Finite Group Databasebeta

← #10858809

Subgroups — #10858809

Order 768.

746 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2439C211yesmaximal, characteristic
2#73468C211yesmaximal, characteristic
3#71794C211yesmaximal, characteristic, complemented
4#71780C211yesmaximal, characteristic, complemented
5#71585C211yesmaximal, characteristic
6#71580C211yesmaximal, characteristic
7#69635C211yesmaximal, characteristic, complemented
8#69632C211yesmaximal, characteristic, complemented
9#69623C211yesmaximal, characteristic, complemented
10#69602C211yesmaximal, characteristic, complemented
11#69559C211yesmaximal, characteristic, complemented
12#69542C211yesmaximal, characteristic, complemented
13#69514C211yesmaximal, characteristic
14#69506C211yesmaximal, characteristic
15#69449C211yesmaximal, characteristic
16#26413—31nomaximal, Sylow
17C2×C3×C4⋊Q8C2211yescharacteristic
18C2×C3×#117C2211yescharacteristic
19C2×#926C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C23×C3⋊C8C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C3×#396—21no—
24C3×#393—21no—
25C3×#391—21no—
26C3×#389C2211yescharacteristic
27C3×#389C2211yescharacteristic
28C3×#387—21no—
29C3×#382C2211yescharacteristic
30C3×#380C2211yescharacteristic
31C3×C4×C22⋊C4C2211yescharacteristic
32C2×#830C2211yescharacteristic
33C2×#810C2211yescharacteristic
34C2×#810C2211yescharacteristic
35C2×#810C2211yescharacteristic
36C2×#810C2211yescharacteristic
37C2×#808C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4945—21no—
40#4944—21no—
41C4×C4⋊Dic3C2211yescharacteristic
42#4929C2211yescharacteristic
43#4928C2211yescharacteristic
44C2×C4⋊(C3⋊C8)C2211yescharacteristic
45C2×C4⋊(C3⋊C8)C2211yescharacteristic
46#4652—21no—
47#4649—21no—
48#4529C2211yescharacteristic
49#4529C2211yescharacteristic
50#4529C2211yescharacteristic
51#4529C2211yescharacteristic
52#4529C2211yescharacteristic
53#4529C2211yescharacteristic
54#4529C2211yescharacteristic
55#4529C2211yescharacteristic
56#4529C2211yescharacteristic
57#4529C2211yescharacteristic
58#4524C2211yescharacteristic
59#4524C2211yescharacteristic
60#2439D311yescharacteristic
61#2362—31no—
62#2101—31no—
63#2087—31no—
64#2068—31no—
65#2024—31no—
66#1988—31no—
67#1920—31no—
68#1892—31no—
69#1873—31no—
70#1865—31no—
71#1854—31no—
72#1850—31no—
73#1805—31no—
74#1735—31no—
75C22×C3×Q8—21no—
76C23×C3×C4C2311yescharacteristic
77C22×Dic6C2311yescharacteristic
78C3×C4⋊Q8—22no—
79C3×C4⋊Q8—41no—
80C3×#117—22no—
81C3×#117—41no—
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C4⋊C4C2311yescharacteristic
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C22⋊C4—21no—
90C2×C3×C22⋊C4—21no—
91C2×C3×C22⋊C4D411yescharacteristic
92C2×C3×C22⋊C4D411yescharacteristic
93C2×C3×C42—21no—
94C2×C3×C42C2311yescharacteristic
95C2×C3×C42C2311yescharacteristic
96C2×#217—21no—
97C2×#217—21no—
98#926—22no—
99#926—41no—
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103#924—22no—
104#924Q814yescomplemented
105C2×Dic3⋊C4—21no—
106C2×Dic3⋊C4—21no—
107C2×Dic3⋊C4—21no—
108C2×C4×Dic3—21no—
109C2×C4×Dic3—21no—
110C2×C4×Dic3—21no—
111C22×C3⋊C8—22no—
112C22×C3⋊C8—41no—
113C22×C3⋊C8—41no—
114C22×C3⋊C8C2311yescharacteristic
115C22×C3⋊C8C2311yescharacteristic
116C22×C3⋊C8C2311yescharacteristic
117C22×C3⋊C8C2311yescharacteristic
118C4×Dic6—22no—
119C4×Dic6—41no—
120C3×#95—21no—
121C3×#95—21no—
122C3×#95—21no—
123C3×#95—21no—
124C3×#95—21no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131#830—22no—
132#810—22no—
133#810—41no—
134#810—41no—
135#810—41no—
136#808—22no—
137#807—41no—
138C4⋊(C3⋊C8)—41no—
139C4⋊(C3⋊C8)—41no—
140C23×C8—31no—
141C2×C4⋊Q8C2×D311yescharacteristic
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C2×#117C2×D311yescharacteristic
146C2×#107—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×C22⋊C8—31no—
155#396—21no—
156#393—21no—
157#391—21no—
158#391—31no—
159#389—31no—
160#389C2×D311yescharacteristic
161#389C2×D311yescharacteristic
