Finite Group Databasebeta

← #29736139

Subgroups — #29736139

Order 15616.

1056 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840473C211yesmaximal, characteristic
2C2×#16840170C211yesmaximal, characteristic, complemented
3C2×#16840041C211yesmaximal, characteristic
4#21451026C211yesmaximal, characteristic, complemented
5#21447156C211yesmaximal, characteristic, complemented
6#21447155C211yesmaximal, characteristic, complemented
7#21447144C211yesmaximal, characteristic, complemented
8#21446775C211yesmaximal, characteristic, complemented
9#21446757C211yesmaximal, characteristic, complemented
10#21446083C211yesmaximal, characteristic
11#21446079C211yesmaximal, characteristic
12#21446067C211yesmaximal, characteristic
13#21446063C211yesmaximal, characteristic
14#21445643C211yesmaximal, characteristic
15C61×#2057C211yesmaximal, characteristic, complemented
16#16840473—24no—
17#16840473—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519768C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519757—21no—
22C2×#14519753—21no—
23C2×#14519753—21no—
24C2×C4×Dic122C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×#14519739C2211yescharacteristic
27C2×#14519739C2211yescharacteristic
28C2×C4⋊C4×D61—21no—
29#16840170—24no—
30#16840041—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519708C2211yescharacteristic
35C2×#14519706C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37C2×#14519699C2211yescharacteristic
38C2×#14519697C2211yescharacteristic
39C2×#14519697C2211yescharacteristic
40C2×#14519687C2211yescharacteristic
41#16839683—21no—
42#16839682—21no—
43#16839666—21no—
44#16839661C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839631—21no—
48C4⋊C4×Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839517C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C4⋊Q8×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#107×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122D411yescharacteristic
65C22×D244C2311yescharacteristic
66C22×C4×D61—21no—
67C22×C4×D61—21no—
68C22×C4×D61—21no—
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519776—22no—
72#14519776—22no—
73#14519768—22no—
74#14519768—22no—
75#14519768—22no—
76#14519768D414yes—
77#14519757—28no—
78#14519753—28no—
79#14519753—28no—
80C4×Dic122—22no—
81C4×Dic122—22no—
82C4×D244—22no—
83C4×D244—22no—
84#14519739—22no—
85#14519739—22no—
86#14519739—22no—
87#14519739D414yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—21no—
91C2×C4⋊Dic61D411yescharacteristic
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
102C4⋊C4×D61—28no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519710—22no—
110#14519710—22no—
111#14519709—22no—
112#14519708—22no—
113#14519706—22no—
114#14519699—22no—
115#14519699Q814yescomplemented
116#14519697—22no—
117#14519697Q814yescomplemented
118#14519687—22no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123C22×Q8×C61—21no—
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C4⋊Q8×C61—22no—
127C4⋊Q8×C61—22no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61—21no—
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#107×C61—22no—
136#107×C61—22no—
137#95×C61—21no—
138C23×D61—21no—
139C2×Q8×C61—24no—
140C2×Q8×C61—81no—
141C2×Q8×C61—81no—
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×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244C2×D412yes—
160C2×D244C2×Q812yes—
161C2×D244C2×Q812yes—
162C2×C4×D61—24no—
163C2×C4×D61—24no—
164C2×C4×D61—24no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—81no—
168C2×C4×D61—81no—
169C2×C4×D61—81no—
170C2×C4×D61—81no—
171C2×Dic122—22no—
172C2×Dic122—22no—
173C2×Dic122—41no—
174C2×Dic122—42no—
175C2×Dic122—81no—
176C2×Dic122—81no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C2×Q812yes—
183C4⋊C4×C61—24no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—81no—
190C4⋊C4×C61C2×D412yes—
191C4⋊C4×C61C2×D412yes—
192C4⋊C4×C61C2×Q812yes—
193C42×C61—21no—
194C42×C61—21no—
195#11758198—24no—
196#11758198—24no—
