Finite Group Databasebeta

← #29730090

Subgroups — #29730090

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840124C211yesmaximal, characteristic
2C2×#16839984C211yesmaximal, characteristic
3#21448888C211yesmaximal, characteristic
4#21447228C211yesmaximal, characteristic, complemented
5#21447223C211yesmaximal, characteristic
6#21447141C211yesmaximal, characteristic, complemented
7#21447134C211yesmaximal, characteristic
8#21446772C211yesmaximal, characteristic, complemented
9#21446543C211yesmaximal, characteristic, complemented
10#21446163C211yesmaximal, characteristic, complemented
11#21446156C211yesmaximal, characteristic
12#21445821C211yesmaximal, characteristic, complemented
13#21445816C211yesmaximal, characteristic
14#21445649C211yesmaximal, characteristic, complemented
15C61×#2567C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16840124—24no—
22#16839984—24no—
23C2×C4⋊#92418C2211yescharacteristic
24C2×C4⋊#92418C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35#16839668C2211yescharacteristic
36#16839664C2211yescharacteristic
37#16839652—21no—
38#16839651—21no—
39#16839633—21no—
40C4×C4⋊Dic61C2211yescharacteristic
41#16839609—21no—
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C4⋊Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#395—21no—
53C61×#395—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389—21no—
57C61×#389—21no—
58C61×#384C2211yescharacteristic
59C61×#384C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C22×D244C2311yescharacteristic
63C22×C4×D61—21no—
64C2×#11758202—22no—
65C2×#11758202—22no—
66C2×#11758202—41no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—41no—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519765—22no—
75#14519765D414yes—
76C4×Dic122—22no—
77C4×Dic122—41no—
78C4×D244—22no—
79C4×D244—41no—
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758198—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×C4×Dic61—21no—
89C2×C4×Dic61—21no—
90C4⋊#92418—22no—
91C4⋊#92418—41no—
92C4⋊#92418—41no—
93#14519701—22no—
94#14519701—22no—
95#14519700—22no—
96#14519700—22no—
97#14519700—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519688—22no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105C22×Q8×C61—21no—
106C22×Q8×C61C2311yescharacteristic
107C4⋊Q8×C61—22no—
108C4⋊Q8×C61—22no—
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61D411yescharacteristic
131#95×C61D411yescharacteristic
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—42no—
139C2×Q8×C61—81no—
140C2×Q8×C61—81no—
141C2×Q8×C61#4112yes—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244C2×D412yes—
162C2×C4×D61—42no—
163C2×C4×D61—42no—
164C2×C4×D61—81no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122C2×D412yes—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—22no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—42no—
181C4⋊C4×C61—81no—
182C4⋊C4×C61—81no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61—81no—
185C42×C61—21no—
186C42×C61—21no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—42no—
190C42×C61C2×D412yes—
191C42×C61#4112yes—
192#11758202—44no—
193#11758201—44no—
194#11758198—24no—
195#11758198—24no—
196#11758198—42no—
197#11758198—42no—
198#11758198—81no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61#4112yes—
203#11758196—42no—
204C4×Dic61—42no—
205C4×Dic61—81no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C23×C61C22×D411yescharacteristic
219C22×D61—22no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—41no—
229Q8×C61—41no—
230Q8×C61—84no—
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—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
