Finite Group Databasebeta

← #29749866

Subgroups — #29749866

Order 15616.

876 conjugacy classes of subgroups.

The subgroup lattice has 688 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840128C211yesmaximal, characteristic
2C2×#16839860C211yesmaximal, characteristic, complemented
3#21449584C211yesmaximal, characteristic
4#21447204C211yesmaximal, characteristic, complemented
5#21447154C211yesmaximal, characteristic, complemented
6#21447134C211yesmaximal, characteristic, complemented
7#21446906C211yesmaximal, characteristic
8#21446676C211yesmaximal, characteristic, complemented
9#21446515C211yesmaximal, characteristic, complemented
10#21446512C211yesmaximal, characteristic, complemented
11#21446501C211yesmaximal, characteristic, complemented
12#21446174C211yesmaximal, characteristic, complemented
13#21446079C211yesmaximal, characteristic, complemented
14#21445688C211yesmaximal, characteristic, complemented
15C61×#2453C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519774C2211yescharacteristic
19C2×#14519762—21no—
20C2×C4×D244C2211yescharacteristic
21#16840128—24no—
22#16839860—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×C4⋊#92418C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519688C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839680—21no—
38#16839679—21no—
39#16839664C2211yescharacteristic
40#16839664C2211yescharacteristic
41#16839664C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839640—21no—
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×#125×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#400—21no—
53C61×#398—21no—
54C61×#391C2211yescharacteristic
55C61×#391C2211yescharacteristic
56C61×#389—21no—
57C61×#389C2211yescharacteristic
58C61×#389C2211yescharacteristic
59C61×#387—21no—
60C4×C4⋊C4×C61C2211yescharacteristic
61C22×D244—21no—
62C22×D244—21no—
63C22×D244C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C2×#11758201—22no—
67C2×#11758201—22no—
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201—22no—
71C2×#11758201—41no—
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519774—22no—
77#14519774—41no—
78#14519762—44no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91#14519710—22no—
92#14519708—41no—
93C4⋊#92418—22no—
94#14519701—22no—
95#14519701—22no—
96#14519697—22no—
97#14519697—22no—
98#14519697—22no—
99#14519697—22no—
100#14519697—22no—
101#14519697D414yes—
102#14519688—22no—
103#14519688—22no—
104C4×#92418—22no—
105#14519684—21no—
106C22×Q8×C61C2311yescharacteristic
107#125×C61—22no—
108#125×C61—41no—
109C4×Q8×C61—22no—
110C4×Q8×C61—41no—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61D411yescharacteristic
117C2×C4⋊C4×C61D411yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C42×C61—21no—
122C2×C42×C61C2311yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125#95×C61—21no—
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61D411yescharacteristic
132#95×C61D411yescharacteristic
133C23×D61—21no—
134C23×D61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61#4112yes—
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×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165C2×D244—42no—
166C2×D244—42no—
167C2×D244—42no—
168C2×D244—42no—
169C2×D244—81no—
170C2×D244—81no—
171C2×D244C2×D412yes—
172C2×C4×D61—42no—
173C2×C4×D61—42no—
174C2×C4×D61—81no—
175C4⋊C4×C61—22no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—41no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—42no—
185C4⋊C4×C61—42no—
186C4⋊C4×C61—42no—
187C4⋊C4×C61C2×D412yes—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—21no—
194C42×C61—42no—
195#11758201—28no—
196#11758201—44no—
197#11758198—42no—
198#11758198—42no—
199#11758198—42no—
200#11758198—42no—
201#11758198—81no—
202#11758198—81no—
203#11758198—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208#11758196—42no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418C2×D412yes—
219C2×#92418#4112yes—
220C2×#92418#4112yes—
221C23×C61C22×D411yescharacteristic
222C22×D61—22no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231C22×D61—41no—
232C22×D61—41no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—41no—
239Q8×C61—41no—
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—22no—
248C2×C4×C61—22no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—21no—
