Finite Group Databasebeta

← #10896407

Subgroups — #10896407

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
1#82953C211yesmaximal, characteristic
2#73627C211yesmaximal, characteristic, complemented
3C3×#1983C211yesmaximal, characteristic
4#71721C211yesmaximal, characteristic
5#70454C211yesmaximal, characteristic, complemented
6#70444C211yesmaximal, characteristic, complemented
7#70252C211yesmaximal, characteristic, complemented
8#70238C211yesmaximal, characteristic, complemented
9#70235C211yesmaximal, characteristic, complemented
10#70231C211yesmaximal, characteristic, complemented
11#70182C211yesmaximal, characteristic
12#70170C211yesmaximal, characteristic
13#70167C211yesmaximal, characteristic
14#70163C211yesmaximal, characteristic
15#69466C211yesmaximal, characteristic, complemented
16#29267—31nomaximal, Sylow
17C2×C3×#124C2211yescharacteristic
18C2×Q8×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C2×#888C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×#103C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C2×C3×#102C2211yescharacteristic
26C3×C4×C4⋊C4C2211yescharacteristic
27#5229C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C2×#835C2211yescharacteristic
30#5221C2211yescharacteristic
31C2×#832C2211yescharacteristic
32C2×#832C2211yescharacteristic
33C2×#817C2211yescharacteristic
34C2×#817C2211yescharacteristic
35#4984C2211yescharacteristic
36#4977C2211yescharacteristic
37#4972C2211yescharacteristic
38C2×#807C2211yescharacteristic
39C2×#807C2211yescharacteristic
40#4951—21no—
41#4951—21no—
42#4950—21no—
43#4950—21no—
44#4935C2211yescharacteristic
45C4×C4⋊Dic3C2211yescharacteristic
46#4933—21no—
47#4932C2211yescharacteristic
48#4929—21no—
49C2×C4⋊(C3⋊C8)C2211yescharacteristic
50#4655—21no—
51#4646—21no—
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4550C2211yescharacteristic
55#4550C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#2331—31no—
61#2326—31no—
62#2097—31no—
63#2092—31no—
64#2024—31no—
65#2023—31no—
66#1983—31no—
67#1983D311yescharacteristic
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×Dic3—21no—
78C22×Dic6—21no—
79C22×C3×C8C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C3×#124—22no—
82C3×#124Q814yescomplemented
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C22⋊C4—21no—
86C2×C3×C22⋊C4—21no—
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92Q8×Dic3—41no—
93Q8×Dic3—41no—
94D4×Dic3—41no—
95D4×Dic3—41no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—21no—
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—
108C2×C4×Dic3C2311yescharacteristic
109C2×C4×Dic3C2311yescharacteristic
110C22×C3⋊C8C2311yescharacteristic
111C22×C3⋊C8C2311yescharacteristic
112#888—22no—
113#888—22no—
114C3×C4⋊C8—22no—
115C3×#103—22no—
116C3×#103—41no—
117C3×#102—22no—
118C3×#102—41no—
119C3×#95—21no—
120#835—22no—
121#835—41no—
122#832—22no—
123#832—41no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95D411yescharacteristic
132C3⋊#95D411yescharacteristic
133C3⋊#95D411yescharacteristic
134C3⋊#95D411yescharacteristic
135#817—41no—
136#817—41no—
137#807—41no—
138#807—41no—
139C4⋊(C3⋊C8)—22no—
140C2×C4⋊Q8—31no—
141C2×#124C2×D311yescharacteristic
142C2×C4×Q8—31no—
143C2×C4×D4—31no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106—31no—
148C2×C4⋊C8—31no—
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103C2×D311yescharacteristic
153C2×#103C2×D311yescharacteristic
154C2×#102—31no—
155C2×#102—31no—
156C2×#102C2×D311yescharacteristic
157C2×#102C2×D311yescharacteristic
158#391—31no—
159#391—61no—
160#391—61no—
161#390—31no—
162#389—31no—
163#389—31no—
164#388—61no—
165#388—61no—
166#387—61no—
167#387—61no—
168#385—61no—
169#385—61no—
170#384—31no—
171#382—31no—
172C4×C4⋊C4—31no—
173C4×C4⋊C4C2×D311yescharacteristic
174#376—31no—
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×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2411yescharacteristic
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—42no—
207C22×Dic3—81no—
208C22×Dic3—81no—
209C22×Dic3C2×D411yescharacteristic
210C22×Dic3C2×D411yescharacteristic
211C22×Dic3C2×D411yescharacteristic
212C22×Dic3C2×D411yescharacteristic
213C2×Dic6—24no—
214C2×Dic6—81no—
215C2×Dic6—81no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8—21no—
219C2×C3×C8—21no—
220C2×C3×C8—21no—
221C2×C3×C8—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4#4112yes—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—24no—
228C3×C42—81no—
229C3×C42C2×Q812yes—
230C3×C42#4112yes—
231#217—42no—
232#217—81no—
233#217—81no—
234C4⋊Dic3—21no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—42no—
242C4⋊Dic3—81no—
243C4⋊Dic3—81no—
244C4⋊Dic3—81no—
245C4⋊Dic3#4112yes—
246Dic3⋊C4—24no—
247Dic3⋊C4—81no—
248C4×Dic3—21no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—42no—
253C4×Dic3—81no—
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—61no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8C22×D311yescharacteristic
