Finite Group Databasebeta

← #29779576

Subgroups — #29779576

Order 15616.

910 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21454170C211yesmaximal, characteristic, complemented
2#21448306C211yesmaximal, characteristic
3#21447231C211yesmaximal, characteristic, complemented
4#21447225C211yesmaximal, characteristic, complemented
5#21446959C211yesmaximal, characteristic, complemented
6#21446953C211yesmaximal, characteristic, complemented
7#21446519C211yesmaximal, characteristic, complemented
8#21446509C211yesmaximal, characteristic, complemented
9#21446508C211yesmaximal, characteristic, complemented
10#21445912C211yesmaximal, characteristic, complemented
11#21445912C211yesmaximal, characteristic, complemented
12#21445756C211yesmaximal, characteristic
13#21445756C211yesmaximal, characteristic
14#21445582C211yesmaximal, characteristic, complemented
15C61×#1936C211yesmaximal, characteristic, complemented
16C22×#11758224C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519755—21no—
19C2×#14519735C2211yescharacteristic
20C2×C42×D61C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519700C2211yescharacteristic
24C2×#14519700C2211yescharacteristic
25C2×#14519698C2211yescharacteristic
26C2×#14519698C2211yescharacteristic
27C2×#14519689C2211yescharacteristic
28C2×#14519689C2211yescharacteristic
29C2×#14519688C2211yescharacteristic
30#16839668C2211yescharacteristic
31#16839667C2211yescharacteristic
32#16839666—21no—
33#16839665C2211yescharacteristic
34#16839660C2211yescharacteristic
35#16839657—21no—
36#16839638—21no—
37#16839638—21no—
38#16839631C2211yescharacteristic
39#16839621—21no—
40#16839619C2211yescharacteristic
41#16839619C2211yescharacteristic
42C22⋊C4×Dic61—21no—
43C22⋊C4×Dic61—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#102×C61C2211yescharacteristic
53C2×#102×C61C2211yescharacteristic
54C2×#97×C61C2211yescharacteristic
55C61×#390C2211yescharacteristic
56C61×#389C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C2×#11758224—22no—
60C2×#11758224—22no—
61C2×#11758224—22no—
62C2×#11758224—22no—
63C2×#11758224—41no—
64C2×#11758224—41no—
65C22×Dic122—21no—
66C22×Dic122C2311yescharacteristic
67C22×C22⋊D61—21no—
68C23×Dic61—21no—
69C22×C4×D61D411yescharacteristic
70C22×C4×D61D411yescharacteristic
71C22×C4×D61C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74#14519776—41no—
75#14519776—41no—
76#14519755—44no—
77#14519735—22no—
78#14519735—22no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×C22⋊Dic61—21no—
87C2×C22⋊Dic61—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61D411yescharacteristic
93C2×C4×Dic61D411yescharacteristic
94C2×C4×Dic61C2311yescharacteristic
95C42×D61—41no—
96C42×D61—41no—
97#14519702—22no—
98#14519702—41no—
99#14519700—22no—
100#14519700—41no—
101#14519698—22no—
102#14519698—41no—
103#14519689—22no—
104#14519689—22no—
105#14519688—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113C22×D4×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×C22⋊C4×C61—21no—
121C2×C22⋊C4×C61—21no—
122C2×C42×C61C2311yescharacteristic
123#102×C61—22no—
124#102×C61—41no—
125#97×C61—41no—
126#95×C61—21no—
127C24×C61—21no—
128C23×D61C2×D411yescharacteristic
129C2×D4×C61—21no—
130C2×D4×C61—21no—
131C2×D4×C61—21no—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61#4112yes—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—24no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—42no—
144C22×Dic61—24no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—42no—
150C22×Dic61—42no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154#11758224—44no—
155#11758224—44no—
156#11758224—44no—
157#11758224—44no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—22no—
163C2×C4×D61—22no—
164C2×C4×D61—41no—
165C2×C4×D61—41no—
166C2×C4×D61—41no—
167C2×C4×D61—41no—
168C2×C4×D61#4112yes—
169C2×C4×D61#4112yes—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—42no—
176C2×Dic122—42no—
177C2×Dic122—42no—
178C2×Dic122#4112yes—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—42no—
192C22⋊C4×C61—42no—
193C22⋊C4×C61—42no—
194C42×C61—21no—
195C42×C61—21no—
196C22⋊Dic61—42no—