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×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61#12112yes—
258C2×C4×C61#12112yes—
259C2×C4×C61#13712yes—
260C2×C4×C61#13712yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61H(5,𝔽2)11yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—84no—
281Dic122—22no—
282Dic122—22no—
283Dic122—41no—
284Dic122—41no—
285#92418—41no—
286#92418—41no—
287#92418—41no—
288#92418—41no—
289#25336—611nomaximal, Sylow
290C22×C61C2×#12111yescharacteristic
291C22×C61#54611yescharacteristic
292C22×C61#54811yescharacteristic
293C22×C61C2×#13711yescharacteristic
294C22×C61C2×#13711yescharacteristic
295C22×C61#57611yescharacteristic
296C22×C61#57711yescharacteristic
297C2×D61—42no—
298C2×D61—42no—
299C2×D61—42no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—81no—
303C2×D61—81no—
304C4×C61—22no—
305C4×C61—22no—
306C4×C61—22no—
307C4×C61—22no—
308C4×C61—42no—
309C4×C61—41no—
310C4×C61—41no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61#52911yescharacteristic
315C4×C61#54211yescharacteristic
316C4×C61#54611yescharacteristic
317C4×C61#54711yescharacteristic
318Dic61—42no—
319Dic61—42no—
320Dic61—81no—
321C2×#489—611no—
322C2×#464—611no—
323#2567D6111yescharacteristic, complemented
324#2367—611no—
325#2072—611no—
326#2071—611no—
327#2056—611no—
328#2055—611no—
329#1988—611no—
330#1940—611no—
331#1878—611no—
332#1877—611no—
333#1803—611no—
334#1802—611no—
335#1763—611no—
336C2×C61#242711yescharacteristic
337C2×C61#303911yescharacteristic
338C2×C61#304511yescharacteristic
339C2×C61#331411yescharacteristic
340C2×C61#331611yescharacteristic
341C2×C61#336911yescharacteristic
342C2×C61#337511yescharacteristic
343D61—82no—
344C22×Q16—611no—
345C22×SD16—611no—
346C2×C4⋊Q8C2×D6111yescharacteristic
347C2×#124—611no—
348C2×C4×Q8—611no—
349C2×C4×Q8C2×D6111yescharacteristic
350C2×C4×D4—611no—
351#489—1224no—
352#464—1224no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×C4⋊C8—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#102—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#97—611no—
365#395—21no—
366#395—21no—
367#391—611no—
368#391C2×D6111yescharacteristic
369#391C2×D6111yescharacteristic
370#390—611no—
371#389—21no—
372#389—21no—
373#388—1221no—
374#388—1221no—
375#385—1221no—
376#384C2×D6111yescharacteristic
377#384C2×D6111yescharacteristic
378C4×C4⋊C4—611no—
379C4×C4⋊C4C2×D6111yescharacteristic
380C4×C22⋊C4—1221no—
381#340—611no—
382#340—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#336—611no—
388#336—611no—
389C61#2533611yescharacteristic, complemented, Sylow
390C22×Q8—21no—
391C22×Q8—611no—
392C22×Q8C22×D6111yescharacteristic
393C22×D4—611no—
394C23×C4—1221no—
395C2×Q16—1222no—
396C2×Q16—1222no—
397C2×Q16—2441no—
398C2×SD16—1222no—
399C2×SD16—1222no—
400C2×SD16—2441no—
401C22×C8—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8—611no—
405C4⋊Q8—22no—
406C4⋊Q8—22no—
407#124—614no—
408#124—1222no—
409C4×Q8—22no—
410C4×Q8—22no—
411C4×Q8—1222no—
412C4×Q8—2441no—
413C4×D4—1222no—
414C4×D4—2441no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—21no—
420C2×C4⋊C4—21no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22⋊D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C42—21no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439C2×C42C22×D6111yescharacteristic
440C4⋊C8—1222no—
441C4⋊C8—2441no—
442C4⋊C8—2441no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#103—1222no—
447#103—1222no—
448#102—1222no—
449#102—1222no—
450#102—1222no—
451#97—1222no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95C22⋊D6111yescharacteristic
460#95C22⋊D6111yescharacteristic
461C24—1221no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—42no—
