Finite Group Databasebeta

← #29737512

Subgroups — #29737512

Order 15616.

928 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840056C211yesmaximal, characteristic, complemented
2C2×#16839772C211yesmaximal, characteristic
3#21449511C211yesmaximal, characteristic
4#21448651C211yesmaximal, characteristic
5#21447364C211yesmaximal, characteristic, complemented
6#21447215C211yesmaximal, characteristic, complemented
7#21447195C211yesmaximal, characteristic, complemented
8#21446959C211yesmaximal, characteristic, complemented
9#21446088C211yesmaximal, characteristic, complemented
10#21446084C211yesmaximal, characteristic, complemented
11#21446016C211yesmaximal, characteristic
12#21446002C211yesmaximal, characteristic
13#21445833C211yesmaximal, characteristic, complemented
14#21445753C211yesmaximal, characteristic
15C61×#1809C211yesmaximal, characteristic
16C23×Dic122C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840056—24no—
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519706C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26#16839772—24no—
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519700C2211yescharacteristic
35C2×C22⋊#92418C2211yescharacteristic
36#16839685—21no—
37#16839674—21no—
38#16839667C2211yescharacteristic
39#16839667C2211yescharacteristic
40#16839666—21no—
41#16839666—21no—
42#16839666—21no—
43#16839661C2211yescharacteristic
44#16839658C2211yescharacteristic
45#16839645—21no—
46#16839627C2211yescharacteristic
47#16839620—21no—
48C22⋊C4×Dic61—21no—
49C4×C22⋊Dic61C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—22no—
62C22×Dic122—22no—
63C22×Dic122—21no—
64C22×Dic122—22no—
65C22×Dic122—21no—
66C22×Dic122—41no—
67C22×Dic122—41no—
68C22×Dic122Q814yescomplemented
69C22×Dic122C2311yescharacteristic
70C22×Dic122C2311yescharacteristic
71C23×Dic61—21no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519776—22no—
75#14519776—41no—
76#14519742—22no—
77#14519742—22no—
78#14519742—41no—
79#14519742—41no—
80#14519736—22no—
81#14519736—41no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×C22⋊Dic61D411yescharacteristic
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97#14519710—22no—
98#14519708—41no—
99#14519706—22no—
100#14519706—41no—
101#14519702—22no—
102#14519702—22no—
103#14519702—22no—
104#14519702D414yes—
105#14519700—22no—
106#14519700—22no—
107#14519700—41no—
108#14519700—41no—
109C22⋊#92418—22no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C23×C4×C61C2311yescharacteristic
121C2×M16×C61—22no—
122C2×M16×C61—22no—
123C2×M16×C61—41no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C22⋊C4×C61—21no—
132#107×C61—22no—
133#106×C61—41no—
134#95×C61—21no—
135#95×C61—21no—
136C24×C61C2×D411yescharacteristic
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×Q812yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—24no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—42no—
155C22×Dic61—42no—
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—28no—
168C2×Dic122—24no—
169C2×Dic122—21no—
170C2×Dic122—44no—
171C2×Dic122—44no—
172C2×Dic122—44no—
173C2×Dic122—42no—
174C2×Dic122—42no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122—81no—
178C2×Dic122C2×D412yes—
179C2×Dic122C2×Q812yes—
180C2×Dic122C2×Q812yes—
181C2×Dic122#4112yes—
182M16×C61—44no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61C2×D412yes—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C22⋊C4×C61—42no—
198C22⋊C4×C61—81no—
199C22⋊Dic61—22no—
200C22⋊Dic61—41no—
201C22⋊Dic61—42no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61C2×D412yes—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210#11758196—81no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—21no—
214C4×Dic61—21no—
215C4×Dic61—42no—
216C4×Dic61—42no—
217C4×Dic61—42no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61C22⋊Q812yes—
231C23×C61C22×D411yescharacteristic
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
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×C61C22⋊Q812yes—
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—21no—
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—48no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61#13712yes—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—22no—
287Dic122—48no—
288Dic122—41no—
289Dic122—41no—
290Dic122—84no—
291Dic122C22⋊Q814yes—
292C8×C61—22no—
293C8×C61—41no—
294#92418—41no—
295#92418—41no—
296#25979—611nomaximal, Sylow
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61C2×C22⋊Q811yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13711yescharacteristic
308C22×C61#57511yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57811yescharacteristic
311C4×C61—22no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61C2×C22⋊Q811yescharacteristic
317C4×C61#54311yescharacteristic
318C4×C61#54611yescharacteristic
319C4×C61#55111yescharacteristic
320Dic61—24no—
321Dic61—22no—
322Dic61—41no—
323Dic61—42no—
324Dic61—41no—
325Dic61—42no—
326Dic61—41no—
327Dic61—81no—
328C2×#477—611no—
329C2×#419—611no—
330#2439—611no—
331Q8×C22⋊C4—611no—
332#2097—611no—
333#2068—611no—
334#2066—611no—
335#2024—611no—
336#1865—611no—
337#1865—611no—
338#1854—611no—
339#1851—611no—
340#1809D6111yescharacteristic
341#1805—611no—
342#1787—611no—
343C2×C61—42no—
344C2×C61#243711yescharacteristic
345C2×C61#307811yescharacteristic
346C2×C61#308211yescharacteristic
347C2×C61#327011yescharacteristic
348C2×C61#327511yescharacteristic