197C22⋊Dic61—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207#11758196—42no—
208#11758196—42no—
209#11758196—42no—
210#11758196—42no—
211C4×Dic61—21no—
212C4×Dic61—21no—
213C4×Dic61—41no—
214C4×Dic61—41no—
215C4×Dic61—42no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61#12412yes—
237D4×C61—22no—
238D4×C61—22no—
239D4×C61—41no—
240D4×C61—41no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
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×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61C2×#4111yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61H(5,𝔽2)11yescharacteristic
261C2×C4×C61H(5,𝔽2)11yescharacteristic
262C22⋊D61—48no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—48no—
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#12412yes—
283C2×Dic61AGL(1,ℤ/8)12yes—
284C2×Dic61#13712yes—
285C4×D61—22no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—44no—
290C4×D61—44no—
291Dic122—22no—
292Dic122—22no—
293Dic122—41no—
294Dic122—41no—
295Dic122—84no—
296C8×C61—41no—
297C8×C61—41no—
298#92418—41no—
299#92418—41no—
300#29113—611nomaximal, Sylow
301C22×C61—42no—
302C22×C61—42no—
303C22×C61—42no—
304C22×C61—81no—
305C22×C61—81no—
306C22×C61—81no—
307C22×C61—81no—
308C22×C61#52511yescharacteristic
309C22×C61C2×#12411yescharacteristic
310C22×C61#53411yescharacteristic
311C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
312C22×C61C2×#13711yescharacteristic
313C22×C61#57511yescharacteristic
314C22×C61#57511yescharacteristic
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—41no—
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—81no—
325C4×C61—81no—
326C4×C61—81no—
327C4×C61#55111yescharacteristic
328C4×C61#55111yescharacteristic
329C4×C61#55111yescharacteristic
330C4×C61#55111yescharacteristic
331Dic61—22no—
332Dic61—22no—
333Dic61—22no—
334Dic61—42no—
335Dic61—81no—
336Dic61—81no—
337Dic61—81no—
338#2857—611no—
339#2297—611no—
340#2072—611no—
341#2071—611no—
342#2024—611no—
343#2023—611no—
344#1936—611no—
345#1936D6111yescharacteristic, complemented
346#1935—611no—
347#1935—611no—
348#1825—611no—
349#1825—611no—
350#1788—611no—
351#1788—611no—
352#1749—611no—
353C2×C61—82no—
354C2×C61#256111yescharacteristic
355C2×C61#311511yescharacteristic
356C2×C61#311611yescharacteristic
357C2×C61#313511yescharacteristic
358C2×C61#314111yescharacteristic
359C2×C61#319411yescharacteristic
360C2×C61#319611yescharacteristic
361D61—42no—
362C22×#41—611no—
363C2×C4⋊Q8—611no—
364C2×C22⋊Q8—1221no—
365C2×#117—611no—
366C22×C42—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#102—611no—
372C2×#102—611no—
373C2×#102C2×D6111yescharacteristic
374C2×#102C2×D6111yescharacteristic
375C2×C22⋊C8—611no—
376C2×C22⋊C8—611no—
377C2×#97—611no—
378C2×#97C2×D6111yescharacteristic
379#391—611no—
380#391—611no—
381#391—1221no—
382#390—611no—
383#390C2×D6111yescharacteristic
384#389—611no—
385#389—1221no—
386#389C2×D6111yescharacteristic
387#386—1221no—
388#386—1221no—
389#384—611no—
390#380—611no—
391#380—611no—
392#380—1221no—
393C4×C22⋊C4—1221no—
394C4×C22⋊C4—1221no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340C2×D6111yescharacteristic
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405C61#2911311yescharacteristic, complemented, Sylow
406C2×#41—1222no—
407C2×#41—1222no—
408C2×#41—1222no—
409C2×#41—1222no—
410C2×#41—2441no—
411C2×#41—2441no—
412C22×Q8—611no—
413C22×Q8—1221no—
414C22×D4—1221no—
415C22×D4C22×D6111yescharacteristic
416C23×C4—611no—
417C23×C4—611no—
418C23×C4—611no—
419C23×C4—1221no—
420C22×C8—611no—
421C22×C8—611no—
422C22×C8C22×D6111yescharacteristic
423C22×C8C22×D6111yescharacteristic
424C4⋊Q8—2441no—
425C4⋊Q8—2441no—
426C22⋊Q8—2444no—
427#117—1222no—
428#117—1222no—
429C2×C4⋊C4—21no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C42—611no—
448C2×C42—611no—
449C2×C42—611no—
450C2×C42—1221no—
451C2×C42—2441no—
452C2×C42—2441no—
453C2×C42C22×D6111yescharacteristic
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#103—2441no—
458#102—22no—
459#102—41no—
460#102—1222no—
461#102—2441no—
462C22⋊C8—1222no—
463C22⋊C8—1222no—
464#97—41no—
465#97—2441no—
466#95—21no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474C24—21no—
