Finite Group Databasebeta

← #29690007

Subgroups — #29690007

Order 15616.

1014 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840370C211yesmaximal, characteristic
2C2×#16839787C211yesmaximal, characteristic
3#21448429C211yesmaximal, characteristic
4#21446775C211yesmaximal, characteristic, complemented
5#21446773C211yesmaximal, characteristic, complemented
6#21446740C211yesmaximal, characteristic, complemented
7#21446740C211yesmaximal, characteristic, complemented
8#21446251C211yesmaximal, characteristic
9#21446249C211yesmaximal, characteristic
10#21446241C211yesmaximal, characteristic
11#21446239C211yesmaximal, characteristic
12#21445831C211yesmaximal, characteristic, complemented
13#21445825C211yesmaximal, characteristic, complemented
14#21445824C211yesmaximal, characteristic, complemented
15C61×#1805C211yesmaximal, characteristic, complemented
16C22×#11758189C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18#16840370—24no—
19#16840370C418yescomplemented
20C2×#14519775C2211yescharacteristic
21C2×#14519772—21no—
22C2×#14519771—21no—
23C2×#14519769—21no—
24C2×#14519768C2211yescharacteristic
25C2×#14519767—21no—
26C2×#14519757C2211yescharacteristic
27C2×#14519753C2211yescharacteristic
28C2×C4×Dic122C2211yescharacteristic
29C2×#14519734C2211yescharacteristic
30#16839787—24no—
31C2×C4⋊#92418C2211yescharacteristic
32C2×#14519706C2211yescharacteristic
33C2×#14519706C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40#16839654—21no—
41#16839654—21no—
42#16839649—21no—
43#16839649—21no—
44#16839632C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C61×#382C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C2×#11758189—22no—
65C2×#11758189—41no—
66C2×#11758189—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C2×C8×D61—22no—
70C2×C8×D61—41no—
71C2×C8×D61—41no—
72#14519775—22no—
73#14519775C2×C414yes—
74#14519772—28no—
75#14519771—28no—
76#14519769—28no—
77#14519768—22no—
78#14519768C2×C414yes—
79#14519767—28no—
80#14519757—22no—
81#14519757C2×C414yes—
82#14519753—22no—
83#14519753C2×C414yes—
84C4×Dic122—22no—
85C4×Dic122C2×C414yes—
86#14519734C2×C414yes—
87#14519734D414yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61—21no—
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196D411yescharacteristic
101C2×#11758196D411yescharacteristic
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198D411yescharacteristic
107C2×#11758198D411yescharacteristic
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110C2×C4×Dic61—21no—
111C2×C4×Dic61C2311yescharacteristic
112C4⋊#92418—22no—
113#14519706—22no—
114#14519706—22no—
115#14519702—41no—
116#14519702—41no—
117#14519701—41no—
118#14519701—41no—
119#14519700—41no—
120#14519700—41no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125C22×Q8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C4×Q8×C61—22no—
129C4×Q8×C61C2×C414yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61—21no—
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—22no—
137#103×C61—41no—
138#103×C61—41no—
139#95×C61—21no—
140#95×C61—21no—
141C23×D61C2×D411yescharacteristic
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C22×C412yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61C22×C412yes—
167C2×C4×D61C2×D412yes—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122C22×C412yes—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C4⋊C4×C61—24no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61C22×C412yes—
187C42×C61—24no—
188C42×C61C22×C412yes—
189C42×C61C2×D412yes—
190#11758198—24no—
191#11758198—24no—
192#11758198—81no—
193#11758198—81no—
194#11758198C22⋊C414yes—
