Finite Group Databasebeta

← #29743425

Subgroups — #29743425

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×#16839865C211yesmaximal, characteristic
2#21449687C211yesmaximal, characteristic
3#21448737C211yesmaximal, characteristic, complemented
4#21447337C211yesmaximal, characteristic
5#21447304C211yesmaximal, characteristic, complemented
6#21447192C211yesmaximal, characteristic
7#21447154C211yesmaximal, characteristic, complemented
8#21446560C211yesmaximal, characteristic, complemented
9#21446380C211yesmaximal, characteristic, complemented
10#21446174C211yesmaximal, characteristic
11#21446172C211yesmaximal, characteristic
12#21446089C211yesmaximal, characteristic, complemented
13#21446085C211yesmaximal, characteristic, complemented
14#21446077C211yesmaximal, characteristic
15C2×C61×#479C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×Q8×Dic61C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×#14519739C2211yescharacteristic
20#16839865—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27C2×#14519697C2211yescharacteristic
28C2×#14519687C2211yescharacteristic
29#16839697—21no—
30#16839684—21no—
31#16839664C2211yescharacteristic
32#16839663C2211yescharacteristic
33#16839661C2211yescharacteristic
34#16839660—21no—
35#16839660C2211yescharacteristic
36#16839658—21no—
37#16839658C2211yescharacteristic
38#16839658C2211yescharacteristic
39#16839633—21no—
40#16839627—21no—
41#16839627C2211yescharacteristic
42#16839619—21no—
43C4×#11758196—21no—
44C4×#11758196C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53C2×#125×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#479—24no—
56C2×#107×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C4×D61—21no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C2×#11758201Q814yes—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68Q8×Dic61—22no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—41no—
72#14519739—22no—
73#14519739—41no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196D411yescharacteristic
84C2×#11758196D411yescharacteristic
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93C2×C4×Dic61C2311yescharacteristic
94#14519710—22no—
95#14519709—22no—
96#14519708—41no—
97#14519701—22no—
98#14519699—22no—
99#14519699—22no—
100#14519697—41no—
101#14519687—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684D411yescharacteristic
109#14519684D411yescharacteristic
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113#125×C61—22no—
114#125×C61—41no—
115C4×Q8×C61—22no—
116C4×Q8×C61—41no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C42×C61—21no—
124C2×C42×C61C2311yescharacteristic
125#107×C61—22no—
126#106×C61—22no—
127C4⋊C8×C61—22no—
128#103×C61—22no—
129#103×C61D414yes—
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×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
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—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D412yes—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—42no—
172C4⋊C4×C61—42no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61C2×D412yes—
175C42×C61—21no—
176C42×C61—21no—
177C42×C61—42no—
178#11758201—28no—
179#11758198—42no—
180#11758198—42no—
181#11758198—81no—
182#11758198—81no—
183#11758198—81no—
184C4⋊Dic61—24no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—42no—
191C4⋊Dic61—81no—
192C4⋊Dic61#4112yes—
193#11758196—41no—
194#11758196—41no—
195#11758196—42no—
196#11758196—42no—
197#11758196—81no—
198C4×Dic61—24no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
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—22no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61C22×D411yescharacteristic
241C2×C4×C61C22×Q811yescharacteristic
242C2×C4×C61C2×#4111yescharacteristic
243C2×C4×C61C2×#4111yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—22no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
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—
272C8×C61—22no—
273C8×C61—41no—
274#92418—22no—
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—81no—
296C4×C61—81no—
297C4×C61C2×C22⋊Q811yescharacteristic
298C4×C61#53911yescharacteristic
299C4×C61#54711yescharacteristic
300C4×C61#55411yescharacteristic
301Dic61—41no—
302Dic61—42no—
303Dic61—41no—
304Dic61—41no—
305Dic61—41no—
306Dic61—81no—
307Dic61—81no—
308C2×#479D6111yescharacteristic, complemented
309C2×#440—611no—
310#2454—611no—
311#2351—611no—
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×#125C2×D6111yescharacteristic
333C2×C4×Q8—611no—
334C2×C4×Q8—611no—
335C2×C4×Q8C2×D6111yescharacteristic
336C2×C4×D4—611no—
337#479—24no—
338#440—1224no—
339C2×#107—611no—
340C2×#107C2×D6111yescharacteristic
341C2×#106—611no—
342C2×#106—611no—
