Finite Group Databasebeta

← #10875986

Subgroups — #10875986

Order 768.

782 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Dic3C211yesmaximal, characteristic
2#76046C211yesmaximal, characteristic, complemented
3C3×#1864C211yesmaximal, characteristic
4#71612C211yesmaximal, characteristic
5#70589C211yesmaximal, characteristic, complemented
6#70558C211yesmaximal, characteristic, complemented
7#70171C211yesmaximal, characteristic
8#70171C211yesmaximal, characteristic
9#70153C211yesmaximal, characteristic
10#70152C211yesmaximal, characteristic
11#69612C211yesmaximal, characteristic, complemented
12#68855C211yesmaximal, characteristic, complemented
13#68842C211yesmaximal, characteristic, complemented
14#68636C211yesmaximal, characteristic, complemented
15#68613C211yesmaximal, characteristic, complemented
16#27573—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×D4×Dic3C2211yescharacteristic
19C2×D4×Dic3C2211yescharacteristic
20C22×#214C2211yescharacteristic
21C22×C4⋊Dic3—21no—
22C2×C3×#106C2211yescharacteristic
23C2×C3×#102C2211yescharacteristic
24C2×C3×#102C2211yescharacteristic
25C3×#389C2211yescharacteristic
26C2×#832C2211yescharacteristic
27#5212—21no—
28C2×#816C2211yescharacteristic
29C2×#816C2211yescharacteristic
30C2×#815C2211yescharacteristic
31C2×Dic3⋊C8C2211yescharacteristic
32#4985—21no—
33#4984C2211yescharacteristic
34#4984C2211yescharacteristic
35#4978—21no—
36#4975C2211yescharacteristic
37Dic3×C4⋊C4C2211yescharacteristic
38#4973C2211yescharacteristic
39#4973C2211yescharacteristic
40C2×#810C2211yescharacteristic
41#4950—21no—
42#4950—21no—
43#4935C2211yescharacteristic
44C4×C4⋊Dic3C2211yescharacteristic
45#4929C2211yescharacteristic
46#4660—21no—
47#4654—21no—
48C3×#340C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4550C2211yescharacteristic
53#4550C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4524C2211yescharacteristic
60#2511—31no—
61D4×C4⋊C4—31no—
62#2100—31no—
63#2099—31no—
64#2072—31no—
65#2066—31no—
66#1935—31no—
67#1920—31no—
68#1917—31no—
69#1877—31no—
70#1877—31no—
71#1877—31no—
72#1864D311yescharacteristic
73#1802—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×Dic3—21no—
78C23×Dic3—21no—
79C22×Dic6C2311yescharacteristic
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C22⋊C4—21no—
89C2×C3×C42C2311yescharacteristic
90C2×#217—21no—
91C2×#217—21no—
92D4×Dic3—22no—
93D4×Dic3—22no—
94D4×Dic3—41no—
95D4×Dic3—41no—
96C2×#214—22no—
97C2×#214—22no—
98C2×#214—22no—
99C2×C4⋊Dic3—21no—
100C2×C4⋊Dic3—21no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—44no—
104C2×C4⋊Dic3D411yescharacteristic
105C2×C4⋊Dic3D411yescharacteristic
106C2×C4⋊Dic3C2311yescharacteristic
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C2×C4×Dic3C2311yescharacteristic
113C2×C4×Dic3C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#106—22no—
117C3×#102—22no—
118C3×#102—22no—
119C3×#95—21no—
120#832—22no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95D411yescharacteristic
131C3⋊#95D411yescharacteristic
132#816—41no—
133#816—41no—
134#815—41no—
135Dic3⋊C8—41no—
136#810—22no—
137C22×SD16—31no—
138C2×C4×D4—31no—
139C2×C4×D4—31no—
140C2×C4×D4C2×D311yescharacteristic
141C22×C4⋊C4—61no—
142C2×#106C2×D311yescharacteristic
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102C2×D311yescharacteristic
149C2×#102C2×D311yescharacteristic
150C2×#97—31no—
151#400—61no—
152#400—61no—
153#391—31no—
154#391—31no—
155#391—31no—
156#389—31no—
157#389—31no—
158#389C2×D311yescharacteristic
159#387—61no—
160#387—61no—
161#385—61no—
162#385—61no—
163#384—31no—
