Finite Group Databasebeta

← #29716225

Subgroups — #29716225

Order 15616.

1012 conjugacy classes of subgroups.

The subgroup lattice has 668 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840390C211yesmaximal, characteristic
2C2×#16840070C211yesmaximal, characteristic, complemented
3C2×#16840058C211yesmaximal, characteristic, complemented
4#21448501C211yesmaximal, characteristic
5#21447158C211yesmaximal, characteristic, complemented
6#21447148C211yesmaximal, characteristic, complemented
7#21446971C211yesmaximal, characteristic, complemented
8#21446961C211yesmaximal, characteristic, complemented
9#21446774C211yesmaximal, characteristic, complemented
10#21446770C211yesmaximal, characteristic, complemented
11#21446129C211yesmaximal, characteristic
12#21446124C211yesmaximal, characteristic
13#21446020C211yesmaximal, characteristic
14#21446009C211yesmaximal, characteristic
15C61×#1798C211yesmaximal, characteristic
16#16840390—24no—
17#16840390—24no—
18C2×#14519777C2211yescharacteristic
19C2×#14519777C2211yescharacteristic
20C2×#14519776—21no—
21C2×#14519769C2211yescharacteristic
22C2×#14519769C2211yescharacteristic
23C2×#14519768—21no—
24C2×#14519756C2211yescharacteristic
25C2×#14519754—21no—
26C2×#14519754—21no—
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519736C2211yescharacteristic
29C22×C4⋊Dic61C2211yescharacteristic
30#16840070—24no—
31#16840058—24no—
32C2×#14519711C2211yescharacteristic
33C2×#14519711C2211yescharacteristic
34C2×#14519711C2211yescharacteristic
35C2×#14519709C2211yescharacteristic
36C2×#14519702C2211yescharacteristic
37C2×#14519702C2211yescharacteristic
38C2×#14519702C2211yescharacteristic
39C2×#14519702C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839660C2211yescharacteristic
42#16839657—21no—
43#16839656—21no—
44#16839650—21no—
45#16839643—21no—
46#16839622C2211yescharacteristic
47C4×C4⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55#16839525C2211yescharacteristic
56C2×#117×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×C22⋊C8×C61C2211yescharacteristic
60C2×C22⋊C8×C61C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122C2311yescharacteristic
66C23×Dic61—21no—
67#14519777—22no—
68#14519777—22no—
69#14519777—22no—
70#14519777D414yes—
71#14519776—28no—
72#14519769—22no—
73#14519769—22no—
74#14519769—22no—
75#14519769D414yes—
76#14519768—28no—
77#14519756—22no—
78#14519756Q814yescomplemented
79#14519754—28no—
80#14519754—28no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83#14519736—22no—
84#14519736—22no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—22no—
89C2×C4⋊Dic61—22no—
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×C4⋊Dic61C2311yescharacteristic
93C2×C4⋊Dic61C2311yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61—21no—
106C2×C22⋊Dic61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61—21no—
110#14519711—22no—
111#14519711—22no—
112#14519711—22no—
113#14519709—22no—
114#14519702—22no—
115#14519702—22no—
116#14519702—22no—
117#14519702—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C23×C4×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128#117×C61—22no—
129#117×C61—22no—
130C2×C4⋊C4×C61D411yescharacteristic
131C2×C4⋊C4×C61D411yescharacteristic
132C2×C22⋊C4×C61D411yescharacteristic
133C2×C22⋊C4×C61D411yescharacteristic
134C2×C42×C61C2311yescharacteristic
135C2×C42×C61C2311yescharacteristic
136C4⋊C8×C61—22no—
137C4⋊C8×C61—22no—
138C22⋊C8×C61—22no—
139C22⋊C8×C61—22no—
140C24×C61C2×D411yescharacteristic
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×Q812yes—
151C22×C4×C61#4112yes—
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—24no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—81no—
163C22×Dic61—81no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—24no—
170C2×Dic122—24no—
171C2×Dic122—81no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122—81no—
175C2×Dic122C2×Q812yes—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C4⋊C4×C61—22no—
