Finite Group Databasebeta

← #29730369

Subgroups — #29730369

Order 15616.

840 conjugacy classes of subgroups.

The subgroup lattice has 680 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840046C211yesmaximal, characteristic, complemented
2C2×#16839831C211yesmaximal, characteristic
3#21450989C211yesmaximal, characteristic
4#21450730C211yesmaximal, characteristic
5#21447256C211yesmaximal, characteristic, complemented
6#21447253C211yesmaximal, characteristic, complemented
7#21447024C211yesmaximal, characteristic, complemented
8#21447015C211yesmaximal, characteristic, complemented
9#21446790C211yesmaximal, characteristic, complemented
10#21446584C211yesmaximal, characteristic, complemented
11#21446252C211yesmaximal, characteristic, complemented
12#21446241C211yesmaximal, characteristic, complemented
13#21445843C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1809C211yesmaximal, characteristic, complemented
16C22×C8×D61C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519757C2211yescharacteristic
19C2×#14519756C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×C4×C22⋊D61C2211yescharacteristic
22#16840046—24no—
23#16839831—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519706C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×#14519689C2211yescharacteristic
33C2×#14519687C2211yescharacteristic
34#16839696—21no—
35#16839677—21no—
36#16839675—21no—
37#16839674—21no—
38#16839662—21no—
39#16839655—21no—
40#16839651—21no—
41#16839627C2211yescharacteristic
42#16839626—21no—
43#16839622C2211yescharacteristic
44#16839619C2211yescharacteristic
45#16839619C2211yescharacteristic
46#16839619C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54C22×M16×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C61×#380C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C22×C22⋊D61D411yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C2×C8×D61—22no—
68C2×C8×D61—22no—
69C2×C8×D61—41no—
70#14519776—22no—
71#14519776—22no—
72#14519757—22no—
73#14519757—41no—
74#14519756—22no—
75#14519756—41no—
76#14519742—22no—
77#14519742—22no—
78C4×C22⋊D61—22no—
79C4×C22⋊D61—22no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196D411yescharacteristic
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61D411yescharacteristic
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×C4×Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519708—22no—
100#14519708—41no—
101#14519706D414yes—
102#14519702—22no—
103#14519702—22no—
104#14519700—22no—
105#14519700—22no—
106C22⋊#92418—22no—
107#14519689—22no—
108#14519687—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116C23×C4×C61C2311yescharacteristic
117C2×M16×C61—22no—
118C2×M16×C61—22no—
119C2×M16×C61—41no—
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128#107×C61—41no—
129#106×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61C2×D411yescharacteristic
133C23×D61C2×D411yescharacteristic
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2411yescharacteristic
146C2×C22⋊D61—22no—
147C2×C22⋊D61—22no—
148C2×C22⋊D61—41no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C22×Dic61C2×D411yescharacteristic
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—42no—
168C2×Dic122—81no—
169C2×Dic122—81no—
170C2×Dic122#4112yes—
171M16×C61—44no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61C2×D412yes—
177C2×C8×C61#4112yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C22⋊C4×C61—42no—
187C22⋊C4×C61—81no—
188C22⋊Dic61—22no—
189C22⋊Dic61—42no—
190C22⋊Dic61—81no—
191#11758198—22no—
192#11758198—42no—
193#11758198—42no—
194#11758198—81no—
195#11758198—81no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—42no—
203#11758196—22no—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61C2×D412yes—
210C4×Dic61#4112yes—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×D412yes—
217C8×D61—44no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61C22×D411yescharacteristic
226C22×D61—21no—
227C22×D61—21no—
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61#13712yes—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
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×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61H(5,𝔽2)11yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C22⋊D61—44no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
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—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61#12112yes—
279C2×Dic61#12112yes—
280C4×D61—41no—
281C4×D61—41no—
282C4×D61—41no—
283C4×D61—41no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122—84no—
289C8×C61—22no—
290C8×C61—22no—
291#92418—22no—
292#92418—41no—
293#25356—611nomaximal, Sylow
294C22×C61—41no—
295C22×C61—41no—
296C22×C61—41no—
297C22×C61—41no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61C2×#12111yescharacteristic
302C22×C61#54611yescharacteristic
303C22×C61#54711yescharacteristic
304C22×C61C2×#13711yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—41no—
312C2×D61—41no—
313C2×D61—41no—
314C2×D61—41no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61—81no—
319C4×C61—81no—
320C4×C61#52911yescharacteristic
321C4×C61#54211yescharacteristic
322C4×C61#54611yescharacteristic
323C4×C61#54811yescharacteristic
324Dic61—22no—
325Dic61—22no—
326Dic61—42no—
327Dic61—41no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331C2×#475—611no—
332C2×#435—611no—
333#2563—611no—
334#2536—611no—
335#2077—611no—
336#2077—611no—
337#2036—611no—
338#2035—611no—
339#1993—611no—
340#1949—611no—
341#1894—611no—
342#1892—611no—
343#1809D6111yescharacteristic, complemented
344#1807—611no—
345#1763—611no—
346C2×C61—81no—
347C2×C61#242711yescharacteristic
