Finite Group Databasebeta

← #29748236

Subgroups — #29748236

Order 15616.

804 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#21450178C211yesmaximal, characteristic
2#21448846C211yesmaximal, characteristic, complemented
3#21447138C211yesmaximal, characteristic, complemented
4#21447022C211yesmaximal, characteristic, complemented
5#21446959C211yesmaximal, characteristic, complemented
6#21446824C211yesmaximal, characteristic
7#21446765C211yesmaximal, characteristic, complemented
8#21446500C211yesmaximal, characteristic, complemented
9#21446437C211yesmaximal, characteristic, complemented
10#21446422C211yesmaximal, characteristic, complemented
11#21446388C211yesmaximal, characteristic, complemented
12#21446175C211yesmaximal, characteristic, complemented
13#21446141C211yesmaximal, characteristic
14#21445638C211yesmaximal, characteristic, complemented
15C61×#1825C211yesmaximal, characteristic, complemented
16C22×#11758190C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519763—21no—
19C2×#14519756C2211yescharacteristic
20C2×#14519751C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519706C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519700C2211yescharacteristic
26C2×#14519700C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519697C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×C22⋊#92418C2211yescharacteristic
31C2×C8×Dic61C2211yescharacteristic
32#16839677—21no—
33#16839675—21no—
34#16839667C2211yescharacteristic
35#16839663C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839658C2211yescharacteristic
38#16839656—21no—
39#16839651—21no—
40#16839650—21no—
41#16839646—21no—
42C4⋊C4×Dic61C2211yescharacteristic
43C4⋊C4×Dic61C2211yescharacteristic
44C4×#11758196—21no—
45C4×C22⋊Dic61C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839516C2211yescharacteristic
51#16839516C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C2×C22⋊C8×C61C2211yescharacteristic
54C61×#380C2211yescharacteristic
55C61×#340C2211yescharacteristic
56C61×#340C2211yescharacteristic
57C61×#340C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C22×Dic122—21no—
60C22×Dic122C2311yescharacteristic
61C22×D244C2311yescharacteristic
62C22×C22⋊D61—21no—
63C2×#11758190—22no—
64C2×#11758190—41no—
65C2×#11758190—41no—
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519776—22no—
69#14519776—41no—
70#14519763—44no—
71#14519756—22no—
72#14519756—22no—
73#14519751—22no—
74#14519751—41no—
75#14519742—22no—
76#14519742—41no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×#11758196—21no—
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×C22⋊Dic61—21no—
84C2×C22⋊Dic61D411yescharacteristic
85C2×C22⋊Dic61D411yescharacteristic
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97#14519706—41no—
98#14519702—22no—
99#14519700—22no—
100#14519700—41no—
101#14519699—22no—
102#14519697—22no—
103#14519697—41no—
104C22⋊#92418—41no—
105C8×Dic61—41no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684D411yescharacteristic
112#14519684D411yescharacteristic
113C23×C4×C61C2311yescharacteristic
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116#117×C61—22no—
117#117×C61—41no—
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C42×C61—21no—
125C22⋊C8×C61—41no—
126#95×C61—21no—
127C24×C61C2×D411yescharacteristic
128C23×D61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2411yescharacteristic
141C2×C22⋊D61—42no—
142C2×C22⋊D61—42no—
143C2×C22⋊D61—81no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61C2×D411yescharacteristic
148C22×Dic61C2×D411yescharacteristic
149C22×Dic61C2×D411yescharacteristic
150C22×Dic61C2×D411yescharacteristic
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—21no—
160C2×Dic122—21no—
161C2×Dic122—21no—
162C2×Dic122—42no—
163C2×Dic122—42no—
164C2×Dic122—81no—
165C2×Dic122#4112yes—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61#4112yes—
180C22⋊C4×C61—42no—
181C22⋊C4×C61—42no—
182C42×C61—42no—
183C22⋊Dic61—24no—
184C22⋊Dic61—22no—
185C22⋊Dic61—22no—
186#11758198—42no—
187#11758198—42no—
188#11758198—42no—
189#11758198—42no—