195#11758198C22⋊C414yes—
196C4⋊Dic61—21no—
197C4⋊Dic61—24no—
198C4⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—81no—
204C4⋊Dic61—81no—
205C4⋊Dic61C22×C412yes—
206#11758196—24no—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196C22⋊C414yes—
212#11758196C22⋊C414yes—
213C4×Dic61—24no—
214C4×Dic61—81no—
215C4×Dic61—81no—
216C4×Dic61C22×C412yes—
217C4×Dic61C2×D412yes—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224#11758189—44no—
225C8×D61—44no—
226C23×C61C22×D411yescharacteristic
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61C2×C22⋊C412yes—
234Q8×C61—22no—
235Q8×C61—22no—
236Q8×C61—41no—
237Q8×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
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×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61C2×C22⋊C412yes—
281C2×Dic61C2×C22⋊C412yes—
282C2×Dic61C2×C22⋊C412yes—
283C4×D61—22no—
284C4×D61—22no—
285C4×D61—22no—
286C4×D61—22no—
287Dic122—22no—
288Dic122—22no—
289Dic122—41no—
290Dic122—41no—
291C8×C61—41no—
292C8×C61—41no—
293#92418—41no—
294#92418—41no—
295#21898—611nomaximal, Sylow
296C22×C61C22×C22⋊C411yescharacteristic
297C22×C61#53711yescharacteristic
298C22×C61#53711yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57811yescharacteristic
302C22×C61#57811yescharacteristic
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—42no—
313C4×C61—42no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61#52011yescharacteristic
317C4×C61#52011yescharacteristic
318C4×C61#53511yescharacteristic
319C4×C61#53611yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—42no—
325Dic61—81no—
326Dic61—81no—
327C2×#541—611no—
328C2×#424—611no—
329#2314—611no—
330#1989—611no—
331#1989—611no—
332#1981—611no—
333#1981—611no—
334#1894—611no—
335#1894—611no—
336#1892—611no—
337#1892—611no—
338#1805—611no—
339#1805D6111yescharacteristic, complemented
340#1803—611no—
341#1803—611no—
342C2×C61#231111yescharacteristic
343C2×C61#289811yescharacteristic
344C2×C61#289811yescharacteristic
345C2×C61#325811yescharacteristic
346C2×C61#325811yescharacteristic
347C2×C61#326011yescharacteristic
348C2×C61#326111yescharacteristic
349D61—42no—
350C22×M16—611no—
351C23×C8—611no—
352#541—618no—
353#541—1224no—
354C2×C4⋊Q8—611no—
355C2×#126—1221no—
356C2×#126—1221no—
357C2×#125—611no—
358C2×#125—1221no—
359C2×#125—1221no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8C2×D6111yescharacteristic
364C2×#117—611no—
365#424—1224no—
366C2×C4⋊C8—611no—
367C2×C4⋊C8—611no—
368C2×C4⋊C8—611no—
369C2×C4⋊C8C2×D6111yescharacteristic
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103C2×D6111yescharacteristic
377C2×#103C2×D6111yescharacteristic
378#388—1221no—
379#388—1221no—
380#387—1221no—
381#387—1221no—
382#384—611no—
383#382C2×D6111yescharacteristic
384#380—611no—
385#380—611no—
386C4×C4⋊C4—611no—
387C4×C4⋊C4—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340C2×D6111yescharacteristic
395#340C2×D6111yescharacteristic
396C61#2189811yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8C22×D6111yescharacteristic
399C23×C4—611no—
400C2×M16—1222no—
401C2×M16—2441no—
402C2×M16—2441no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8—1222no—
406C22×C8—2441no—
407C22×C8—2441no—
408C22×C8C22×D6111yescharacteristic
409C22×C8C22×D6111yescharacteristic
410C4⋊Q8—614no—
411C4⋊Q8—1222no—
412#126—1228no—
413#126—1228no—
414#125—614no—
415#125—1228no—
416#125—1222no—
417#125—1228no—
418C22⋊Q8—614no—
419C22⋊Q8—614no—
420C22⋊Q8—1222no—
421C22⋊Q8—1222no—
422C4×Q8—22no—
423C4×Q8—614no—
424C4×Q8—1222no—
425C4×Q8C4×D6114yes—
426#117—614no—
