Finite Group Databasebeta

← #29743429

Subgroups — #29743429

Order 15616.

814 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840069C211yesmaximal, characteristic, complemented
2C2×#16839860C211yesmaximal, characteristic
3#21449691C211yesmaximal, characteristic
4#21447343C211yesmaximal, characteristic
5#21447301C211yesmaximal, characteristic, complemented
6#21447190C211yesmaximal, characteristic
7#21447157C211yesmaximal, characteristic, complemented
8#21446559C211yesmaximal, characteristic, complemented
9#21446385C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446174C211yesmaximal, characteristic
12#21446087C211yesmaximal, characteristic, complemented
13#21446087C211yesmaximal, characteristic, complemented
14#21446079C211yesmaximal, characteristic
15C61×#2351C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519769C2211yescharacteristic
18C2×C4×D244C2211yescharacteristic
19#16840069—24no—
20#16839860—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519688C2211yescharacteristic
34#16839697—21no—
35#16839686—21no—
36#16839664C2211yescharacteristic
37#16839664C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839633—21no—
41#16839621—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839517C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C61×#389—21no—
54C61×#389—21no—
55C61×#389C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#382—21no—
58C61×#382C2211yescharacteristic
59C4×C4⋊C4×C61—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201Q814yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519769—22no—
71#14519769—41no—
72C4×D244—22no—
73C4×D244—41no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758198—21no—
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86#14519710—22no—
87#14519710—22no—
88#14519708—22no—
89#14519708—22no—
90#14519708—41no—
91C4⋊#92418—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701D414yes—
95#14519697—22no—
96#14519697—22no—
97#14519697—41no—
98#14519688—22no—
99#14519684—21no—
100#14519684—21no—
101C22×Q8×C61C2311yescharacteristic
102C4×Q8×C61—22no—
103C4×Q8×C61—22no—
104C4×Q8×C61—22no—
105C4×Q8×C61—22no—
106C4×Q8×C61—41no—
107C4×Q8×C61—41no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61—21no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C42×C61—21no—
119C2×C42×C61—21no—
120C2×C42×C61—21no—
121C2×C42×C61C2311yescharacteristic
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61D411yescharacteristic
129#95×C61D411yescharacteristic
130C23×D61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61C2×D412yes—
135C2×Q8×C61C2×Q812yes—
136C2×Q8×C61#4112yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244C2×Q812yes—
157C2×C4×D61—42no—
158C2×C4×D61—42no—
159C2×C4×D61—81no—
160C4⋊C4×C61—24no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—41no—
167C4⋊C4×C61—41no—
168C4⋊C4×C61—42no—
169C4⋊C4×C61—42no—
170C4⋊C4×C61—42no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61#4112yes—
173C42×C61—24no—
174C42×C61—21no—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—42no—
181C42×C61—42no—
182#11758201—28no—
183#11758198—42no—
184#11758198—42no—
185#11758198—81no—
186#11758198—81no—
187#11758198—81no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61C2×D412yes—
194#11758196—42no—
195#11758196—42no—
196#11758196—81no—
197#11758196—81no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418C2×D412yes—
209C2×#92418C2×Q812yes—
210C23×C61C22×D411yescharacteristic
211C22×D61—22no—
212C22×D61—41no—
213C22×D61—41no—
214C22×D61—41no—
215C22×D61—41no—
216C22×D61—41no—
217C22×D61—41no—
218Q8×C61—22no—