468C2×Q8—81no—
469C2×Q8—81no—
470C2×Q8—612no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8#1175822612yes—
477C2×D4—612no—
478C2×D4—1221no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—4881no—
491C22×C4—2442no—
492C22×C4—2442no—
493C22×C4C2×C22⋊D6111yescharacteristic
494C22×C4C2×C22⋊D6111yescharacteristic
495C22×C4C2×C22⋊D6111yescharacteristic
496C22×C4C2×C22⋊D6111yescharacteristic
497C22×C4C2×C22⋊D6111yescharacteristic
498C22×C4C2×C22⋊D6111yescharacteristic
499C22×C4C23×D6111yescharacteristic
500Q16—2444no—
501SD16—2444no—
502C2×C8—1221no—
503C2×C8—1221no—
504C2×C8—1221no—
505C2×C8—1221no—
506C2×C8—1221no—
507C2×C8—1221no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C4⋊C4—22no—
515C4⋊C4—22no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—42no—
523C4⋊C4—42no—
524C4⋊C4—81no—
525C4⋊C4—81no—
526C4⋊C4—81no—
527C4⋊C4—81no—
528C4⋊C4—612no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2442no—
533C22⋊C4—1224no—
534C22⋊C4—1224no—
535C22⋊C4—4881no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C42—21no—
539C42—21no—
540C42—21no—
541C42—21no—
542C42—42no—
543C42—4881no—
544C42—2442no—
545C42D4×D6112yes—
546C42#1175822412yes—
547C23—1222no—
548C23—2441no—
549C23—2441no—
550C23—2441no—
551C23—2441no—
552C23—2441no—
553C23—2441no—
554C23C22×C22⋊D6111yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—84no—
560Q8—1222no—
561Q8—1222no—
562Q8—2441no—
563Q8—2441no—
564D4—1222no—
565D4—1222no—
566D4—2441no—
567D4—2441no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—4884no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4#1451974812yes—
605C2×C4#1451974812yes—
606C2×C4#1451980812yes—
607C2×C4#1451980812yes—
608C2×C4C2×D4×D6111yescharacteristic
609C2×C4#41×D6111yescharacteristic
610C2×C4C2×#1175822411yescharacteristic
611C2×C4C2×#1175822611yescharacteristic
612C2×C4#1451982311yescharacteristic
613C2×C4#1451982511yescharacteristic
614C8—2441no—
615C8—2441no—
616C8—2441no—
617C8—2441no—
618C22—4881no—
619C22—4881no—
620C22—4881no—
621C22—4881no—
622C22—2442no—
623C22—2442no—
624C22—2442no—
625C22C2×#1451974811yescharacteristic
626C22#1684040011yescharacteristic
627C22#1684042311yescharacteristic
628C22C2×#1451980811yescharacteristic
629C22C2×#1451980811yescharacteristic
630C22#1684056511yescharacteristic
631C22#1684057311yescharacteristic
632C4—22no—
633C4—22no—
634C4—22no—
635C4—22no—
636C4—41no—
637C4—42no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—4881no—
643C4—2442no—
644C4—2442no—
645C4#1684029811yescharacteristic
646C4#1684038511yescharacteristic
647C4#1684040311yescharacteristic
648C4#1684041411yescharacteristic
649C2—4882no—
650C2#2144939211yescharacteristic
651C2#2145530211yescharacteristic
652C2#2145535411yescharacteristic
653C2#2145789411yescharacteristic
654C2#2145792411yescharacteristic
655C2#2145855911yescharacteristic
656C2#2145863311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2567C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2567D61
hypocenterC61#25336
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2567

Complements

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

NH ≅ G/Ntype
4. #21447228C2semidirect
6. #21447141C2semidirect
8. #21446772C2semidirect
9. #21446543C2semidirect
10. #21446163C2semidirect
12. #21445821C2semidirect
14. #21445649C2semidirect
15. C61×#2567C2semidirect
323. #2567D61semidirect
389. C61#25336semidirect

Maximal subgroups

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

subgroupindex
C2×#168401242
C2×#168399842
#214488882
#214472282
#214472232
#214471412
#214471342
#214467722
#214465432
#214461632
#214461562
#214458212
#214458162
#214456492
C61×#25672
#2533661