Finite Group Databasebeta

← #29721399

Subgroups — #29721399

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840069C211yesmaximal, characteristic, complemented
2C2×#16839822C211yesmaximal, characteristic
3#21451652C211yesmaximal, characteristic
4#21448495C211yesmaximal, characteristic
5#21447258C211yesmaximal, characteristic
6#21447246C211yesmaximal, characteristic
7#21447020C211yesmaximal, characteristic
8#21447020C211yesmaximal, characteristic
9#21446774C211yesmaximal, characteristic, complemented
10#21446575C211yesmaximal, characteristic, complemented
11#21446556C211yesmaximal, characteristic, complemented
12#21446551C211yesmaximal, characteristic, complemented
13#21446511C211yesmaximal, characteristic, complemented
14#21446497C211yesmaximal, characteristic, complemented
15C61×#1803C211yesmaximal, characteristic, complemented
16C2×#14519769C2211yescharacteristic
17C2×#14519753C2211yescharacteristic
18C2×#14519734C2211yescharacteristic
19#16840069—24no—
20#16839822—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519687C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839692—21no—
33#16839692—21no—
34#16839684—21no—
35#16839684—21no—
36#16839669C2211yescharacteristic
37#16839661C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839657—21no—
41#16839653—21no—
42#16839652—21no—
43#16839646—21no—
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×C4×D61C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63C22×#92418C2311yescharacteristic
64#14519769—22no—
65#14519769—22no—
66#14519753—22no—
67#14519753—22no—
68#14519734—22no—
69#14519734—22no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×#11758196—21no—
77C2×#11758196—21no—
78C2×#11758196—21no—
79C2×#11758196—21no—
80C2×#11758196D411yescharacteristic
81C2×#11758196D411yescharacteristic
82C2×#11758198—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198D411yescharacteristic
87C2×#11758198D411yescharacteristic
88C2×C4×Dic61—21no—
89C2×C4×Dic61C2311yescharacteristic
90#14519710—22no—
91#14519710—22no—
92#14519710—22no—
93#14519708—22no—
94C4⋊#92418D414yes—
95#14519701—22no—
96#14519701—22no—
97#14519689—22no—
98#14519689—22no—
99#14519687—22no—
100C8×Dic61—22no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110C22×Q8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C4×Q8×C61—22no—
114C4×Q8×C61—22no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C4⋊C8×C61—22no—
124#103×C61—22no—
125#103×C61—22no—
126C23×D61C2×D411yescharacteristic
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61#4112yes—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2411yescharacteristic
140C22×Dic61—21no—
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61C2×D411yescharacteristic
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C2×C4×D61—21no—
148C2×C4×D61—21no—
149C2×C4×D61—21no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C8×C61—21no—
154C2×C8×C61—21no—
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—21no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61—81no—
168C42×C61—42no—
169C42×C61C2×D412yes—
170C42×C61#4112yes—
171#11758198—24no—
172#11758198—22no—
173#11758198—22no—
174#11758198—81no—
175#11758198—81no—
176#11758198—81no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—81no—
185C4⋊Dic61—81no—
186C4⋊Dic61#4112yes—
187#11758196—24no—
188#11758196—22no—
189#11758196—22no—
190#11758196—42no—
191#11758196—42no—
192#11758196—81no—
193C4×Dic61—21no—
194C4×Dic61—21no—
195C4×Dic61—81no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418C2×D412yes—
201C2×#92418C2×D412yes—
202C23×C61C22×D411yescharacteristic
203C22×D61—21no—
