Finite Group Databasebeta

← #10817039

Subgroups — #10817039

Order 768.

846 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4871C211yesmaximal, characteristic
2#76203C211yesmaximal, characteristic
3#73661C211yesmaximal, characteristic, complemented
4C3×#1920C211yesmaximal, characteristic, complemented
5#70560C211yesmaximal, characteristic, complemented
6#69619C211yesmaximal, characteristic, complemented
7C4×#809C211yesmaximal, characteristic
8#68894C211yesmaximal, characteristic
9#68821C211yesmaximal, characteristic, complemented
10#68804C211yesmaximal, characteristic, complemented
11#68797C211yesmaximal, characteristic, complemented
12#68780C211yesmaximal, characteristic
13#68780C211yesmaximal, characteristic
14#68671C211yesmaximal, characteristic
15#68651C211yesmaximal, characteristic, complemented
16#22966—31nomaximal, Sylow
17C2×#893C2211yescharacteristic
18C2×#893C2211yescharacteristic
19C2×#890C2211yescharacteristic
20C2×C4×D12C2211yescharacteristic
21C2×C3×#106C2211yescharacteristic
22C3×#389C2211yescharacteristic
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#821C2211yescharacteristic
25C2×#821C2211yescharacteristic
26C2×#817C2211yescharacteristic
27C2×Dic3⋊C8C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4989—21no—
31#4987—21no—
32#4984C2211yescharacteristic
33#4984C2211yescharacteristic
34#4977C2211yescharacteristic
35Dic3×C4⋊C4C2211yescharacteristic
36Dic3×C4⋊C4C2211yescharacteristic
37C2×#809C2211yescharacteristic
38C2×#809C2211yescharacteristic
39C2×#807C2211yescharacteristic
40C2×#807C2211yescharacteristic
41C4×C4⋊Dic3—21no—
42#4933—21no—
43C4×Dic3⋊C4C2211yescharacteristic
44#4928C2211yescharacteristic
45C2×C4×C3⋊C8C2211yescharacteristic
46#4871C418yescomplemented
47#4668—21no—
48D3×#95—21no—
49#4654—21no—
50#4647—21no—
51C3×#340C2211yescharacteristic
52C3×#340C2211yescharacteristic
53C3×#340C2211yescharacteristic
54C3×#336C2211yescharacteristic
55#4551C2211yescharacteristic
56#4550C2211yescharacteristic
57#4529C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61C2×#478—31no—
62#2370—31no—
63#2349—31no—
64#1991—31no—
65#1949—31no—
66#1942—31no—
67#1941—31no—
68#1921—31no—
69#1920D311yescharacteristic, complemented
70#1864—31no—
71#1862—31no—
72C8×C4⋊C4—31no—
73C4×#102—31no—
74C4×#102—31no—
75C4×#102—31no—
76C22×D12C2311yescharacteristic
77C22×C4×D3—21no—
78C22×C4×D3—21no—
79C22×C4×D3—21no—
80C22×C3×C8C2311yescharacteristic
81C22×C3×C8C2311yescharacteristic
82C2×C3×C4⋊C4—21no—
83C2×C3×C4⋊C4C2311yescharacteristic
84C2×C3×C4⋊C4C2311yescharacteristic
85C2×C3×C4⋊C4C2311yescharacteristic
86C2×C3×C4⋊C4C2311yescharacteristic
87C2×C3×C42—21no—
88C2×C3×C42C2311yescharacteristic
89C2×(C2×D3)⋊C4—21no—
90C2×(C2×D3)⋊C4—21no—
91C2×(C2×D3)⋊C4—21no—
92C2×C4⋊Dic3—21no—
93C2×C4⋊Dic3—21no—
94C2×C4⋊Dic3—21no—
95C2×C4⋊Dic3C2311yescharacteristic
96C2×C4⋊Dic3C2311yescharacteristic
97C2×C4⋊Dic3C2311yescharacteristic
98C2×Dic3⋊C4—21no—
99C2×Dic3⋊C4—21no—
100C2×Dic3⋊C4D411yescharacteristic
101C2×Dic3⋊C4D411yescharacteristic
102C2×C4×Dic3—21no—
103C2×C4×Dic3—21no—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C22×C3⋊C8C2311yescharacteristic
107C22×C3⋊C8C2311yescharacteristic