197#11758198—24no—
198C4⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—22no—
202C4⋊Dic61—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—81no—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61C2×Q812yes—
208#11758196—24no—
209#11758196—24no—
210#11758196—22no—
211#11758196—22no—
212#11758196—42no—
213#11758196—81no—
214#11758196—81no—
215C4×Dic61—21no—
216C4×Dic61—21no—
217C4×Dic61—42no—
218C4×Dic61C2×D412yes—
219C4×Dic61C2×D412yes—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418C2×Q812yes—
226C23×C61C22×D411yescharacteristic
227C22×D61—22no—
228C22×D61—22no—
229C22×D61—22no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234Q8×C61—84no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
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×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61#13712yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C22×Q811yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61#12012yes—
279C2×Dic61#12012yes—
280D244—22no—
281D244—22no—
282D244—22no—
283D244C4⋊Q814yes—
284C4×D61—48no—
285C4×D61—84no—
286C4×D61—84no—
287Dic122—44no—
288Dic122—84no—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—41no—
293#25850—611nomaximal, Sylow
294C22×C61C2×#12011yescharacteristic
295C22×C61D4×Q811yescharacteristic
296C22×C61D4×Q811yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57611yescharacteristic
300C22×C61#57611yescharacteristic
301C2×D61—24no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—42no—
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61C2×C4⋊Q811yescharacteristic
315C4×C61#53611yescharacteristic
316C4×C61D4×Q811yescharacteristic
317C4×C61D4×Q811yescharacteristic
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—42no—
323Dic61—41no—
324Dic61—81no—
325C2×#554—611no—
326C2×#501—611no—
327C2×#474—611no—
328#2567—611no—
329#2059—611no—
330#2059—611no—
331#2057—611no—
332#2057D6111yescharacteristic, complemented
333#1989—611no—
334#1985—611no—
335#1864—611no—
336#1864—611no—
337#1862—611no—
338#1862—611no—
339#1762—611no—
340C2×C61#243411yescharacteristic
341C2×C61#300711yescharacteristic
342C2×C61#300811yescharacteristic
343C2×C61#338511yescharacteristic
344C2×C61#338511yescharacteristic
345C2×C61#338911yescharacteristic
346C2×C61#338911yescharacteristic
347D61—44no—
348#554—1224no—
349#554—1224no—
350C2×C4⋊Q8—611no—
351C2×C4⋊Q8C2×D6111yescharacteristic
352C2×#125—611no—
353C2×#125—611no—
354C2×C22⋊Q8—1221no—
355C2×C22⋊Q8—1221no—
356C2×C22⋊Q8—1221no—
357C2×C4×Q8—611no—
358C2×C4×D4—611no—
359C2×C4×D4—611no—
360C2×C4×D4—611no—
361C22×C4⋊C4—1221no—
362#501—1224no—
363#474—1224no—
364C2×#107—611no—
365C2×#107—611no—
366C2×#107C2×D6111yescharacteristic
367C2×#107C2×D6111yescharacteristic
368C2×#106—611no—
369C2×#106—611no—
370C2×C4⋊C8—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#97—611no—
376#395—1221no—
377#395—1221no—
378#391—1221no—
379#389—611no—
380#389—611no—
381#389—611no—
382#389C2×D6111yescharacteristic
383#384—1221no—
384C4×C4⋊C4—611no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340C2×D6111yescharacteristic
390#340C2×D6111yescharacteristic
391#336—611no—
392#336—611no—
393#336—611no—
394#336C2×D6111yescharacteristic
395C61#2585011yescharacteristic, complemented, Sylow
396C22×Q8—21no—
397C22×Q8—611no—
398C22×Q8—1221no—
399C22×D4—611no—
400C23×C4—1221no—
401C23×C4—1221no—
402C23×C4—1221no—
403C22×C8—611no—
404C22×C8—611no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C4⋊Q8—22no—
408C4⋊Q8—22no—
409C4⋊Q8—1222no—
410C4⋊Q8—1222no—
411#125—614no—
412#125—1222no—
413#125—1222no—
414#125—1222no—
415C22⋊Q8—1228no—
416C22⋊Q8—1228no—
417C22⋊Q8—1228no—
418C4×Q8—1222no—
419C4×Q8—1222no—
420C4×D4—614no—
421C4×D4—1222no—
422C4×D4—1222no—
423C4×D4—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1228no—
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×C4⋊C4C22×D6111yescharacteristic
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—611no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#107—22no—
