Finite Group Databasebeta

← #10901365

Subgroups — #10901365

Order 768.

1100 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
1C2×C3×#529C211yesmaximal, characteristic
2C2×#5020C211yesmaximal, characteristic, complemented
3C2×#5007C211yesmaximal, characteristic, complemented
4#82076C211yesmaximal, characteristic, complemented
5#71739C212yesmaximal, complemented
6#71722C212yesmaximal, complemented
7#71695C212yesmaximal, complemented
8#71680C212yesmaximal
9#71626C211yesmaximal, characteristic, complemented
10#71604C211yesmaximal, characteristic, complemented
11#71530C211yesmaximal, characteristic, complemented
12#29689—31nomaximal, Sylow
13C3×#529—24no—
14C3×#529—24no—
15C2×C3×#126—21no—
16C2×C3×#125C2211yescharacteristic
17C2×C3×#124C2211yescharacteristic
18C2×C3×#123—21no—
19C2×C3×C22⋊Q8C2211yescharacteristic
20C2×C3×#121C2211yescharacteristic
21C2×C3×C4×Q8C2211yescharacteristic
22C2×C3×C4×D4—21no—
23C2×C3×C4×D4C2211yescharacteristic
24C2×C3×#117—21no—
25C22×C3×C42C2211yescharacteristic
26C2×#924—21no—
27C2×#869C2211yescharacteristic
28C2×#835C2212yes—
29C2×#835C2212yes—
30C2×#832C2212yes—
31C2×#832C2212yes—
32#5212—21no—
33#5212C2212yes—
34C2×#830C2212yes—
35#5020—24no—
36#5007—24no—
37C2×#808C2212yes—
38C2×#807C2212yes—
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42#4928C2212yes—
43C2×C4⋊(C3⋊C8)C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45#4529C2212yes—
46#4529C2212yes—
47#4524C2212yes—
48C2×#529D311yescharacteristic
49C2×#475—31no—
50C2×#474—31no—
51#2857—31no—
52#2059—32no—
53#2042—32no—
54#2036—32no—
55#2034—32no—
56#1803—31no—
57#1802—31no—
58#1797—31no—
59C22×C3×Q8C2311yescharacteristic
60C22×C3×D4—21no—
61C22×C3×D4C2311yescharacteristic
62C23×C3×C4—21no—
63C23×C3×C4D411yescharacteristic
64C23×C3×C4D411yescharacteristic
65C23×C3×C4C2311yescharacteristic
66C22×Dic6—21no—
67C3×#126—28no—
68C3×#125—22no—
69C3×#125—22no—
70C3×#124—22no—
71C3×#124—22no—
72C3×#123—28no—
73C3×C22⋊Q8—22no—
74C3×C22⋊Q8—22no—
75C3×#121—22no—
76C3×#121—22no—
77C3×C4×Q8—22no—
78C3×C4×Q8D414yes—
79C3×C4×D4—28no—
80C3×C4×D4—22no—
81C3×C4×D4D414yes—
82C3×#117—28no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—21no—
86C2×C3×C4⋊C4—21no—
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×C3×C22⋊C4—21no—
94C2×C3×C42—21no—
95C2×C3×C42—22no—
96C2×C3×C42—22no—
97C2×C3×C42D411yescharacteristic
98C2×C3×C42D411yescharacteristic
99C2×C3×C42C2311yescharacteristic
100C2×C3×C42C2311yescharacteristic
101C2×#217—22no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3C2312yes—
106C2×C4⋊Dic3C2312yes—
107#924—44no—
108C2×Dic3⋊C4—22no—
109C22×C3⋊C8C2312yes—
110C22×C3⋊C8C2312yes—
111#869—22no—
112#869—22no—
113#835—24no—
114#835—42no—
115#832—24no—
116#832—42no—
117C3⋊#95—22no—
118C3⋊#95—22no—
119#830—42no—
120#808—24no—
121#807—24no—
122C4⋊(C3⋊C8)—22no—
123C4⋊(C3⋊C8)—22no—
124C2×C4⋊Q8—31no—
125#529—24no—
126#529—24no—
127C2×#126—21no—
128C2×#125C2×D311yescharacteristic
129C2×#124C2×D311yescharacteristic
130C2×#123—21no—
131C2×C22⋊Q8—61no—
132C2×C22⋊Q8C2×D311yescharacteristic
133C2×#121C2×D311yescharacteristic
134C2×C4×Q8C2×D311yescharacteristic
135C2×C4×D4—21no—
136C2×C4×D4C2×D311yescharacteristic
137C2×#117—21no—
138C22×C42C2×D311yescharacteristic
139#475—64no—
140#474—64no—
141C2×#107—32no—
142C2×#106—32no—
143C2×C4⋊C8—31no—
144C2×C4⋊C8—31no—
145C2×#103—32no—
146C2×#103—32no—
147C2×#102—32no—
148C2×#102—32no—
149C2×C22⋊C8—32no—
150#391—61no—
151#389—32no—
152#389—61no—
153#384—31no—
154#380—32no—
155#380—61no—
156#340—32no—
157#340—32no—
158#336—32no—
159C24×C3—21no—
160C24×C3C2×D411yescharacteristic
161C2×C3×Q8—21no—
162C2×C3×Q8—21no—
163C2×C3×Q8—21no—
164C2×C3×Q8—21no—
165C2×C3×Q8—21no—
166C2×C3×Q8C2×D412yes—
167C2×C3×D4—24no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—42no—
174C2×C3×D4—42no—
175C2×C3×D4C2×D412yes—
176C22×C3×C4—22no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—21no—