164#384—31no—
165#380—61no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4—31no—
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340C2×D311yescharacteristic
176#336—31no—
177#336—31no—
178#336—31no—
179#336C2×D311yescharacteristic
180C24×C3—21no—
181C2×C3×D4—21no—
182C2×C3×D4—21no—
183C2×C3×D4—21no—
184C2×C3×D4#4112yes—
185C2×C3×D4#4112yes—
186C2×C3×D4#4112yes—
187C22×C3×C4—24no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—42no—
192C22×C3×C4—42no—
193C22×C3×C4C2×D411yescharacteristic
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—42no—
201C22×Dic3—42no—
202C22×Dic3—42no—
203C22×Dic3—42no—
204C22×Dic3—81no—
205C22×Dic3—81no—
206C22×Dic3C2×D411yescharacteristic
207C22×Dic3C2×D411yescharacteristic
208C22×Dic3C2×D411yescharacteristic
209C22×Dic3C2×D411yescharacteristic
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
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—21no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—81no—
228C3×C4⋊C4#4112yes—
229C3×C22⋊C4—24no—
230C3×C42—21no—
231C3×C42—21no—
232#217—42no—
233#217—42no—
234#214—44no—
235C4⋊Dic3—22no—
236C4⋊Dic3—22no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—42no—
244C4⋊Dic3—42no—
245C4⋊Dic3—42no—
246C4⋊Dic3—81no—
247Dic3⋊C4—42no—
248Dic3⋊C4—81no—
249C4×Dic3—21no—
250C4×Dic3—21no—
251C4×Dic3—21no—
252C4×Dic3—21no—
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—31no—
261C22×D4C22×D311yescharacteristic
262C23×C4—21no—
263C23×C4—61no—
264C23×C4—61no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—62no—
275C4×D4—62no—
276C4×D4—121no—
277C4×D4—121no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—61no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—124no—
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C42—31no—
298C2×C42—31no—
299C2×C42—61no—
300C2×C42C22×D311yescharacteristic
301#106—22no—
302C4⋊C8—121no—
303#103—62no—
304#103—121no—
305#103—121no—
306#102—22no—
307#102—22no—
308#102—62no—
309#97—121no—
310#95—21no—
311#95—31no—
312#95—31no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3—22no—
323C23×C3—22no—
324C23×C3—22no—
325C23×C3—41no—
326C23×C3—41no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3C22×D411yescharacteristic
330C3×D4—22no—
331C3×D4—22no—
332C3×D4—22no—
333C3×D4#12614yes—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—22no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—48no—
341C2×C3×C4—41no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4—41no—
349C2×C3×C4#13712yes—
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354C2×C3×C4H(5,𝔽2)11yescharacteristic
355C2×C3×C42−1+411yescharacteristic
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—21no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—21no—
365C2×Dic3—21no—
366C2×Dic3—21no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3—41no—
373C2×Dic3—41no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—84no—
380C2×Dic3—84no—
381C2×Dic3AGL(1,ℤ/8)12yes—
382Dic6—41no—
383Dic6—41no—
384Dic6—41no—
385Dic6—41no—
386C3×C8—41no—
387C3×C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C24—21no—
391C2×Q8—61no—
392C2×Q8—61no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4#23512yes—
401C2×D4#23712yes—
402C2×D4#23912yes—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—122no—
418C22×C4—122no—
419C22×C4—122no—
420C22×C4—122no—
421C22×C4—241no—
422C22×C4—241no—
423C22×C4D3×D411yescharacteristic
424C22×C4D3×D411yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—124no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—61no—
434C2×C8—61no—