190#11758198—81no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61#4112yes—
194C4⋊Dic61#4112yes—
195#11758196—24no—
196#11758196—42no—
197#11758196—42no—
198#11758196—42no—
199C4×Dic61—21no—
200C4×Dic61—21no—
201C4×Dic61—21no—
202C4×Dic61—21no—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212#11758190—44no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61—21no—
216C23×C61—21no—
217C23×C61—21no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61C22×D411yescharacteristic
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C2×C4×C61—21no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
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×C61C2×#4111yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C61H(5,𝔽2)11yescharacteristic
253C22⋊D61—84no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61#13512yes—
274C2×Dic61AGL(1,ℤ/8)12yes—
275D244—41no—
276D244—41no—
277D244—41no—
278D244—41no—
279Dic122—22no—
280Dic122—22no—
281Dic122—41no—
282Dic122—41no—
283Dic122—84no—
284C8×C61—41no—
285C8×C61—41no—
286#92418—41no—
287#92418—41no—
288#26733—611nomaximal, Sylow
289C22×C61—41no—
290C22×C61—41no—
291C22×C61—41no—
292C22×C61—41no—
293C22×C61—41no—
294C22×C61—41no—
295C22×C61—41no—
296C22×C61#52511yescharacteristic
297C22×C61#54611yescharacteristic
298C22×C61#54611yescharacteristic
299C22×C61C2×#13511yescharacteristic
300C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C2×D61—81no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C2×D61—81no—
309C2×D61—81no—
310C4×C61—41no—
311C4×C61—42no—
312C4×C61—81no—
313C4×C61—81no—
314C4×C61—81no—
315C4×C61#52911yescharacteristic
316C4×C61#53811yescharacteristic
317C4×C61#55511yescharacteristic
318C4×C61#55911yescharacteristic
319Dic61—22no—
320Dic61—22no—
321Dic61—42no—
322Dic61—42no—
323Dic61—41no—
324Dic61—41no—
325Dic61—42no—
326#2491—611no—
327#2362—611no—
328#2055—611no—
329#2036—611no—
330#2024—611no—
331#2000—611no—
332#1987—611no—
333#1934—611no—
334#1925—611no—
335#1923—611no—
336#1918—611no—
337#1880—611no—
338#1875—611no—
339#1825D6111yescharacteristic, complemented
340C4×#103—611no—
341C2×C61—81no—
342C2×C61#245911yescharacteristic
343C2×C61#310611yescharacteristic
344C2×C61#312111yescharacteristic
345C2×C61#330011yescharacteristic
346C2×C61#330311yescharacteristic
347C2×C61#330811yescharacteristic
348C2×C61#332411yescharacteristic
349D61—161no—
350C22×SD16—611no—
351C2×C4⋊Q8—611no—
352C2×#124—1221no—
353C2×C22⋊Q8—611no—
354C2×#121—611no—
355C2×C4×Q8—611no—
356C2×#117C2×D6111yescharacteristic
357C2×#107—611no—
358C2×C4⋊C8—611no—
359C2×#103—611no—
360C2×#103—611no—
361C2×#103—611no—
362C2×#102—611no—
363C2×#102—611no—
364C2×#102—611no—
365C2×C22⋊C8—611no—
366C2×C22⋊C8C2×D6111yescharacteristic
367C2×C4×C8—611no—
368#394—1221no—
369#393—1221no—
370#391—611no—
371#390—611no—
372#390—611no—
373#389—611no—
374#388—1221no—
375#388—1221no—
376#388—1221no—
377#387—1221no—
378#380C2×D6111yescharacteristic
379C4×C4⋊C4—611no—
380C4×C4⋊C4—611no—
381C4×C4⋊C4—1221no—
382C4×C22⋊C4—611no—
383#340—611no—
384#340—611no—
385#340—611no—
386#340—611no—
387#340C2×D6111yescharacteristic
388#340C2×D6111yescharacteristic
389#340C2×D6111yescharacteristic
390#340C2×D6111yescharacteristic
391#336—611no—
392#336—611no—
393C61#2673311yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×D4—611no—
397C22×D4—1221no—
398C23×C4C22×D6111yescharacteristic
399C2×SD16—1222no—
400C2×SD16—2441no—
401C2×SD16—2441no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—1222no—
407C4⋊Q8—2441no—
408#124—2444no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411#121—1222no—
412#121—2441no—
413C4×Q8—1222no—
414C4×Q8—2441no—
415#117—22no—
416#117—41no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C4⋊C4C22×D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C22⋊C4—21no—
428C2×C22⋊C4—21no—
429C2×C22⋊C4—611no—
430C2×C22⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C22⋊C4—1221no—
434C2×C22⋊C4—1221no—
435C2×C22⋊C4—1221no—
436C2×C22⋊C4—1221no—
437C2×C42—21no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443#107—1222no—
