Finite Group Databasebeta

← #10828852

Subgroups — #10828852

Order 768.

1052 conjugacy classes of subgroups.

The subgroup lattice has 706 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5542C211yesmaximal, characteristic
2C3×#2485C211yesmaximal, characteristic
3C2×#4996C211yesmaximal, characteristic, complemented
4C2×#4998C211yesmaximal, characteristic, complemented
5#71610C211yesmaximal, characteristic
6#69632C211yesmaximal, characteristic, complemented
7#69631C211yesmaximal, characteristic, complemented
8#69605C211yesmaximal, characteristic, complemented
9#69604C211yesmaximal, characteristic, complemented
10#69527C211yesmaximal, characteristic, complemented
11#69526C211yesmaximal, characteristic, complemented
12#69514C211yesmaximal, characteristic, complemented
13#69506C211yesmaximal, characteristic, complemented
14#69506C211yesmaximal, characteristic, complemented
15#69503C211yesmaximal, characteristic, complemented
16#24096—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#936—21no—
20C2×#926C2211yescharacteristic
21C2×#926C2211yescharacteristic
22C2×#924C2211yescharacteristic
23C2×#924C2211yescharacteristic
24C23×C3⋊C8C2211yescharacteristic
25#5542—24no—
26#5542—24no—
27C2×#887—21no—
28C2×#879—21no—
29C2×#879—21no—
30C2×#869C2211yescharacteristic
31C2×C4×Dic6C2211yescharacteristic
32C3×#397—21no—
33C3×#394—21no—
34C3×#389C2211yescharacteristic
35C3×#389C2211yescharacteristic
36C3×#388—21no—
37C3×#387—21no—
38C3×#380C2211yescharacteristic
39C3×#380C2211yescharacteristic
40C3×#380C2211yescharacteristic
41C2×#830C2211yescharacteristic
42C2×#830C2211yescharacteristic
43C2×#830C2211yescharacteristic
44#4998—24no—
45#4996—24no—
46C2×#810C2211yescharacteristic
47C2×#810C2211yescharacteristic
48C2×#810C2211yescharacteristic
49C2×#810C2211yescharacteristic
50C2×#808C2211yescharacteristic
51C2×#808C2211yescharacteristic
52C2×#808C2211yescharacteristic
53C2×#807C2211yescharacteristic
54C2×#807C2211yescharacteristic
55C2×#807C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62#4524C2211yescharacteristic
63#4524C2211yescharacteristic
64C2×#556—31no—
65C2×#485—31no—
66C2×#427—31no—
67#2485D311yescharacteristic
68#2058—31no—
69#2057—31no—
70#2036—31no—
71#2035—31no—
72#1892—31no—
73#1869—31no—
74#1865—31no—
75#1863—31no—
76#1854—31no—
77#1854—31no—
78#1825—31no—
79C22×C3×D4D411yescharacteristic
80C23×C3×C4C2311yescharacteristic
81C23×C3×C4C2311yescharacteristic
82C22×Dic6—21no—
83C22×Dic6C2311yescharacteristic
84C3×C4×D4—22no—
85C3×C4×D4—22no—
86C3×#117—22no—
87C3×#117—22no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4D411yescharacteristic
91C2×C3×C4⋊C4C2311yescharacteristic
92C2×C3×C4⋊C4C2311yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C22⋊C4—21no—
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C22⋊C4D411yescharacteristic
