Finite Group Databasebeta

← #10828825

Subgroups — #10828825

Order 768.

1100 conjugacy classes of subgroups.

The subgroup lattice has 714 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5694C211yesmaximal, characteristic
2C2×D3×#107C211yesmaximal, characteristic, complemented
3C2×#4862C211yesmaximal, characteristic, complemented
4#76231C211yesmaximal, characteristic
5C3×#2059C211yesmaximal, characteristic, complemented
6#71739C211yesmaximal, characteristic, complemented
7#70635C211yesmaximal, characteristic, complemented
8#70580C211yesmaximal, characteristic, complemented
9#69631C211yesmaximal, characteristic, complemented
10#68882C211yesmaximal, characteristic, complemented
11#68877C211yesmaximal, characteristic, complemented
12#68871C211yesmaximal, characteristic, complemented
13#68825C211yesmaximal, characteristic, complemented
14#68683C211yesmaximal, characteristic, complemented
15#68668C211yesmaximal, characteristic
16#24094—31nomaximal, Sylow
17C2×C3×#125C2211yescharacteristic
18C22×D3×C8C2211yescharacteristic
19#5694—24no—
20#5694—24no—
21C2×#898—21no—
22C2×#897C2211yescharacteristic
23C2×#897C2211yescharacteristic
24C2×#894—21no—
25C2×#894—21no—
26C2×#892—21no—
27C2×#892C2211yescharacteristic
28C2×D3×C4⋊C4C2211yescharacteristic
29C2×D3×C4⋊C4C2211yescharacteristic
30C2×#890—21no—
31C2×C4×D12C2211yescharacteristic
32C2×C4×Dic6C2211yescharacteristic
33C2×C3×#107C2211yescharacteristic
34C2×C3×#106C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C2×(C2×D3)⋊C8C2211yescharacteristic
37C2×(C2×D3)⋊C8C2211yescharacteristic
38C2×(C2×D3)⋊C8C2211yescharacteristic
39C2×#818C2211yescharacteristic
40C2×#816C2211yescharacteristic
41C2×#816C2211yescharacteristic
42#4987C2211yescharacteristic
43#4987C2211yescharacteristic
44#4987C2211yescharacteristic
45#4982—21no—
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#808C2211yescharacteristic
49C2×#807C2211yescharacteristic
50C2×#807C2211yescharacteristic
51#4928C2211yescharacteristic
52D3×#107—24no—
53#4862—24no—
54#4675—21no—
55#4664—21no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58C3×#336C2211yescharacteristic
59#4551C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4529C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#552—31no—
65C2×#483—31no—
66C22×#107—31no—
67#2486—31no—
68#2059—31no—
69#2059D311yescharacteristic, complemented
70#2036—31no—
71#2036—31no—
72#1892—31no—
73#1869—31no—
74#1863—31no—
75#1863—31no—
76#1854—31no—
77#1854—31no—
78#1823—31no—
79C22×D12D411yescharacteristic
80C22×C4×D3—21no—
81C22×C4×D3—21no—
82C22×C4×D3C2311yescharacteristic
83C22×C4×D3C2311yescharacteristic
84C22×Dic6C2311yescharacteristic
85C22×C3×C8C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C3×#125—22no—
88C3×#125—22no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—21no—
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C42C2311yescharacteristic
97C2×(C2×D3)⋊C4—21no—
98C2×(C2×D3)⋊C4—21no—
99C2×(C2×D3)⋊C4—21no—
100C2×(C2×D3)⋊C4—21no—
101C2×(C2×D3)⋊C4—21no—
102C2×(C2×D3)⋊C4—21no—
103C2×(C2×D3)⋊C4D411yescharacteristic
104C2×(C2×D3)⋊C4D411yescharacteristic
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3D411yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C22×C3⋊C8C2311yescharacteristic
118C22×C3⋊C8C2311yescharacteristic
119C2×D3×C8—22no—
120C2×D3×C8—22no—
121C2×D3×C8—22no—
122#898—28no—
123#897—22no—
124#897—22no—
125#897—22no—
126#897D414yes—
127#894—28no—
128#894—44no—
129#892—28no—
130#892—22no—
131#892—22no—
132D3×C4⋊C4—22no—
133D3×C4⋊C4—22no—
134D3×C4⋊C4—22no—
135D3×C4⋊C4D414yes—
136#890—28no—
137C4×D12—22no—
138C4×D12—22no—
