Finite Group Databasebeta

← #10859096

Subgroups — #10859096

Order 768.

748 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
1#73494C211yesmaximal, characteristic
2#73465C211yesmaximal, characteristic
3C3×#1851C211yesmaximal, characteristic
4#71989C211yesmaximal, characteristic
5#71985C211yesmaximal, characteristic
6#70489C211yesmaximal, characteristic, complemented
7#70485C211yesmaximal, characteristic, complemented
8#70463C211yesmaximal, characteristic, complemented
9#70461C211yesmaximal, characteristic, complemented
10#69931C211yesmaximal, characteristic
11#69927C211yesmaximal, characteristic
12#68690C211yesmaximal, characteristic, complemented
13#68683C211yesmaximal, characteristic, complemented
14#68642C211yesmaximal, characteristic, complemented
15#68621C211yesmaximal, characteristic, complemented
16#26432—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×#926C2211yescharacteristic
19C22×C4⋊Dic3C2211yescharacteristic
20C2×#924C2211yescharacteristic
21C2×#870C2211yescharacteristic
22C2×C4×Dic6C2211yescharacteristic
23C2×C3×C4⋊C8C2211yescharacteristic
24C2×C3×C4⋊C8C2211yescharacteristic
25C2×C3×C22⋊C8C2211yescharacteristic
26C3×C4×C22⋊C4C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#818C2211yescharacteristic
30C2×#816C2211yescharacteristic
31C2×#816C2211yescharacteristic
32C2×#816C2211yescharacteristic
33C2×#816C2211yescharacteristic
34#4949—21no—
35#4948—21no—
36#4946—21no—
37Dic3×C22⋊C4—21no—
38#4935C2211yescharacteristic
39C4×C4⋊Dic3C2211yescharacteristic
40#4929C2211yescharacteristic
41#4928C2211yescharacteristic
42#4928C2211yescharacteristic
43#4928C2211yescharacteristic
44#4655—21no—
45#4654—21no—
46#4647—21no—
47#4647—21no—
48C3×#336C2211yescharacteristic
49C3×#336C2211yescharacteristic
50#4551C2211yescharacteristic
51#4551C2211yescharacteristic
52#4551C2211yescharacteristic
53#4551C2211yescharacteristic
54#4551C2211yescharacteristic
55#4551C2211yescharacteristic
56#4551C2211yescharacteristic
57#4551C2211yescharacteristic
58#4551C2211yescharacteristic
59#4551C2211yescharacteristic
60#2441—31no—
61#2360—31no—
62#2099—31no—
63#2087—31no—
64#2068—31no—
65#2026—31no—
66#1989—31no—
67#1918—31no—
68#1894—31no—
69#1873—31no—
70#1863—31no—
71#1852—31no—
72#1851D311yescharacteristic
73#1803—31no—
74#1736—31no—
75C23×C3×C4C2311yescharacteristic
76C23×Dic3—21no—
77C22×Dic6C2311yescharacteristic
78C2×C3×M16—22no—
79C2×C3×M16—41no—
80C2×C3×M16—41no—
81C22×C3×C8C2311yescharacteristic
82C22×C3×C8C2311yescharacteristic
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C2×C3×C22⋊C4D411yescharacteristic
86C2×C3×C22⋊C4D411yescharacteristic
87C2×C3×C42C2311yescharacteristic
88C2×C3×C42C2311yescharacteristic
89C2×#217—21no—
90C2×#217—21no—
91C2×#217—21no—
92#926—22no—
93#926—41no—
94C2×C4⋊Dic3—22no—
95C2×C4⋊Dic3—21no—
96C2×C4⋊Dic3—21no—
97C2×C4⋊Dic3—21no—
98C2×C4⋊Dic3—41no—
99C2×C4⋊Dic3C2311yescharacteristic
100C2×C4⋊Dic3C2311yescharacteristic
101C2×C4⋊Dic3C2311yescharacteristic
102C2×C4⋊Dic3C2311yescharacteristic
103C2×C4⋊Dic3C2311yescharacteristic
104C2×C4⋊Dic3C2311yescharacteristic
105C2×C4⋊Dic3C2311yescharacteristic
106#924—22no—