255C2×C4×C61—21no—
256C2×C4×C61—21no—
257C2×C4×C61—21no—
258C2×C4×C61—21no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61AGL(1,ℤ/8)12yes—
270C2×C4×C61C22×D411yescharacteristic
271C2×C4×C61C2×#4111yescharacteristic
272C2×C4×C61C2×#4111yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61H(5,𝔽2)11yescharacteristic
275C2×C4×C612−1+411yescharacteristic
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282D244—22no—
283D244—22no—
284D244—41no—
285D244—41no—
286D244—41no—
287D244—41no—
288D244—41no—
289D244—41no—
290D244—84no—
291D244—84no—
292C4×D61—84no—
293#92418—22no—
294#92418—22no—
295#92418—22no—
296#92418—41no—
297#26841—611nomaximal, Sylow
298C22×C61#53211yescharacteristic
299C22×C61D4211yescharacteristic
300C22×C61#54811yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61#57511yescharacteristic
303C22×C61#57711yescharacteristic
304C22×C61#57711yescharacteristic
305C2×D61—42no—
306C2×D61—42no—
307C2×D61—42no—
308C2×D61—81no—
309C2×D61—81no—
310C2×D61—81no—
311C2×D61—81no—
312C2×D61—81no—
313C2×D61—81no—
314C2×D61—81no—
315C2×D61—81no—
316C2×D61—81no—
317C2×D61—81no—
318C2×D61—81no—
319C4×C61—22no—
320C4×C61—41no—
321C4×C61—42no—
322C4×C61—41no—
323C4×C61—41no—
324C4×C61—42no—
325C4×C61—41no—
326C4×C61—41no—
327C4×C61—81no—
328C4×C61—81no—
329C4×C61#52511yescharacteristic
330C4×C61#54211yescharacteristic
331C4×C61#54711yescharacteristic
332C4×C61#55111yescharacteristic
333Dic61—81no—
334Dic61—81no—
335C2×#490—611no—
336C2×#440—611no—
337#2453D6111yescharacteristic, complemented
338#2445—611no—
339#2067—611no—
340#2059—611no—
341#2055—611no—
342#2014—611no—
343#1968—611no—
344#1936—611no—
345#1935—611no—
346#1934—611no—
347#1880—611no—
348#1864—611no—
349#1770—611no—
350C2×C61#246111yescharacteristic
351C2×C61#317411yescharacteristic
352C2×C61#318011yescharacteristic
353C2×C61#327211yescharacteristic
354C2×C61#327511yescharacteristic
355C2×C61#336811yescharacteristic
356C2×C61#337111yescharacteristic
357D61—82no—
358D61—161no—
359C22×SD16—611no—
360C22×SD16—611no—
361C2×#127—611no—
362C2×#125C2×D6111yescharacteristic
363C2×#123—1221no—
364C2×C4×Q8C2×D6111yescharacteristic
365C2×C4×D4—611no—
366#490—1224no—
367#440—1224no—
368C2×#107—611no—
369C2×#106—611no—
370C2×C4⋊C8—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#102—611no—
374C2×#102—611no—
375C2×#102—611no—
376C2×#102—611no—
377C2×#102—611no—
378C2×#102—611no—
379C2×#97—611no—
380C2×#97—611no—
381C2×C4×C8—611no—
382#400—21no—
383#398—21no—
384#394—1221no—
385#394—1221no—
386#391C2×D6111yescharacteristic
387#391C2×D6111yescharacteristic
388#390—611no—
389#390—611no—
390#390—611no—
391#389—21no—
392#389—611no—
393#389C2×D6111yescharacteristic
394#389C2×D6111yescharacteristic
395#387—21no—
396#386—1221no—
397C4×C4⋊C4C2×D6111yescharacteristic
398#340—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#336—611no—
403#336—611no—
404C61#2684111yescharacteristic, complemented, Sylow
405C22×Q8C22×D6111yescharacteristic
406C22×D4—611no—
407C22×D4—611no—
408C22×D4—1221no—
409C22×D4—1221no—
410C23×C4—1221no—
411C2×SD16—1222no—
412C2×SD16—1222no—
413C2×SD16—1222no—
414C2×SD16—1222no—
415C2×SD16—1222no—
416C2×SD16—2441no—
417C22×C8—611no—
418C22×C8—611no—
419C22×C8—611no—
420C22×C8—611no—
421#127—1222no—
422#127—2441no—
423#125—22no—
424#125—41no—
425#123—2444no—
426C4×Q8—22no—
427C4×Q8—41no—
428C4×D4—1222no—
429C4×D4—2441no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4C22⋊D6111yescharacteristic
439C2×C4⋊C4C22⋊D6111yescharacteristic
440C2×C4⋊C4C22×D6111yescharacteristic
441C2×C4⋊C4C22×D6111yescharacteristic
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42C22×D6111yescharacteristic
452C2×C42C22×D6111yescharacteristic
453C2×C42C22×D6111yescharacteristic
454#107—1222no—
455#106—2441no—
456C4⋊C8—1222no—
457#103—1222no—
458#103—1222no—
459#102—614no—
460#102—1222no—
461#102—1222no—
462#102—1222no—
463#102—1222no—
464#102—1222no—
465#97—1222no—
466#97—1222no—
467C4×C8—1222no—
468#95—21no—
469#95—21no—
470#95—21no—
471#95—21no—
472#95—21no—
473#95—21no—
474#95—1221no—
475#95C22⋊D6111yescharacteristic
476#95C22⋊D6111yescharacteristic
477C24—1221no—
478C24—1221no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8#1175822612yes—
485C2×D4—612no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C2×D4—1221no—