343C2×#106C2×D6111yescharacteristic
344C2×C4⋊C8C2×D6111yescharacteristic
345C2×#103—611no—
346C2×#103C2×D6111yescharacteristic
347C2×#103C2×D6111yescharacteristic
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—611no—
357#389—611no—
358#389—611no—
359#389—611no—
360#389—1221no—
361#389—1221no—
362#385—1221no—
363#382—611no—
364#382—1221no—
365#380—1221no—
366C4×C4⋊C4—611no—
367C4×C4⋊C4—1221no—
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×C8C22×D6111yescharacteristic
386C22×C8C22×D6111yescharacteristic
387#125—22no—
388#125—41no—
389C4×Q8—22no—
390C4×Q8—41no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—2441no—
395C4×D4—1222no—
396C4×D4—2441no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—611no—
401C2×C4⋊C4—611no—
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⋊C4—1221no—
410C2×C4⋊C4—1221no—
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—611no—
421C2×C42—611no—
422C2×C42—1221no—
423C2×C42—1221no—
424C2×C42C22×D6111yescharacteristic
425#107—22no—
426#107—1222no—
427#106—22no—
428#106—1222no—
429#106—2441no—
430C4⋊C8—22no—
431#103—22no—
432#103—1222no—
433#103C22⋊D6114yes—
434#102—1222no—
435#102—1222no—
436#102—2441no—
437#97—1222no—
438#95—611no—
439#95—611no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445#95—1221no—
446C24—1221no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8Q8×D6112yes—
451C2×Q8#1175822612yes—
452C2×Q8C2×C22⋊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—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—4881no—
471C22×C4—2442no—
472C22×C4—2442no—
473C22×C4D4×D6111yescharacteristic
474C22×C4D4×D6111yescharacteristic
475C22×C4C23×D6111yescharacteristic
476SD16—1228no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C2×C8C2×C22⋊D6112yes—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—612no—
498C4⋊C4—1224no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—2441no—
507C4⋊C4—2441no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4C2×C22⋊D6112yes—
512C22⋊C4—4881no—
513C22⋊C4—4881no—
514C22⋊C4—4881no—
515C22⋊C4—2442no—
516C22⋊C4—2442no—
517C42—21no—
518C42—21no—
519C42—42no—
520C42—1224no—
521C42—1221no—
522C42—1221no—
523C42—1221no—
524C42—1221no—
525C42—2442no—
526C23—1222no—
527C23—2441no—
528C23—2441no—
529C23—2441no—
530C23—2441no—
531C23—2441no—
532C23—2441no—
533C23C2×D4×D6111yescharacteristic
534Q8—22no—
535Q8—22no—
536Q8—22no—
537Q8#1451975314yes—
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—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—1221no—
561C2×C4—1222no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1221no—
569C2×C4—1221no—
570C2×C4—1221no—
571C2×C4—1221no—
572C2×C4—1221no—
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×C4C22×C22⋊D6111yescharacteristic
583C2×C4C2×Q8×D6111yescharacteristic
584C2×C4#41×D6111yescharacteristic
585C2×C4C2×#1175822611yescharacteristic
586C2×C4#1451982211yescharacteristic
587C2×C4#1451982411yescharacteristic
588C8—22no—
589C8—41no—
590C8—1222no—
591C8—2441no—
592C22—4881no—
593C22—4881no—
594C22—4881no—
595C22—4881no—
596C22—2442no—
597C22—2442no—
598C22—2442no—
599C22#1684027511yescharacteristic
600C22D4×C22⋊D6111yescharacteristic
601C22#1684042511yescharacteristic
602C22#1684055911yescharacteristic
603C22#1684056111yescharacteristic
604C22#1684056911yescharacteristic
605C22#1684057111yescharacteristic
606C4—24no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—81no—
611C4—81no—
612C4—4881no—
613C4—4881no—
614C4—2441no—
615C4—2441no—
616C4—2442no—
617C4—2441no—
618C4—2441no—
619C4C2×#1451975311yescharacteristic
620C4#1684036111yescharacteristic
621C4#1684041111yescharacteristic
622C4#1684046611yescharacteristic
623C2—4882no—
624C2#2144962411yescharacteristic
625C2#2145593511yescharacteristic
626C2#2145611211yescharacteristic
627C2#2145740411yescharacteristic
628C2#2145744411yescharacteristic
629C2#2145876211yescharacteristic
630C2#2145876711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#479C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenterC2×#479D61
hypocenterC61#26399
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#479

Complements

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

NH ≅ G/Ntype
3. #21448737C2semidirect
5. #21447304C2semidirect
7. #21447154C2semidirect
8. #21446560C2semidirect
9. #21446380C2semidirect
12. #21446089C2semidirect
13. #21446085C2semidirect
15. C2×C61×#479C2semidirect
308. C2×#479D61semidirect
376. C61#26399semidirect

Maximal subgroups

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

subgroupindex
C2×#168398652
#214496872
#214487372
#214473372
#214473042
#214471922
#214471542
#214465602
#214463802
#214461742
#214461722
#214460892
#214460852
#214460772
C2×C61×#4792
#2639961