162#388—61no—
163#388—61no—
164#387—21no—
165#387—61no—
166#382C2×D311yescharacteristic
167#380C2×D311yescharacteristic
168C4×C4⋊C4—31no—
169C4×C4⋊C4—61no—
170C4×C22⋊C4C2×D311yescharacteristic
171#340—31no—
172#340—31no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340—31no—
179#340—31no—
180#340—31no—
181#336—31no—
182#336—31no—
183C24×C3C2×D411yescharacteristic
184C2×C3×Q8—42no—
185C2×C3×Q8—42no—
186C2×C3×Q8—81no—
187C22×C3×C4—21no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4—21no—
191C22×C3×C4—21no—
192C22×C3×C4—21no—
193C22×C3×C4—21no—
194C22×C3×C4—21no—
195C22×C3×C4C2×D411yescharacteristic
196C22×C3×C4C2×D411yescharacteristic
197C22×C3×C4C2×D411yescharacteristic
198C22×C3×C4C2×D411yescharacteristic
199C22×C3×C4C2×D411yescharacteristic
200C22×C3×C4C2×Q812yes—
201C22×C3×C4#4112yes—
202C22×C3×C4C2411yescharacteristic
203C22×Dic3—21no—
204C22×Dic3—21no—
205C22×Dic3—21no—
206C22×Dic3—21no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—42no—
221C3×C4⋊C4—42no—
222C3×C4⋊C4—81no—
223C3×C4⋊C4#4112yes—
224C3×C22⋊C4—22no—
225C3×C22⋊C4—41no—
226C3×C22⋊C4—42no—
227C3×C22⋊C4—81no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232C3×C42—42no—
233#217—24no—
234#217—42no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3C2×Q812yes—
240C4⋊Dic3#4112yes—
241Dic3⋊C4—24no—
242Dic3⋊C4—42no—
243Dic3⋊C4—42no—
244C4×Dic3—42no—
245C4×Dic3—42no—
246C4×Dic3—42no—
247C2×C3⋊C8—21no—
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—44no—
260C22×Q8—21no—
261C22×Q8—31no—
262C23×C4C22×D311yescharacteristic
263C22×C8—31no—
264C22×C8—31no—
265C22×C8—31no—
266C22×C8—31no—
267C22×C8—62no—
268C22×C8—121no—
269C22×C8—121no—
270C4⋊Q8—22no—
271C4⋊Q8—41no—
272C22⋊Q8—34no—
273C22⋊Q8—62no—
274C4×Q8—62no—
275C4×Q8—121no—
276#117—22no—
277#117—41no—
278#117—62no—
279#117—121no—
280C2×C4⋊C4—21no—
281C2×C4⋊C4—21no—
282C2×C4⋊C4—21no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—61no—
287C2×C4⋊C4—61no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4C22×D311yescharacteristic
290C2×C4⋊C4C22×D311yescharacteristic
291C2×C4⋊C4C22×D311yescharacteristic
292C2×C4⋊C4C22×D311yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—61no—
296C2×C22⋊C4—61no—
297C2×C22⋊C4#6711yescharacteristic
298C2×C22⋊C4#6711yescharacteristic
299C2×C42—21no—
300C2×C42—61no—
301C2×C42—61no—
302C2×C42—61no—
303C2×C42C22×D311yescharacteristic
304C2×C42C22×D311yescharacteristic
305#107—121no—
306#106—62no—
307C4⋊C8—121no—
308C4⋊C8—121no—
309#103—62no—
310#103—121no—
311#103—121no—
312#103—121no—
313C22⋊C8—62no—
314#95—21no—
315#95—21no—
316#95—21no—
317#95—21no—
318#95—21no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95#6711yescharacteristic
324#95#6711yescharacteristic
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3C22⋊Q812yes—
332C23×C3C22×D411yescharacteristic
333C3×Q8—84no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—22no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
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—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—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4C22⋊Q812yes—
364C2×C3×C4#13512yes—
365C2×C3×C4C22×Q811yescharacteristic
366C2×C3×C4C2×#4111yescharacteristic
367C2×C3×C4C2×#4111yescharacteristic
368C2×C3×C4C2×#4111yescharacteristic
369C2×C3×C4H(5,𝔽2)11yescharacteristic
370C2×C3×C4H(5,𝔽2)11yescharacteristic
371C2×Dic3—22no—
372C2×Dic3—22no—
373C2×Dic3—22no—
374C2×Dic3—41no—
375C2×Dic3—41no—
376C2×Dic3—41no—
377C2×Dic3—41no—
378C2×Dic3—41no—
379C2×Dic3—41no—
380C2×Dic3—41no—
381C2×Dic3—41no—
382C2×Dic3—41no—
383Dic6—22no—
384Dic6—22no—
385Dic6—41no—
386Dic6—41no—
387C3⋊C8—41no—
388C3⋊C8—41no—
389C3⋊C8—41no—
390C3⋊C8—41no—
391C24C2×#6711yescharacteristic
392C2×Q8—32no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C2×Q8—81no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—61no—
413C22×C4D3×Q812yes—
414C22×C4#23912yes—
