Finite Group Databasebeta

← #10896403

Subgroups — #10896403

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2326C211yesmaximal, characteristic
2#73391C211yesmaximal, characteristic, complemented
3#71721C211yesmaximal, characteristic
4#71721C211yesmaximal, characteristic
5#71629C211yesmaximal, characteristic, complemented
6#71628C211yesmaximal, characteristic, complemented
7#71607C211yesmaximal, characteristic
8#71606C211yesmaximal, characteristic
9#69686C211yesmaximal, characteristic, complemented
10#69668C211yesmaximal, characteristic, complemented
11#69654C211yesmaximal, characteristic
12#69638C211yesmaximal, characteristic
13#69568C211yesmaximal, characteristic, complemented
14#69567C211yesmaximal, characteristic, complemented
15#69437C211yesmaximal, characteristic, complemented
16#29267—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C2×#869C2211yescharacteristic
21C3×#391C2211yescharacteristic
22C3×#390C2211yescharacteristic
23C3×#388—21no—
24C3×#388—21no—
25C3×#385—21no—
26C3×#385—21no—
27C3×#384C2211yescharacteristic
28C3×#376C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#835C2211yescharacteristic
31C2×#835C2211yescharacteristic
32C2×#835C2211yescharacteristic
33C2×#832C2211yescharacteristic
34C2×#832C2211yescharacteristic
35C2×#832C2211yescharacteristic
36C2×#832C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#808C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41#4936C2211yescharacteristic
42C4×C4⋊Dic3C2211yescharacteristic
43C4×C4⋊Dic3C2211yescharacteristic
44#4928C2211yescharacteristic
45#4928C2211yescharacteristic
46C2×C4⋊(C3⋊C8)C2211yescharacteristic
47C2×C4⋊(C3⋊C8)C2211yescharacteristic
48#4654—21no—
49#4654—21no—
50#4646—21no—
51#4646—21no—
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2331—31no—
61#2326D311yescharacteristic
62#2097—31no—
63#2092—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983—31no—
68#1948—31no—
69#1947—31no—
70#1865—31no—
71#1864—31no—
72#1863—31no—
73#1862—31no—
74#1832—31no—
75C22×C3×Q8C2311yescharacteristic
76C22×C3×D4C2311yescharacteristic
77C23×C3×C4—21no—
78C22×Dic6—21no—
79C3×#124—22no—
80C3×#124Q814yescomplemented
81C3×C4×Q8—41no—
82C3×C4×Q8—41no—
83C3×C4×D4—41no—
84C3×C4×D4—41no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C22⋊C4—21no—
95C2×C3×C42—21no—
96C2×C3×C42C2311yescharacteristic
97C2×C3×C42C2311yescharacteristic
98C2×C3×C42C2311yescharacteristic
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×Dic3⋊C4—21no—
105C2×Dic3⋊C4—21no—
106C2×C4×Dic3—21no—
107C2×C4×Dic3—21no—
108C22×C3⋊C8C2311yescharacteristic
109C22×C3⋊C8C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#869—22no—
113#869—22no—
114C3×#95—21no—
115C3×#95—21no—
116C3×#95D411yescharacteristic
117C3×#95D411yescharacteristic
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120#835—22no—
121#835—22no—
122#835—41no—
123#835—41no—
124#832—22no—
125#832—22no—
126#832—41no—
127#832—41no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#808—41no—
135#808—41no—
136#807—41no—
137#807—41no—
138C4⋊(C3⋊C8)—22no—
139C4⋊(C3⋊C8)—22no—
140C2×C4⋊Q8—31no—
141C2×#124C2×D311yescharacteristic
142C2×C4×Q8C2×D311yescharacteristic
143C2×C4×D4C2×D311yescharacteristic
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8—31no—
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×#102—31no—
155C2×#102—31no—
156C2×#102—31no—
157C2×#102—31no—
158#391—61no—
159#391—61no—
160#391C2×D311yescharacteristic
161#390C2×D311yescharacteristic
162#389—31no—
163#389—31no—
164#388—21no—
165#388—21no—
166#387—61no—
167#387—61no—
168#385—21no—
169#385—21no—
170#384C2×D311yescharacteristic
171#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4—31no—
174#376C2×D311yescharacteristic
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#336—31no—
180#336—31no—
181#336—31no—
182#336—31no—
183C24×C3—21no—
184C2×C3×Q8—21no—
185C2×C3×Q8—21no—
186C2×C3×Q8—21no—
187C2×C3×Q8—21no—
188C2×C3×Q8—21no—
189C2×C3×Q8C2×Q812yes—
190C2×C3×D4—21no—
191C2×C3×D4—21no—
192C2×C3×D4—21no—
193C2×C3×D4—21no—
194C2×C3×D4—21no—
195C2×C3×D4C2×Q812yes—
196C22×C3×C4—21no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—42no—
200C22×C3×C4—81no—
201C22×C3×C4—81no—
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2×D411yescharacteristic
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×D411yescharacteristic
208C22×C3×C4C2411yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C2×Dic6—24no—
214C2×Dic6—81no—
215C2×Dic6—81no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—81no—
222C3×C4⋊C4—81no—
223C3×C22⋊C4—24no—
224C3×C22⋊C4—24no—
225C3×C22⋊C4—42no—
226C3×C22⋊C4—81no—
227C3×C22⋊C4—81no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—42no—
233C3×C42C2×Q812yes—
234C3×C42#4112yes—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3#4112yes—
243C4⋊Dic3#4112yes—
244Dic3⋊C4—81no—
245Dic3⋊C4—81no—
246C4×Dic3—81no—
247C4×Dic3—81no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—21no—
251C2×C3⋊C8—21no—
252C2×C3⋊C8—21no—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—21no—
260C22×Q8—61no—
261C22×Q8C22×D311yescharacteristic
262C22×D4C22×D311yescharacteristic
263C23×C4—21no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—31no—
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#124—22no—
271#124Dic614yescomplemented
272C4×Q8—41no—
273C4×Q8—41no—
274C4×D4—41no—
275C4×D4—41no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4C22×D311yescharacteristic
