Finite Group Databasebeta

← #10828607

Subgroups — #10828607

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5603C211yesmaximal, characteristic
2C2×#5035C211yesmaximal, characteristic
3#82156C211yesmaximal, characteristic, complemented
4C2×#4723C211yesmaximal, characteristic, complemented
5C3×#1874C211yesmaximal, characteristic
6#71559C211yesmaximal, characteristic, complemented
7#70178C211yesmaximal, characteristic, complemented
8#70178C211yesmaximal, characteristic, complemented
9#70170C211yesmaximal, characteristic, complemented
10#70168C211yesmaximal, characteristic, complemented
11#69911C211yesmaximal, characteristic, complemented
12#68907C211yesmaximal, characteristic, complemented
13#68906C211yesmaximal, characteristic, complemented
14#68827C211yesmaximal, characteristic, complemented
15#68826C211yesmaximal, characteristic, complemented
16#24075—31nomaximal, Sylow
17C2×C3×#121C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C2×#936—21no—
20C2×#936C2211yescharacteristic
21C2×#935—21no—
22C2×D4×Dic3C2211yescharacteristic
23C2×#926C2211yescharacteristic
24C2×#926C2211yescharacteristic
25C2×#924C2211yescharacteristic
26#5603—24no—
27#5603—24no—
28C2×#888C2211yescharacteristic
29C2×#887C2211yescharacteristic
30C2×#879—21no—
31C2×#878—21no—
32C2×C3×#102C2211yescharacteristic
33C2×C3×#102C2211yescharacteristic
34C2×C3×C22⋊C8C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37#5212—21no—
38#5212—21no—
39C2×#830C2211yescharacteristic
40C2×#818C2211yescharacteristic
41C2×#817C2211yescharacteristic
42C2×#816C2211yescharacteristic
43C2×#816C2211yescharacteristic
44#5035—24no—
45#4984—21no—
46#4984—21no—
47#4984C2211yescharacteristic
48#4984C2211yescharacteristic
49#4984C2211yescharacteristic
50#4984C2211yescharacteristic
51C2×#810C2211yescharacteristic
52C2×#810C2211yescharacteristic
53C2×#808C2211yescharacteristic
54C2×#807C2211yescharacteristic
55#4723—24no—
56C3×#340C2211yescharacteristic
57C3×#340C2211yescharacteristic
58#4550C2211yescharacteristic
59#4550C2211yescharacteristic
60#4550C2211yescharacteristic
61#4550C2211yescharacteristic
62#4529C2211yescharacteristic
63#4529C2211yescharacteristic
64C2×#542—31no—
65C2×#485—31no—
66C2×#482—31no—
67#2316—31no—
68#2058—31no—
69#2057—31no—
70#2042—31no—
71#2042—31no—
72#1874D311yescharacteristic
73#1872—31no—
74#1865—31no—
75#1864—31no—
76#1863—31no—
77#1862—31no—
78#1818—31no—
79C22×C3×D4—21no—
80C22×C3×D4C2311yescharacteristic
81C23×C3×C4—21no—
82C23×C3×C4C2311yescharacteristic
83C23×Dic3—21no—
84C22×Dic6—21no—
85C22×Dic6C2311yescharacteristic
86C22×C3×C8C2311yescharacteristic
87C22×C3×C8C2311yescharacteristic
88C3×#121—22no—
89C3×#121D414yes—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4—22no—
92C2×C3×C4⋊C4—22no—
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×#217—21no—
98C2×#217—21no—