219Q8×C61—22no—
220Q8×C61—22no—
221Q8×C61C22⋊Q814yes—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C22×Q811yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267D244—22no—
268D244—22no—
269D244—41no—
270D244—41no—
271C4×D61—84no—
272#92418—22no—
273#92418—22no—
274#92418—41no—
275#92418—41no—
276#26399—611nomaximal, Sylow
277C22×C61#52511yescharacteristic
278C22×C61D4211yescharacteristic
279C22×C61D4×Q811yescharacteristic
280C22×C61#57511yescharacteristic
281C22×C61#57511yescharacteristic
282C22×C61#57711yescharacteristic
283C22×C61#57711yescharacteristic
284C2×D61—42no—
285C2×D61—42no—
286C2×D61—42no—
287C2×D61—81no—
288C2×D61—81no—
289C2×D61—81no—
290C2×D61—81no—
291C4×C61—24no—
292C4×C61—41no—
293C4×C61—42no—
294C4×C61—41no—
295C4×C61—41no—
296C4×C61—41no—
297C4×C61—41no—
298C4×C61—41no—
299C4×C61—81no—
300C4×C61—81no—
301C4×C61C2×C22⋊Q811yescharacteristic
302C4×C61#53911yescharacteristic
303C4×C61#54711yescharacteristic
304C4×C61#55411yescharacteristic
305Dic61—42no—
306Dic61—81no—
307Dic61—81no—
308C2×#479—611no—
309C2×#440—611no—
310#2454—611no—
311#2351D6111yescharacteristic, complemented
312#2092—611no—
313#2085—611no—
314#2065—611no—
315#2059—611no—
316#1944—611no—
317#1917—611no—
318#1880—611no—
319#1880—611no—
320#1865—611no—
321#1865—611no—
322#1864—611no—
323C2×C61#244911yescharacteristic
324C2×C61#311111yescharacteristic
325C2×C61#312611yescharacteristic
326C2×C61#327111yescharacteristic
327C2×C61#327511yescharacteristic
328C2×C61#338811yescharacteristic
329C2×C61#338811yescharacteristic
330D61—82no—
331C22×SD16—611no—
332C2×#125—611no—
333C2×C4×Q8C2×D6111yescharacteristic
334C2×C4×Q8C2×D6111yescharacteristic
335C2×C4×Q8C2×D6111yescharacteristic
336C2×C4×D4—611no—
337#479—1224no—
338#440—1224no—
339C2×#107—611no—
340C2×#107—611no—
341C2×#106—611no—
342C2×#106—611no—
343C2×#106—611no—
344C2×C4⋊C8—611no—
345C2×#103—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#102—611no—
349C2×#102—611no—
350C2×#102—611no—
351C2×#97—611no—
352#399—1221no—
353#396—1221no—
354#390—611no—
355#390—611no—
356#389—21no—
357#389—21no—
358#389—611no—
359#389—611no—
360#389C2×D6111yescharacteristic
361#389C2×D6111yescharacteristic
362#385—1221no—
363#382—21no—
364#382C2×D6111yescharacteristic
365#380—1221no—
366C4×C4⋊C4—21no—
367C4×C4⋊C4C2×D6111yescharacteristic
368#340—611no—
369#340—611no—
370#340—611no—
371#340—611no—
372#336—611no—
373#336—611no—
374#336—611no—
375#336—611no—
376C61#2639911yescharacteristic, complemented, Sylow
377C22×Q8C22×D6111yescharacteristic
378C22×D4—611no—
379C23×C4—1221no—
380C2×SD16—614no—
381C2×SD16—1222no—
382C2×SD16—1222no—
383C22×C8—611no—
384C22×C8—611no—
385C22×C8—611no—
386C22×C8—611no—
387#125—1222no—
388#125—2441no—
389C4×Q8—22no—
390C4×Q8—22no—
391C4×Q8—22no—
392C4×Q8—22no—
393C4×Q8—41no—
394C4×Q8—41no—
395C4×D4—1222no—
396C4×D4—2441no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—611no—
403C2×C4⋊C4—611no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22⋊D6111yescharacteristic
410C2×C4⋊C4C22⋊D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C4⋊C4C22×D6111yescharacteristic
414C2×C22⋊C4—1221no—
415C2×C22⋊C4—1221no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C22⋊C4—1221no—
419C2×C42—21no—
420C2×C42—21no—
421C2×C42—21no—
422C2×C42C22×D6111yescharacteristic
423C2×C42C22×D6111yescharacteristic
424C2×C42C22×D6111yescharacteristic
425#107—1222no—
426#107—1222no—
427#106—1222no—
428#106—1222no—
429#106—2441no—
430C4⋊C8—1222no—
431#103—614no—
432#103—1222no—
433#103—1222no—
434#102—1222no—
435#102—1222no—
436#102—2441no—
