Finite Group Databasebeta

← #10858054

Subgroups — #10858054

Order 768.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5121C211yesmaximal, characteristic
2#82747C211yesmaximal, characteristic
3C2×#4799C211yesmaximal, characteristic, complemented
4#73900C211yesmaximal, characteristic
5C3×#1894C211yesmaximal, characteristic
6#71995C211yesmaximal, characteristic
7#71693C211yesmaximal, characteristic
8#70249C211yesmaximal, characteristic, complemented
9#70247C211yesmaximal, characteristic, complemented
10#70236C211yesmaximal, characteristic, complemented
11#69504C211yesmaximal, characteristic
12#68838C211yesmaximal, characteristic, complemented
13#68828C211yesmaximal, characteristic, complemented
14#68816C211yesmaximal, characteristic, complemented
15#68795C211yesmaximal, characteristic, complemented
16#26362—31nomaximal, Sylow
17C22×C3×M16C2211yescharacteristic
18C2×C3×C22⋊Q8C2211yescharacteristic
19C2×Q8×Dic3C2211yescharacteristic
20C2×#926C2211yescharacteristic
21C22×C4⋊Dic3C2211yescharacteristic
22C2×#890C2211yescharacteristic
23C2×C3×#103C2211yescharacteristic
24C2×C3×#103C2211yescharacteristic
25C3×#380C2211yescharacteristic
26#5229C2211yescharacteristic
27C2×#835C2211yescharacteristic
28C2×#835C2211yescharacteristic
29C4×#217C2211yescharacteristic
30C2×#830C2211yescharacteristic
31#5121—24no—
32C2×#818C2211yescharacteristic
33C2×#818C2211yescharacteristic
34C2×#817C2211yescharacteristic
35C2×#817C2211yescharacteristic
36C2×Dic3⋊C8C2211yescharacteristic
37C2×Dic3⋊C8C2211yescharacteristic
38#4984—21no—
39#4984C2211yescharacteristic
40#4984C2211yescharacteristic
41#4980—21no—
42#4975C2211yescharacteristic
43Dic3×C4⋊C4C2211yescharacteristic
44#4972C2211yescharacteristic
45C2×#807C2211yescharacteristic
46C2×#807C2211yescharacteristic
47#4950—21no—
48#4949—21no—
49#4948—21no—
50Dic3×C22⋊C4—21no—
51#4799—24no—
52#4655—21no—
53#4647—21no—
54C3×#340C2211yescharacteristic
55C3×#340C2211yescharacteristic
56#4550C2211yescharacteristic
57#4550C2211yescharacteristic
58#4529C2211yescharacteristic
59#4529C2211yescharacteristic
60#4529C2211yescharacteristic
61#4529C2211yescharacteristic
62C2×#479—31no—
63C2×#428—31no—
64#2441—31no—
65#2360—31no—
66#2095—31no—
67#2086—31no—
68#2066—31no—
69#2059—31no—
70#1947—31no—
71#1919—31no—
72#1894D311yescharacteristic
73#1873—31no—
74#1863—31no—
75#1852—31no—
76#1851—31no—
77C22×C3×Q8C2311yescharacteristic
78C23×C3×C4C2311yescharacteristic
79C23×Dic3—21no—
80C2×C3×M16—22no—
81C2×C3×M16—22no—
82C2×C3×M16—22no—
83C22×C3×C8C2311yescharacteristic
84C22×C3×C8C2311yescharacteristic
85C3×C22⋊Q8—22no—
86C3×C22⋊Q8Q814yes—
87C2×C3×C4⋊C4—21no—
88C2×C3×C4⋊C4C2311yescharacteristic
89C2×C3×C4⋊C4C2311yescharacteristic
90C2×C3×C4⋊C4C2311yescharacteristic
91C2×C3×C22⋊C4—21no—
92C2×C3×C22⋊C4—21no—
93C2×#217—21no—
94C2×#217D411yescharacteristic
95C2×#217D411yescharacteristic
96Q8×Dic3—22no—
97Q8×Dic3—41no—
98#926—22no—
99#926—41no—
100C2×C4⋊Dic3—22no—
101C2×C4⋊Dic3—21no—
102C2×C4⋊Dic3—21no—
103C2×C4⋊Dic3—21no—
104C2×C4⋊Dic3—21no—
105C2×C4⋊Dic3—21no—
106C2×C4⋊Dic3—41no—
107C2×C4⋊Dic3C2311yescharacteristic
108C2×C4⋊Dic3C2311yescharacteristic
109C2×C4⋊Dic3C2311yescharacteristic
110C2×C4⋊Dic3C2311yescharacteristic
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—