180C22×C3×C4—22no—
181C22×C3×C4—22no—
182C22×C3×C4—24no—
183C22×C3×C4—22no—
184C22×C3×C4—22no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—41no—
192C22×C3×C4—42no—
193C22×C3×C4—42no—
194C22×C3×C4C2×D411yescharacteristic
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2411yescharacteristic
200C22×Dic3—22no—
201C22×Dic3—22no—
202C2×Dic6—42no—
203C2×Dic6—42no—
204C2×Dic6—42no—
205C3×C4⋊C4—24no—
206C3×C4⋊C4—24no—
207C3×C4⋊C4—24no—
208C3×C4⋊C4—24no—
209C3×C4⋊C4—21no—
210C3×C4⋊C4—21no—
211C3×C4⋊C4C2×D412yes—
212C3×C4⋊C4C2×D412yes—
213C3×C22⋊C4—24no—
214C3×C22⋊C4—24no—
215C3×C22⋊C4—24no—
216C3×C22⋊C4—24no—
217C3×C22⋊C4—24no—
218C3×C42—22no—
219C3×C42—22no—
220C3×C42—24no—
221C3×C42—21no—
222C3×C42—21no—
223C3×C42C2×D412yes—
224C3×C42C2×D412yes—
225#217—44no—
226C4⋊Dic3—22no—
227C4⋊Dic3—22no—
228C4⋊Dic3—22no—
229C4⋊Dic3—22no—
230C4⋊Dic3—42no—
231C4⋊Dic3—42no—
232C4⋊Dic3—42no—
233Dic3⋊C4—44no—
234C2×C3⋊C8—22no—
235C2×C3⋊C8—22no—
236C2×C3⋊C8—22no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C22×Q8—61no—
241C22×Q8C22×D311yescharacteristic
242C22×D4—21no—
243C22×D4C22×D311yescharacteristic
244C23×C4—21no—
245C23×C4#6711yescharacteristic
246C23×C4#6711yescharacteristic
247C23×C4C22×D311yescharacteristic
248C22×C8—32no—
249C22×C8—32no—
250C4⋊Q8—62no—
251C4⋊Q8—62no—
252#126—28no—
253#125—22no—
254#125—22no—
255#124—22no—
256#124—22no—
257#123—28no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—124no—
261#121—22no—
262#121—22no—
263C4×Q8—22no—
264C4×Q8#6714yes—
265C4×D4—28no—
266C4×D4—22no—
267C4×D4#6714yes—
268#117—28no—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—32no—
274C2×C4⋊C4—32no—
275C2×C4⋊C4—61no—
276C2×C4⋊C4—62no—
277C2×C4⋊C4—61no—
278C2×C4⋊C4—61no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—21no—
284C2×C22⋊C4—21no—
285C2×C22⋊C4—21no—
286C2×C22⋊C4—62no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—22no—
290C2×C42#6711yescharacteristic
291C2×C42#6711yescharacteristic
292C2×C42C22×D311yescharacteristic
293C2×C42C22×D311yescharacteristic
294#107—64no—
295#106—64no—
296C4⋊C8—62no—
297C4⋊C8—62no—
298#103—64no—
299#103—122no—
300#102—64no—
301#102—122no—
302C22⋊C8—122no—
303#95—62no—
304#95—62no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3—21no—
308C23×C3—21no—
309C23×C3—21no—
310C23×C3—21no—
311C23×C3—22no—
312C23×C3—41no—
313C23×C3—41no—
314C23×C3—41no—
315C23×C3—41no—
316C23×C3—41no—
317C23×C3—41no—
318C23×C3#13712yes—
319C23×C3C22×D411yescharacteristic
320C3×Q8—22no—
321C3×Q8—22no—
322C3×Q8—41no—
323C3×Q8—41no—
324C3×D4—22no—
325C3×D4—22no—
326C3×D4—48no—
327C3×D4—41no—
328C3×D4—41no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—44no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
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×C4—44no—
355C2×C3×C4—48no—
356C2×C3×C4#12012yes—
357C2×C3×C4#12012yes—
358C2×C3×C4#13512yes—
359C2×C3×C4AGL(1,ℤ/8)12yes—
360C2×C3×C4C22×D411yescharacteristic
361C2×C3×C4C22×D411yescharacteristic
362C2×C3×C4H(5,𝔽2)11yescharacteristic
363C2×C3×C42−1+411yescharacteristic
364C2×C3×C42−1+411yescharacteristic
365C2×C3×C42−1+411yescharacteristic
366C2×Dic3—42no—
367C2×Dic3—42no—
368C2×Dic3—42no—
369C2×Dic3—42no—
370C2×Dic3—42no—
371C2×Dic3—42no—
372Dic6—84no—
373C3⋊C8—42no—
374C3⋊C8—42no—
375C24—21no—
376C24C2×#6711yescharacteristic
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—122no—
383C2×Q8—122no—
384C2×Q8—122no—
385C2×Q8C2×#6712yes—
386C2×D4—24no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—42no—
393C2×D4—42no—
394C2×D4C2×#6712yes—
395C22×C4—22no—
396C22×C4—21no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—62no—
414C22×C4—62no—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—62no—
422C2×C8—62no—
423C2×C8—62no—
424C2×C8—62no—
425C2×C8—62no—
426C2×C8—62no—
427C4⋊C4—24no—
428C4⋊C4—24no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—62no—