139C4×Dic6—22no—
140C4×Dic6—22no—
141C3×#107—22no—
142C3×#106—22no—
143C3×#95—21no—
144C3⋊#95—21no—
145C3⋊#95—21no—
146C3⋊#95—21no—
147C3⋊#95—21no—
148(C2×D3)⋊C8—22no—
149(C2×D3)⋊C8—22no—
150(C2×D3)⋊C8—22no—
151#818—22no—
152#816—22no—
153#816—22no—
154#810—22no—
155#810—22no—
156#808—22no—
157#807—22no—
158#807—22no—
159C23×C8—31no—
160#552—64no—
161#552—64no—
162C2×#126—61no—
163C2×#125—61no—
164C2×#125C2×D311yescharacteristic
165C2×#123—61no—
166C2×#123—61no—
167C2×C22⋊Q8—31no—
168C2×C22⋊Q8—31no—
169C2×C4×Q8—31no—
170C2×C4×D4—31no—
171C2×#117—31no—
172C2×#117—61no—
173C22×C4⋊C4—31no—
174C22×C4⋊C4—31no—
175#483—64no—
176C2×#107—31no—
177C2×#107—31no—
178C2×#107—31no—
179C2×#107—64no—
180C2×#107C2×D311yescharacteristic
181C2×#106—31no—
182C2×#106C2×D311yescharacteristic
183C2×#103—31no—
184C2×#103—31no—
185C2×#103—31no—
186C2×#103—31no—
187C2×C22⋊C8—31no—
188C2×C22⋊C8—31no—
189C2×C22⋊C8—31no—
190#398—61no—
191#397—61no—
192#389—31no—
193#389C2×D311yescharacteristic
194#381—61no—
195#380—31no—
196#380—31no—
197#380—31no—
198#340—31no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340C2×D311yescharacteristic
203#340C2×D311yescharacteristic
204#336—31no—
205#336C2×D311yescharacteristic
206C23×D3C2×D411yescharacteristic
207C23×D3C2×D411yescharacteristic
208C22×C3×C4—21no—
209C22×C3×C4—21no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
212C22×C3×C4C2×D411yescharacteristic
213C22×C3×C4C2×D411yescharacteristic
214C22×C3×C4C2411yescharacteristic
215C22×Dic3—21no—
216C22×Dic3—21no—
217C22×Dic3—21no—
218C22×Dic3—21no—
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C2×D12—22no—
222C2×D12—22no—
223C2×D12—22no—
224C2×C4×D3—21no—
225C2×C4×D3—21no—
226C2×C4×D3—21no—
227C2×C4×D3—21no—
228C2×C4×D3—24no—
229C2×C4×D3—21no—
230C2×C4×D3—21no—
231C2×C4×D3—21no—
232C2×C4×D3—21no—
233C2×C4×D3—21no—
234C2×C4×D3—21no—
235C2×C4×D3—42no—
236C2×C4×D3—42no—
237C2×C4×D3—42no—
238C2×C4×D3—42no—
239C2×C4×D3—42no—
240C2×C4×D3C2×D412yes—
241C2×C4×D3C2×D412yes—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—21no—
247C2×Dic6C2×D412yes—
248C2×C3×C8—21no—
249C2×C3×C8—21no—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8#4112yes—
254C3×C4⋊C4—24no—
255C3×C4⋊C4—24no—
256C3×C4⋊C4—21no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4—21no—
261C3×C4⋊C4—21no—
262C3×C4⋊C4—42no—
263C3×C4⋊C4C2×D412yes—
264C3×C4⋊C4C2×D412yes—
265C3×C42—21no—
266C3×C42—21no—
267(C2×D3)⋊C4—24no—
268(C2×D3)⋊C4—24no—
269(C2×D3)⋊C4—22no—
270(C2×D3)⋊C4—24no—
271(C2×D3)⋊C4—22no—
272(C2×D3)⋊C4—42no—
273(C2×D3)⋊C4—42no—
274(C2×D3)⋊C4—42no—
275C4⋊Dic3—24no—
276C4⋊Dic3—21no—
277C4⋊Dic3—22no—
278C4⋊Dic3—24no—
279C4⋊Dic3—21no—
280C4⋊Dic3—21no—
281C4⋊Dic3—21no—
282C4⋊Dic3—21no—
283C4⋊Dic3C2×D412yes—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286Dic3⋊C4—42no—
287Dic3⋊C4—42no—
288C4×Dic3—24no—
289C4×Dic3—24no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8#4112yes—
296D3×C8—28no—
297C22×Q8—31no—
298C22×D4—31no—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4—61no—
302C23×C4—61no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—62no—
306C22×C8—62no—
307C22×C8—62no—
308C22×C8C22×D311yescharacteristic
309C22×C8C22×D311yescharacteristic
310#126—68no—
311#125—22no—
312#125—22no—
313#125—68no—
314#123—68no—
315#123—124no—
316C22⋊Q8—34no—
317C22⋊Q8—62no—
318C22⋊Q8—62no—
319C22⋊Q8—62no—
320C4×Q8—62no—
321C4×Q8—62no—
322C4×D4—62no—
323C4×D4—62no—
324#117—62no—
325#117—68no—
326#117—62no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—34no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—62no—
336C2×C4⋊C4—62no—
337C2×C4⋊C4—62no—
338C2×C4⋊C4—61no—
