31no—
202#336—31no—
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C2×C3×Q8—21no—
207C2×C3×Q8—21no—
208C2×C3×Q8—21no—
209C2×C3×Q8—21no—
210C2×C3×Q8—21no—
211C2×C3×Q8C2×D412yes—
212C22×C3×C4—21no—
213C22×C3×C4—21no—
214C22×C3×C4—21no—
215C22×C3×C4—21no—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4C2×D412yes—
221C22×C3×C4C2×D411yescharacteristic
222C22×C3×C4C2×D412yes—
223C22×C3×C4C2411yescharacteristic
224C22×Dic3—22no—
225C22×Dic3—22no—
226C22×Dic3—22no—
227C22×Dic3—22no—
228C22×Dic3—21no—
229C22×Dic3—21no—
230C22×Dic3—22no—
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—41no—
234C22×Dic3C2×D411yescharacteristic
235C22×Dic3C2×D411yescharacteristic
236C22×Dic3C2×D411yescharacteristic
237C22×Dic3C2×D411yescharacteristic
238C2×Dic6—21no—
239C2×Dic6—21no—
240C2×Dic6—21no—
241C2×Dic6—21no—
242C2×Dic6—24no—
243C2×Dic6—21no—
244C2×Dic6—42no—
245C2×Dic6—42no—
246C2×Dic6C2×D412yes—
247C2×C3×C8—21no—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C3×C4⋊C4—24no—
254C3×C4⋊C4—21no—
255C3×C4⋊C4—21no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4C2×D412yes—
258C3×C4⋊C4#4112yes—
259C3×C4⋊C4#4112yes—
260C3×C22⋊C4—24no—
261C3×C22⋊C4—24no—
262C3×C42—24no—
263#217—24no—
264#217—42no—
265C4⋊Dic3—24no—
266C4⋊Dic3—21no—
267C4⋊Dic3—21no—
268C4⋊Dic3—21no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—42no—
272C4⋊Dic3—42no—
273C4⋊Dic3—42no—
274C4⋊Dic3—42no—
275C4⋊Dic3C2×D412yes—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278Dic3⋊C4—24no—
279Dic3⋊C4—24no—
280Dic3⋊C4—24no—
281C4×Dic3—24no—
282C4×Dic3—24no—
283C4×Dic3—22no—
284C4×Dic3—22no—
285C4×Dic3—21no—
286C4×Dic3—21no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—42no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C22×Q8—31no—
297C22×Q8—61no—
298C22×Q8C22×D311yescharacteristic
299C23×C4—31no—
300C23×C4—31no—
301C23×C4C22×D311yescharacteristic
302C22×C8—31no—
303C22×C8—31no—
304C22×C8C22×D311yescharacteristic
305C22×C8C22×D311yescharacteristic
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308C4⋊Q8—62no—
309C4⋊Q8—62no—
310#125—68no—
311C22⋊Q8—22no—
312C22⋊Q8—34no—
313C22⋊Q8—68no—
314C22⋊Q8—62no—
315C22⋊Q8D1214yes—
316C4×Q8—62no—
317C4×Q8—68no—
318C4×Q8—62no—
319C4×Q8—62no—
320C4×Q8—62no—
321#117—22no—
322#117—22no—
323#117—68no—
324#117—62no—
325#117—62no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—31no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—61no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4—61no—
336C2×C4⋊C4—61no—
337C2×C4⋊C4—61no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4C22×D311yescharacteristic
341C2×C4⋊C4C22×D311yescharacteristic
342C2×C4⋊C4C22×D311yescharacteristic
343C2×C22⋊C4—21no—
344C2×C22⋊C4—21no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C42—21no—
348C2×C42—31no—
349C2×C42—31no—
350C2×C42—31no—
351C2×C42—31no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42—62no—
355C2×C42—62no—
356C2×C42—61no—
357#107—62no—
358#107—62no—
359#106—62no—
360#106—62no—
361#103—22no—
362#103—22no—
363#103—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#103—62no—
368#103—62no—
369C22⋊C8—22no—
370C22⋊C8—62no—
371#95—61no—
372#95—61no—
373#95—61no—
374#95—61no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3#12012yes—
382C23×C3C22×D411yescharacteristic
383C3×Q8—22no—
384C3×Q8—22no—
385C3×Q8—41no—
386C3×Q8—41no—
387C2×C3×C4—21no—
388C2×C3×C4—21no—
389C2×C3×C4—22no—
390C2×C3×C4—22no—
391C2×C3×C4—22no—
392C2×C3×C4—22no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—41no—
398C2×C3×C4—41no—
399C2×C3×C4—41no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4#12012yes—
406C2×C3×C4C22×D411yescharacteristic
407C2×C3×C4C22×D411yescharacteristic
408C2×C3×C4C2×#4111yescharacteristic
409C2×C3×C4C2×#4111yescharacteristic
410C2×C3×C42−1+411yescharacteristic
411C2×C3×C42−1+411yescharacteristic
412C2×Dic3—28no—
413C2×Dic3—21no—
414C2×Dic3—21no—
415C2×Dic3—21no—
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—21no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—22no—
422C2×Dic3—21no—
423C2×Dic3—21no—
424C2×Dic3—44no—
425C2×Dic3—41no—
426C2×Dic3—41no—
427C2×Dic3—41no—
428C2×Dic3—41no—
429C2×Dic3—41no—
430C2×Dic3—41no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3#13512yes—
436C2×Dic3#13712yes—
437C2×Dic3#13712yes—
438Dic6—22no—
439Dic6—22no—
440Dic6—48no—
441Dic6—41no—
442Dic6—41no—
443C3×C8—41no—
444C3×C8—41no—
445C3⋊C8—41no—
446C3⋊C8—41no—
447C24D3×D411yescharacteristic
