Finite Group Databasebeta

← #29779646

Subgroups — #29779646

Order 15616.

850 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21451640C211yesmaximal, characteristic, complemented
2#21448407C211yesmaximal, characteristic
3#21447239C211yesmaximal, characteristic, complemented
4#21447234C211yesmaximal, characteristic, complemented
5#21446961C211yesmaximal, characteristic, complemented
6#21446956C211yesmaximal, characteristic, complemented
7#21446516C211yesmaximal, characteristic, complemented
8#21446509C211yesmaximal, characteristic, complemented
9#21446502C211yesmaximal, characteristic, complemented
10#21446495C211yesmaximal, characteristic, complemented
11#21445910C211yesmaximal, characteristic, complemented
12#21445906C211yesmaximal, characteristic, complemented
13#21445760C211yesmaximal, characteristic
14#21445760C211yesmaximal, characteristic
15C61×#1750C211yesmaximal, characteristic, complemented
16C22×#11758222C2211yescharacteristic
17C2×#14519777C2211yescharacteristic
18C22×C4⋊Dic61C2211yescharacteristic
19C2×#14519702C2211yescharacteristic
20C2×#14519702C2211yescharacteristic
21C2×#14519702C2211yescharacteristic
22C2×#14519702C2211yescharacteristic
23C2×#14519699C2211yescharacteristic
24C2×#14519699C2211yescharacteristic
25C2×#14519699C2211yescharacteristic
26C2×#14519699C2211yescharacteristic
27#16839701—21no—
28#16839698—21no—
29#16839682—21no—
30#16839674—21no—
31#16839668C2211yescharacteristic
32#16839668C2211yescharacteristic
33#16839664C2211yescharacteristic
34#16839664C2211yescharacteristic
35#16839660C2211yescharacteristic
36#16839660C2211yescharacteristic
37#16839639—21no—
38#16839639—21no—
39#16839628C2211yescharacteristic
40#16839622C2211yescharacteristic
41#16839622C2211yescharacteristic
42#16839619—21no—
43#16839619—21no—
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×#97×C61C2211yescharacteristic
56C2×C4×C8×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C2×#11758222—22no—
60C2×#11758222—22no—
61C2×#11758222—22no—
62C2×#11758222—22no—
63C2×#11758222—41no—
64C2×#11758222—41no—
65C22×Dic122—21no—
66C22×Dic122C2311yescharacteristic
67C22×D244C2311yescharacteristic
68C22×C22⋊D61—21no—
69C23×Dic61—21no—
70C22×C4×D61—21no—
71#14519777—41no—
72#14519777—41no—
73C2×C4⋊Dic61—22no—
74C2×C4⋊Dic61—22no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×C22⋊Dic61—21no—
86C2×C22⋊Dic61—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91#14519702—22no—
92#14519702—22no—
93#14519702—41no—
94#14519702—41no—
95#14519699—22no—
96#14519699—22no—
97#14519699—41no—
98#14519699—41no—
99#14519684—21no—
100#14519684—21no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109C23×C4×C61C2311yescharacteristic
110C22×C8×C61C2311yescharacteristic
111C22×C8×C61C2311yescharacteristic
112C22×C8×C61C2311yescharacteristic
113C22×C8×C61C2311yescharacteristic
114#117×C61—41no—
115#117×C61—41no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C22⋊C4×C61D411yescharacteristic
119C2×C22⋊C4×C61D411yescharacteristic
120C2×C42×C61C2311yescharacteristic
121C2×C42×C61C2311yescharacteristic
122C22⋊C8×C61—22no—
123C22⋊C8×C61—22no—
124#97×C61—41no—
125C4×C8×C61—41no—
126C24×C61C2×D411yescharacteristic
127C23×D61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61C2×D411yescharacteristic
133C22×C4×C61C2×D411yescharacteristic
134C22×C4×C61C2×D411yescharacteristic
135C22×C4×C61C2×D411yescharacteristic
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61#4112yes—
138C22×C4×C61#4112yes—
139C22×C4×C61C2411yescharacteristic
140C2×C22⋊D61—24no—
141C2×C22⋊D61—42no—
142C2×C22⋊D61—42no—
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—42no—
151C22×Dic61—81no—
152C22×Dic61—81no—
153#11758222—44no—
154#11758222—44no—
155#11758222—44no—
156#11758222—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244#4112yes—
163C2×C4×D61—24no—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—42no—
172C2×Dic122—81no—
173C2×Dic122—81no—