99C2×#217—21no—
100C2×#217—21no—
101C2×#217—21no—
102C2×#217D411yescharacteristic
103C2×#217D411yescharacteristic
104#936—28no—
105#936—22no—
106#936—22no—
107#935—28no—
108D4×Dic3—22no—
109D4×Dic3—22no—
110#926—22no—
111#926—22no—
112#926—22no—
113#926—22no—
114C2×C4⋊Dic3—21no—
115C2×C4⋊Dic3—21no—
116C2×C4⋊Dic3—21no—
117C2×C4⋊Dic3—21no—
118C2×C4⋊Dic3D411yescharacteristic
119C2×C4⋊Dic3D411yescharacteristic
120C2×C4⋊Dic3C2311yescharacteristic
121C2×C4⋊Dic3C2311yescharacteristic
122C2×C4⋊Dic3C2311yescharacteristic
123#924—22no—
124#924D414yes—
125C2×Dic3⋊C4—21no—
126C2×Dic3⋊C4—21no—
127C2×Dic3⋊C4—21no—
128C2×C4×Dic3—21no—
129C2×C4×Dic3—21no—
130C2×C4×Dic3C2311yescharacteristic
131C2×C4×Dic3C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133C22×C3⋊C8C2311yescharacteristic
134#888—22no—
135#888—22no—
136#887—22no—
137#887—22no—
138#879—28no—
139#878—28no—
140C3×#102—22no—
141C3×#102—22no—
142C3×C22⋊C8—22no—
143#832—22no—
144#832—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#542—64no—
159#542—64no—
160C2×C4⋊Q8—31no—
161C2×#126—61no—
162C2×#124—31no—
163C2×#124—61no—
164C2×#123—61no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—61no—
167C2×#121C2×D311yescharacteristic
168C2×C4×Q8—31no—
169C2×C4×D4—31no—
170C2×#117—31no—
171C2×#117—31no—
172C22×C4⋊C4C2×D311yescharacteristic
173#485—64no—
174#482—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×#102—31no—
184C2×#102—31no—
185C2×#102C2×D311yescharacteristic
186C2×#102C2×D311yescharacteristic
187C2×C22⋊C8—31no—
188C2×C22⋊C8C2×D311yescharacteristic
189#389—31no—
190#389—31no—
191#389—31no—
192#389—31no—
193#389—61no—
194#389—61no—
195#380—61no—
196#380—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×C3—21no—
206C24×C3C2×D411yescharacteristic
207C2×C3×D4—24no—
208C2×C3×D4—21no—
209C2×C3×D4—21no—
210C2×C3×D4—21no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—81no—
214C2×C3×D4—81no—
215C2×C3×D4C2×D412yes—
216C22×C3×C4—21no—
217C22×C3×C4—21no—
218C22×C3×C4—21no—
219C22×C3×C4—21no—
220C22×C3×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—24no—
224C22×C3×C4—81no—
225C22×C3×C4—81no—
226C22×C3×C4C2×D412yes—
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D412yes—
229C22×C3×C4C2411yescharacteristic
230C22×Dic3—21no—
231C22×Dic3—21no—
232C22×Dic3—24no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C22×Dic3—81no—
236C22×Dic3—81no—
237C22×Dic3C2×D411yescharacteristic
238C22×Dic3C2×D411yescharacteristic
239C22×Dic3C2×D411yescharacteristic
240C22×Dic3C2×D411yescharacteristic
241C2×Dic6—21no—
242C2×Dic6—21no—
243C2×Dic6—21no—
244C2×Dic6—21no—
245C2×Dic6—21no—
246C2×Dic6—24no—
247C2×Dic6—81no—
248C2×Dic6—81no—
249C2×Dic6C2×D412yes—
250C2×C3×C8—21no—
251C2×C3×C8—21no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C3×C4⋊C4—24no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—21no—