287C2×C4⋊C4C22×D311yescharacteristic
288C2×C22⋊C4—21no—
289C2×C22⋊C4—21no—
290C2×C22⋊C4—21no—
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C42—21no—
294C2×C42—61no—
295C2×C42—61no—
296C2×C42C22×D311yescharacteristic
297C2×C42C22×D311yescharacteristic
298C2×C42C22×D311yescharacteristic
299#107—121no—
300#107—121no—
301#106—121no—
302#106—121no—
303C4⋊C8—62no—
304C4⋊C8—62no—
305#103—62no—
306#103—62no—
307#103—121no—
308#103—121no—
309#102—62no—
310#102—62no—
311#102—121no—
312#102—121no—
313#95—21no—
314#95—21no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95#6711yescharacteristic
322#95#6711yescharacteristic
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3—22no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3C22×D411yescharacteristic
333C3×Q8—22no—
334C3×Q8—22no—
335C3×Q8—41no—
336C3×Q8—41no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—41no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—22no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—21no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—21no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—84no—
367C2×C3×C4C22⋊Q812yes—
368C2×C3×C4C22⋊Q812yes—
369C2×C3×C4C22×Q811yescharacteristic
370C2×C3×C4C2×#4111yescharacteristic
371C2×C3×C4C2×#4111yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4H(5,𝔽2)11yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—41no—
386C2×Dic3—41no—
387Dic6—84no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C3⋊C8—41no—
392C24—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—64no—
399C2×Q8—241no—
400C2×Q8—241no—
401C2×Q8C2×Dic612yes—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4C2×Dic612yes—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—42no—
412C22×C4—61no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—81no—
417C22×C4—81no—
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—61no—
426C2×C8—61no—
427C2×C8—61no—
428C2×C8—61no—
429C2×C8—61no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—61no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—42no—
444C4⋊C4—64no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—81no—
452C4⋊C4—81no—
453C4⋊C4—241no—
454C4⋊C4—241no—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—42no—
458C22⋊C4—81no—
459C22⋊C4—81no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—42no—
465C42—241no—
466C42—241no—
467C42C2×Dic612yes—
468C42#23512yes—
469C22×C3—42no—
470C22×C3—42no—
471C22×C3—42no—
472C22×C3—81no—
473C22×C3—81no—
474C22×C3—81no—
475C22×C3—81no—
476C22×C3C2×C22⋊Q811yescharacteristic
477C22×C3#52511yescharacteristic
478C22×C3#53511yescharacteristic
479C22×C3#57611yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57711yescharacteristic
482C22×C3#57811yescharacteristic
483C3×C4—22no—
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—81no—
491C3×C4—81no—
492C3×C4#55411yescharacteristic
493C3×C4#55411yescharacteristic
494C3×C4#55911yescharacteristic
495C3×C4#55911yescharacteristic
496Dic3—42no—
497Dic3—42no—
498Dic3—81no—
499Dic3—81no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C22×#6711yescharacteristic
508Q8—22no—
509Q8—22no—
510Q8—41no—
511Q8—41no—
512Q8—244no—
513D4—22no—
514D4—22no—
515D4—41no—
516D4—41no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—84no—
545C2×C4—121no—
546C2×C4—121no—
547C2×C4—121no—
548C2×C4—121no—
549C2×C4—121no—
550C2×C4—121no—
551C2×C4—121no—
552C2×C4—121no—
553C2×C4—121no—
554C2×C4—121no—
555C2×C4#92412yes—
556C2×C4#92412yes—
557C2×C4C22×Dic611yescharacteristic
558C2×C4C2×#23511yescharacteristic
559C2×C4#100411yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4D3×#4111yescharacteristic
563C8—121no—
564C8—121no—
565C8—121no—
566C8—121no—
567C2×C3—82no—
568C2×C3#258011yescharacteristic
569C2×C3#308311yescharacteristic
570C2×C3#308411yescharacteristic
571C2×C3#311811yescharacteristic
572C2×C3#312111yescharacteristic
573C2×C3#322411yescharacteristic
574C2×C3#322511yescharacteristic
575C22—42no—
576C22—42no—
577C22—42no—
578C22—81no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22C2×#92411yescharacteristic
583C22#580511yescharacteristic
584C22#582811yescharacteristic
585C22#583511yescharacteristic
586C22#583611yescharacteristic
587C22#583711yescharacteristic
588C22#583811yescharacteristic
589C4—22no—
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—122no—
599C4—122no—
600C4—241no—
601C4—241no—
602C4#553911yescharacteristic
603C4#553911yescharacteristic
604C4#555911yescharacteristic
605C4#555911yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—82no—
608C2#8238811yescharacteristic
609C2#8255911yescharacteristic
610C2#8256011yescharacteristic
611C2#8338911yescharacteristic
612C2#8341011yescharacteristic
613C2#8346011yescharacteristic
614C2#8346711yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2326C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2326D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
2. #73391C2semidirect
5. #71629C2semidirect
6. #71628C2semidirect
9. #69686C2semidirect
10. #69668C2semidirect
13. #69568C2semidirect
14. #69567C2semidirect
15. #69437C2semidirect
80. C3×#124Q8semidirect
271. #124Dic6semidirect
606. C3#29267semidirect

Maximal subgroups

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

subgroupindex
C3×#23262
#733912
#717212
#716292
#716282
#716072
#716062
#696862
#696682
#696542
#696382
#695682
#695672
#694372
#292673