118C2×C4×Dic3C2311yescharacteristic
119C2×C4×Dic3C2311yescharacteristic
120C22×C3⋊C8C2311yescharacteristic
121C22×C3⋊C8C2311yescharacteristic
122#890—22no—
123#890—41no—
124C3×#103—22no—
125C3×#103—22no—
126C3×#95—21no—
127#835—22no—
128#835—41no—
129C3⋊#95—21no—
130C3⋊#95—21no—
131C3⋊#95—21no—
132C3⋊#95—21no—
133C3⋊#95—21no—
134C3⋊#95—21no—
135C3⋊#95D411yescharacteristic
136C3⋊#95D411yescharacteristic
137#830—22no—
138#818—22no—
139#818—22no—
140#817—22no—
141#817D414yes—
142Dic3⋊C8—22no—
143Dic3⋊C8—22no—
144#807—22no—
145#807—41no—
146C22×M16C2×D311yescharacteristic
147C2×#125—31no—
148C2×C22⋊Q8C2×D311yescharacteristic
149C2×C4×Q8—31no—
150C2×#117—31no—
151C22×C4⋊C4—31no—
152#479—64no—
153#428—64no—
154C2×#107—31no—
155C2×#107—31no—
156C2×#107—31no—
157C2×#107—31no—
158C2×#106—31no—
159C2×#106—31no—
160C2×C4⋊C8—31no—
161C2×C4⋊C8—31no—
162C2×#103—31no—
163C2×#103—31no—
164C2×#103C2×D311yescharacteristic
165C2×#103C2×D311yescharacteristic
166C2×C22⋊C8—31no—
167#399—61no—
168#396—61no—
169#391—31no—
170#389—31no—
171#389—31no—
172#389—31no—
173#389—61no—
174#389—61no—
175#387—61no—
176#387—61no—
177#385—61no—
178#384—31no—
179#380C2×D311yescharacteristic
180C4×C4⋊C4—31no—
181C4×C22⋊C4—31no—
182C4×C22⋊C4—61no—
183#340—31no—
184#340—31no—
185#340—31no—
186#340—31no—
187#340C2×D311yescharacteristic
188#340C2×D311yescharacteristic
189#336—31no—
190#336—31no—
191C24×C3C2×D411yescharacteristic
192C2×C3×Q8—21no—
193C2×C3×Q8—21no—
194C2×C3×Q8—21no—
195C2×C3×Q8—21no—
196C2×C3×Q8—21no—
197C2×C3×Q8C2×Q812yes—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—21no—
201C22×C3×C4—21no—
202C22×C3×C4—21no—
203C22×C3×C4—21no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4C2×D411yescharacteristic
207C22×C3×C4C2×Q812yes—
208C22×C3×C4#4112yes—
209C22×C3×C4C2411yescharacteristic
210C22×Dic3—21no—
211C22×Dic3—21no—
212C22×Dic3—21no—
213C22×Dic3—21no—
214C22×Dic3—42no—
215C22×Dic3—42no—
216C22×Dic3—81no—
217C22×Dic3C2×D411yescharacteristic
218C22×Dic3C2×D411yescharacteristic
219C22×Dic3C2×D411yescharacteristic
220C22×Dic3C2×D411yescharacteristic
221C3×M16—28no—
222C2×C3×C8—21no—
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—21no—
226C2×C3×C8C2×D412yes—
227C2×C3×C8#4112yes—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—21no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4C2×Q812yes—
235C3×C22⋊C4—24no—
236C3×C22⋊C4—81no—
237#217—22no—
238#217—41no—
239#217—42no—
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—42no—
249C4⋊Dic3—81no—
250C4⋊Dic3—81no—
251C4⋊Dic3C2×D412yes—
252C4⋊Dic3C2×D412yes—
253Dic3⋊C4—42no—
254Dic3⋊C4—42no—
255Dic3⋊C4—42no—
256Dic3⋊C4—81no—
257C4×Dic3—21no—
258C4×Dic3—21no—
259C4×Dic3—21no—
260C4×Dic3—21no—
261C4×Dic3—42no—
262C4×Dic3—42no—
263C4×Dic3—81no—
264C2×C3⋊C8—21no—
265C2×C3⋊C8—21no—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C22×Q8C22×D311yescharacteristic
271C23×C4—61no—
272C23×C4C22×D311yescharacteristic
273C2×M16—22no—
274C2×M16—22no—
275C2×M16—22no—
276C22×C8—31no—
277C22×C8—31no—
278C22×C8C22×D311yescharacteristic
279C22×C8C22×D311yescharacteristic
280#125—62no—
281#125—121no—
282C22⋊Q8—22no—
283C22⋊Q8Dic614yes—
284C4×Q8—62no—
285C4×Q8—121no—
286#117—62no—