108#893—22no—
109#893—22no—
110#893—41no—
111#893C2×C414yes—
112#890—22no—
113#890C2×C414yescomplemented
114C4×D12—22no—
115C4×D12—41no—
116C3×#106C2×C414yescomplemented
117C3×#95—21no—
118C3×#95—21no—
119C3⋊#95—21no—
120C3⋊#95—21no—
121C3⋊#95—21no—
122C3⋊#95—21no—
123C3⋊#95—21no—
124C3⋊#95—21no—
125C3⋊#95—21no—
126C3⋊#95D411yescharacteristic
127C3⋊#95D411yescharacteristic
128#821—22no—
129#821C2×C414yes—
130#817—41no—
131Dic3⋊C8C2×C414yescomplemented
132C8×Dic3—22no—
133C8×Dic3—41no—
134#809C2×C414yes—
135#809Q814yes—
136#807—41no—
137#807C2×C414yescomplemented
138C4×C3⋊C8—41no—
139C2×#125—31no—
140C2×C4×D4—31no—
141C2×C4×D4—31no—
142C2×C4×D4—31no—
143#478—38no—
144C2×#107—31no—
145C2×#107—31no—
146C2×#106—31no—
147C2×#106C2×D311yescharacteristic
148C2×C4⋊C8—31no—
149C2×#102—31no—
150C2×#102—31no—
151C2×#102—31no—
152C2×#102—31no—
153C2×C4×C8—31no—
154C2×C4×C8—31no—
155C2×C4×C8—31no—
156#389—31no—
157#389—31no—
158#389—31no—
159#389—61no—
160#389C2×D311yescharacteristic
161#387—61no—
162#387—61no—
163#385—61no—
164#385—61no—
165#382—31no—
166#380—61no—
167C4×C4⋊C4—31no—
168C4×C4⋊C4—31no—
169C4×C4⋊C4—31no—
170C4×C4⋊C4—61no—
171C4×C4⋊C4C2×D311yescharacteristic
172C2×#95—61no—
173#340—31no—
174#340—31no—
175#340—31no—
176#340—31no—
177#340—31no—
178#340C2×D311yescharacteristic
179#340C2×D311yescharacteristic
180#340C2×D311yescharacteristic
181#336—31no—
182#336C2×D311yescharacteristic
183C23×D3—21no—
184C22×C3×C4—21no—
185C22×C3×C4—21no—
186C22×C3×C4—21no—
187C22×C3×C4—21no—
188C22×C3×C4C2×D411yescharacteristic
189C22×C3×C4C2×D411yescharacteristic
190C22×C3×C4C2411yescharacteristic
191C22×Dic3—21no—
192C22×Dic3—21no—
193C22×Dic3—21no—
194C22×Dic3C2×D411yescharacteristic
195C22×Dic3C2×D411yescharacteristic
196C22×Dic3C2×D411yescharacteristic
197C22×Dic3C2×D411yescharacteristic
198C2×D12—21no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12C22×C412yes—
202C2×D12C2×Q812yes—
203C2×D12#4112yes—
204C2×C4×D3—24no—
205C2×C4×D3—42no—
206C2×C4×D3—42no—
207C2×C4×D3—42no—
208C2×C4×D3—42no—
209C2×C4×D3—81no—
210C2×C4×D3—81no—
211C2×C4×D3—81no—
212C2×C4×D3—81no—
213C2×C3×C8—21no—
214C2×C3×C8—21no—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—21no—
218C2×C3×C8C22×C412yes—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—81no—
225C3×C4⋊C4C22×C412yes—
226C3×C4⋊C4C22×C412yes—
227C3×C4⋊C4C2×Q812yes—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—42no—
231(C2×D3)⋊C4—24no—
232(C2×D3)⋊C4—42no—
233(C2×D3)⋊C4—42no—
234C4⋊Dic3—21no—
235C4⋊Dic3—24no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—42no—
241C4⋊Dic3—81no—
242C4⋊Dic3C22×C412yes—
243Dic3⋊C4—22no—
244Dic3⋊C4—24no—
245Dic3⋊C4—22no—
246Dic3⋊C4—81no—
247C4×Dic3—21no—
248C4×Dic3—21no—
249C4×Dic3—81no—
250C4×Dic3—81no—
251C4×Dic3C22×C412yes—
252C4×Dic3#4112yes—
253C2×C3⋊C8—21no—
254C2×C3⋊C8—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C2×C3⋊C8C22×C412yes—
258C2×C3⋊C8C2×Q812yes—
259C22×D4—31no—
260C23×C4—61no—
261C23×C4—61no—
262C23×C4—61no—
263C22×C8—31no—
264C22×C8—31no—
265C22×C8C22×D311yescharacteristic
266C22×C8C22×D311yescharacteristic
267#125—34no—
268#125—62no—
269C4×D4—34no—
270C4×D4—62no—