348C2×C61#305611yescharacteristic
349C2×C61#305711yescharacteristic
350C2×C61#331611yescharacteristic
351C2×C61#332111yescharacteristic
352C2×C61#334411yescharacteristic
353C2×C61#334811yescharacteristic
354D61—42no—
355C22×M16C2×D6111yescharacteristic
356C23×C8—611no—
357C2×C4⋊Q8—611no—
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C4×Q8—611no—
361C2×C4×D4—611no—
362#475—1224no—
363#435—1224no—
364C2×#107C2×D6111yescharacteristic
365C2×#106—611no—
366C2×#106—611no—
367C2×#106C2×D6111yescharacteristic
368C2×C4⋊C8—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×C22⋊C8—611no—
374C2×C22⋊C8—611no—
375C2×#97—611no—
376#398—1221no—
377#394—1221no—
378#393—1221no—
379#393—1221no—
380#390—1221no—
381#388—1221no—
382#388—1221no—
383#382—611no—
384#382—1221no—
385#380—611no—
386#380—611no—
387#380—611no—
388#380—611no—
389#380C2×D6111yescharacteristic
390#380C2×D6111yescharacteristic
391C4×C4⋊C4—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#340C2×D6111yescharacteristic
400C61#2535611yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×Q8—1221no—
403C22×D4—611no—
404C23×C4—611no—
405C23×C4C22×D6111yescharacteristic
406C2×M16—22no—
407C2×M16—22no—
408C2×M16—41no—
409C22×C8—611no—
410C22×C8—611no—
411C22×C8—1222no—
412C22×C8—1222no—
413C22×C8—2441no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—1222no—
417C4⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C22⋊Q8—2441no—
421C22⋊Q8—2441no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424C4×D4—1222no—
425C4×D4—1222no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—611no—
441C2×C22⋊C4—611no—
442C2×C22⋊C4—1221no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C42—611no—
449C2×C42—1221no—
450C2×C42—1221no—
451#107—41no—
452#106—22no—
453#106—1222no—
454#106—2441no—
455C4⋊C8—614no—
456#103—1222no—
457#103—1222no—
458#103—1222no—
459#103—1222no—
460C22⋊C8—1222no—
461C22⋊C8—1222no—
462#97—1222no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470#95—1221no—
471#95—1221no—
472C24—611no—
473C24D4×D6111yescharacteristic
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—4881no—
481C2×Q8—4881no—
482C2×Q8—2442no—
483C2×D4—1222no—
484C2×D4—1222no—
485C2×D4—2441no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
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—1221no—
508C22×C4—1221no—
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511M16—44no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—2444no—
523C2×C8#1175822212yes—
524C2×C8D4×D6112yes—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—21no—
532C4⋊C4—21no—
533C4⋊C4—1222no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—2442no—
541C4⋊C4—2442no—
542C4⋊C4—2442no—
543C4⋊C4—2442no—
544C22⋊C4—42no—
545C22⋊C4—81no—
546C22⋊C4—1222no—
547C22⋊C4—1222no—
548C22⋊C4—4881no—
549C22⋊C4—4881no—
550C22⋊C4—4881no—
551C22⋊C4—2442no—
552C22⋊C4—2442no—
553C22⋊C4—2442no—
554C42—612no—
555C42—612no—
556C42—2442no—
557C42—2442no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—612no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23—1221no—
571C23—1221no—
572C23C2×D4×D6111yescharacteristic
573Q8—1222no—
574Q8—1222no—
575Q8—4884no—
576Q8—2441no—
577Q8—2441no—
578D4—2444no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
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—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—612no—
599C2×C4—612no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1222no—
606C2×C4—1221no—
607C2×C4—1221no—
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×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4#41×D6111yescharacteristic
626C2×C4C2×#1175822211yescharacteristic
627C2×C4#1451982211yescharacteristic
628C2×C4#1451982511yescharacteristic
629C8—22no—
630C8—22no—
631C8—1222no—
632C8—2441no—
633C22—41no—
634C22—41no—
635C22—41no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22—41no—
640C22—1222no—
641C22—1222no—
642C22—1222no—
643C22—2441no—
644C22—2441no—
645C22—2441no—
646C22—2441no—
647C22#121×D6111yescharacteristic
648C22#1684040111yescharacteristic
649C22#1684041311yescharacteristic
650C22#137×D6111yescharacteristic
651C22#1684055811yescharacteristic
652C22#1684057111yescharacteristic
653C22#1684057211yescharacteristic
654C4—41no—
655C4—81no—
656C4—81no—
657C4—81no—
658C4—81no—
659C4—1222no—
660C4—1222no—
661C4—4881no—
662C4—4881no—
663C4—4881no—
664C4—2441no—
665C4—2442no—
666C4#1684029011yescharacteristic
667C4#1684038211yescharacteristic
668C4#1684040211yescharacteristic
669C4#1684042211yescharacteristic
670C2—81no—
671C2—2442no—
672C2#2144939511yescharacteristic
673C2#2145544411yescharacteristic
674C2#2145545611yescharacteristic
675C2#2145792511yescharacteristic
676C2#2145800011yescharacteristic
677C2#2145827511yescharacteristic
678C2#2145833511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1809C2
socleC23×C61C22×D4
prime radicalC22×C22⋊D61D4
hypercenter#1809D61
hypocenterC61#25356
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1809

Complements

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

NH ≅ G/Ntype
1. C2×#16840046C2semidirect
5. #21447256C2semidirect
6. #21447253C2semidirect
7. #21447024C2semidirect
8. #21447015C2semidirect
9. #21446790C2semidirect
10. #21446584C2semidirect
11. #21446252C2semidirect
12. #21446241C2semidirect
13. #21445843C2semidirect
14. #21445648C2semidirect
15. C61×#1809C2semidirect
343. #1809D61semidirect
400. C61#25356semidirect

Maximal subgroups

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

subgroupindex
C2×#168400462
C2×#168398312
#214509892
#214507302
#214472562
#214472532
#214470242
#214470152
#214467902
#214465842
#214462522
#214462412
#214458432
#214456482
C61×#18092
#2535661