174C2×Dic122#4112yes—
175C2×C8×C61—21no—
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—
187C4⋊C4×C61—41no—
188C4⋊C4×C61—41no—
189C22⋊C4×C61—41no—
190C22⋊C4×C61—41no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195C22⋊Dic61—81no—
196C22⋊Dic61—81no—
197#11758198—81no—
198#11758198—81no—
199#11758198—81no—
200#11758198—81no—
201C4⋊Dic61—21no—
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⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—42no—
214C4⋊Dic61—42no—
215#11758196—81no—
216#11758196—81no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61#12412yes—
224C23×C61C22×D411yescharacteristic
225C22×D61—22no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—22no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
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—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61#12412yes—
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C22⋊D61—48no—
258C2×Dic61—22no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
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—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—84no—
280D244—22no—
281D244—22no—
282D244—41no—
283D244—41no—
284C4×D61—48no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289Dic122—84no—
290C8×C61—41no—
291C8×C61—41no—
292C8×C61—41no—
293C8×C61—41no—
294#29120—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61#52511yescharacteristic
303C22×C61C2×#12411yescharacteristic
304C22×C61#53411yescharacteristic
305C22×C61#57611yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57811yescharacteristic
309C2×D61—42no—
310C2×D61—42no—
311C2×D61—42no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C4×C61—22no—
317C4×C61—41no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61#55111yescharacteristic
322C4×C61#55111yescharacteristic
323C4×C61#55111yescharacteristic
324C4×C61#55111yescharacteristic
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332#2628—611no—
333#2311—611no—
334#2074—611no—
335#2073—611no—
336#2024—611no—
337#2023—611no—
338#1936—611no—
339#1935—611no—
340#1934—611no—
341#1933—611no—
342#1824—611no—
343#1823—611no—
344#1788—611no—
345#1788—611no—
346#1750D6111yescharacteristic, complemented
347C2×C61—42no—
348C2×C61#256111yescharacteristic
349C2×C61#312411yescharacteristic
350C2×C61#312511yescharacteristic
351C2×C61#313611yescharacteristic
352C2×C61#313711yescharacteristic
353C2×C61#319811yescharacteristic
354C2×C61#320111yescharacteristic
355D61—82no—
356C22×#41—611no—
357C2×C4⋊Q8—611no—
358C2×#117C2×D6111yescharacteristic
359C22×C4⋊C4—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#102—611no—
365C2×#102—611no—
366C2×#102—611no—
367C2×#102—611no—
368C2×C22⋊C8C2×D6111yescharacteristic
369C2×C22⋊C8C2×D6111yescharacteristic
370C2×#97C2×D6111yescharacteristic
371C2×C4×C8C2×D6111yescharacteristic
372#400—1221no—
373#399—1221no—
374#395—1221no—
375#393—1221no—
376#391—611no—
377#391—611no—
378#390—611no—
379#390—611no—
380#389—611no—
381#389—611no—
382#386—1221no—
383#386—1221no—
384#382—611no—
385#380—611no—
386#380—611no—
387#380—1221no—
388#380—1221no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#336C2×D6111yescharacteristic
398#336C2×D6111yescharacteristic
399C61#2912011yescharacteristic, complemented, Sylow
400C2×#41—1222no—
401C2×#41—1222no—
402C2×#41—1222no—
403C2×#41—1222no—
404C2×#41—2441no—
405C2×#41—2441no—
406C22×Q8—611no—
407C22×Q8—1221no—
408C22×D4—611no—
409C22×D4—1221no—
410C23×C4—1221no—
411C23×C4—1221no—
412C23×C4C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C22×C8C22×D6111yescharacteristic
417C4⋊Q8—2441no—
418C4⋊Q8—2441no—
419#117—41no—
420#117—41no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1222no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4D24411yescharacteristic
434C2×C4⋊C4D24411yescharacteristic
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4D24411yescharacteristic
442C2×C22⋊C4D24411yescharacteristic