349C2×C61#331311yescharacteristic
350C2×C61#332911yescharacteristic
351C23×Q8—611no—
352C22×M16C2×D6111yescharacteristic
353C2×C4⋊Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×Q8—611no—
356C2×#117—611no—
357#477—1224no—
358C2×#107—611no—
359C2×#107C2×D6111yescharacteristic
360C2×#106—611no—
361C2×#106C2×D6111yescharacteristic
362C2×C4⋊C8—611no—
363C2×C4⋊C8—611no—
364#419—1224no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×C22⋊C8—611no—
374#396—1221no—
375#393—1221no—
376#391—611no—
377#391—611no—
378#391—1221no—
379#391—1221no—
380#391—1221no—
381#389—611no—
382#389—611no—
383#387—1221no—
384#382—611no—
385#380—1221no—
386#380C2×D6111yescharacteristic
387#380C2×D6111yescharacteristic
388C4×C22⋊C4—611no—
389C4×C22⋊C4—1221no—
390#340—611no—
391#340—611no—
392#340C2×D6111yescharacteristic
393#340C2×D6111yescharacteristic
394#336—611no—
395#336—611no—
396C61#2597911yescharacteristic, complemented, Sylow
397C22×Q8—614no—
398C22×Q8—611no—
399C22×Q8—611no—
400C22×Q8—1221no—
401C22×Q8—1222no—
402C22×Q8—1222no—
403C22×Q8—1222no—
404C22×Q8—1221no—
405C22×Q8—2441no—
406C22×Q8—2441no—
407C23×C4—1221no—
408C23×C4C22×D6111yescharacteristic
409C2×M16—22no—
410C2×M16—22no—
411C2×M16—41no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—1222no—
417C4⋊Q8—2441no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×Q8—2441no—
421C4×Q8—2441no—
422#117—1222no—
423#117—2441no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22×D6111yescharacteristic
432C2×C4⋊C4C22×D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C22⋊C4—21no—
436C2×C22⋊C4—21no—
437C2×C22⋊C4—611no—
438C2×C22⋊C4—611no—
439C2×C22⋊C4—1221no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445#107—22no—
446#107—1222no—
447#106—41no—
448#106—2441no—
449C4⋊C8—1222no—
450C4⋊C8—2441no—
451#103—614no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#103—2441no—
458#103—2441no—
459C22⋊C8—1222no—
460#95—21no—
461#95—21no—
462#95—611no—
463#95—611no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24D4×D6111yescharacteristic
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—612no—
476C2×Q8—612no—
477C2×Q8—1228no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1224no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—4881no—
488C2×Q8—2442no—
489C2×Q8—2442no—
490C2×Q8—2444no—
491C2×Q8—2444no—
492C2×Q8—2444no—
493C2×Q8—2442no—
494C2×Q8—2442no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—611no—
504C22×C4—611no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1224no—
511C22×C4—1221no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4#1175822212yes—
515C22×C4D4×D6111yescharacteristic
516C22×C4Q8×D6112yes—
517C22×C4C23×D6111yescharacteristic
518M16—44no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8D4×D6112yes—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—612no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C22⋊C4—42no—
549C22⋊C4—81no—
550C22⋊C4—1222no—
551C22⋊C4—2441no—
552C22⋊C4—2442no—
553C42—1221no—
554C42—1221no—
555C42—1221no—
556C42—1221no—
557C42—2442no—
558C42—2442no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975512yes—
567C23C2×D4×D6111yescharacteristic
568Q8—614no—
569Q8—1222no—
570Q8—1222no—
571Q8—1222no—
572Q8—1222no—
573Q8—1222no—
574Q8—4884no—
575Q8—2448no—
576Q8—2441no—
577Q8—2441no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
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—41no—
595C2×C4—41no—
596C2×C4—612no—
597C2×C4—1222no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—2448no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4#1451975512yes—
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×Q8×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4#1451982211yescharacteristic
627C2×C4#1451982311yescharacteristic
628C8—22no—
629C8—41no—
630C8—2441no—
631C8—2441no—
632C22—22no—
633C22—22no—
634C22—22no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22C2×#1451975511yescharacteristic
640C22#1684039611yescharacteristic
641C22#1684040111yescharacteristic
642C22#137×D6111yescharacteristic
643C22#1684055811yescharacteristic
644C22#1684057211yescharacteristic
645C22#1684057811yescharacteristic
646C4—22no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—81no—
651C4—1224no—
652C4—1222no—
653C4—4881no—
654C4—2442no—
655C4—2441no—
656C4—2442no—
657C4—2441no—
658C4—2441no—
659C4C22⋊Q8×D6111yescharacteristic
660C4#1684038711yescharacteristic
661C4#1684040211yescharacteristic
662C4#1684043611yescharacteristic
663C2—42no—
664C2#2144948711yescharacteristic
665C2#2145562311yescharacteristic
666C2#2145565911yescharacteristic
667C2#2145739611yescharacteristic
668C2#2145745711yescharacteristic
669C2#2145788011yescharacteristic
670C2#2145810811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1809C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1809D61
hypocenterC61#25979
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
1. C2×#16840056C2semidirect
5. #21447364C2semidirect
6. #21447215C2semidirect
7. #21447195C2semidirect
8. #21446959C2semidirect
9. #21446088C2semidirect
10. #21446084C2semidirect
13. #21445833C2semidirect
68. C22×Dic122Q8semidirect
396. C61#25979semidirect

Maximal subgroups

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

subgroupindex
C2×#168400562
C2×#168397722
#214495112
#214486512
#214473642
#214472152
#214471952
#214469592
#214460882
#214460842
#214460162
#214460022
#214458332
#214457532
C61×#18092
#2597961