107#924Q814yes—
108C2×Dic3⋊C4—21no—
109C2×Dic3⋊C4—21no—
110C2×Dic3⋊C4—21no—
111C2×Dic3⋊C4—21no—
112C2×Dic3⋊C4—21no—
113C2×Dic3⋊C4—21no—
114C2×Dic3⋊C4—21no—
115C2×C4×Dic3—21no—
116C2×C4×Dic3—21no—
117C2×C4×Dic3—21no—
118#870—22no—
119#870—41no—
120C4×Dic6—22no—
121C4×Dic6—41no—
122C3×C4⋊C8—41no—
123C3×C4⋊C8—41no—
124C3×C22⋊C8—22no—
125C3×#95D411yescharacteristic
126C3×#95D411yescharacteristic
127C3⋊#95—21no—
128C3⋊#95—21no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134#818—22no—
135#818—41no—
136#816—22no—
137#816—41no—
138#816—41no—
139#816—41no—
140C22×M16C2×D311yescharacteristic
141C2×#125—31no—
142C2×C22⋊Q8—31no—
143C2×C4×Q8—31no—
144C2×#117—31no—
145C22×C4⋊C4—31no—
146C2×#107—31no—
147C2×#107—31no—
148C2×C4⋊C8C2×D311yescharacteristic
149C2×C4⋊C8C2×D311yescharacteristic
150C2×#103—31no—
151C2×#103—31no—
152C2×#103—31no—
153C2×#103—31no—
154C2×C22⋊C8C2×D311yescharacteristic
155#399—61no—
156#396—61no—
157#391—31no—
158#389—31no—
159#389—31no—
160#389—31no—
161#389—61no—
162#389—61no—
163#387—61no—
164#387—61no—
165#385—61no—
166#384—31no—
167#380—31no—
168C4×C4⋊C4—31no—
169C4×C22⋊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#336C2×D311yescharacteristic
182#336C2×D311yescharacteristic
183C24×C3C2×D411yescharacteristic
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×C4C2×D411yescharacteristic
191C22×C3×C4C2×D411yescharacteristic
192C22×C3×C4C2×D411yescharacteristic
193C22×C3×C4C2×Q812yes—
194C22×C3×C4#4112yes—
195C22×C3×C4C2411yescharacteristic
196C22×Dic3—21no—
197C22×Dic3—21no—
198C22×Dic3—21no—
199C22×Dic3—21no—
200C22×Dic3—21no—
201C22×Dic3—21no—
202C22×Dic3—21no—
203C22×Dic3—21no—
204C22×Dic3—42no—
205C22×Dic3—42no—
206C22×Dic3—81no—
207C2×Dic6—21no—
208C2×Dic6—21no—
209C2×Dic6—21no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6C2×Q812yes—
213C3×M16—44no—
214C2×C3×C8—21no—
215C2×C3×C8—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—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C3×C22⋊C4—22no—
227C3×C22⋊C4—41no—
228C3×C42—21no—
229C3×C42—21no—
230C3×C42—21no—
231C3×C42—21no—
232#217—24no—
233#217—42no—
234#217—81no—
235C4⋊Dic3—21no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—21no—
239C4⋊Dic3—21no—
240C4⋊Dic3—21no—
241C4⋊Dic3—21no—
242C4⋊Dic3—21no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—42no—
247C4⋊Dic3—42no—
248C4⋊Dic3—81no—
249C4⋊Dic3C2×Q812yes—
250C4⋊Dic3#4112yes—
251C4⋊Dic3#4112yes—
252Dic3⋊C4—24no—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—42no—
257Dic3⋊C4—81no—
258Dic3⋊C4—81no—
259C4×Dic3—42no—
260C4×Dic3—42no—
261C4×Dic3—81no—
262C22×Q8—31no—
263C23×C4—61no—
264C23×C4C22×D311yescharacteristic
265C2×M16—22no—
266C2×M16—41no—
267C2×M16—41no—
268C22×C8C22×D311yescharacteristic
269C22×C8C22×D311yescharacteristic
270C22×C8C22×D311yescharacteristic
271C22×C8C22×D311yescharacteristic
272#125—62no—
273#125—121no—
274C22⋊Q8—34no—
275C22⋊Q8—62no—
276C4×Q8—62no—
277C4×Q8—121no—
278#117—62no—
279#117—121no—
280C2×C4⋊C4—31no—
281C2×C4⋊C4—31no—
282C2×C4⋊C4—31no—
283C2×C4⋊C4—31no—
284C2×C4⋊C4—31no—
285C2×C4⋊C4—31no—
286C2×C4⋊C4—31no—
287C2×C4⋊C4—62no—
288C2×C4⋊C4—61no—