443C2×C42C22×D6111yescharacteristic
444C2×C42C22×D6111yescharacteristic
445#103—1222no—
446#103—1222no—
447#103—2441no—
448#103—2441no—
449#102—1222no—
450#102—1222no—
451#102—2441no—
452#102—2441no—
453C22⋊C8—22no—
454C22⋊C8—22no—
455#97—41no—
456C4×C8—41no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—1221no—
468C24C2×D24411yescharacteristic
469#41—2444no—
470#41—2444no—
471#41—2444no—
472#41—2444no—
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—4881no—
480C2×Q8—4881no—
481C2×Q8—2442no—
482C2×D4—612no—
483C2×D4—1224no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—2442no—
490C2×D4—2442no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—1224no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—4881no—
504C22×C4—4881no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4—2442no—
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C2×D24411yescharacteristic
511C22×C4C2×D24411yescharacteristic
512C22×C4C2×D24411yescharacteristic
513C22×C4#1175822212yes—
514C22×C4#1175822212yes—
515C22×C4C23×D6111yescharacteristic
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—21no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C22⋊C4—41no—
547C22⋊C4—41no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—4881no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C42—21no—
555C42—21no—
556C42—21no—
557C42—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—1222no—
565C23—2441no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23#1451976512yes—
572C23C22×D24411yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—4884no—
576Q8—2441no—
577Q8—2441no—
578D4—1222no—
579D4—1222no—
580D4—2448no—
581D4—2441no—
582D4—2441no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—1222no—
602C2×C4—4884no—
603C2×C4—2448no—
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—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4#1451976512yes—
625C2×C4#41×D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4C2×#1175822211yescharacteristic
628C2×C4C2×#1175822211yescharacteristic
629C2×C4#1451982211yescharacteristic
630C2×C4#1451982211yescharacteristic
631C8—41no—
632C8—41no—
633C8—41no—
634C8—41no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—2442no—
647C22—2442no—
648C22—2442no—
649C22#1684027411yescharacteristic
650C22C2×#1451976511yescharacteristic
651C22#1684032411yescharacteristic
652C22#1684056711yescharacteristic
653C22#1684057411yescharacteristic
654C22#1684057411yescharacteristic
655C22#1684057911yescharacteristic
656C4—22no—
657C4—41no—
658C4—41no—
659C4—41no—
660C4—41no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—2442no—
668C4#1684044111yescharacteristic
669C4#1684044111yescharacteristic
670C4#1684044111yescharacteristic
671C4#1684044111yescharacteristic
672C2—42no—
673C2—4882no—
674C2#2145097211yescharacteristic
675C2#2145608311yescharacteristic
676C2#2145609811yescharacteristic
677C2#2145617511yescharacteristic
678C2#2145618611yescharacteristic
679C2#2145667311yescharacteristic
680C2#2145670311yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1750C2
socleC23×C61C22×D4
prime radicalC22×#11758222C22
hypercenter#1750D61
hypocenterC61#29120
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1750

Complements

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

NH ≅ G/Ntype
1. #21451640C2semidirect
3. #21447239C2semidirect
4. #21447234C2semidirect
5. #21446961C2semidirect
6. #21446956C2semidirect
7. #21446516C2semidirect
8. #21446509C2semidirect
9. #21446502C2semidirect
10. #21446495C2semidirect
11. #21445910C2semidirect
12. #21445906C2semidirect
15. C61×#1750C2semidirect
346. #1750D61semidirect
399. C61#29120semidirect

Maximal subgroups

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

subgroupindex
#214516402
#214484072
#214472392
#214472342
#214469612
#214469562
#214465162
#214465092
#214465022
#214464952
#214459102
#214459062
#214457602
C61×#17502
#2912061