100C2×C3×C22⋊C4D411yescharacteristic
101C2×C3×C42—21no—
102C2×C3×C42C2311yescharacteristic
103C2×#217—21no—
104C2×#217—21no—
105C2×#217—21no—
106C2×#217—21no—
107#936—28no—
108#926—22no—
109#926—22no—
110#926—22no—
111#926D414yes—
112C2×C4⋊Dic3—21no—
113C2×C4⋊Dic3C2311yescharacteristic
114C2×C4⋊Dic3C2311yescharacteristic
115C2×C4⋊Dic3C2311yescharacteristic
116#924—22no—
117#924—22no—
118#924—22no—
119#924D414yes—
120C2×Dic3⋊C4—21no—
121C2×Dic3⋊C4—21no—
122C2×Dic3⋊C4—21no—
123C2×C4×Dic3—21no—
124C2×C4×Dic3—21no—
125C2×C4×Dic3—21no—
126C22×C3⋊C8—22no—
127C22×C3⋊C8—22no—
128C22×C3⋊C8—22no—
129C22×C3⋊C8C2311yescharacteristic
130C22×C3⋊C8C2311yescharacteristic
131C22×C3⋊C8C2311yescharacteristic
132C22×C3⋊C8C2311yescharacteristic
133#887—28no—
134#879—28no—
135#879—28no—
136#869—22no—
137#869—22no—
138C4×Dic6—22no—
139C4×Dic6—22no—
140C3×#95—21no—
141C3×#95—21no—
142C3×#95—21no—
143C3×#95—21no—
144C3×#95—21no—
145#830—22no—
146#830—22no—
147#830—22no—
148#810—22no—
149#810—22no—
150#810—22no—
151#810—22no—
152#808—22no—
153#808—22no—
154#808—22no—
155#807—22no—
156#807—22no—
157#807—22no—
158C23×C8—31no—
159#556—64no—
160#556—64no—
161C2×C4⋊Q8—31no—
162C2×#126—61no—
163C2×#126—61no—
164C2×#124—61no—
165C2×C22⋊Q8—31no—
166C2×C22⋊Q8—31no—
167C2×C4×Q8—31no—
168C2×C4×Q8—61no—
169C2×C4×D4C2×D311yescharacteristic
170C2×#117—31no—
171C2×#117—31no—
172C2×#117C2×D311yescharacteristic
173#485—64no—
174#427—64no—
175C2×#107—31no—
176C2×#107—31no—
177C2×#107—31no—
178C2×#106—31no—
179C2×#106—31no—
180C2×#106—31no—
181C2×#103—31no—
182C2×#103—31no—
183C2×#103—31no—
184C2×#103—31no—
185C2×C22⋊C8—31no—
186C2×C22⋊C8—31no—
187C2×C22⋊C8—31no—
188#397—21no—
189#394—21no—
190#389C2×D311yescharacteristic
191#389C2×D311yescharacteristic
192#388—21no—
193#387—21no—
194#380C2×D311yescharacteristic
195#380C2×D311yescharacteristic
196#380C2×D311yescharacteristic
197#340—31no—
198#340—31no—
199#340—31no—
200#340—31no—
201#340—31no—
202#340—31no—
203#336—31no—
204#336—31no—
205C24×C3C2×D411yescharacteristic
206C24×C3C2×D411yescharacteristic
207C2×C3×D4—22no—
208C2×C3×D4—22no—
209C2×C3×D4—41no—
210C22×C3×C4—21no—
211C22×C3×C4—21no—
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×C4—21no—
221C22×C3×C4—21no—
222C22×C3×C4—21no—
223C22×C3×C4—21no—
224C22×C3×C4C2×D412yes—
225C22×C3×C4C2×D412yes—
226C22×C3×C4C2×D411yescharacteristic
227C22×C3×C4C2×D411yescharacteristic
228C22×C3×C4C2×D411yescharacteristic
229C22×C3×C4C2×D411yescharacteristic
230C22×C3×C4C2411yescharacteristic
231C22×Dic3—21no—
232C22×Dic3—21no—
233C22×Dic3—21no—
234C22×Dic3—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—21no—
238C2×Dic6—21no—
239C2×Dic6—24no—
240C2×Dic6—21no—
241C2×Dic6—42no—
242C2×Dic6—42no—
243C2×Dic6C2×D412yes—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—21no—
246C3×C4⋊C4—21no—
247C3×C4⋊C4—21no—