339C2×C4⋊C4—61no—
340C2×C4⋊C4—61no—
341C2×C4⋊C4—61no—
342C2×C4⋊C4—61no—
343C2×C4⋊C4—61no—
344C2×C4⋊C4C22×D311yescharacteristic
345C2×C4⋊C4C22×D311yescharacteristic
346C2×C4⋊C4C22×D311yescharacteristic
347C2×C4⋊C4C22×D311yescharacteristic
348C2×C22⋊C4—31no—
349C2×C22⋊C4—31no—
350C2×C22⋊C4—61no—
351C2×C22⋊C4—61no—
352C2×C22⋊C4—61no—
353C2×C22⋊C4—61no—
354C2×C22⋊C4—61no—
355C2×C22⋊C4—61no—
356C2×C42—61no—
357C2×C42—61no—
358C2×C42C22×D311yescharacteristic
359#107—22no—
360#107—62no—
361#107—62no—
362#107—62no—
363#106—22no—
364#106—62no—
365#103—62no—
366#103—62no—
367#103—62no—
368#103—62no—
369C22⋊C8—62no—
370C22⋊C8—62no—
371C22⋊C8—62no—
372#95—21no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3C22×D411yescharacteristic
378C22×D3—21no—
379C22×D3—21no—
380C22×D3—21no—
381C22×D3—21no—
382C22×D3—21no—
383C22×D3—21no—
384C22×D3—21no—
385C22×D3—21no—
386C22×D3—21no—
387C22×D3—21no—
388C22×D3—21no—
389C22×D3#12012yes—
390C22×D3#13512yes—
391C22×D3AGL(1,ℤ/8)12yes—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—21no—
395C2×C3×C4—22no—
396C2×C3×C4—21no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—41no—
401C2×C3×C4—41no—
402C2×C3×C4—41no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—41no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4C22×D411yescharacteristic
411C2×C3×C4C22×D411yescharacteristic
412C2×C3×C4C2×#4111yescharacteristic
413C2×C3×C4C2×#4111yescharacteristic
414C2×C3×C42−1+411yescharacteristic
415C2×C3×C42−1+411yescharacteristic
416C2×Dic3—21no—
417C2×Dic3—21no—
418C2×Dic3—22no—
419C2×Dic3—21no—
420C2×Dic3—22no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—41no—
424C2×Dic3—41no—
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#12012yes—
434D12—44no—
435C4×D3—22no—
436C4×D3—22no—
437C4×D3—22no—
438C4×D3—22no—
439C4×D3—48no—
440C4×D3—41no—
441C4×D3—41no—
442C4×D3—41no—
443C4×D3—41no—
444C4×D3—84no—
445Dic6—22no—
446Dic6—22no—
447Dic6—41no—
448Dic6—41no—
449C3×C8—22no—
450C3×C8—41no—
451C3⋊C8—22no—
452C3⋊C8—41no—
453C24—31no—
454C24—31no—
455C2×Q8—32no—
456C2×Q8—61no—
457C2×Q8—61no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×D4—62no—
462C2×D4—62no—
463C2×D4—62no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—31no—
469C22×C4—32no—
470C22×C4—32no—
471C22×C4—31no—
472C22×C4—64no—
473C22×C4—61no—
474C22×C4—61no—
475C22×C4—61no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4—61no—
481C22×C4—61no—
482C22×C4—61no—
483C22×C4—61no—
484C22×C4—61no—
485C22×C4—61no—
486C22×C4—61no—
487C22×C4—122no—
488C22×C4—122no—
489C22×C4—122no—
490C22×C4—122no—
491C22×C4—122no—
492C22×C4D3×D411yescharacteristic
493C22×C4D3×D411yescharacteristic
494C22×C4C23×D311yescharacteristic
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—32no—
501C2×C8—68no—
502C2×C8—61no—
503C2×C8—61no—
504C2×C8—61no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8#23912yes—
508C4⋊C4—24no—
509C4⋊C4—24no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—32no—
517C4⋊C4—42no—
518C4⋊C4—64no—
519C4⋊C4—62no—
520C4⋊C4—64no—
521C4⋊C4—64no—
522C4⋊C4—64no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—61no—
527C4⋊C4—61no—
528C4⋊C4—122no—
529C4⋊C4—122no—
530C4⋊C4D3×D412yes—
531C4⋊C4D3×D412yes—
532C22⋊C4—62no—
533C22⋊C4—62no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—122no—
538C22⋊C4—122no—
539C22⋊C4—122no—
540C42—21no—
541C42—21no—
542C42—64no—
543C42—64no—
544C22×C3C2×#12011yescharacteristic
545C22×C3#54711yescharacteristic
546C22×C3#54711yescharacteristic
547C22×C3C2×#13511yescharacteristic
548C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
549C22×C3#57811yescharacteristic
