Finite Group Databasebeta

← #29737151

Subgroups — #29737151

Order 15616.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840045C211yesmaximal, characteristic, complemented
2C2×#103×D61C211yesmaximal, characteristic
3#21449524C211yesmaximal, characteristic
4Q8×#11758198C211yesmaximal, characteristic
5#21447363C211yesmaximal, characteristic, complemented
6#21447201C211yesmaximal, characteristic, complemented
7#21447199C211yesmaximal, characteristic, complemented
8#21446959C211yesmaximal, characteristic, complemented
9#21446087C211yesmaximal, characteristic, complemented
10#21446085C211yesmaximal, characteristic, complemented
11#21446013C211yesmaximal, characteristic
12#21445995C211yesmaximal, characteristic
13#21445843C211yesmaximal, characteristic
14#21445752C211yesmaximal, characteristic
15C61×#1803C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×C8×D61C2211yescharacteristic
18C2×#14519776C2211yescharacteristic
19C2×#14519755—21no—
20C2×Q8×Dic61C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22#16840045—24no—
23C2×#14519709C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34#103×D61—24no—
35C2×#14519689C2211yescharacteristic
36#16839693—21no—
37#16839682—21no—
38#16839668—21no—
39#16839668C2211yescharacteristic
40#16839667—21no—
41#16839667C2211yescharacteristic
42#16839660C2211yescharacteristic
43#16839658C2211yescharacteristic
44#16839633—21no—
45#16839619—21no—
46#16839619C2211yescharacteristic
47#16839619C2211yescharacteristic
48C4×#11758198—21no—
49C4×#11758198C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#384C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122—21no—
63C22×Dic122C2311yescharacteristic
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—41no—
68C2×Q8×D61—41no—
69C2×Q8×D61Q814yescomplemented
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75C2×C8×D61—22no—
76C2×C8×D61—22no—
77C2×C8×D61—41no—
78#14519776—22no—
79#14519776—41no—
80#14519755—44no—
81Q8×Dic61—22no—
82Q8×Dic61—41no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C4⋊C4×D61—22no—
92C4⋊C4×D61—41no—
93C2×#11758198—21no—
94C2×#11758198—21no—
95C2×#11758198D411yescharacteristic
96C2×#11758198D411yescharacteristic
97C2×C4×Dic61—21no—
98C2×C4×Dic61C2311yescharacteristic
99#14519709—22no—
100#14519708—22no—
101#14519708—41no—
102#14519708—41no—
103#14519706—22no—
104#14519702—22no—
105#14519702—41no—
106#14519701—22no—
107#14519701D414yes—
108#14519700—22no—
109#14519700—41no—
110#14519689—22no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×Q8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×Q8×C61—22no—
125C4×Q8×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—41no—
135#103×C61—22no—
136#103×C61—22no—
137C23×D61C2×D411yescharacteristic
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61C2×Q812yes—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2411yescharacteristic
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158Q8×D61—28no—
159Q8×D61—44no—
160Q8×D61—44no—
161Q8×D61—44no—
162C2×C4×D61—21no—
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—24no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172C2×C4×D61C2×Q812yes—
173C2×C4×D61#4112yes—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—24no—
180C2×Dic122—42no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122—42no—
184C2×Dic122—81no—
185C2×Dic122C2×Q812yes—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—42no—
199C4⋊C4×C61—42no—
200C4⋊C4×C61—81no—
201C42×C61—21no—
202C42×C61—21no—
203C42×C61—42no—
204#11758198—22no—
205#11758198—42no—
206#11758198—42no—
207#11758198SD1614yes—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—42no—
214C4⋊Dic61C2×D412yes—
215#11758196—42no—
216#11758196—42no—