189C4⋊C4×C61—22no—
190C22⋊C4×C61—22no—
191C22⋊C4×C61—22no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61C2×D412yes—
195C42×C61C2×D412yes—
196C22⋊Dic61—24no—
197C22⋊Dic61—24no—
198C22⋊Dic61—81no—
199C4⋊Dic61—24no—
200C4⋊Dic61—24no—
201C4⋊Dic61—24no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61C2×D412yes—
212C4⋊Dic61C2×D412yes—
213C4⋊Dic61C2×D412yes—
214C4⋊Dic61C2×D412yes—
215C4⋊Dic61C2×Q812yes—
216#11758196—24no—
217#11758196—24no—
218#11758196—24no—
219#11758196—24no—
220#11758196—24no—
221#11758196—42no—
222#11758196—81no—
223C4×Dic61—24no—
224C4×Dic61—42no—
225C4×Dic61—81no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61C22⋊Q812yes—
233C23×C61C22×D411yescharacteristic
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—22no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61#12012yes—
251C2×C4×C61#12012yes—
252C2×C4×C61C22⋊Q812yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—22no—
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
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×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—84no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122—84no—
289Dic122—84no—
290C8×C61—41no—
291C8×C61—41no—
292C8×C61—41no—
293C8×C61—41no—
294#24311—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61C2×#12011yescharacteristic
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61#53611yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57811yescharacteristic
308C22×C61#57811yescharacteristic
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61#54811yescharacteristic
315C4×C61#54811yescharacteristic
316C4×C61D4×Q811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—42no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326C2×#544—611no—
327C2×#479—611no—
328C2×#477—611no—
329#2323—611no—
330#2059—611no—
331#2057—611no—
332#2026—611no—
333#2024—611no—
334#1989—611no—
335#1988—611no—
336#1873—611no—
337#1872—611no—
338#1854—611no—
339#1852—611no—
340#1798D6111yescharacteristic
341C2×C61—42no—
342C2×C61#240911yescharacteristic
343C2×C61#301111yescharacteristic
344C2×C61#301211yescharacteristic
345C2×C61#308211yescharacteristic
346C2×C61#308311yescharacteristic
347C2×C61#324211yescharacteristic
348C2×C61#324311yescharacteristic
349#544—1224no—
350#544—1224no—
351C2×C4⋊Q8—611no—
352C2×C4⋊Q8—611no—
353C2×C4⋊Q8—1221no—
354C2×#125—611no—
355C2×#125—611no—
356C2×#125—1221no—
357C2×C22⋊Q8—611no—
358C2×C22⋊Q8—1221no—
359C2×C22⋊Q8—1221no—
360C2×C4×Q8—611no—
361C2×#117—611no—
362C2×#117C2×D6111yescharacteristic
363C22×C4⋊C4—611no—
364#479—1224no—
365#477—1224no—
366C2×#107—611no—
367C2×#107—611no—
368C2×#107—611no—
369C2×#106—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
371C2×C4⋊C8C2×D6111yescharacteristic
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×C22⋊C8C2×D6111yescharacteristic
377C2×C22⋊C8C2×D6111yescharacteristic
378#391—611no—
379#389—611no—
380#389—1221no—
381#388—1221no—
382#388—1221no—
383#387—1221no—
384#380—611no—
385C4×C4⋊C4—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#336C2×D6111yescharacteristic
395#336C2×D6111yescharacteristic
396C61#2431111yescharacteristic, complemented, Sylow
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×Q8—1221no—
400C23×C4—1221no—
401C23×C4C22×D6111yescharacteristic
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—614no—
407C4⋊Q8—1222no—
408C4⋊Q8—1222no—
409C4⋊Q8—1228no—
410C4⋊Q8—1222no—
411#125—614no—
412#125—1222no—
413#125—1222no—
414#125—1228no—
415#125—1222no—
416C22⋊Q8—614no—
417C22⋊Q8—1228no—
418C22⋊Q8—1228no—
419C22⋊Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1222no—
422#117—22no—
423#117—22no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1222no—
434C2×C4⋊C4—1222no—
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⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4D24411yescharacteristic
446C2×C4⋊C4D24411yescharacteristic