287#117—121no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—31no—
290C2×C4⋊C4—31no—
291C2×C4⋊C4—31no—
292C2×C4⋊C4—31no—
293C2×C4⋊C4—62no—
294C2×C4⋊C4—61no—
295C2×C4⋊C4—61no—
296C2×C4⋊C4—61no—
297C2×C4⋊C4—61no—
298C2×C4⋊C4—61no—
299C2×C4⋊C4—61no—
300C2×C4⋊C4—61no—
301C2×C4⋊C4—61no—
302C2×C4⋊C4—61no—
303C2×C4⋊C4—121no—
304C2×C4⋊C4C22×D311yescharacteristic
305C2×C4⋊C4C22×D311yescharacteristic
306C2×C4⋊C4C22×D311yescharacteristic
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—31no—
310C2×C22⋊C4—31no—
311C2×C22⋊C4—61no—
312C2×C42—31no—
313C2×C42—31no—
314C2×C42—61no—
315C2×C42—61no—
316C2×C42—61no—
317#107—62no—
318#107—62no—
319#107—62no—
320#107—121no—
321#106—34no—
322#106—62no—
323C4⋊C8—62no—
324C4⋊C8—62no—
325#103—22no—
326#103—22no—
327#103—62no—
328#103—121no—
329C22⋊C8—62no—
330#95—21no—
331#95—31no—
332#95—31no—
333#95—61no—
334#95—61no—
335#95—61no—
336#95—61no—
337#95—61no—
338#95—61no—
339C23×C3—21no—
340C23×C3—21no—
341C23×C3—21no—
342C23×C3—21no—
343C23×C3—21no—
344C23×C3—21no—
345C23×C3C22⋊Q812yes—
346C23×C3C22×D411yescharacteristic
347C3×Q8—22no—
348C3×Q8—22no—
349C3×Q8—41no—
350C3×Q8—41no—
351C2×C3×C4—21no—
352C2×C3×C4—21no—
353C2×C3×C4—22no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—22no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—41no—
360C2×C3×C4—41no—
361C2×C3×C4—41no—
362C2×C3×C4—41no—
363C2×C3×C4—41no—
364C2×C3×C4—41no—
365C2×C3×C4—41no—
366C2×C3×C4—41no—
367C2×C3×C4—41no—
368C2×C3×C4—41no—
369C2×C3×C4C22⋊Q812yes—
370C2×C3×C4C22×D411yescharacteristic
371C2×C3×C4C22×Q811yescharacteristic
372C2×C3×C4C2×#4111yescharacteristic
373C2×C3×C4C2×#4111yescharacteristic
374C2×C3×C4H(5,𝔽2)11yescharacteristic
375C2×C3×C42−1+411yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—21no—
378C2×Dic3—21no—
379C2×Dic3—21no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—41no—
390C2×Dic3—41no—
391C2×Dic3—41no—
392C2×Dic3—41no—
393C2×Dic3—41no—
394C2×Dic3—41no—
395C2×Dic3—41no—
396C2×Dic3—41no—
397C2×Dic3—41no—
398C2×Dic3—41no—
399C2×Dic3—84no—
400C2×Dic3#13712yes—
401C3×C8—22no—
402C3×C8—22no—
403C3⋊C8—41no—
404C3⋊C8—41no—
405C24D3×D411yescharacteristic
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—21no—
411C2×Q8C2×Dic612yes—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—61no—
425C22×C4—61no—
426C22×C4—61no—
427C22×C4—61no—
428C22×C4—122no—
429C22×C4—122no—
430C22×C4—241no—
431C22×C4C2×Dic612yes—
432C22×C4D3×D411yescharacteristic
433C22×C4#23712yes—
434C22×C4C23×D311yescharacteristic
435M16—28no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—61no—
441C2×C8—61no—
442C2×C8—61no—
443C2×C8—61no—
444C2×C8—61no—
445C2×C8—61no—
446C2×C8D3×D412yes—
447C2×C8#23712yes—
448C4⋊C4—24no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—32no—
455C4⋊C4—32no—
456C4⋊C4—61no—
457C4⋊C4—61no—
458C4⋊C4—61no—
459C4⋊C4—61no—
460C4⋊C4—61no—
461C4⋊C4—61no—
462C4⋊C4—122no—
463C4⋊C4—122no—
464C4⋊C4—122no—
465C4⋊C4—122no—
466C4⋊C4—122no—
467C4⋊C4—122no—
468C4⋊C4—241no—
469C4⋊C4—241no—
470C4⋊C4—241no—
471C4⋊C4C2×Dic612yes—
472C22⋊C4—24no—
473C22⋊C4—62no—
474C22⋊C4—81no—
475C22⋊C4—121no—
476C22⋊C4—122no—
477C42—61no—
478C42—61no—
479C42—61no—
480C42—61no—
481C42—122no—
482C42—122no—
483C42—241no—
484C22×C3—22no—
485C22×C3—22no—
486C22×C3—22no—
487C22×C3—41no—
