Finite Group Databasebeta

← #29779560

Subgroups — #29779560

Order 15616.

906 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
1#21454170C211yesmaximal, characteristic
2#391×D61C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447230C211yesmaximal, characteristic, complemented
5#21446959C211yesmaximal, characteristic, complemented
6#21446958C211yesmaximal, characteristic, complemented
7#21446512C211yesmaximal, characteristic, complemented
8#21446509C211yesmaximal, characteristic, complemented
9#21446508C211yesmaximal, characteristic, complemented
10#21445904C211yesmaximal, characteristic
11#21445904C211yesmaximal, characteristic
12#21445752C211yesmaximal, characteristic
13#21445752C211yesmaximal, characteristic
14#21445582C211yesmaximal, characteristic
15C61×#1935C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519755—21no—
19C2×C4⋊C4×D61C2211yescharacteristic
20C2×C42×D61C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519701C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839669C2211yescharacteristic
31#16839668C2211yescharacteristic
32#16839667C2211yescharacteristic
33#16839666—21no—
34#16839666—21no—
35#16839666—21no—
36#16839660C2211yescharacteristic
37#16839657—21no—
38#16839631C2211yescharacteristic
39#16839621—21no—
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42#95×D61—21no—
43#95×D61—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#103×C61C2211yescharacteristic
53C2×#103×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122—21no—
61C22×Dic122C2311yescharacteristic
62C2×Q8×D61—22no—
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—41no—
67C2×Q8×D61—41no—
68C22×C4×D61—21no—
69C22×C4×D61—21no—
70C22×C4×D61D411yescharacteristic
71C22×C4×D61D411yescharacteristic
72C22×C4×D61C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519776—41no—
76#14519776—41no—
77#14519755—44no—
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C4⋊C4×D61—22no—
87C4⋊C4×D61—22no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×C4×Dic61D411yescharacteristic
91C2×C4×Dic61D411yescharacteristic
92C2×C4×Dic61C2311yescharacteristic
93C42×D61—41no—
94C42×D61—41no—
95#14519702—22no—
96#14519702—41no—
97#14519701—22no—
98#14519701—41no—
99#14519700—22no—
100#14519700—41no—
101#14519689—22no—
102#14519689—22no—
103#14519688—41no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×Q8×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#103×C61—22no—
122#103×C61—41no—
123#97×C61—41no—
124#95×C61—21no—
125#95×C61—21no—
126#95×C61—21no—
127C23×D61C2×D411yescharacteristic
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61—21no—
132C2×Q8×C61—21no—
133C2×Q8×C61#4112yes—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C22×Dic61—21no—
142C22×Dic61—21no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61C2×D411yescharacteristic
146C22×Dic61C2×D411yescharacteristic
147C22×Dic61C2×D411yescharacteristic
148Q8×D61—44no—
149Q8×D61—44no—
150Q8×D61—44no—
151Q8×D61—44no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—21no—
155C2×C4×D61—21no—
156C2×C4×D61—24no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—22no—
160C2×C4×D61—22no—
161C2×C4×D61—41no—
162C2×C4×D61—41no—
163C2×C4×D61—42no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—42no—
167C2×C4×D61—42no—
168C2×C4×D61#4112yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—24no—
175C2×Dic122—21no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122—42no—
179C2×Dic122—42no—
180C2×Dic122—42no—
181C2×Dic122#4112yes—
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—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—42no—
195C42×C61—21no—
196C42×C61—21no—
197#11758198—42no—
198#11758198—42no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210C4×Dic61—21no—