267C22×C8C22×D311yescharacteristic
268C4⋊Q8—62no—
269C4⋊Q8—62no—
270#124—22no—
271#124Dic614yescomplemented
272C4×Q8—121no—
273C4×Q8—121no—
274C4×D4—121no—
275C4×D4—121no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—61no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
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—61no—
291C2×C22⋊C4—61no—
292C2×C22⋊C4—61no—
293C2×C42—21no—
294C2×C42—31no—
295C2×C42—31no—
296C2×C42—61no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299#107—121no—
300#107—121no—
301#106—121no—
302#106—121no—
303C4⋊C8—22no—
304C4⋊C8—62no—
305#103—22no—
306#103—41no—
307#103—62no—
308#103—121no—
309#102—22no—
310#102—41no—
311#102—62no—
312#102—121no—
313#95—21no—
314#95—31no—
315#95—31no—
316#95—31no—
317#95—31no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
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—22no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4—41no—
350C2×C3×C4—41no—
351C2×C3×C4—41no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C22⋊Q812yes—
355C2×C3×C4C22⋊Q812yes—
356C2×C3×C4C22×Q811yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4C2×#4111yescharacteristic
359C2×C3×C4C2×#4111yescharacteristic
360C2×C3×C4H(5,𝔽2)11yescharacteristic
361C2×C3×C4H(5,𝔽2)11yescharacteristic
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—22no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—21no—
368C2×Dic3—21no—
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—21no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—21no—
375C2×Dic3—41no—
376C2×Dic3—41no—
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—84no—
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—31no—
412C22×C4—31no—
413C22×C4—31no—
414C22×C4—31no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—61no—
418C22×C4—61no—
419C22×C4—122no—
420C22×C4—241no—
421C22×C4—241no—
422C22×C4D3×D411yescharacteristic
423C22×C4D3×D411yescharacteristic
424C22×C4C23×D311yescharacteristic
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
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—32no—
441C4⋊C4—64no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—122no—
450C4⋊C4—241no—
451C4⋊C4—241no—
452C4⋊C4—241no—
453C4⋊C4—241no—
454C4⋊C4#23712yes—
455C22⋊C4—24no—
456C22⋊C4—24no—
457C22⋊C4—122no—
458C22⋊C4—241no—
459C22⋊C4—241no—
460C42—61no—
461C42—61no—
462C42—61no—
463C42—61no—
464C42—81no—
465C42—122no—
466C42—241no—
467C42C2×Dic612yes—
468C42#23712yes—
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—42no—
487C3×C4—81no—
488C3×C4#55411yescharacteristic
489C3×C4#55411yescharacteristic
490C3×C4#55911yescharacteristic
491C3×C4#55911yescharacteristic
492Dic3—41no—
493Dic3—42no—
494Dic3—41no—
495Dic3—41no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500C23—22no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23C2×D3×D411yescharacteristic
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—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—62no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—121no—
544C2×C4—121no—
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—244no—
555C2×C4#87812yes—
556C2×C4#87812yes—
557C2×C4C22×Dic611yescharacteristic
558C2×C4C2×#23711yescharacteristic
559C2×C4C2×#23711yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4D3×#4111yescharacteristic
562C2×C4#100911yescharacteristic
563C8—41no—
564C8—41no—
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×#87811yescharacteristic
583C22#560611yescharacteristic
584C22#560711yescharacteristic
585C22#577011yescharacteristic
586C22#577011yescharacteristic
587C22#577411yescharacteristic
588C22#577911yescharacteristic
589C4—22no—
590C4—22no—
591C4—42no—
592C4—42no—
593C4—81no—
594C4—121no—
595C4—122no—
596C4—121no—
597C4—121no—
598C4—121no—
599C4—241no—
600C4—241no—
601C4—241no—
602C4#553911yescharacteristic
603C4#554311yescharacteristic
604C4#568111yescharacteristic
605C4#571311yescharacteristic
606C3#2926711yescharacteristic, complemented, Sylow
607C2—82no—
608C2#7636411yescharacteristic
609C2#7882411yescharacteristic
610C2#7887211yescharacteristic
611C2#8050911yescharacteristic
612C2#8054611yescharacteristic
613C2#8057211yescharacteristic
614C2#8061111yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1983C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1983D3
hypocenterC3#29267
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1983

Complements

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

NH ≅ G/Ntype
2. #73627C2semidirect
5. #70454C2semidirect
6. #70444C2semidirect
7. #70252C2semidirect
8. #70238C2semidirect
9. #70235C2semidirect
10. #70231C2semidirect
15. #69466C2semidirect
82. 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
#829532
#736272
C3×#19832
#717212
#704542
#704442
#702522
#702382
#702352
#702312
#701822
#701702
#701672
#701632
#694662
#292673