248C3×C4⋊C4—21no—
249C3×C4⋊C4—21no—
250C3×C4⋊C4—21no—
251C3×C4⋊C4—21no—
252C3×C4⋊C4—21no—
253C3×C4⋊C4—42no—
254C3×C4⋊C4—81no—
255C3×C22⋊C4—22no—
256C3×C22⋊C4—22no—
257C3×C22⋊C4—42no—
258C3×C22⋊C4—42no—
259C3×C22⋊C4—81no—
260C3×C22⋊C4—81no—
261C3×C42—21no—
262C3×C42—21no—
263C3×C42—42no—
264#217—24no—
265#217—24no—
266#217—24no—
267#217—24no—
268C4⋊Dic3—24no—
269C4⋊Dic3—21no—
270C4⋊Dic3—21no—
271C4⋊Dic3—21no—
272C4⋊Dic3C2×D412yes—
273C4⋊Dic3C2×D412yes—
274C4⋊Dic3C2×D412yes—
275Dic3⋊C4—24no—
276Dic3⋊C4—24no—
277Dic3⋊C4—24no—
278C4×Dic3—24no—
279C4×Dic3—24no—
280C4×Dic3—24no—
281C2×C3⋊C8—28no—
282C2×C3⋊C8—21no—
283C2×C3⋊C8—21no—
284C2×C3⋊C8—21no—
285C2×C3⋊C8—21no—
286C2×C3⋊C8—21no—
287C2×C3⋊C8—21no—
288C2×C3⋊C8—21no—
289C2×C3⋊C8—21no—
290C2×C3⋊C8—21no—
291C2×C3⋊C8—21no—
292C2×C3⋊C8#4112yes—
293C2×C3⋊C8#4112yes—
294C22×Q8—31no—
295C22×Q8—61no—
296C22×D4#6711yescharacteristic
297C23×C4C22×D311yescharacteristic
298C23×C4C22×D311yescharacteristic
299C22×C8—31no—
300C22×C8—31no—
301C22×C8—31no—
302C22×C8—31no—
303C22×C8—62no—
304C22×C8—62no—
305C22×C8—62no—
306C4⋊Q8—62no—
307C4⋊Q8—62no—
308#126—68no—
309#126—68no—
310#124—68no—
311C22⋊Q8—34no—
312C22⋊Q8—62no—
313C22⋊Q8—62no—
314C22⋊Q8—62no—
315C4×Q8—62no—
316C4×Q8—62no—
317C4×Q8—68no—
318C4×D4—22no—
319C4×D4—22no—
320#117—22no—
321#117—22no—
322#117—34no—
323#117—62no—
324#117—62no—
325#117—62no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—21no—
328C2×C4⋊C4—31no—
329C2×C4⋊C4—31no—
330C2×C4⋊C4—31no—
331C2×C4⋊C4—61no—
332C2×C4⋊C4—61no—
333C2×C4⋊C4—61no—
334C2×C4⋊C4—61no—
335C2×C4⋊C4#6711yescharacteristic
336C2×C4⋊C4C22×D311yescharacteristic
337C2×C4⋊C4C22×D311yescharacteristic
338C2×C4⋊C4C22×D311yescharacteristic
339C2×C4⋊C4C22×D311yescharacteristic
340C2×C22⋊C4—21no—
341C2×C22⋊C4—21no—
342C2×C22⋊C4—21no—
343C2×C22⋊C4—21no—
344C2×C22⋊C4—61no—
345C2×C22⋊C4—61no—
346C2×C22⋊C4—61no—
347C2×C22⋊C4—61no—
348C2×C22⋊C4#6711yescharacteristic
349C2×C22⋊C4#6711yescharacteristic
350C2×C42—21no—
351C2×C42—61no—
352C2×C42—61no—
353C2×C42—61no—
354C2×C42C22×D311yescharacteristic
355#107—62no—
356#107—62no—
357#107—62no—
358#106—62no—
359#106—62no—
360#106—62no—
361#103—62no—
362#103—62no—
363#103—62no—
364#103—62no—
365C22⋊C8—62no—
366C22⋊C8—62no—
367C22⋊C8—62no—
368#95—21no—
369#95—21no—
370#95—21no—
371#95—21no—
372#95—21no—
373C23×C3—21no—
374C23×C3—21no—
375C23×C3—21no—
376C23×C3—21no—
377C23×C3—21no—
378C23×C3—21no—
379C23×C3—21no—
380C23×C3—21no—
381C23×C3—21no—
382C23×C3—21no—
383C23×C3—21no—
384C23×C3—21no—
385C23×C3—21no—
386C23×C3#12012yes—
387C23×C3C22×D411yescharacteristic
388C3×D4—44no—
389C2×C3×C4—21no—
390C2×C3×C4—21no—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
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×C4—41no—
411C2×C3×C4—41no—
412C2×C3×C4—41no—
413C2×C3×C4—41no—
414C2×C3×C4—41no—