259C3×C4⋊C4—21no—
260C3×C4⋊C4C2×D412yes—
261C3×C4⋊C4#4112yes—
262C3×C4⋊C4#4112yes—
263C3×C22⋊C4—24no—
264#217—24no—
265#217—24no—
266#217—24no—
267#217—24no—
268#217—22no—
269#217—22no—
270#217—42no—
271C4⋊Dic3—22no—
272C4⋊Dic3—22no—
273C4⋊Dic3—21no—
274C4⋊Dic3—21no—
275C4⋊Dic3—21no—
276C4⋊Dic3—21no—
277C4⋊Dic3—21no—
278C4⋊Dic3—81no—
279C4⋊Dic3—81no—
280C4⋊Dic3—81no—
281C4⋊Dic3—81no—
282C4⋊Dic3C2×D412yes—
283Dic3⋊C4—24no—
284Dic3⋊C4—24no—
285Dic3⋊C4—24no—
286C4×Dic3—24no—
287C4×Dic3—21no—
288C4×Dic3—21no—
289C4×Dic3—21no—
290C4×Dic3—21no—
291C4×Dic3—42no—
292C2×C3⋊C8—21no—
293C2×C3⋊C8—21no—
294C2×C3⋊C8—21no—
295C2×C3⋊C8—21no—
296C2×C3⋊C8—21no—
297C2×C3⋊C8—21no—
298C22×Q8—31no—
299C22×Q8—61no—
300C22×D4—21no—
301C22×D4C22×D311yescharacteristic
302C23×C4—21no—
303C23×C4—61no—
304C23×C4C22×D311yescharacteristic
305C22×C8—31no—
306C22×C8—31no—
307C22×C8C22×D311yescharacteristic
308C22×C8C22×D311yescharacteristic
309C4⋊Q8—62no—
310C4⋊Q8—62no—
311#126—68no—
312#124—62no—
313#124—68no—
314#124—62no—
315#123—68no—
316C22⋊Q8—34no—
317C22⋊Q8—68no—
318C22⋊Q8—62no—
319#121—22no—
320#121D1214yes—
321C4×Q8—62no—
322C4×Q8—62no—
323C4×D4—62no—
324C4×D4—62no—
325#117—62no—
326#117—62no—
327#117—62no—
328#117—62no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—22no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—31no—
333C2×C4⋊C4—31no—
334C2×C4⋊C4—31no—
335C2×C4⋊C4—31no—
336C2×C4⋊C4—31no—
337C2×C4⋊C4—61no—
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×C22⋊C4—21no—
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×C42—31no—
356C2×C42—31no—
357C2×C42—61no—
358C2×C42—61no—
359#107—62no—
360#107—62no—
361#106—62no—
362#106—62no—
363#103—62no—
364#103—62no—
365#103—62no—
366#103—62no—
367#102—22no—
368#102—22no—
369#102—62no—
370#102—62no—
371C22⋊C8—22no—
372C22⋊C8—62no—
373#95—61no—
374#95—61no—
375#95—61no—
376#95—61no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—22no—
384C23×C3—41no—
385C23×C3—41no—
386C23×C3—41no—
387C23×C3—41no—
388C23×C3—41no—
389C23×C3—41no—
390C23×C3#12012yes—
391C23×C3C22×D411yescharacteristic
392C3×D4—22no—
393C3×D4—22no—
394C3×D4—41no—
395C3×D4—41no—
396C3×D4—84no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—22no—
400C2×C3×C4—22no—
401C2×C3×C4—22no—
402C2×C3×C4—22no—
403C2×C3×C4—22no—
404C2×C3×C4—22no—
405C2×C3×C4—22no—
406C2×C3×C4—41no—
407C2×C3×C4—41no—
408C2×C3×C4—41no—
409C2×C3×C4—41no—
410C2×C3×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—84no—
413C2×C3×C4#12012yes—
414C2×C3×C4C22×D411yescharacteristic
415C2×C3×C4C22×D411yescharacteristic
416C2×C3×C4C2×#4111yescharacteristic
417C2×C3×C4C2×#4111yescharacteristic
418C2×C3×C42−1+411yescharacteristic
419C2×C3×C42−1+411yescharacteristic
420C2×Dic3—21no—
421C2×Dic3—21no—
422C2×Dic3—21no—
423C2×Dic3—22no—
424C2×Dic3—21no—
425C2×Dic3—22no—
426C2×Dic3—21no—