427#117—614no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C42—21no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458C4⋊C8—22no—
459C4⋊C8—1222no—
460C4⋊C8—1222no—
461C4⋊C8—1222no—
462#103—41no—
463#103—41no—
464#103—2441no—
465#103—2441no—
466#103—2441no—
467#103—2441no—
468#103—2441no—
469#103—2441no—
470#95—21no—
471#95—21no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476C24—611no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—612no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—612no—
497C22×C4—612no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4D4×D6111yescharacteristic
507C22×C4D4×D6111yescharacteristic
508C22×C4C23×D6111yescharacteristic
509M16—2444no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—2444no—
523C4⋊C4—21no—
524C4⋊C4—24no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—614no—
530C4⋊C4—614no—
531C4⋊C4—612no—
532C4⋊C4—1224no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—4881no—
545C4⋊C4—4881no—
546C4⋊C4C2×C4×D6112yes—
547C22⋊C4—614no—
548C22⋊C4—614no—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C42—24no—
554C42—612no—
555C42—612no—
556C42—1224no—
557C42—4881no—
558C42—4881no—
559C42C2×C4×D6112yes—
560C42D4×D6112yes—
561C23—612no—
562C23—1221no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23C2×D4×D6111yescharacteristic
569Q8—22no—
570Q8—22no—
571Q8—41no—
572Q8—41no—
573Q8—1222no—
574Q8—1222no—
575Q8—2441no—
576Q8—2441no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
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—612no—
594C2×C4—612no—
595C2×C4—612no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1222no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4C22⋊C4×D6112yes—
619C2×C4C22⋊C4×D6112yes—
620C2×C4C22×C4×D6111yescharacteristic
621C2×C4C2×D4×D6111yescharacteristic
622C2×C4#1451982211yescharacteristic
623C2×C4#1451982311yescharacteristic
624C2×C4#1451982411yescharacteristic
625C2×C4#1451982511yescharacteristic
626C8—41no—
627C8—41no—
628C8—2441no—
629C8—2441no—
630C22—1222no—
631C22—1222no—
632C22—1222no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22C2×C22⋊C4×D6111yescharacteristic
638C22#1684034111yescharacteristic
639C22#1684034211yescharacteristic
640C22#1684056911yescharacteristic
641C22#1684057111yescharacteristic
642C22#1684057611yescharacteristic
643C22#1684057811yescharacteristic
644C4—22no—
645C4—22no—
646C4—42no—
647C4—42no—
648C4—81no—
649C4—81no—
650C4—1222no—
651C4—1222no—
652C4—1222no—
653C4—4881no—
654C4—4881no—
655C4—2442no—
656C4—2442no—
657C4#1684024211yescharacteristic
658C4#1684024311yescharacteristic
659C4#1684032811yescharacteristic
660C4#1684033511yescharacteristic
661C2—2442no—
662C2#2144840811yescharacteristic
663C2#2145434811yescharacteristic
664C2#2145435111yescharacteristic
665C2#2145724211yescharacteristic
666C2#2145725011yescharacteristic
667C2#2145726411yescharacteristic
668C2#2145727211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1805C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
cosocleC22×C4×C61C24
hypercenter#1805D61
hypocenterC61#21898
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
4. #21446775C2semidirect
5. #21446773C2semidirect
6. #21446740C2semidirect
7. #21446740C2semidirect
12. #21445831C2semidirect
13. #21445825C2semidirect
14. #21445824C2semidirect
15. C61×#1805C2semidirect
19. #16840370C4semidirect
339. #1805D61semidirect
396. C61#21898semidirect

Maximal subgroups

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

subgroupindex
C2×#168403702
C2×#168397872
#214484292
#214467752
#214467732
#214467402
#214462512
#214462492
#214462412
#214462392
#214458312
#214458252
#214458242
C61×#18052
#2189861