204C22×D61—21no—
205C22×D61—21no—
206C22×D61—21no—
207C22×D61—21no—
208C22×D61—21no—
209C22×D61—21no—
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—22no—
219C2×C4×C61—41no—
220C2×C4×C61—41no—
221C2×C4×C61—41no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61#12112yes—
231C2×C4×C61#12112yes—
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C2×#4111yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C612−1+411yescharacteristic
237C2×C4×C612−1+411yescharacteristic
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—22no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—41no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C8×C61—41no—
264C8×C61—41no—
265#92418—22no—
266#92418—22no—
267#24715—611nomaximal, Sylow
268C22×C61C2×#12111yescharacteristic
269C22×C61#52511yescharacteristic
270C22×C61#53611yescharacteristic
271C22×C61#57611yescharacteristic
272C22×C61#57711yescharacteristic
273C22×C61#57811yescharacteristic
274C22×C61#57811yescharacteristic
275C2×D61—41no—
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C2×D61—41no—
280C2×D61—41no—
281C2×D61—41no—
282C4×C61—22no—
283C4×C61—22no—
284C4×C61—42no—
285C4×C61—81no—
286C4×C61—81no—
287C4×C61—81no—
288C4×C61#54711yescharacteristic
289C4×C61#54811yescharacteristic
290C4×C61#55211yescharacteristic
291C4×C61#55611yescharacteristic
292Dic61—42no—
293Dic61—41no—
294Dic61—41no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299C2×#479—611no—
300C2×#433—611no—
301#2629—611no—
302#2323—611no—
303#2077—611no—
304#2075—611no—
305#2036—611no—
306#2036—611no—
307#1989—611no—
308#1947—611no—
309#1943—611no—
310#1942—611no—
311#1935—611no—
312#1933—611no—
313#1803D6111yescharacteristic, complemented
314C2×C61#241511yescharacteristic
315C2×C61#306111yescharacteristic
316C2×C61#306311yescharacteristic
317C2×C61#311911yescharacteristic
318C2×C61#312711yescharacteristic
319C2×C61#324211yescharacteristic
320C2×C61#324311yescharacteristic
321D61—81no—
322C2×#125—611no—
323C2×C22⋊Q8—611no—
324C2×C4×Q8C2×D6111yescharacteristic
325C2×#117—611no—
326#479—1224no—
327#433—1224no—
328C2×#107—611no—
329C2×#107—611no—
330C2×#107—611no—
331C2×#106—611no—
332C2×C4⋊C8—611no—
333C2×C4⋊C8C2×D6111yescharacteristic
334C2×#103—611no—
335C2×#103—611no—
336C2×#103C2×D6111yescharacteristic
337C2×#103C2×D6111yescharacteristic
338C2×C22⋊C8—611no—
339C2×C22⋊C8—611no—
340C2×#97—611no—
341C2×C4×C8—611no—
342#398—1221no—
343#398—1221no—
344#396—1221no—
345#396—1221no—
346#391—611no—
347#389—611no—
348#389—611no—
349#389—611no—
350#389—1221no—
351#388—1221no—
352#388—1221no—
353#387—1221no—
354#384C2×D6111yescharacteristic
355#380—611no—
356#380—611no—
357#380—611no—
358C4×C4⋊C4—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340C2×D6111yescharacteristic
366#340C2×D6111yescharacteristic
367C61#2471511yescharacteristic, complemented, Sylow
368C22×Q8C22×D6111yescharacteristic
369C23×C4—611no—
370C22×C8—611no—
371C22×C8—611no—
372C22×C8C22×D6111yescharacteristic
373C22×C8C22×D6111yescharacteristic
374#125—1222no—
375#125—1222no—
376C22⋊Q8—1222no—
377C22⋊Q8—1222no—
378C4×Q8—22no—
379C4×Q8—22no—
380#117—1222no—
381#117—1222no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—611no—
390C2×C4⋊C4—611no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4—1221no—
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C4⋊C4C22×D6111yescharacteristic
399C2×C4⋊C4C22×D6111yescharacteristic
400C2×C22⋊C4—611no—
401C2×C22⋊C4—611no—
402C2×C22⋊C4—1221no—
403C2×C22⋊C4—1221no—
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C42—21no—
407C2×C42—611no—
408C2×C42—1221no—
409C2×C42C22×D6111yescharacteristic