211C4×Dic61—21no—
212C4×Dic61—41no—
213C4×Dic61—41no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C23×C61C22×D411yescharacteristic
221C22×D61—21no—
222C22×D61—21no—
223C22×D61—21no—
224C22×D61—21no—
225C22×D61#12412yes—
226C22×D61AGL(1,ℤ/8)12yes—
227C22×D61#13712yes—
228Q8×C61—22no—
229Q8×C61—22no—
230Q8×C61—41no—
231Q8×C61—41no—
232C2×C4×C61—22no—
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—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×C61C2×#4111yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—41no—
264C2×Dic61—41no—
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#12412yes—
275C2×Dic61#13712yes—
276C2×Dic61#13712yes—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—22no—
280C4×D61—22no—
281C4×D61—44no—
282C4×D61—44no—
283C4×D61—48no—
284C4×D61—84no—
285Dic122—22no—
286Dic122—22no—
287Dic122—48no—
288Dic122—41no—
289Dic122—41no—
290Dic122—84no—
291C8×C61—41no—
292C8×C61—41no—
293#92418—41no—
294#92418—41no—
295#29111—611nomaximal, Sylow
296C22×C61#52511yescharacteristic
297C22×C61C2×#12411yescharacteristic
298C22×C61#53411yescharacteristic
299C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
300C22×C61C2×#13711yescharacteristic
301C22×C61C2×#13711yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C2×D61—22no—
304C2×D61—22no—
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61#48914yes—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61—81no—
316C4×C61#55111yescharacteristic
317C4×C61#55111yescharacteristic
318C4×C61#55111yescharacteristic
319C4×C61#55111yescharacteristic
320Dic61—22no—
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—81no—
325Dic61—81no—
326Dic61—81no—
327#2857—611no—
328C2×#391—611no—
329#2072—611no—
330#2072—611no—
331#2024—611no—
332#2024—611no—
333#1935—611no—
334#1935—611no—
335#1935—611no—
336#1935D6111yescharacteristic, complemented
337#1823—611no—
338#1823—611no—
339#1787—611no—
340#1787—611no—
341#1749—611no—
342C2×C61#256111yescharacteristic
343C2×C61#311211yescharacteristic
344C2×C61#311311yescharacteristic
345C2×C61C2×#48911yescharacteristic
346C2×C61#313911yescharacteristic
347C2×C61#319311yescharacteristic
348C2×C61#319511yescharacteristic
349D61—24no—
350C23×Q8—611no—
351C2×C4⋊Q8—611no—
352C2×C22⋊Q8—1221no—
353C22×C4⋊C4—611no—
354C22×C42—611no—
355C2×#103—611no—
356C2×#103—611no—
357C2×#103—611no—
358C2×#103—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103C2×D6111yescharacteristic
362C2×#103C2×D6111yescharacteristic
363C2×C22⋊C8—611no—
364C2×C22⋊C8—611no—
365C2×#97—611no—
366C2×#97C2×D6111yescharacteristic
367#391—611no—
368#391—611no—
369#391—611no—
370#391—1221no—
371#391—1221no—
372#391—1221no—
373#391C2×D6111yescharacteristic
374#389—611no—
375#389—1221no—
376#389C2×D6111yescharacteristic
377#384—611no—
378#380—611no—
379#380—611no—
380#380—1221no—
381C2×#95—1221no—
382C2×#95—1221no—
383#340—611no—
384#340—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—
393C61#2911111yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8—1221no—
397C22×Q8—1222no—
398C22×Q8—1222no—
399C22×Q8—1222no—
400C22×Q8—1222no—
401C22×Q8—2441no—
402C22×Q8—2441no—
403C22×Q8C22×D6111yescharacteristic
404C23×C4—611no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—1221no—
408C23×C4—1221no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C4⋊Q8—2441no—
414C4⋊Q8—2441no—
415C22⋊Q8—2444no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—1222no—
421C2×C4⋊C4—1222no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—2441no—
436C2×C42—2441no—
437C2×C42C22×D6111yescharacteristic
438#103—22no—
439#103—41no—
440#103—1222no—
441#103—1222no—
442#103—1222no—
443#103—2441no—
444#103—2441no—
445#103—2441no—
446C22⋊C8—1222no—
447C22⋊C8—1222no—