415C2×C3×C4—41no—
416C2×C3×C4—41no—
417C2×C3×C4—41no—
418C2×C3×C4—41no—
419C2×C3×C4#12012yes—
420C2×C3×C4#13512yes—
421C2×C3×C4C22×D411yescharacteristic
422C2×C3×C4C22×D411yescharacteristic
423C2×C3×C4C2×#4111yescharacteristic
424C2×C3×C4C2×#4111yescharacteristic
425C2×C3×C42−1+411yescharacteristic
426C2×C3×C42−1+411yescharacteristic
427C2×Dic3—22no—
428C2×Dic3—22no—
429C2×Dic3—22no—
430C2×Dic3—22no—
431C2×Dic3—41no—
432C2×Dic3—41no—
433C2×Dic3—41no—
434C2×Dic3—41no—
435C2×Dic3—41no—
436C2×Dic3—41no—
437C2×Dic3—41no—
438C2×Dic3—41no—
439Dic6—22no—
440Dic6—22no—
441Dic6—48no—
442Dic6—41no—
443Dic6—41no—
444C3⋊C8—22no—
445C3⋊C8—22no—
446C3⋊C8—41no—
447C3⋊C8—41no—
448C24C2×#6711yescharacteristic
449C24C2×#6711yescharacteristic
450C2×Q8—32no—
451C2×Q8—64no—
452C2×Q8—61no—
453C2×Q8—61no—
454C2×Q8—61no—
455C2×Q8—61no—
456C2×Q8—61no—
457C2×Q8—122no—
458C2×Q8—122no—
459C2×D4—22no—
460C2×D4—22no—
461C2×D4—41no—
462C22×C4—21no—
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—21no—
472C22×C4—21no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—61no—
477C22×C4—61no—
478C22×C4—61no—
479C22×C4—61no—
480C22×C4D3×D412yes—
481C22×C4D3×D412yes—
482C22×C4C2×#6711yescharacteristic
483C22×C4C2×#6711yescharacteristic
484C22×C4C2×#6711yescharacteristic
485C22×C4C2×#6711yescharacteristic
486C22×C4C23×D311yescharacteristic
487C2×C8—32no—
488C2×C8—32no—
489C2×C8—68no—
490C2×C8—61no—
491C2×C8—61no—
492C2×C8—61no—
493C2×C8—61no—
494C2×C8—61no—
495C2×C8—61no—
496C2×C8—61no—
497C2×C8—61no—
498C2×C8—61no—
499C2×C8—61no—
500C4⋊C4—22no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—32no—
510C4⋊C4—32no—
511C4⋊C4—32no—
512C4⋊C4—42no—
513C4⋊C4—64no—
514C4⋊C4—64no—
515C4⋊C4—64no—
516C4⋊C4—64no—
517C4⋊C4—61no—
518C4⋊C4—61no—
519C4⋊C4—61no—
520C4⋊C4—81no—
521C22⋊C4—22no—
522C22⋊C4—22no—
523C22⋊C4—42no—
524C22⋊C4—42no—
525C22⋊C4—64no—
526C22⋊C4—64no—
527C22⋊C4—64no—
528C22⋊C4—64no—
529C22⋊C4—81no—
530C22⋊C4—81no—
531C42—21no—
532C42—21no—
533C42—42no—
534C42—64no—
535C42—64no—
536C42—64no—
537C22×C3—22no—
538C22×C3—22no—
539C22×C3—22no—
540C22×C3—41no—
541C22×C3—41no—
542C22×C3—41no—
543C22×C3—41no—
544C22×C3—41no—
545C22×C3—41no—
546C22×C3—41no—
547C22×C3—41no—
548C22×C3—41no—
549C22×C3—41no—
550C22×C3—41no—
551C22×C3C2×#12011yescharacteristic
552C22×C3#54711yescharacteristic
553C22×C3#54711yescharacteristic
554C22×C3C2×#13511yescharacteristic
555C22×C3#57511yescharacteristic
556C22×C3#57711yescharacteristic
557C22×C3#57811yescharacteristic
558C3×C4—22no—
559C3×C4—22no—
560C3×C4—41no—
561C3×C4—41no—
562C3×C4—81no—
563C3×C4—81no—
564C3×C4—81no—
565C3×C4—81no—
566C3×C4#52511yescharacteristic
567C3×C4#53611yescharacteristic
568C3×C4#54711yescharacteristic
569C3×C4#54711yescharacteristic
570Dic3—42no—
571Dic3—42no—
572Dic3—42no—
573Dic3—42no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23—21no—