427C2×Dic3—21no—
428C2×Dic3—21no—
429C2×Dic3—21no—
430C2×Dic3—21no—
431C2×Dic3—21no—
432C2×Dic3—21no—
433C2×Dic3—21no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439C2×Dic3—41no—
440C2×Dic3—41no—
441C2×Dic3—41no—
442C2×Dic3—41no—
443C2×Dic3—41no—
444C2×Dic3—84no—
445Dic6—22no—
446Dic6—22no—
447Dic6—41no—
448Dic6—41no—
449Dic6—84no—
450C3×C8—41no—
451C3×C8—41no—
452C3⋊C8—41no—
453C3⋊C8—41no—
454C24—21no—
455C24D3×D411yescharacteristic
456C2×Q8—32no—
457C2×Q8—64no—
458C2×Q8—61no—
459C2×Q8—61no—
460C2×Q8—61no—
461C2×Q8—61no—
462C2×Q8—61no—
463C2×Q8—241no—
464C2×Q8—241no—
465C2×D4—24no—
466C2×D4—21no—
467C2×D4—21no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—81no—
472C2×D4—81no—
473C2×D4C2×D1212yes—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—24no—
482C22×C4—31no—
483C22×C4—31no—
484C22×C4—31no—
485C22×C4—31no—
486C22×C4—61no—
487C22×C4—61no—
488C22×C4—64no—
489C22×C4—61no—
490C22×C4—61no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—241no—
494C22×C4—241no—
495C22×C4C2×D1212yes—
496C22×C4D3×D411yescharacteristic
497C22×C4D3×D412yes—
498C22×C4C23×D311yescharacteristic
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—61no—
506C2×C8—61no—
507C2×C8—61no—
508C2×C8—61no—
509C2×C8—61no—
510C2×C8—61no—
511C4⋊C4—24no—
512C4⋊C4—21no—
513C4⋊C4—21no—
514C4⋊C4—21no—
515C4⋊C4—32no—
516C4⋊C4—64no—
517C4⋊C4—64no—
518C4⋊C4—62no—
519C4⋊C4—62no—
520C4⋊C4—64no—
521C4⋊C4—61no—
522C4⋊C4—61no—
523C4⋊C4—61no—
524C4⋊C4—61no—
525C4⋊C4—61no—
526C4⋊C4—241no—
527C4⋊C4—241no—
528C4⋊C4—241no—
529C4⋊C4—241no—
530C4⋊C4C2×D1212yes—
531C4⋊C4#23712yes—
532C4⋊C4#23712yes—
533C22⋊C4—24no—
534C22⋊C4—64no—
535C22⋊C4—64no—
536C22⋊C4—64no—
537C22⋊C4—62no—
538C22⋊C4—62no—
539C22⋊C4—64no—
540C22⋊C4—122no—
541C42—64no—
542C42—61no—
543C42—61no—
544C42—61no—
545C42—61no—
546C42—122no—
547C22×C3—22no—
548C22×C3—22no—
549C22×C3—22no—
550C22×C3—41no—
551C22×C3—41no—
552C22×C3—41no—
553C22×C3—41no—
554C22×C3—42no—
555C22×C3—42no—
556C22×C3—42no—
557C22×C3—81no—
558C22×C3—81no—
559C22×C3—81no—
560C22×C3—81no—
561C22×C3C2×#12011yescharacteristic
562C22×C3#52511yescharacteristic
563C22×C3#53611yescharacteristic
564C22×C3#57611yescharacteristic
565C22×C3#57711yescharacteristic
566C22×C3#57711yescharacteristic
567C22×C3#57811yescharacteristic
568C3×C4—22no—
569C3×C4—42no—
570C3×C4—42no—
571C3×C4—42no—
572C3×C4#54711yescharacteristic
573C3×C4#54711yescharacteristic
574C3×C4#54711yescharacteristic
575C3×C4#54711yescharacteristic
576Dic3—42no—
577Dic3—41no—
578Dic3—42no—
579Dic3—41no—
580Dic3—41no—
581Dic3—41no—
582Dic3—81no—
583Dic3—81no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23—21no—
588C23—21no—
589C23—21no—
590C23—22no—
591C23—41no—
592C23—41no—
593C23—41no—
594C23—41no—
595C23—41no—
596C23—41no—
597C23#88212yes—
598C23C2×D3×D411yescharacteristic
599Q8—62no—
600Q8—62no—