488C22×C3—41no—
489C22×C3—41no—
490C22×C3—41no—
491C22×C3C2×C22⋊Q811yescharacteristic
492C22×C3D4211yescharacteristic
493C22×C3#54711yescharacteristic
494C22×C3C2×#13711yescharacteristic
495C22×C3#57511yescharacteristic
496C22×C3#57611yescharacteristic
497C22×C3#57811yescharacteristic
498C3×C4—22no—
499C3×C4—42no—
500C3×C4—42no—
501C3×C4—81no—
502C3×C4—81no—
503C3×C4#52511yescharacteristic
504C3×C4#53911yescharacteristic
505C3×C4D4×Q811yescharacteristic
506C3×C4#55411yescharacteristic
507Dic3—22no—
508Dic3—41no—
509Dic3—41no—
510Dic3—42no—
511Dic3—41no—
512Dic3—42no—
513Dic3—81no—
514Dic3—81no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23—21no—
520C23—21no—
521C23#87812yes—
522C23C2×D3×D411yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—32no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—61no—
547C2×C4—61no—
548C2×C4—61no—
549C2×C4—61no—
550C2×C4—61no—
551C2×C4—61no—
552C2×C4—61no—
553C2×C4—61no—
554C2×C4—61no—
555C2×C4—61no—
556C2×C4—62no—
557C2×C4—62no—
558C2×C4—61no—
559C2×C4—121no—
560C2×C4—121no—
561C2×C4—121no—
562C2×C4—121no—
563C2×C4—121no—
564C2×C4—121no—
565C2×C4—121no—
566C2×C4—121no—
567C2×C4—121no—
568C2×C4—121no—
569C2×C4—244no—
570C2×C4#87812yes—
571C2×C4C22×Dic611yescharacteristic
572C2×C4C2×D3×D411yescharacteristic
573C2×C4C2×#23711yescharacteristic
574C2×C4C2×#23711yescharacteristic
575C2×C4#100911yescharacteristic
576C2×C4#101011yescharacteristic
577C8—22no—
578C8—22no—
579C8—121no—
580C8—121no—
581C2×C3—42no—
582C2×C3#244911yescharacteristic
583C2×C3#307811yescharacteristic
584C2×C3#308311yescharacteristic
585C2×C3#326811yescharacteristic
586C2×C3#327611yescharacteristic
587C2×C3#334611yescharacteristic
588C2×C3#335211yescharacteristic
589C22—22no—
590C22—22no—
591C22—22no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22C2×#87811yescharacteristic
597C22#560911yescharacteristic
598C22#561111yescharacteristic
599C22#577411yescharacteristic
600C22D3×#13711yescharacteristic
601C22#577711yescharacteristic
602C22#577911yescharacteristic
603C4—22no—
604C4—42no—
605C4—42no—
606C4—62no—
607C4—81no—
608C4—81no—
609C4—121no—
610C4—121no—
611C4—122no—
612C4—121no—
613C4—122no—
614C4—241no—
615C4—241no—
616C4D4×Dic611yescharacteristic
617C4#554311yescharacteristic
618C4#558611yescharacteristic
619C4#566511yescharacteristic
620C3#2636211yescharacteristic, complemented, Sylow
621C2—42no—
622C2#7471011yescharacteristic
623C2#7887211yescharacteristic
624C2#7894911yescharacteristic
625C2#8083011yescharacteristic
626C2#8083611yescharacteristic
627C2#8090011yescharacteristic
628C2#8092311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1894C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1894D3
hypocenterC3#26362
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1894

Complements

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

NH ≅ G/Ntype
3. C2×#4799C2semidirect
8. #70249C2semidirect
9. #70247C2semidirect
10. #70236C2semidirect
12. #68838C2semidirect
13. #68828C2semidirect
14. #68816C2semidirect
15. #68795C2semidirect
620. C3#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#51212
#827472
C2×#47992
#739002
C3×#18942
#719952
#716932
#702492
#702472
#702362
#695042
#688382
#688282
#688162
#687952
#263623