271C4×D4—62no—
272C4×D4—62no—
273C4×D4—121no—
274C4×D4—121no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—31no—
277C2×C4⋊C4—31no—
278C2×C4⋊C4—31no—
279C2×C4⋊C4—31no—
280C2×C4⋊C4—31no—
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×C4⋊C4C22×D311yescharacteristic
289C2×C4⋊C4C22×D311yescharacteristic
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—34no—
300#107—121no—
301#106—121no—
302#106C4×D314yescomplemented
303C4⋊C8—34no—
304#102—34no—
305#102—34no—
306#102—34no—
307#102—62no—
308C4×C8—62no—
309C4×C8—121no—
310C4×C8—121no—
311#95—21no—
312#95—21no—
313#95—31no—
314#95—31no—
315#95—61no—
316#95—61no—
317#95—61no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322C23×C3C22×D411yescharacteristic
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—22no—
326C22×D3—41no—
327C22×D3—41no—
328C22×D3—41no—
329C22×D3—41no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—22no—
333C2×C3×C4—22no—
334C2×C3×C4—21no—
335C2×C3×C4—22no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—41no—
340C2×C3×C4—41no—
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#13512yes—
348C2×C3×C4C23×C411yescharacteristic
349C2×C3×C4C22×Q811yescharacteristic
350C2×C3×C4C2×#4111yescharacteristic
351C2×C3×C4C2×#4111yescharacteristic
352C2×C3×C4H(5,𝔽2)11yescharacteristic
353C2×C3×C42−1+411yescharacteristic
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357C2×Dic3—21no—
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—21no—
364C2×Dic3—41no—
365C2×Dic3—41no—
366C2×Dic3—41no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—41no—
370C2×Dic3—41no—
371C2×Dic3—41no—
372C2×Dic3C4×D412yes—
373C2×Dic3C4×D412yes—
374C2×Dic3#13512yes—
375D12—22no—
376D12—22no—
377D12—22no—
378D12C4×Q814yes—
379C4×D3—84no—
380C4×D3—84no—
381C4×D3—84no—
382C3×C8—22no—
383C3×C8—41no—
384C3⋊C8—22no—
385C3⋊C8—22no—
386C24—61no—
387C2×D4—32no—
388C2×D4—32no—
389C2×D4—32no—
390C2×D4—61no—
391C2×D4—61no—
392C2×D4—61no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—31no—
398C22×C4—31no—
399C22×C4—31no—
400C22×C4—31no—
401C22×C4—64no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4—61no—
405C22×C4—122no—
406C22×C4—122no—
407C22×C4—122no—
408C22×C4—122no—
409C22×C4—241no—
410C22×C4—241no—
411C22×C4—241no—
412C22×C4—241no—
413C22×C4D3×D411yescharacteristic
414C22×C4D3×D411yescharacteristic
415C22×C4C23×D311yescharacteristic
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—21no—
420C2×C8—21no—
421C2×C8—32no—
422C2×C8—32no—
423C2×C8—61no—
424C2×C8—61no—
425C2×C8—61no—
426C2×C8—61no—
427C2×C8C2×C4×D312yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—32no—
434C4⋊C4—62no—
435C4⋊C4—64no—
436C4⋊C4—64no—
437C4⋊C4—62no—
438C4⋊C4—61no—
439C4⋊C4—61no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—81no—
444C4⋊C4—122no—
445C4⋊C4—241no—
446C4⋊C4—241no—
447C4⋊C4C2×C4×D312yes—
448C4⋊C4C2×C4×D312yes—
449C4⋊C4D3×Q812yes—
450C22⋊C4—64no—
451C22⋊C4—122no—
452C22⋊C4—122no—
453C42—21no—
454C42—21no—
455C42—32no—
456C42—32no—
457C42—42no—
458C42—61no—
459C42—61no—
460C42—241no—
461C42—241no—
462C22×C3C2×C4×D411yescharacteristic
463C22×C3#54611yescharacteristic
464C22×C3D4×Q811yescharacteristic