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—32no—
454C2×Q8—64no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—122no—
461C2×Q8—122no—
462C2×Q8C2×D1212yes—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—31no—
472C22×C4—31no—
473C22×C4—31no—
474C22×C4—31no—
475C22×C4—62no—
476C22×C4—62no—
477C22×C4—62no—
478C22×C4—62no—
479C22×C4—62no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—121no—
485C22×C4C2×D1212yes—
486C22×C4D3×D411yescharacteristic
487C22×C4D3×D412yes—
488C22×C4C23×D311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C2×C8—61no—
501C4⋊C4—24no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—32no—
506C4⋊C4—64no—
507C4⋊C4—64no—
508C4⋊C4—64no—
509C4⋊C4—64no—
510C4⋊C4—64no—
511C4⋊C4—64no—
512C4⋊C4—61no—
513C4⋊C4—61no—
514C4⋊C4—61no—
515C4⋊C4—61no—
516C4⋊C4—61no—
517C4⋊C4—122no—
518C4⋊C4—122no—
519C4⋊C4—122no—
520C4⋊C4—122no—
521C4⋊C4C2×D1212yes—
522C4⋊C4#23712yes—
523C4⋊C4#23712yes—
524C22⋊C4—24no—
525C22⋊C4—24no—
526C22⋊C4—64no—
527C22⋊C4—122no—
528C42—24no—
529C42—64no—
530C42—64no—
531C42—62no—
532C42—62no—
533C42—61no—
534C42—61no—
535C42—61no—
536C42—61no—
537C42—122no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—22no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3C2×#12011yescharacteristic
546C22×C3#52511yescharacteristic
547C22×C3#53611yescharacteristic
548C22×C3C2×#13511yescharacteristic
549C22×C3C2×#13711yescharacteristic
550C22×C3C2×#13711yescharacteristic
551C22×C3#57511yescharacteristic
552C3×C4—22no—
553C3×C4—42no—
554C3×C4—42no—
555C3×C4—42no—
556C3×C4—42no—
557C3×C4#54711yescharacteristic
558C3×C4#54711yescharacteristic
559C3×C4#54711yescharacteristic
560C3×C4#54711yescharacteristic
561Dic3—22no—
562Dic3—22no—
563Dic3—22no—
564Dic3—42no—
565Dic3—42no—
566Dic3—41no—
567Dic3—81no—
568Dic3—81no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#88212yes—
576C23C2×D3×D411yescharacteristic
577Q8—22no—
578Q8—22no—
579Q8—41no—
580Q8—41no—
581Q8—62no—
582Q8—62no—
583Q8—128no—
584Q8—121no—
585Q8—121no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—32no—
597C2×C4—32no—
598C2×C4—32no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—68no—
608C2×C4—61no—
609C2×C4—61no—
610C2×C4—61no—
611C2×C4—61no—
612C2×C4—61no—
613C2×C4—61no—
614C2×C4—61no—
615C2×C4—62no—
616C2×C4—62no—
617C2×C4—61no—
618C2×C4—61no—
619C2×C4—124no—
620C2×C4—121no—
621C2×C4—121no—
622C2×C4—121no—
623C2×C4—121no—
624C2×C4—121no—
625C2×C4—121no—
626C2×C4—121no—
627C2×C4—121no—
628C2×C4—121no—
629C2×C4—121no—
630C2×C4#88212yes—
631C2×C4C22×D1211yescharacteristic
632C2×C4C2×D3×D411yescharacteristic
633C2×C4C2×#23711yescharacteristic
634C2×C4C2×#23711yescharacteristic
635C2×C4#101011yescharacteristic
636C2×C4#101011yescharacteristic
637C8—41no—
638C8—41no—
639C8—121no—
640C8—121no—
641C2×C3—42no—
642C2×C3#240611yescharacteristic
643C2×C3#300211yescharacteristic
644C2×C3#300411yescharacteristic
645C2×C3#310711yescharacteristic
646C2×C3#311611yescharacteristic
647C2×C3#323111yescharacteristic
648C2×C3#323711yescharacteristic
649C22—22no—
650C22—22no—
651C22—22no—
652C22—41no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22C2×#88211yescharacteristic
657C22#558611yescharacteristic
658C22#564811yescharacteristic
659C22D3×#13511yescharacteristic
660C22#576811yescharacteristic
661C22D3×#13711yescharacteristic
662C22D3×#13711yescharacteristic
663C4—22no—
664C4—42no—
665C4—42no—
666C4—42no—
667C4—42no—
668C4—62no—
669C4—62no—
670C4—62no—
671C4—122no—
672C4—121no—
673C4—122no—
674C4—241no—
675C4—241no—
676C4#555511yescharacteristic
677C4#555511yescharacteristic
678C4#561111yescharacteristic
679C4#561111yescharacteristic
680C3#2406511yescharacteristic, complemented, Sylow
681C2—42no—
682C2#7468711yescharacteristic
683C2#7883311yescharacteristic
684C2#7895811yescharacteristic
685C2#8021311yescharacteristic
686C2#8021511yescharacteristic
687C2#8150011yescharacteristic
688C2#8150211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1873C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1873D3
hypocenterC3#24065
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
7. #70246C2semidirect
8. #70246C2semidirect
9. #70238C2semidirect
10. #70236C2semidirect
12. #68907C2semidirect
13. #68906C2semidirect
14. #68827C2semidirect
15. #68826C2semidirect
680. C3#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#56222
C2×#50452
#822272
C2×#47232
C3×#18732
#715852
#702462
#702382
#702362
#699102
#689072
#689062
#688272
#688262
#240653