475C24—611no—
476#41—2444no—
477#41—2444no—
478#41—2444no—
479#41—2444no—
480C2×Q8—612no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×Q8—1221no—
484C2×Q8—1221no—
485C2×Q8—1221no—
486C2×Q8—2442no—
487C2×Q8—2442no—
488C2×Q8—2442no—
489C2×D4—21no—
490C2×D4—21no—
491C2×D4—21no—
492C2×D4—21no—
493C2×D4—21no—
494C2×D4—1224no—
495C2×D4—2442no—
496C2×D4—2442no—
497C2×D4#1175822212yes—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—611no—
502C22×C4—611no—
503C22×C4—611no—
504C22×C4—612no—
505C22×C4—612no—
506C22×C4—1222no—
507C22×C4—1222no—
508C22×C4—1221no—
509C22×C4—1221no—
510C22×C4—1221no—
511C22×C4—1221no—
512C22×C4—1224no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—2441no—
518C22×C4—2441no—
519C22×C4—2441no—
520C22×C4—2441no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4D4×D6111yescharacteristic
524C22×C4D4×D6111yescharacteristic
525C22×C4C23×D6111yescharacteristic
526C2×C8—21no—
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—1221no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—42no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—1221no—
550C4⋊C4—1221no—
551C4⋊C4—2442no—
552C4⋊C4—2442no—
553C4⋊C4—2442no—
554C4⋊C4—2442no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—2442no—
558C22⋊C4—2442no—
559C22⋊C4—2442no—
560C22⋊C4—2442no—
561C22⋊C4—2442no—
562C42—21no—
563C42—21no—
564C42—1221no—
565C42—1221no—
566C42—2441no—
567C42—2441no—
568C42—2442no—
569C23—22no—
570C23—41no—
571C23—41no—
572C23—41no—
573C23—41no—
574C23—41no—
575C23—41no—
576C23—612no—
577C23—1221no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23—1221no—
582C23—1221no—
583C23C2×D4×D6111yescharacteristic
584Q8—1222no—
585Q8—1222no—
586Q8—4884no—
587Q8—2441no—
588Q8—2441no—
589D4—22no—
590D4—22no—
591D4—41no—
592D4—41no—
593D4—2448no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1222no—
621C2×C4—1222no—
622C2×C4—1222no—
623C2×C4—2444no—
624C2×C4—2444no—
625C2×C4—2448no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4#41×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4C2×#1175822211yescharacteristic
641C2×C4#1451982211yescharacteristic
642C2×C4#1451982311yescharacteristic
643C8—41no—
644C8—41no—
645C8—2441no—
646C8—2441no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—81no—
651C22—81no—
652C22—81no—
653C22—81no—
654C22—1222no—
655C22—1222no—
656C22—1222no—
657C22—2441no—
658C22—2441no—
659C22—2441no—
660C22—2441no—
661C22#1684027111yescharacteristic
662C22#124×D6111yescharacteristic
663C22#1684032011yescharacteristic
664C22AGL(1,ℤ/8)×D6111yescharacteristic
665C22#137×D6111yescharacteristic
666C22#1684055811yescharacteristic
667C22#1684055811yescharacteristic
668C4—41no—
669C4—41no—
670C4—81no—
671C4—81no—
672C4—81no—
673C4—1222no—
674C4—1222no—
675C4—1222no—
676C4—4881no—
677C4—4881no—
678C4—4881no—
679C4—2442no—
680C4#1684043611yescharacteristic
681C4#1684043711yescharacteristic
682C4#1684044011yescharacteristic
683C4#1684044111yescharacteristic
684C2—82no—
685C2—2442no—
686C2#2145097511yescharacteristic
687C2#2145596611yescharacteristic
688C2#2145597711yescharacteristic
689C2#2145616311yescharacteristic
690C2#2145621111yescharacteristic
691C2#2145662811yescharacteristic
692C2#2145664411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1936C2
socleC23×C61C22×D4
prime radicalC22×#11758224C22
hypercenter#1936D61
hypocenterC61#29113
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1936

Complements

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

NH ≅ G/Ntype
1. #21454170C2semidirect
3. #21447231C2semidirect
4. #21447225C2semidirect
5. #21446959C2semidirect
6. #21446953C2semidirect
7. #21446519C2semidirect
8. #21446509C2semidirect
9. #21446508C2semidirect
10. #21445912C2semidirect
11. #21445912C2semidirect
14. #21445582C2semidirect
15. C61×#1936C2semidirect
345. #1936D61semidirect
405. C61#29113semidirect

Maximal subgroups

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

subgroupindex
#214541702
#214483062
#214472312
#214472252
#214469592
#214469532
#214465192
#214465092
#214465082
#214459122
#214457562
#214455822
C61×#19362
#2911361