448#97—41no—
449#97—2441no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—611no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—612no—
469C2×Q8—1224no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—2442no—
476C2×Q8—2442no—
477C2×Q8—2442no—
478C2×Q8—2442no—
479C2×Q8—2442no—
480C2×Q8—2444no—
481C2×Q8—2444no—
482C2×Q8—2444no—
483C2×Q8—2444no—
484C2×Q8#1175822212yes—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—612no—
493C22×C4—612no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1224no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—2441no—
508C22×C4—2441no—
509C22×C4—2442no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4—2442no—
513C22×C4—2442no—
514C22×C4D4×D6111yescharacteristic
515C22×C4D4×D6111yescharacteristic
516C22×C4C23×D6111yescharacteristic
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—1221no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—21no—
534C4⋊C4—21no—
535C4⋊C4—42no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C22⋊C4—2442no—
548C22⋊C4—2442no—
549C42—21no—
550C42—21no—
551C42—1221no—
552C42—1221no—
553C42—2441no—
554C42—2441no—
555C23—612no—
556C23—612no—
557C23—612no—
558C23—1221no—
559C23—1221no—
560C23—1221no—
561C23—1221no—
562C23C2×D4×D6111yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—1222no—
568Q8—1222no—
569Q8—4884no—
570Q8—2448no—
571Q8—2441no—
572Q8—2441no—
573C2×C4—22no—
574C2×C4—21no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—41no—
581C2×C4—41no—
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—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—1222no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—4884no—
606C2×C4—2444no—
607C2×C4—2444no—
608C2×C4—2448no—
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×C4—2441no—
619C2×C4—2441no—
620C2×C4#41×D6111yescharacteristic
621C2×C4#41×D6111yescharacteristic
622C2×C4#41×D6111yescharacteristic
623C2×C4C2×#1175822211yescharacteristic
624C2×C4#1451982211yescharacteristic
625C2×C4#1451982311yescharacteristic
626C8—41no—
627C8—41no—
628C8—2441no—
629C8—2441no—
630C22—614no—
631C22—1222no—
632C22—1222no—
633C22—1222no—
634C22—1222no—
635C22—1222no—
636C22—1222no—
637C22#1684027111yescharacteristic
638C22#124×D6111yescharacteristic
639C22#1684032011yescharacteristic
640C22AGL(1,ℤ/8)×D6111yescharacteristic
641C22#137×D6111yescharacteristic
642C22#137×D6111yescharacteristic
643C22#137×D6111yescharacteristic
644C4—42no—
645C4—41no—
646C4—41no—
647C4—81no—
648C4—81no—
649C4—81no—
650C4—1222no—
651C4—1222no—
652C4—1222no—
653C4—4881no—
654C4—4881no—
655C4—4881no—
656C4—2442no—
657C4#1684043611yescharacteristic
658C4#1684043711yescharacteristic
659C4#1684044011yescharacteristic
660C4#1684044111yescharacteristic
661C2—1224no—
662C2#2145097511yescharacteristic
663C2#2145593911yescharacteristic
664C2#2145594711yescharacteristic
665C2#489×D6111yescharacteristic
666C2#2145619811yescharacteristic
667C2#2145662011yescharacteristic
668C2#2145663911yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1935C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1935D61
hypocenterC61#29111
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1935

Complements

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

NH ≅ G/Ntype
3. #21447231C2semidirect
4. #21447230C2semidirect
5. #21446959C2semidirect
6. #21446958C2semidirect
7. #21446512C2semidirect
8. #21446509C2semidirect
9. #21446508C2semidirect
15. C61×#1935C2semidirect
336. #1935D61semidirect
393. C61#29111semidirect

Maximal subgroups

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

subgroupindex
#214541702
#391×D612
#214472312
#214472302
#214469592
#214469582
#214465122
#214465092
#214465082
#214459042
#214457522
#214455822
C61×#19352
#2911161