465C22×C3C2×#13511yescharacteristic
466C22×C3C2×#13511yescharacteristic
467C22×C3#57611yescharacteristic
468C22×C3#57711yescharacteristic
469C2×D3—24no—
470C2×D3—42no—
471C2×D3—42no—
472C2×D3—42no—
473C2×D3—42no—
474C2×D3—42no—
475C2×D3—42no—
476C3×C4—22no—
477C3×C4—42no—
478C3×C4—42no—
479C3×C4—42no—
480C3×C4—41no—
481C3×C4—81no—
482C3×C4C2×C4×Q811yescharacteristic
483C3×C4#52011yescharacteristic
484C3×C4#55211yescharacteristic
485C3×C4#55411yescharacteristic
486Dic3—22no—
487Dic3—22no—
488Dic3—22no—
489Dic3—42no—
490Dic3—41no—
491Dic3—81no—
492Dic3—81no—
493C23—62no—
494C23—62no—
495C23—62no—
496C23—121no—
497C23—121no—
498C23—121no—
499C23—121no—
500C23C2×D3×D411yescharacteristic
501D4—34no—
502D4—62no—
503D4—62no—
504D4—62no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—32no—
514C2×C4—32no—
515C2×C4—32no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—121no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—244no—
544C2×C4—244no—
545C2×C4—244no—
546C2×C4#91612yes—
547C2×C4C22×C4×D311yescharacteristic
548C2×C4C2×D3×Q811yescharacteristic
549C2×C4D3×#4111yescharacteristic
550C2×C4D3×#4111yescharacteristic
551C2×C4#100911yescharacteristic
552C2×C4#101011yescharacteristic
553C8—22no—
554C8—41no—
555C8—62no—
556C8—62no—
557C2×C3#236011yescharacteristic
558C2×C3C4×#13511yescharacteristic
559C2×C3#294611yescharacteristic
560C2×C3#329611yescharacteristic
561C2×C3#329711yescharacteristic
562C2×C3#338011yescharacteristic
563C2×C3#338111yescharacteristic
564D3—44no—
565C22—64no—
566C22—122no—
567C22—122no—
568C22—122no—
569C22—122no—
570C22—122no—
571C22—122no—
572C22C4×D3×D411yescharacteristic
573C22#563211yescharacteristic
574C22#565411yescharacteristic
575C22C2×#91611yescharacteristic
576C22D3×#13511yescharacteristic
577C22#577411yescharacteristic
578C22#577811yescharacteristic
579C4—22no—
580C4—42no—
581C4—42no—
582C4—42no—
583C4—41no—
584C4—62no—
585C4—62no—
586C4—62no—
587C4—81no—
588C4—121no—
589C4—122no—
590C4—241no—
591C4—241no—
592C4#552311yescharacteristic
593C4C4×D3×Q811yescharacteristic
594C4#569011yescharacteristic
595C4#570811yescharacteristic
596C3#2296611yescharacteristic, complemented, Sylow
597C2—124no—
598C2#7496511yescharacteristic
599C2C4×#91611yescharacteristic
600C2#7977511yescharacteristic
601C2#8105411yescharacteristic
602C2#8122111yescharacteristic
603C2#8138211yescharacteristic
604C2#8144511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1920C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1920D3
hypocenterC3#22966
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
3. #73661C2semidirect
4. C3×#1920C2semidirect
5. #70560C2semidirect
6. #69619C2semidirect
9. #68821C2semidirect
10. #68804C2semidirect
11. #68797C2semidirect
15. #68651C2semidirect
46. #4871C4semidirect
69. #1920D3semidirect
113. #890C2×C4semidirect
116. C3×#106C2×C4semidirect
131. Dic3⋊C8C2×C4semidirect
137. #807C2×C4semidirect
302. #106C4×D3semidirect
596. C3#22966semidirect

Maximal subgroups

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

subgroupindex
C2×#48712
#762032
#736612
C3×#19202
#705602
#696192
C4×#8092
#688942
#688212
#688042
#687972
#687802
#686712
#686512
#229663