289C2×C4⋊C4—61no—
290C2×C4⋊C4—61no—
291C2×C4⋊C4—61no—
292C2×C4⋊C4—61no—
293C2×C4⋊C4—61no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—121no—
299C2×C22⋊C4—61no—
300C2×C22⋊C4—61no—
301C2×C22⋊C4—61no—
302C2×C22⋊C4D1211yescharacteristic
303C2×C22⋊C4D1211yescharacteristic
304C2×C42—61no—
305C2×C42—61no—
306C2×C42—61no—
307C2×C42C22×D311yescharacteristic
308C2×C42C22×D311yescharacteristic
309#107—62no—
310#107—121no—
311C4⋊C8—41no—
312C4⋊C8—41no—
313#103—62no—
314#103—121no—
315#103—121no—
316#103—121no—
317C22⋊C8—22no—
318#95—61no—
319#95—61no—
320#95—61no—
321#95—61no—
322#95—61no—
323#95—61no—
324#95—61no—
325#95D1211yescharacteristic
326#95D1211yescharacteristic
327C23×C3—21no—
328C23×C3—21no—
329C23×C3—21no—
330C23×C3—21no—
331C23×C3—21no—
332C23×C3—21no—
333C23×C3C22⋊Q812yes—
334C23×C3C22×D411yescharacteristic
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—22no—
340C2×C3×C4—22no—
341C2×C3×C4—22no—
342C2×C3×C4—22no—
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—21no—
352C2×C3×C4C22⋊Q812yes—
353C2×C3×C4#13712yes—
354C2×C3×C4C22×Q811yescharacteristic
355C2×C3×C4C2×#4111yescharacteristic
356C2×C3×C4C2×#4111yescharacteristic
357C2×C3×C4C2×#4111yescharacteristic
358C2×C3×C4H(5,𝔽2)11yescharacteristic
359C2×C3×C4H(5,𝔽2)11yescharacteristic
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
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×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—
382C2×Dic3—41no—
383C2×Dic3—41no—
384C2×Dic3—84no—
385Dic6—22no—
386Dic6—22no—
387Dic6—41no—
388Dic6—41no—
389C3×C8—41no—
390C3×C8—41no—
391C3×C8—41no—
392C3×C8—41no—
393C24C2×D1211yescharacteristic
394C2×Q8—32no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×Q8—61no—
399C2×Q8—61no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—21no—
403C22×C4—21no—
404C22×C4—61no—
405C22×C4—61no—
406C22×C4—61no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4—61no—
410C22×C4—61no—
411C22×C4—61no—
412C22×C4—122no—
413C22×C4—122no—
414C22×C4—241no—
415C22×C4C2×D1211yescharacteristic
416C22×C4C2×D1211yescharacteristic
417C22×C4C2×D1211yescharacteristic
418C22×C4C2×D1211yescharacteristic
419C22×C4C2×D1211yescharacteristic
420C22×C4#23712yes—
421C22×C4D3×Q812yes—
422C22×C4C23×D311yescharacteristic
423M16—44no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8—21no—
428C2×C8—21no—
429C2×C8—21no—
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—21no—
436C4⋊C4—32no—
437C4⋊C4—32no—
438C4⋊C4—32no—
439C4⋊C4—64no—
440C4⋊C4—61no—
441C4⋊C4—61no—
442C4⋊C4—61no—
443C4⋊C4—61no—
444C4⋊C4—61no—
445C4⋊C4—61no—
446C4⋊C4—61no—
447C4⋊C4—61no—
448C4⋊C4—61no—
449C4⋊C4—61no—
450C4⋊C4—61no—
451C4⋊C4—122no—
452C4⋊C4—122no—
453C4⋊C4—122no—
454C4⋊C4—122no—
455C4⋊C4—122no—
456C4⋊C4—122no—
457C4⋊C4—241no—
458C4⋊C4—241no—
459C4⋊C4—241no—
460C22⋊C4—22no—
461C22⋊C4—41no—
462C22⋊C4—64no—
463C22⋊C4—122no—
464C22⋊C4—241no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—122no—
470C42—122no—
471C42—241no—
472C22×C3—22no—
473C22×C3—22no—
474C22×C3—22no—
475C22×C3—41no—
476C22×C3—41no—
477C22×C3—41no—
478C22×C3—41no—
479C22×C3C2×C22⋊Q811yescharacteristic