415C22×C4C2×#6711yescharacteristic
416C22×C4C2×#6711yescharacteristic
417C22×C4C2×#6711yescharacteristic
418C22×C4C2×#6711yescharacteristic
419C22×C4C2×#6711yescharacteristic
420C22×C4C23×D311yescharacteristic
421C2×C8—61no—
422C2×C8—61no—
423C2×C8—61no—
424C2×C8—61no—
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—124no—
434C4⋊C4—21no—
435C4⋊C4—21no—
436C4⋊C4—21no—
437C4⋊C4—21no—
438C4⋊C4—21no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—32no—
442C4⋊C4—32no—
443C4⋊C4—42no—
444C4⋊C4—42no—
445C4⋊C4—64no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—81no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4#23712yes—
454C22⋊C4—22no—
455C22⋊C4—41no—
456C22⋊C4—42no—
457C22⋊C4—64no—
458C22⋊C4—81no—
459C22⋊C4—122no—
460C42—21no—
461C42—21no—
462C42—21no—
463C42—21no—
464C42—42no—
465C42—122no—
466C42—122no—
467C42—122no—
468C22×C3—22no—
469C22×C3—22no—
470C22×C3—22no—
471C22×C3—41no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—41no—
475C22×C3C2×C22⋊Q811yescharacteristic
476C22×C3#54611yescharacteristic
477C22×C3#54611yescharacteristic
478C22×C3C2×#13511yescharacteristic
479C22×C3#57511yescharacteristic
480C22×C3#57711yescharacteristic
481C22×C3#57811yescharacteristic
482C3×C4—22no—
483C3×C4—22no—
484C3×C4—42no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—81no—
489C3×C4—81no—
490C3×C4—81no—
491C3×C4#52911yescharacteristic
492C3×C4#53811yescharacteristic
493C3×C4#55411yescharacteristic
494C3×C4#55411yescharacteristic
495Dic3—42no—
496Dic3—42no—
497Dic3—42no—
498Dic3—81no—
499C23—21no—
500C23—21no—
501C23—21no—
502C23—21no—
503C23—21no—
504C23—21no—
505C23#94612yes—
506C23C22×#6711yescharacteristic
507Q8—62no—
508Q8—62no—
509Q8—84no—
510Q8—121no—
511Q8—121no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
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—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
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#94612yes—
554C2×C4#95012yes—
555C2×C4C2×#23711yescharacteristic
556C2×C4#100411yescharacteristic
557C2×C4C2×D3×Q811yescharacteristic
558C2×C4C2×#23911yescharacteristic
559C2×C4D3×#4111yescharacteristic
560C2×C4#100911yescharacteristic
561C8—121no—
562C8—121no—
563C8—121no—
564C8—121no—
565C2×C3—42no—
566C2×C3#245011yescharacteristic
567C2×C3#307611yescharacteristic
568C2×C3#308211yescharacteristic
569C2×C3#329611yescharacteristic
570C2×C3#330211yescharacteristic
571C2×C3#331311yescharacteristic
572C2×C3#332111yescharacteristic
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22#580611yescharacteristic
581C22C2×#94611yescharacteristic
582C22C2×#95011yescharacteristic
583C22#582211yescharacteristic
584C22#582911yescharacteristic
585C22#583611yescharacteristic
586C22#583811yescharacteristic
587C4—22no—
588C4—22no—
589C4—42no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—122no—
597C4—122no—
598C4—122no—
599C4—241no—
600C4#562411yescharacteristic
601C4#563511yescharacteristic
602C4#563611yescharacteristic
603C4#564411yescharacteristic
604C3#2641311yescharacteristic, complemented, Sylow
605C2—42no—
606C2#8276511yescharacteristic
607C2#8334911yescharacteristic
608C2#8348011yescharacteristic
609C2#8349411yescharacteristic
610C2#8353711yescharacteristic
611C2#8365611yescharacteristic
612C2#8366411yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2439C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#2439D3
hypocenterC3#26413
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
3. #71794C2semidirect
4. #71780C2semidirect
7. #69635C2semidirect
8. #69632C2semidirect
9. #69623C2semidirect
10. #69602C2semidirect
11. #69559C2semidirect
12. #69542C2semidirect
104. #924Q8semidirect
604. C3#26413semidirect

Maximal subgroups

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

subgroupindex
C3×#24392
#734682
#717942
#717802
#715852
#715802
#696352
#696322
#696232
#696022
#695592
#695422
#695142
#695062
#694492
#264133