455#107—22no—
456#107—1222no—
457#107—1222no—
458#106—1222no—
459#106—1222no—
460C4⋊C8—1222no—
461#102—614no—
462#102—614no—
463#102—1222no—
464#102—1222no—
465#97—1222no—
466#95—21no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471C24—1221no—
472C2×Q8—24no—
473C2×Q8—81no—
474C2×Q8—81no—
475C2×Q8—1222no—
476C2×Q8—1222no—
477C2×Q8—4881no—
478C2×Q8—4881no—
479C2×Q8—2441no—
480C2×Q8—2442no—
481C2×D4—612no—
482C2×D4—612no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—1224no—
496C22×C4—1224no—
497C22×C4—1224no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—4881no—
502C22×C4—4881no—
503C22×C4—4881no—
504C22×C4—4881no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4D4×D6111yescharacteristic
508C22×C4D4×D6111yescharacteristic
509C22×C4C23×D6111yescharacteristic
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—612no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8Q8×D6112yes—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—81no—
529C4⋊C4—612no—
530C4⋊C4—612no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1222no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—4881no—
544C4⋊C4—4881no—
545C4⋊C4—2442no—
546C4⋊C4D4×D6112yes—
547C4⋊C4D4×D6112yes—
548C4⋊C4Q8×D6112yes—
549C22⋊C4—1224no—
550C22⋊C4—1224no—
551C22⋊C4—1224no—
552C42—21no—
553C42—21no—
554C42—612no—
555C42—612no—
556C42—1221no—
557C42—1221no—
558C42—2442no—
559C23—1222no—
560C23—1222no—
561C23—1222no—
562C23—2441no—
563C23—2441no—
564C23—2441no—
565C23—2441no—
566C23C2×D4×D6111yescharacteristic
567Q8—84no—
568Q8—4884no—
569Q8—2444no—
570D4—614no—
571D4—1222no—
572D4—1222no—
573D4—1222no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—41no—
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—612no—
592C2×C4—612no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1222no—
605C2×C4—4884no—
606C2×C4—4884no—
607C2×C4—2448no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451980912yes—
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×Q8×D6111yescharacteristic
619C2×C4C2×Q8×D6111yescharacteristic
620C2×C4#1451982411yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—22no—
623C8—41no—
624C8—1222no—
625C8—2441no—
626C22—1224no—
627C22—2442no—
628C22—2442no—
629C22—2442no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22#120×D6111yescharacteristic
634C22#1684042511yescharacteristic
635C22#1684042711yescharacteristic
636C22C2×#1451980911yescharacteristic
637C22#1684056111yescharacteristic
638C22#1684056311yescharacteristic
639C22#1684056611yescharacteristic
640C4—22no—
641C4—42no—
642C4—41no—
643C4—42no—
644C4—42no—
645C4—81no—
646C4—1222no—
647C4—1222no—
648C4—4881no—
649C4—2442no—
650C4—2441no—
651C4—2442no—
652C4—2441no—
653C4C4⋊Q8×D6111yescharacteristic
654C4#1684033511yescharacteristic
655C4#1684042511yescharacteristic
656C4#1684042711yescharacteristic
657C2—2444no—
658C2#2144945811yescharacteristic
659C2#2145507811yescharacteristic
660C2#2145508211yescharacteristic
661C2#2145872811yescharacteristic
662C2#2145873411yescharacteristic
663C2#2145877211yescharacteristic
664C2#2145877411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2057C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#2057D61
hypocenterC61#25850
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2057

Complements

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

NH ≅ G/Ntype
2. C2×#16840170C2semidirect
4. #21451026C2semidirect
5. #21447156C2semidirect
6. #21447155C2semidirect
7. #21447144C2semidirect
8. #21446775C2semidirect
9. #21446757C2semidirect
15. C61×#2057C2semidirect
115. #14519699Q8semidirect
117. #14519697Q8semidirect
332. #2057D61semidirect
395. C61#25850semidirect

Maximal subgroups

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

subgroupindex
C2×#168404732
C2×#168401702
C2×#168400412
#214510262
#214471562
#214471552
#214471442
#214467752
#214467572
#214460832
#214460792
#214460672
#214460632
#214456432
C61×#20572
#2585061