437#97—1222no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1221no—
443#95—1221no—
444#95C22⋊D6111yescharacteristic
445#95C22⋊D6111yescharacteristic
446C24—1221no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8#1175822212yes—
451C2×Q8D4×D6112yes—
452C2×Q8Q8×D6112yes—
453C2×D4—612no—
454C2×D4—1221no—
455C2×D4—1221no—
456C2×D4—1221no—
457C2×D4—1221no—
458C2×D4—1221no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—1221no—
464C22×C4—1221no—
465C22×C4—1221no—
466C22×C4—4881no—
467C22×C4—2442no—
468C22×C4—2442no—
469C22×C4C2×C22⋊D6111yescharacteristic
470C22×C4C2×C22⋊D6111yescharacteristic
471C22×C4C2×C22⋊D6111yescharacteristic
472C22×C4C2×C22⋊D6111yescharacteristic
473C22×C4C2×C22⋊D6111yescharacteristic
474C22×C4C2×C22⋊D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476SD16—1228no—
477C2×C8—612no—
478C2×C8—612no—
479C2×C8—1221no—
480C2×C8—1221no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8—1221no—
489C4⋊C4—24no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—42no—
500C4⋊C4—42no—
501C4⋊C4—612no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4—1221no—
507C4⋊C4—4881no—
508C4⋊C4—4881no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4#1175822212yes—
512C22⋊C4—4881no—
513C22⋊C4—4881no—
514C22⋊C4—4881no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—24no—
518C42—21no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—42no—
526C23—1222no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23C22×C22⋊D6111yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1451975514yes—
538D4—1222no—
539D4—1222no—
540D4—2441no—
541D4—2441no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—22no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—1222no—
573C2×C4—4884no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4C2×D4×D6111yescharacteristic
583C2×C4C2×Q8×D6111yescharacteristic
584C2×C4C2×#1175822211yescharacteristic
585C2×C4C2×#1175822211yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982511yescharacteristic
588C8—1222no—
589C8—1222no—
590C8—2441no—
591C8—2441no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22#1684027311yescharacteristic
600C22D4×C22⋊D6111yescharacteristic
601C22Q8×C22⋊D6111yescharacteristic
602C22#1684056011yescharacteristic
603C22#1684056011yescharacteristic
604C22#1684057011yescharacteristic
605C22#1684057011yescharacteristic
606C4—24no—
607C4—41no—
608C4—41no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—41no—
613C4—41no—
614C4—81no—
615C4—81no—
616C4—4881no—
617C4—4881no—
618C4—2442no—
619C4C2×#1451975511yescharacteristic
620C4#1684035911yescharacteristic
621C4#1684041311yescharacteristic
622C4#1684046711yescharacteristic
623C2—4882no—
624C2#2144962811yescharacteristic
625C2#2145592811yescharacteristic
626C2#2145611011yescharacteristic
627C2#2145740511yescharacteristic
628C2#2145745111yescharacteristic
629C2#2145875811yescharacteristic
630C2#2145875811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2351C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2351D61
hypocenterC61#26399
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C2×#16840069C2semidirect
5. #21447301C2semidirect
7. #21447157C2semidirect
8. #21446559C2semidirect
9. #21446385C2semidirect
12. #21446087C2semidirect
13. #21446087C2semidirect
15. C61×#2351C2semidirect
311. #2351D61semidirect
376. C61#26399semidirect

Maximal subgroups

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

subgroupindex
C2×#168400692
C2×#168398602
#214496912
#214473432
#214473012
#214471902
#214471572
#214465592
#214463852
#214461742
#214460872
#214460792
C61×#23512
#2639961