601Q8—121no—
602Q8—121no—
603Q8—244no—
604D4—22no—
605D4—22no—
606D4—41no—
607D4—41no—
608D4—84no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—22no—
617C2×C4—22no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—61no—
625C2×C4—61no—
626C2×C4—62no—
627C2×C4—61no—
628C2×C4—61no—
629C2×C4—62no—
630C2×C4—61no—
631C2×C4—61no—
632C2×C4—61no—
633C2×C4—61no—
634C2×C4—61no—
635C2×C4—61no—
636C2×C4—61no—
637C2×C4—61no—
638C2×C4—84no—
639C2×C4—121no—
640C2×C4—121no—
641C2×C4—121no—
642C2×C4—121no—
643C2×C4—121no—
644C2×C4—121no—
645C2×C4—121no—
646C2×C4—121no—
647C2×C4—121no—
648C2×C4—121no—
649C2×C4—244no—
650C2×C4#88212yes—
651C2×C4C22×D1211yescharacteristic
652C2×C4C2×D3×D411yescharacteristic
653C2×C4C2×#23711yescharacteristic
654C2×C4C2×#23711yescharacteristic
655C2×C4#101011yescharacteristic
656C2×C4#101011yescharacteristic
657C8—41no—
658C8—41no—
659C8—121no—
660C8—121no—
661C2×C3—42no—
662C2×C3—82no—
663C2×C3#240611yescharacteristic
664C2×C3#301211yescharacteristic
665C2×C3#301311yescharacteristic
666C2×C3#311811yescharacteristic
667C2×C3#312111yescharacteristic
668C2×C3#324311yescharacteristic
669C2×C3#324411yescharacteristic
670C22—22no—
671C22—22no—
672C22—22no—
673C22—41no—
674C22—41no—
675C22—41no—
676C22—41no—
677C22—42no—
678C22—42no—
679C22—42no—
680C22—81no—
681C22—81no—
682C22—81no—
683C22—81no—
684C22C2×#88211yescharacteristic
685C22#558611yescharacteristic
686C22#564811yescharacteristic
687C22#575711yescharacteristic
688C22#576211yescharacteristic
689C22#577511yescharacteristic
690C22#577911yescharacteristic
691C4—22no—
692C4—42no—
693C4—42no—
694C4—42no—
695C4—121no—
696C4—121no—
697C4—121no—
698C4—121no—
699C4—122no—
700C4—122no—
701C4—241no—
702C4—241no—
703C4#555511yescharacteristic
704C4#555511yescharacteristic
705C4#561111yescharacteristic
706C4#561111yescharacteristic
707C3#2407511yescharacteristic, complemented, Sylow
708C2—42no—
709C2—82no—
710C2#7468711yescharacteristic
711C2#7874711yescharacteristic
712C2#7888711yescharacteristic
713C2#7999211yescharacteristic
714C2#8003211yescharacteristic
715C2#8127911yescharacteristic
716C2#8131711yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1874C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenter#1874D3
hypocenterC3#24075
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
3. #82156C2semidirect
4. C2×#4723C2semidirect
6. #71559C2semidirect
7. #70178C2semidirect
8. #70178C2semidirect
9. #70170C2semidirect
10. #70168C2semidirect
11. #69911C2semidirect
12. #68907C2semidirect
13. #68906C2semidirect
14. #68827C2semidirect
15. #68826C2semidirect
707. C3#24075semidirect

Maximal subgroups

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

subgroupindex
C2×#56032
C2×#50352
#821562
C2×#47232
C3×#18742
#715592
#701782
#701702
#701682
#699112
#689072
#689062
#688272
#688262
#240753