480C22×C3#54611yescharacteristic
481C22×C3#54611yescharacteristic
482C22×C3C2×#13711yescharacteristic
483C22×C3#57511yescharacteristic
484C22×C3#57611yescharacteristic
485C22×C3#57811yescharacteristic
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—41no—
491C3×C4#52911yescharacteristic
492C3×C4#53811yescharacteristic
493C3×C4#55411yescharacteristic
494C3×C4#55411yescharacteristic
495Dic3—42no—
496Dic3—42no—
497Dic3—42no—
498Dic3—42no—
499Dic3—81no—
500Dic3—81no—
501Dic3—81no—
502Dic3—81no—
503C23—21no—
504C23—21no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23#89712yes—
510C23C22×D1211yescharacteristic
511Q8—62no—
512Q8—62no—
513Q8—121no—
514Q8—121no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
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—62no—
533C2×C4—62no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—121no—
537C2×C4—121no—
538C2×C4—121no—
539C2×C4—121no—
540C2×C4—121no—
541C2×C4—121no—
542C2×C4—121no—
543C2×C4—121no—
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#89712yes—
558C2×C4#90912yes—
559C2×C4C2×#23711yescharacteristic
560C2×C4#100411yescharacteristic
561C2×C4#100411yescharacteristic
562C2×C4C2×D3×Q811yescharacteristic
563C2×C4D3×#4111yescharacteristic
564C2×C4D3×#4111yescharacteristic
565C8—41no—
566C8—41no—
567C8—41no—
568C8—41no—
569C2×C3—42no—
570C2×C3#245011yescharacteristic
571C2×C3#307811yescharacteristic
572C2×C3#308311yescharacteristic
573C2×C3#331211yescharacteristic
574C2×C3#331511yescharacteristic
575C2×C3#332211yescharacteristic
576C2×C3#333311yescharacteristic
577C22—22no—
578C22—22no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22C2×#89711yescharacteristic
585C22#555411yescharacteristic
586C22#555411yescharacteristic
587C22C2×#90911yescharacteristic
588C22#574811yescharacteristic
589C22#575211yescharacteristic
590C22#575311yescharacteristic
591C4—22no—
592C4—22no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—122no—
597C4—122no—
598C4—122no—
599C4—122no—
600C4—241no—
601C4—241no—
602C4—241no—
603C4—241no—
604C4#559311yescharacteristic
605C4#563611yescharacteristic
606C4#563611yescharacteristic
607C4#564911yescharacteristic
608C3#2643211yescharacteristic, complemented, Sylow
609C2—42no—
610C2#7442411yescharacteristic
611C2#7906811yescharacteristic
612C2#7909111yescharacteristic
613C2#7950411yescharacteristic
614C2#7951111yescharacteristic
615C2#7955311yescharacteristic
616C2#7956611yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1851C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1851D3
hypocenterC3#26432
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1851

Complements

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

NH ≅ G/Ntype
6. #70489C2semidirect
7. #70485C2semidirect
8. #70463C2semidirect
9. #70461C2semidirect
12. #68690C2semidirect
13. #68683C2semidirect
14. #68642C2semidirect
15. #68621C2semidirect
608. C3#26432semidirect

Maximal subgroups

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

subgroupindex
#734942
#734652
C3×#18512
#719892
#719852
#704892
#704852
#704632
#704612
#699312
#699272
#686902
#686832
#686422
#686212
#264323