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C22⋊C4D24411yescharacteristic
451C2×C22⋊C4D24411yescharacteristic
452C2×C42—1221no—
453C2×C42—1221no—
454C2×C42—1221no—
455C2×C42C22×D6111yescharacteristic
456C2×C42C22×D6111yescharacteristic
457#107—1222no—
458#107—1222no—
459#107—1222no—
460#106—1222no—
461C4⋊C8—22no—
462C4⋊C8—22no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467C22⋊C8—22no—
468C22⋊C8—22no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24C2×D24411yescharacteristic
475C2×Q8—612no—
476C2×Q8—1224no—
477C2×Q8—1221no—
478C2×Q8—1224no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—4881no—
484C2×Q8—4881no—
485C2×Q8—4881no—
486C2×Q8—4881no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—1224no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4881no—
501C22×C4—4881no—
502C22×C4C2×D24411yescharacteristic
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4C2×D24411yescharacteristic
506C22×C4C2×D24411yescharacteristic
507C22×C4#1175822412yes—
508C22×C4Q8×D6112yes—
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C4⋊C4—22no—
523C4⋊C4—22no—
524C4⋊C4—612no—
525C4⋊C4—612no—
526C4⋊C4—612no—
527C4⋊C4—612no—
528C4⋊C4—612no—
529C4⋊C4—1224no—
530C4⋊C4—1224no—
531C4⋊C4—1224no—
532C4⋊C4—1224no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1224no—
541C4⋊C4—1224no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—1221no—
546C4⋊C4—4881no—
547C4⋊C4—2442no—
548C22⋊C4—22no—
549C22⋊C4—22no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C22⋊C4—4881no—
553C42—21no—
554C42—21no—
555C42—1224no—
556C42—4881no—
557C42—2442no—
558C42D4×D6112yes—
559C42D4×D6112yes—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23#1451975712yes—
567C23C22×D24411yescharacteristic
568Q8—1222no—
569Q8—1222no—
570Q8—4884no—
571Q8—4884no—
572Q8—2441no—
573Q8—2441no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—22no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—1222no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
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—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451974712yes—
616C2×C4#1451974712yes—
617C2×C4#1451975712yes—
618C2×C4C2×D4×D6111yescharacteristic
619C2×C4C2×D4×D6111yescharacteristic
620C2×C4C2×Q8×D6111yescharacteristic
621C2×C4C2×#1175822411yescharacteristic
622C2×C4#1451982411yescharacteristic
623C2×C4#1451982411yescharacteristic
624C8—41no—
625C8—41no—
626C8—41no—
627C8—41no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—41no—
632C22—41no—
633C22—41no—
634C22—41no—
635C22C2×#1451974711yescharacteristic
636C22C2×#1451975711yescharacteristic
637C22#1684033811yescharacteristic
638C22#1684056711yescharacteristic
639C22#1684057411yescharacteristic
640C22#1684057911yescharacteristic
641C22#1684057911yescharacteristic
642C4—22no—
643C4—22no—
644C4—22no—
645C4—41no—
646C4—41no—
647C4—4881no—
648C4—4881no—
649C4—2442no—
650C4—2442no—
651C4—2442no—
652C4—2442no—
653C4—2442no—
654C4—2442no—
655C4#1684041811yescharacteristic
656C4#1684041811yescharacteristic
657C4#1684042511yescharacteristic
658C4#1684042511yescharacteristic
659C2—42no—
660C2#2144918511yescharacteristic
661C2#2145511611yescharacteristic
662C2#2145512611yescharacteristic
663C2#2145566211yescharacteristic
664C2#2145567211yescharacteristic
665C2#2145708911yescharacteristic
666C2#2145710411yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1798C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1798D61
hypocenterC61#24311
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#16840070C2semidirect
3. C2×#16840058C2semidirect
5. #21447158C2semidirect
6. #21447148C2semidirect
7. #21446971C2semidirect
8. #21446961C2semidirect
9. #21446774C2semidirect
10. #21446770C2semidirect
78. #14519756Q8semidirect
396. C61#24311semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
C2×#168400702
C2×#168400582
#214485012
#214471582
#214471482
#214469712
#214469612
#214467742
#214467702
#214461292
#214461242
#214460202
#214460092
C61×#17982
#2431161