490C2×D4—1221no—
491C2×D4—1221no—
492C2×D4—1221no—
493C2×D4—1221no—
494C2×D4—1221no—
495C2×D4—1221no—
496C2×D4—1221no—
497C2×D4—4881no—
498C2×D4—4881no—
499C2×D4—2442no—
500C2×D4—2442no—
501C2×D4—2442no—
502C2×D4—2442no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—1221no—
508C22×C4—1221no—
509C22×C4—4881no—
510C22×C4—2442no—
511C22×C4—2442no—
512C22×C4C2×C22⋊D6111yescharacteristic
513C22×C4C2×C22⋊D6111yescharacteristic
514C22×C4C2×C22⋊D6111yescharacteristic
515C22×C4C2×C22⋊D6111yescharacteristic
516C22×C4C2×C22⋊D6111yescharacteristic
517C22×C4C2×C22⋊D6111yescharacteristic
518C22×C4C23×D6111yescharacteristic
519SD16—1228no—
520SD16—2444no—
521C2×C8—612no—
522C2×C8—612no—
523C2×C8—612no—
524C2×C8—1221no—
525C2×C8—1221no—
526C2×C8—1221no—
527C2×C8—1221no—
528C2×C8—1221no—
529C2×C8—1221no—
530C2×C8—1221no—
531C2×C8—1221no—
532C2×C8—1221no—
533C4⋊C4—21no—
534C4⋊C4—22no—
535C4⋊C4—21no—
536C4⋊C4—21no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—41no—
540C4⋊C4—42no—
541C4⋊C4—42no—
542C4⋊C4—42no—
543C4⋊C4—42no—
544C4⋊C4—42no—
545C4⋊C4—1221no—
546C4⋊C4—1221no—
547C4⋊C4—1221no—
548C4⋊C4—1221no—
549C4⋊C4—2442no—
550C4⋊C4D4×D6112yes—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—4881no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C22⋊C4—2442no—
558C42—21no—
559C42—21no—
560C42—21no—
561C42—21no—
562C42—21no—
563C42—21no—
564C42—42no—
565C23—1222no—
566C23—2441no—
567C23—2441no—
568C23—2441no—
569C23—2441no—
570C23—2441no—
571C23—2441no—
572C23—2441no—
573C23—2441no—
574C23—2441no—
575C23—2441no—
576C23—2441no—
577C23—2441no—
578C23—2441no—
579C23C22×C22⋊D6111yescharacteristic
580Q8—22no—
581Q8—22no—
582Q8—41no—
583Q8—41no—
584D4—1222no—
585D4—1222no—
586D4—4884no—
587D4—4884no—
588D4—2441no—
589D4—2441no—
590D4—2441no—
591D4—2441no—
592D4—2441no—
593D4—2441no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—21no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—22no—
612C2×C4—22no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—4884no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4#1451980212yes—
631C2×C4C2×D4×D6111yescharacteristic
632C2×C4#41×D6111yescharacteristic
633C2×C4#41×D6111yescharacteristic
634C2×C4C2×#1175822611yescharacteristic
635C2×C4#1451982211yescharacteristic
636C2×C4#1451982511yescharacteristic
637C8—1222no—
638C8—1222no—
639C8—1222no—
640C8—2441no—
641C22—4881no—
642C22—4881no—
643C22—4881no—
644C22—4881no—
645C22—4881no—
646C22—4881no—
647C22—4881no—
648C22—4881no—
649C22—4881no—
650C22—4881no—
651C22—4881no—
652C22—2442no—
653C22—2442no—
654C22—2442no—
655C22#1684030811yescharacteristic
656C22D4×C22⋊D6111yescharacteristic
657C22#1684042311yescharacteristic
658C22C2×#1451980211yescharacteristic
659C22#1684056011yescharacteristic
660C22#1684057011yescharacteristic
661C22#1684057011yescharacteristic
662C4—22no—
663C4—41no—
664C4—41no—
665C4—42no—
666C4—41no—
667C4—41no—
668C4—41no—
669C4—42no—
670C4—81no—
671C4—81no—
672C4—4881no—
673C4—4881no—
674C4#1684027511yescharacteristic
675C4#1684038211yescharacteristic
676C4#1684041411yescharacteristic
677C4#1684043911yescharacteristic
678C2—9761no—
679C2—4882no—
680C2#2144979311yescharacteristic
681C2#2145644811yescharacteristic
682C2#2145651811yescharacteristic
683C2#2145741911yescharacteristic
684C2#2145745111yescharacteristic
685C2#2145854811yescharacteristic
686C2#2145858711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2453C2
socleC23×C61C22×D4
prime radicalC2×#14519774C22
hypercenter#2453D61
hypocenterC61#26841
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2453

Complements

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

NH ≅ G/Ntype
2. C2×#16839860C2semidirect
4. #21447204C2semidirect
5. #21447154C2semidirect
6. #21447134C2semidirect
8. #21446676C2semidirect
9. #21446515C2semidirect
10. #21446512C2semidirect
11. #21446501C2semidirect
12. #21446174C2semidirect
13. #21446079C2semidirect
14. #21445688C2semidirect
15. C61×#2453C2semidirect
337. #2453D61semidirect
404. C61#26841semidirect

Maximal subgroups

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

subgroupindex
C2×#168401282
C2×#168398602
#214495842
#214472042
#214471542
#214471342
#214469062
#214466762
#214465152
#214465122
#214465012
#214461742
#214460792
#214456882
C61×#24532
#2684161