Finite Group Databasebeta

← #10875983

Subgroups — #10875983

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
1C3×D4×C4⋊C4C211yesmaximal, characteristic
2#73690C211yesmaximal, characteristic, complemented
3#71612C211yesmaximal, characteristic
4#71612C211yesmaximal, characteristic
5#71612C211yesmaximal, characteristic
6#71609C211yesmaximal, characteristic
7#71607C211yesmaximal, characteristic
8#71604C211yesmaximal, characteristic
9#69634C211yesmaximal, characteristic, complemented
10#69616C211yesmaximal, characteristic, complemented
11#69611C211yesmaximal, characteristic, complemented
12#69602C211yesmaximal, characteristic, complemented
13#69589C211yesmaximal, characteristic, complemented
14#69572C211yesmaximal, characteristic, complemented
15#69483C211yesmaximal, characteristic, complemented
16#27573—31nomaximal, Sylow
17C2×C3×C4×D4C2211yescharacteristic
18C2×C3×C4×D4C2211yescharacteristic
19C2×C3×C4×D4C2211yescharacteristic
20C22×C3×C4⋊C4—21no—
21C22×#214C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×#389C2211yescharacteristic
24C3×#385—21no—
25C3×#385—21no—
26C3×#384C2211yescharacteristic
27C3×#380—21no—
28C3×C4×C4⋊C4C2211yescharacteristic
29C2×#832C2211yescharacteristic
30C2×#832C2211yescharacteristic
31C2×#832C2211yescharacteristic
32#4986—21no—
33#4980—21no—
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4929C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×#803C2211yescharacteristic
46#4660—21no—
47#4654—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4524C2211yescharacteristic
57#4524C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2511—31no—
61D4×C4⋊C4D311yescharacteristic
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#1864—31no—
73#1802—31no—
74#1762—31no—
75C22×C3×D4C2311yescharacteristic
76C23×C3×C4—21no—
77C23×C3×C4—21no—
78C23×C3×C4—21no—
79C22×Dic6C2311yescharacteristic
80C3×C4×D4—22no—
81C3×C4×D4—22no—
82C3×C4×D4—22no—
83C3×C4×D4—22no—
84C3×C4×D4—41no—
85C3×C4×D4—41no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4—21no—
89C2×C3×C4⋊C4—21no—
90C2×C3×C4⋊C4—44no—
91C2×C3×C4⋊C4D411yescharacteristic
92C2×C3×C4⋊C4D411yescharacteristic
93C2×C3×C4⋊C4C2311yescharacteristic
94C2×C3×C4⋊C4C2311yescharacteristic
95C2×C3×C4⋊C4C2311yescharacteristic
96C2×C3×C22⋊C4—21no—
97C2×C3×C22⋊C4—21no—
98C2×C3×C22⋊C4—21no—
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×C3×C42C2311yescharacteristic
102C2×#214—22no—
103C2×#214—22no—
104C2×#214—22no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—21no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3—21no—
112C22×C3⋊C8C2311yescharacteristic
113C22×C3⋊C8C2311yescharacteristic
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C3×#95—21no—
117C3×#95—21no—
118C3×#95D411yescharacteristic
119C3×#95D411yescharacteristic
120#832—22no—
121#832—22no—
122#832—22no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95—21no—
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#810—22no—
132#810—41no—
133#810—41no—
134#808—22no—
135C4⋊(C3⋊C8)—41no—
136#803—41no—
137C22×SD16—31no—
138C2×C4×D4C2×D311yescharacteristic
139C2×C4×D4C2×D311yescharacteristic
140C2×C4×D4C2×D311yescharacteristic
141C22×C4⋊C4—21no—
142C2×#106—31no—
143C2×C4⋊C8—31no—
144C2×#103—31no—
145C2×#103—31no—
146C2×#103—31no—
147C2×#102—31no—
148C2×#102—31no—
149C2×#102—31no—
150C2×#97—31no—
151#400—61no—
152#400—61no—
153#391—31no—
154#391—31no—
155#391—31no—
156#389—31no—
157#389C2×D311yescharacteristic
158#389C2×D311yescharacteristic
159#387—61no—
160#387—61no—
161#385—21no—
162#385—21no—
163#384—31no—
164#384C2×D311yescharacteristic
165#380—21no—
166C4×C4⋊C4—31no—
167C4×C4⋊C4C2×D311yescharacteristic
168#340—31no—
169#340—31no—
170#340—31no—
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#336—31no—
177#336—31no—
178#336—31no—
179#336—31no—
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×C4—42no—
194C22×C3×C4—42no—
195C22×C3×C4—42no—
196C22×C3×C4—42no—
197C22×C3×C4—81no—
198C22×C3×C4—81no—
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×D411yescharacteristic
201C22×C3×C4C2×D411yescharacteristic
202C22×C3×C4C2×D411yescharacteristic
203C22×C3×C4C2×D411yescharacteristic
204C22×C3×C4C2×D411yescharacteristic
205C22×C3×C4C2411yescharacteristic
206C22×Dic3—21no—
207C22×Dic3—21no—
208C22×Dic3—21no—
209C22×Dic3—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C3×C4⋊C4—22no—
217C3×C4⋊C4—22no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—42no—
225C3×C4⋊C4—42no—
226C3×C4⋊C4—81no—
227C3×C4⋊C4#4112yes—
228C3×C22⋊C4—24no—
229C3×C22⋊C4—42no—
230C3×C22⋊C4—42no—
231C3×C42—21no—
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—21no—
236C3×C42—21no—
237#214—44no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—42no—
245C4⋊Dic3—81no—
246Dic3⋊C4—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—31no—
261C22×D4C22×D311yescharacteristic
262C23×C4—21no—
263C23×C4—21no—
264C23×C4—21no—
265C2×SD16—62no—
266C2×SD16—62no—