434C4⋊C4—62no—
435C4⋊C4—62no—
436C4⋊C4—62no—
437C4⋊C4—124no—
438C4⋊C4—122no—
439C4⋊C4—122no—
440C4⋊C4—122no—
441C4⋊C4C2×#6712yes—
442C4⋊C4C2×#6712yes—
443C22⋊C4—24no—
444C22⋊C4—24no—
445C22⋊C4—24no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—124no—
449C42—22no—
450C42—22no—
451C42—24no—
452C42—21no—
453C42—21no—
454C42C2×#6712yes—
455C42C2×#6712yes—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—41no—
460C22×C3—41no—
461C22×C3—41no—
462C22×C3—41no—
463C22×C3—42no—
464C22×C3—42no—
465C22×C3—42no—
466C22×C3—81no—
467C22×C3—81no—
468C22×C3—81no—
469C22×C3—81no—
470C22×C3C2×#12011yescharacteristic
471C22×C3#53611yescharacteristic
472C22×C3#53711yescharacteristic
473C22×C3C2×#13511yescharacteristic
474C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
475C22×C3C2×#13711yescharacteristic
476C22×C3#57511yescharacteristic
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—42no—
482C3×C4—41no—
483C3×C4—42no—
484C3×C4#44914yes—
485C3×C4#53611yescharacteristic
486C3×C4#53611yescharacteristic
487C3×C4#53711yescharacteristic
488C3×C4#53711yescharacteristic
489Dic3—82no—
490Dic3—82no—
491C23—21no—
492C23—21no—
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—22no—
498C23—41no—
499C23—41no—
500C23—41no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23#95112yes—
505C23C22×#6711yescharacteristic
506Q8—22no—
507Q8—22no—
508Q8—41no—
509Q8—41no—
510Q8—244no—
511D4—22no—
512D4—22no—
513D4—48no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—44no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—44no—
538C2×C4—48no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—122no—
544C2×C4—122no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4#94912yes—
550C2×C4#95012yes—
551C2×C4#95312yes—
552C2×C4#95312yes—
553C2×C4#100911yescharacteristic
554C2×C4#101011yescharacteristic
555C2×C4#101011yescharacteristic
556C2×C4#101011yescharacteristic
557C2×C4C22×#6711yescharacteristic
558C2×C4C22×#6711yescharacteristic
559C8—122no—
560C8—122no—
561C2×C3—42no—
562C2×C3—82no—
563C2×C3#261611yescharacteristic
564C2×C3C2×#44911yescharacteristic
565C2×C3#300711yescharacteristic
566C2×C3#323311yescharacteristic
567C2×C3#323911yescharacteristic
568C2×C3#324811yescharacteristic
569C2×C3#325111yescharacteristic
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—42no—
578C22—42no—
579C22—42no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22—81no—
584C22C2×#94911yescharacteristic
585C22C2×#95011yescharacteristic
586C22#582211yescharacteristic
587C22C2×#95111yescharacteristic
588C22#582511yescharacteristic
589C22#583311yescharacteristic
590C22C2×#95311yescharacteristic
591C4—22no—
592C4—41no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—41no—
597C4—42no—
598C4—242no—
599C4—242no—
600C4#523814yes—
601C4#582511yescharacteristic
602C4#582511yescharacteristic
603C4#583311yescharacteristic
604C4#583311yescharacteristic
605C3#2968911yescharacteristic, complemented, Sylow
606C2—42no—
607C2—82no—
608C2#8312211yescharacteristic
609C2#8312411yescharacteristic
610C2#8315011yescharacteristic
611C2#8315511yescharacteristic
612C2C2×#523811yescharacteristic
613C2#8408411yescharacteristic
614C2#8409411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#529C2
socleC23×C3C22×D4
prime radicalC2×C3×#121C22
hypercenterC2×#529D3
hypocenterC3#29689
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#529

Complements

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

NH ≅ G/Ntype
2. C2×#5020C2semidirect
3. C2×#5007C2semidirect
4. #82076C2semidirect
5. #71739C2semidirect
6. #71722C2semidirect
7. #71695C2semidirect
9. #71626C2semidirect
10. #71604C2semidirect
11. #71530C2semidirect
605. C3#29689semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5292
C2×#50202
C2×#50072
#820762
#717392
#717222
#716952
#716802
#716262
#716042
#715302
#296893