435C2×C8—61no—
436C2×C8—61no—
437C2×C8—61no—
438C2×C8—61no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—62no—
445C4⋊C4—62no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—81no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4#23712yes—
460C22⋊C4—24no—
461C22⋊C4—122no—
462C22⋊C4—122no—
463C42—21no—
464C42—21no—
465C42—61no—
466C42—61no—
467C42—61no—
468C42—61no—
469C42—241no—
470C22×C3—24no—
471C22×C3—42no—
472C22×C3—42no—
473C22×C3—42no—
474C22×C3—42no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3#52511yescharacteristic
478C22×C3#54611yescharacteristic
479C22×C3#54811yescharacteristic
480C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
481C22×C3C2×#13711yescharacteristic
482C22×C3#57711yescharacteristic
483C22×C3#57711yescharacteristic
484C3×C4—22no—
485C3×C4—42no—
486C3×C4—41no—
487C3×C4—81no—
488C3×C4—81no—
489C3×C4C2×#12611yescharacteristic
490C3×C4#53911yescharacteristic
491C3×C4#55211yescharacteristic
492C3×C4#55311yescharacteristic
493Dic3—22no—
494Dic3—41no—
495Dic3—41no—
496Dic3—41no—
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C2×D3×D411yescharacteristic
509Q8—121no—
510Q8—121no—
511Q8—121no—
512Q8—121no—
513D4—22no—
514D4—22no—
515D4—22no—
516D4#89814yes—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—32no—
524C2×C4—48no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—61no—
534C2×C4—61no—
535C2×C4—61no—
536C2×C4—61no—
537C2×C4—61no—
538C2×C4—61no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—61no—
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—121no—
555C2×C4—121no—
556C2×C4—244no—
557C2×C4—244no—
558C2×C4#91512yes—
559C2×C4C2×#23511yescharacteristic
560C2×C4C2×#23711yescharacteristic
561C2×C4C2×#23711yescharacteristic
562C2×C4C2×#23911yescharacteristic
563C2×C4#100911yescharacteristic
564C2×C4#101011yescharacteristic
565C8—41no—
566C8—41no—
567C8—121no—
568C8—121no—
569C2×C3—44no—
570C2×C3#248111yescharacteristic
571C2×C3#311011yescharacteristic
572C2×C3#312511yescharacteristic
573C2×C3#331011yescharacteristic
574C2×C3#333111yescharacteristic
575C2×C3#337211yescharacteristic
576C2×C3#337311yescharacteristic
577C22—24no—
578C22—42no—
579C22—42no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22#555211yescharacteristic
585C22#558611yescharacteristic
586C22#563011yescharacteristic
587C22C2×#91511yescharacteristic
588C22#577011yescharacteristic
589C22D3×AGL(1,ℤ/8)11yescharacteristic
590C22#577511yescharacteristic
591C4—22no—
592C4—42no—
593C4—41no—
594C4—62no—
595C4—81no—
596C4—81no—
597C4—121no—
598C4—121no—
599C4—121no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4—241no—
604C4C2×#89811yescharacteristic
605C4#556511yescharacteristic
606C4#564211yescharacteristic
607C4#566511yescharacteristic
608C3#2757311yescharacteristic, complemented, Sylow
609C2—44no—
610C2#7445411yescharacteristic
611C2#7962811yescharacteristic
612C2#7985311yescharacteristic
613C2#8007511yescharacteristic
614C2#8013411yescharacteristic
615C2#8108011yescharacteristic
616C2#8118911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1864C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#1864D3
hypocenterC3#27573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. #76046C2semidirect
5. #70589C2semidirect
6. #70558C2semidirect
11. #69612C2semidirect
12. #68855C2semidirect
13. #68842C2semidirect
14. #68636C2semidirect
15. #68613C2semidirect
608. C3#27573semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Dic32
#760462
C3×#18642
#716122
#705892
#705582
#701712
#701532
#701522
#696122
#688552
#688422
#686362
#686132
#275733