410#107—1222no—
411#107—1222no—
412#107—1222no—
413#106—1222no—
414C4⋊C8—22no—
415C4⋊C8—614no—
416#103—22no—
417#103—22no—
418#103—1222no—
419#103—1222no—
420C22⋊C8—1222no—
421C22⋊C8—1222no—
422#97—1222no—
423C4×C8—1222no—
424#95—1221no—
425#95—1221no—
426#95—1221no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—611no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8#1175822412yes—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—611no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4—1221no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4D4×D6111yescharacteristic
458C22×C4D4×D6111yescharacteristic
459C22×C4C23×D6111yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—612no—
467C2×C8—612no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—42no—
479C4⋊C4—81no—
480C4⋊C4—81no—
481C4⋊C4—612no—
482C4⋊C4—1224no—
483C4⋊C4—1221no—
484C4⋊C4—1222no—
485C4⋊C4—1222no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—1221no—
491C4⋊C4—1221no—
492C4⋊C4—4881no—
493C4⋊C4—4881no—
494C4⋊C4—4881no—
495C4⋊C4—2442no—
496C4⋊C4—2442no—
497C22⋊C4—1224no—
498C22⋊C4—1222no—
499C22⋊C4—1222no—
500C22⋊C4—4881no—
501C22⋊C4—4881no—
502C22⋊C4—4881no—
503C42—42no—
504C42—1221no—
505C42—1221no—
506C42—4881no—
507C42#1175822212yes—
508C42D4×D6112yes—
509C23—1221no—
510C23—1221no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—1221no—
516C23C2×D4×D6111yescharacteristic
517Q8—22no—
518Q8—22no—
519Q8—41no—
520Q8—41no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—1221no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1222no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—2441no—
548C2×C4—2441no—
549C2×C4—2441no—
550C2×C4—2441no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4#1451975012yes—
563C2×C4#1451975012yes—
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4#41×D6111yescharacteristic
566C2×C4C2×#1175822411yescharacteristic
567C2×C4C2×#1175822211yescharacteristic
568C2×C4#1451982411yescharacteristic
569C2×C4#1451982511yescharacteristic
570C8—41no—
571C8—41no—
572C8—1222no—
573C8—1222no—
574C22—2441no—
575C22—2441no—
576C22—2441no—
577C22—2441no—
578C22—2441no—
579C22—2441no—
580C22—2441no—
581C22C2×#1451975011yescharacteristic
582C22#1684027211yescharacteristic
583C22#1684033511yescharacteristic
584C22#1684056611yescharacteristic
585C22#1684056811yescharacteristic
586C22#1684057511yescharacteristic
587C22#1684057811yescharacteristic
588C4—22no—
589C4—22no—
590C4—42no—
591C4—81no—
592C4—81no—
593C4—81no—
594C4—4881no—
595C4—4881no—
596C4—4881no—
597C4—2441no—
598C4—2441no—
599C4—2442no—
600C4—2441no—
601C4#1684041411yescharacteristic
602C4#1684041811yescharacteristic
603C4#1684044711yescharacteristic
604C4#1684048011yescharacteristic
605C2—4881no—
606C2#2144925011yescharacteristic
607C2#2145548811yescharacteristic
608C2#2145550811yescharacteristic
609C2#2145601711yescharacteristic
610C2#2145611711yescharacteristic
611C2#2145709411yescharacteristic
612C2#2145710711yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1803D61
hypocenterC61#24715
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#16840069C2semidirect
9. #21446774C2semidirect
10. #21446575C2semidirect
11. #21446556C2semidirect
12. #21446551C2semidirect
13. #21446511C2semidirect
14. #21446497C2semidirect
15. C61×#1803C2semidirect
313. #1803D61semidirect
367. C61#24715semidirect

Maximal subgroups

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

subgroupindex
C2×#168400692
C2×#168398222
#214516522
#214484952
#214472582
#214472462
#214470202
#214467742
#214465752
#214465562
#214465512
#214465112
#214464972
C61×#18032
#2471561