444C4⋊C8—2441no—
445#103—1222no—
446#103—1222no—
447#103—2441no—
448#102—1222no—
449#102—1222no—
450#102—2441no—
451C22⋊C8—41no—
452C22⋊C8—2441no—
453C4×C8—2441no—
454#95—21no—
455#95—611no—
456#95—611no—
457#95—1221no—
458#95—1221no—
459#95—1221no—
460#95—1221no—
461#95—1221no—
462C24—1221no—
463C24D4×D6111yescharacteristic
464C2×Q8—612no—
465C2×Q8—1221no—
466C2×Q8—1221no—
467C2×Q8—1221no—
468C2×Q8—1221no—
469C2×Q8—1221no—
470C2×Q8—4881no—
471C2×Q8—2442no—
472C2×Q8—2442no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C2×D4—1221no—
477C2×D4—1221no—
478C2×D4—1221no—
479C2×D4—4881no—
480C2×D4—2442no—
481C2×D4—2442no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4D4×D6111yescharacteristic
500C22×C4C23×D6111yescharacteristic
501SD16—2444no—
502C2×C8—21no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—1221no—
509C2×C8—1221no—
510C2×C8—1221no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—612no—
522C4⋊C4—612no—
523C4⋊C4—1224no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2442no—
528C4⋊C4—2442no—
529C4⋊C4#1175822212yes—
530C22⋊C4—42no—
531C22⋊C4—42no—
532C22⋊C4—1222no—
533C22⋊C4—1222no—
534C22⋊C4—1224no—
535C22⋊C4—4881no—
536C22⋊C4—2442no—
537C22⋊C4—2442no—
538C22⋊C4—2442no—
539C22⋊C4—2442no—
540C42—42no—
541C42—1221no—
542C42—1221no—
543C42—1221no—
544C42—1221no—
545C42—2442no—
546C42—2442no—
547C42—2442no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—2441no—
556C23—2441no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23C2×D4×D6111yescharacteristic
563Q8—1222no—
564Q8—1222no—
565Q8—4884no—
566Q8—2441no—
567Q8—2441no—
568D4—4884no—
569D4—2441no—
570D4—2441no—
571D4—2441no—
572D4—2441no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—21no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
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—41no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1222no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4#41×D6111yescharacteristic
614C2×C4#41×D6111yescharacteristic
615C2×C4#41×D6111yescharacteristic
616C2×C4C2×#1175822211yescharacteristic
617C2×C4#1451982211yescharacteristic
618C2×C4#1451982311yescharacteristic
619C8—41no—
620C8—41no—
621C8—2441no—
622C8—2441no—
623C22—41no—
624C22—41no—
625C22—41no—
626C22—41no—
627C22—41no—
628C22—41no—
629C22—41no—
630C22—4881no—
631C22—4881no—
632C22—4881no—
633C22—4881no—
634C22—4881no—
635C22—4881no—
636C22—4881no—
637C22#1684027011yescharacteristic
638C22#1684040211yescharacteristic
639C22#1684040411yescharacteristic
640C22#135×D6111yescharacteristic
641C22AGL(1,ℤ/8)×D6111yescharacteristic
642C22#1684056911yescharacteristic
643C22#1684057811yescharacteristic
644C4—41no—
645C4—42no—
646C4—81no—
647C4—81no—
648C4—81no—
649C4—1222no—
650C4—1222no—
651C4—2442no—
652C4—2442no—
653C4—2441no—
654C4—2441no—
655C4—2442no—
656C4#1684029011yescharacteristic
657C4#1684034811yescharacteristic
658C4#1684047511yescharacteristic
659C4#1684049611yescharacteristic
660C2—81no—
661C2—9761no—
662C2#2144975311yescharacteristic
663C2#2145587411yescharacteristic
664C2#2145605111yescharacteristic
665C2#2145770911yescharacteristic
666C2#2145774311yescharacteristic
667C2#2145781111yescharacteristic
668C2#2145804811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1825C2
socleC23×C61C22×D4
prime radicalC2×#14519751C22
hypercenter#1825D61
hypocenterC61#26733
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
2. #21448846C2semidirect
3. #21447138C2semidirect
4. #21447022C2semidirect
5. #21446959C2semidirect
7. #21446765C2semidirect
8. #21446500C2semidirect
9. #21446437C2semidirect
10. #21446422C2semidirect
11. #21446388C2semidirect
12. #21446175C2semidirect
14. #21445638C2semidirect
15. C61×#1825C2semidirect
339. #1825D61semidirect
393. C61#26733semidirect

Maximal subgroups

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

subgroupindex
#214501782
#214488462
#214471382
#214470222
#214469592
#214468242
#214467652
#214465002
#214464372
#214464222
#214463882
#214461752
#214461412
#214456382
C61×#18252
#2673361