267C2×SD16—62no—
268C22×C8—31no—
269C22×C8—31no—
270C22×C8—31no—
271C22×C8—31no—
272C4×D4—22no—
273C4×D4—22no—
274C4×D4—22no—
275C4×D4—22no—
276C4×D4—41no—
277C4×D4—41no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4—21no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—44no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4#6711yescharacteristic
290C2×C4⋊C4#6711yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C4⋊C4C22×D311yescharacteristic
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C42—61no—
298C2×C42C22×D311yescharacteristic
299C2×C42C22×D311yescharacteristic
300C2×C42C22×D311yescharacteristic
301#106—62no—
302C4⋊C8—121no—
303#103—62no—
304#103—121no—
305#103—121no—
306#102—62no—
307#102—62no—
308#102—62no—
309#97—121no—
310#95—21no—
311#95—21no—
312#95—61no—
313#95—61no—
314#95—61no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95#6711yescharacteristic
321#95#6711yescharacteristic
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—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—21no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—48no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4—41no—
355C2×C3×C4—41no—
356C2×C3×C4—41no—
357C2×C3×C4—41no—
358C2×C3×C4—41no—
359C2×C3×C4—41no—
360C2×C3×C4—84no—
361C2×C3×C4—84no—
362C2×C3×C4AGL(1,ℤ/8)12yes—
363C2×C3×C4#13712yes—
364C2×C3×C4C2×#4111yescharacteristic
365C2×C3×C4C2×#4111yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4H(5,𝔽2)11yescharacteristic
369C2×C3×C42−1+411yescharacteristic
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—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
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—42no—
408C22×C4—42no—
409C22×C4—42no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—81no—
418C22×C4—81no—
419C22×C4C2×#6711yescharacteristic
420C22×C4C2×#6711yescharacteristic
421C22×C4C2×#6711yescharacteristic
422C22×C4C2×#6711yescharacteristic
423C22×C4C2×#6711yescharacteristic
424C22×C4C2×#6711yescharacteristic
425C22×C4C23×D311yescharacteristic
426SD16—124no—
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—
437C2×C8—61no—
438C2×C8—61no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4—42no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—61no—
452C4⋊C4—61no—
453C4⋊C4—61no—
454C4⋊C4—61no—
455C4⋊C4—81no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4#23512yes—
460C22⋊C4—24no—
461C22⋊C4—42no—
462C22⋊C4—42no—
463C42—21no—
464C42—21no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
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—22no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—42no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4—81no—
492C3×C4—81no—
493C3×C4C2×#12611yescharacteristic
494C3×C4#53911yescharacteristic
495C3×C4#55211yescharacteristic
496C3×C4#55311yescharacteristic
497Dic3—81no—
498Dic3—81no—
499Dic3—81no—
500Dic3—81no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23C22×#6711yescharacteristic
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—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—48no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—84no—
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—121no—
556C2×C4—121no—
557C2×C4#93212yes—
558C2×C4#95112yes—
559C2×C4C2×#23511yescharacteristic
560C2×C4C2×#23511yescharacteristic
561C2×C4C2×#23711yescharacteristic
562C2×C4#100411yescharacteristic
563C2×C4C2×#23911yescharacteristic
564C2×C4#100711yescharacteristic
565C8—121no—
566C8—121no—
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#579111yescharacteristic
585C22C2×#93211yescharacteristic
586C22#580611yescharacteristic
587C22#581711yescharacteristic
588C22C2×#95111yescharacteristic
589C22#583611yescharacteristic
590C22#583711yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—42no—
597C4—41no—
598C4—81no—
599C4—81no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4—241no—
604C4C2×#89811yescharacteristic
605C4#551311yescharacteristic
606C4#551411yescharacteristic
607C4#551511yescharacteristic
608C3#2757311yescharacteristic, complemented, Sylow
609C2—44no—
610C2#8217511yescharacteristic
611C2#8231511yescharacteristic
612C2#8257611yescharacteristic
613C2#8348411yescharacteristic
614C2#8351511yescharacteristic
615C2#8356911yescharacteristic
616C2#8364411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×D4×C4⋊C4C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenterD4×C4⋊C4D3
hypocenterC3#27573
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #73690C2semidirect
9. #69634C2semidirect
10. #69616C2semidirect
11. #69611C2semidirect
12. #69602C2semidirect
13. #69589C2semidirect
14. #69572C2semidirect
15. #69483C2semidirect
608. C3#27573semidirect

Maximal subgroups

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

subgroupindex
C3×D4×C4⋊C42
#736902
#716122
#716092
#716072
#716042
#696342
#696162
#696112
#696022
#695892
#695722
#694832
#275733