550C22×C3#57811yescharacteristic
551C2×D3—22no—
552C2×D3—22no—
553C2×D3—22no—
554C2×D3—22no—
555C2×D3—22no—
556C2×D3—22no—
557C2×D3—41no—
558C2×D3—41no—
559C2×D3—41no—
560C2×D3—41no—
561C2×D3—41no—
562C2×D3—41no—
563C2×D3—41no—
564C2×D3#44914yes—
565C3×C4—42no—
566C3×C4—41no—
567C3×C4—41no—
568C3×C4—42no—
569C3×C4—81no—
570C3×C4—81no—
571C3×C4#52511yescharacteristic
572C3×C4#53611yescharacteristic
573C3×C4#54711yescharacteristic
574C3×C4#54711yescharacteristic
575Dic3—22no—
576Dic3—42no—
577Dic3—41no—
578Dic3—42no—
579Dic3—81no—
580Dic3—81no—
581C23—32no—
582C23—32no—
583C23—32no—
584C23—61no—
585C23—61no—
586C23—61no—
587C23—61no—
588C23—61no—
589C23—61no—
590C23—61no—
591C23—61no—
592C23—61no—
593C23—61no—
594C23—61no—
595C23C2×D3×D411yescharacteristic
596Q8—62no—
597Q8—62no—
598Q8—121no—
599Q8—121no—
600D4—124no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—22no—
608C2×C4—22no—
609C2×C4—32no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—61no—
621C2×C4—61no—
622C2×C4—62no—
623C2×C4—62no—
624C2×C4—62no—
625C2×C4—62no—
626C2×C4—61no—
627C2×C4—62no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—61no—
631C2×C4—128no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4—121no—
638C2×C4—121no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—244no—
647C2×C4C2×D3×D411yescharacteristic
648C2×C4C2×D3×D411yescharacteristic
649C2×C4C2×#23911yescharacteristic
650C2×C4#100711yescharacteristic
651C2×C4#100711yescharacteristic
652C2×C4D3×#4111yescharacteristic
653C8—22no—
654C8—41no—
655C8—62no—
656C8—121no—
657C2×C3#240611yescharacteristic
658C2×C3C2×#44911yescharacteristic
659C2×C3#301111yescharacteristic
660C2×C3#333611yescharacteristic
661C2×C3#333611yescharacteristic
662C2×C3#334311yescharacteristic
663C2×C3#334311yescharacteristic
664D3—24no—
665D3—81no—
666C22—34no—
667C22—62no—
668C22—62no—
669C22—62no—
670C22—62no—
671C22—62no—
672C22—62no—
673C22—121no—
674C22—121no—
675C22—121no—
676C22—121no—
677C22—121no—
678C22—121no—
679C22—121no—
680C22D3×#12011yescharacteristic
681C22#563111yescharacteristic
682C22#572611yescharacteristic
683C22D3×#13511yescharacteristic
684C22#577111yescharacteristic
685C22D3×AGL(1,ℤ/8)11yescharacteristic
686C22#577911yescharacteristic
687C4—41no—
688C4—42no—
689C4—41no—
690C4—42no—
691C4—62no—
692C4—81no—
693C4—81no—
694C4—121no—
695C4—122no—
696C4—122no—
697C4—241no—
698C4—241no—
699C4#562711yescharacteristic
700C4#563111yescharacteristic
701C4#572411yescharacteristic
702C4#572611yescharacteristic
703C3#2409411yescharacteristic, complemented, Sylow
704C2—64no—
705C2—241no—
706C2#7724911yescharacteristic
707C2D3×#44911yescharacteristic
708C2#8023011yescharacteristic
709C2#8119611yescharacteristic
710C2#8125311yescharacteristic
711C2#8192411yescharacteristic
712C2#8199011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2059C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#2059D3
hypocenterC3#24094
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2059

Complements

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

NH ≅ G/Ntype
2. C2×D3×#107C2semidirect
3. C2×#4862C2semidirect
5. C3×#2059C2semidirect
6. #71739C2semidirect
7. #70635C2semidirect
8. #70580C2semidirect
9. #69631C2semidirect
10. #68882C2semidirect
11. #68877C2semidirect
12. #68871C2semidirect
13. #68825C2semidirect
14. #68683C2semidirect
69. #2059D3semidirect
703. C3#24094semidirect

Maximal subgroups

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

subgroupindex
C2×#56942
C2×D3×#1072
C2×#48622
#762312
C3×#20592
#717392
#706352
#705802
#696312
#688822
#688772
#688712
#688252
#686832
#686682
#240943