585C23—21no—
586C23—21no—
587C23#93712yes—
588C23C22×#6711yescharacteristic
589Q8—62no—
590Q8—62no—
591Q8—128no—
592Q8—121no—
593Q8—121no—
594D4—44no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
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—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—62no—
626C2×C4—62no—
627C2×C4—62no—
628C2×C4—62no—
629C2×C4—121no—
630C2×C4—121no—
631C2×C4—121no—
632C2×C4—121no—
633C2×C4—121no—
634C2×C4—121no—
635C2×C4—121no—
636C2×C4—121no—
637C2×C4#93712yes—
638C2×C4#95012yes—
639C2×C4C2×D3×D411yescharacteristic
640C2×C4C2×D3×D411yescharacteristic
641C2×C4D3×#4111yescharacteristic
642C2×C4D3×#4111yescharacteristic
643C2×C4#101011yescharacteristic
644C2×C4#101011yescharacteristic
645C8—62no—
646C8—62no—
647C8—121no—
648C8—121no—
649C2×C3—42no—
650C2×C3—81no—
651C2×C3#240611yescharacteristic
652C2×C3#300211yescharacteristic
653C2×C3#301311yescharacteristic
654C2×C3#334011yescharacteristic
655C2×C3#334211yescharacteristic
656C2×C3#334911yescharacteristic
657C2×C3#335011yescharacteristic
658C22—22no—
659C22—22no—
660C22—22no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—41no—
665C22—41no—
666C22—41no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—41no—
671C22—41no—
672C22C2×#93711yescharacteristic
673C22C2×#95011yescharacteristic
674C22#582211yescharacteristic
675C22#582411yescharacteristic
676C22#582411yescharacteristic
677C22#583611yescharacteristic
678C22#583811yescharacteristic
679C4—22no—
680C4—22no—
681C4—41no—
682C4—41no—
683C4—81no—
684C4—81no—
685C4—81no—
686C4—81no—
687C4—122no—
688C4—122no—
689C4—122no—
690C4—122no—
691C4#564811yescharacteristic
692C4#565511yescharacteristic
693C4#565511yescharacteristic
694C4#565611yescharacteristic
695C3#2409611yescharacteristic, complemented, Sylow
696C2—42no—
697C2—81no—
698C2#8284111yescharacteristic
699C2#8292611yescharacteristic
700C2#8342711yescharacteristic
701C2#8359511yescharacteristic
702C2#8361111yescharacteristic
703C2#8364611yescharacteristic
704C2#8365911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2485C2
socleC23×C3C22×D4
prime radicalC22×C3×D4D4
hypercenter#2485D3
hypocenterC3#24096
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2485

Complements

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

NH ≅ G/Ntype
3. C2×#4996C2semidirect
4. C2×#4998C2semidirect
6. #69632C2semidirect
7. #69631C2semidirect
8. #69605C2semidirect
9. #69604C2semidirect
10. #69527C2semidirect
11. #69526C2semidirect
12. #69514C2semidirect
13. #69506C2semidirect
14. #69506C2semidirect
15. #69503C2semidirect
695. C3#24096semidirect

Maximal subgroups

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

subgroupindex
C2×#55422
C3×#24852
C2×#49962
C2×#49982
#716102
#696322
#696312
#696052
#696042
#695272
#695262
#695142
#695062
#695032
#240963