217#11758196—42no—
218#11758196—42no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—42no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418C2×D412yes—
228C8×D61—44no—
229C23×C61C22×D411yescharacteristic
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61C22⋊Q812yes—
235C22×D61C2×SD1612yes—
236C22×D61#13712yes—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61C22⋊Q814yescomplemented
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
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×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61C2×SD1612yes—
259C2×C4×C61C22×D411yescharacteristic
260C2×C4×C61C22×Q811yescharacteristic
261C2×C4×C61C2×#4111yescharacteristic
262C2×C4×C61C2×#4111yescharacteristic
263C2×C4×C61H(5,𝔽2)11yescharacteristic
264C2×C4×C61H(5,𝔽2)11yescharacteristic
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—22no—
272C2×Dic61—21no—
273C2×Dic61—21no—
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—41no—
283C2×Dic61—41no—
284C2×Dic61C22⋊Q812yes—
285C2×Dic61C2×SD1612yes—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—48no—
291C4×D61—84no—
292Dic122—22no—
293Dic122—22no—
294Dic122—48no—
295Dic122—41no—
296Dic122—41no—
297Dic122—84no—
298C8×C61—41no—
299C8×C61—41no—
300#92418—22no—
301#92418—41no—
302#25955—611nomaximal, Sylow
303C22×C61C2×C22⋊Q811yescharacteristic
304C22×C61D4211yescharacteristic
305C22×C61#54611yescharacteristic
306C22×C61C22×SD1611yescharacteristic
307C22×C61C2×#13711yescharacteristic
308C22×C61#57711yescharacteristic
309C22×C61#57811yescharacteristic
310C2×D61—22no—
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61#47514yescomplemented
317C4×C61—24no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—81no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61C2×C22⋊Q811yescharacteristic
324C4×C61#54311yescharacteristic
325C4×C61#54611yescharacteristic
326C4×C61#55111yescharacteristic
327Dic61—22no—
328Dic61—22no—
329Dic61—42no—
330Dic61—41no—
331Dic61—42no—
332Dic61—81no—
333Dic61—81no—
334C2×#475—611no—
335C22×#103—611no—
336#2440—611no—
337Q8×C22⋊C4—611no—
338#2096—611no—
339#2066—611no—
340#2066—611no—
341#2024—611no—
342#1865—611no—
343#1865—611no—
344#1853—611no—
345#1850—611no—
346#1807—611no—
347#1803D6111yescharacteristic, complemented
348#1787—611no—
349C2×C61#243711yescharacteristic
350C2×C61C2×#47511yescharacteristic
351C2×C61#308211yescharacteristic
352C2×C61#327011yescharacteristic
353C2×C61D4×SD1611yescharacteristic
354C2×C61#329811yescharacteristic
355C2×C61#331611yescharacteristic
356D61—24no—
357C23×Q8—611no—
358C23×C8—611no—
359C2×C4⋊Q8—611no—
360C2×C22⋊Q8—1221no—
361C2×C4×Q8—611no—
362C2×C4×Q8C2×D6111yescharacteristic
363C22×C4⋊C4—611no—
364#475—1224no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×C4⋊C8C2×D6111yescharacteristic
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103—1224no—
378C2×#103C2×D6111yescharacteristic
379C2×#103C2×D6111yescharacteristic
380C2×C22⋊C8—611no—
381#398—1221no—
382#395—1221no—
383#391—611no—
384#391—611no—
385#391—1221no—
386#391—1221no—
387#389—611no—
388#389—611no—
389#385—1221no—
390#384C2×D6111yescharacteristic
391#380—611no—
392#380—611no—
393#380—1221no—
394C4×C22⋊C4—611no—
395C4×C22⋊C4—1221no—
396#340—611no—
397#340—611no—
398#340C2×D6111yescharacteristic
399#340C2×D6111yescharacteristic
400#336—611no—
401#336—611no—
402C61#2595511yescharacteristic, complemented, Sylow
403C22×Q8—614no—
404C22×Q8—611no—
405C22×Q8—1221no—
406C22×Q8—1221no—
407C22×Q8—1222no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—2441no—
411C22×Q8—2441no—
412C22×Q8C22×D6111yescharacteristic
413C23×C4—611no—
414C23×C4—1221no—
415C23×C4—1221no—
416C22×C8—611no—
417C22×C8—611no—
418C22×C8—1222no—
419C22×C8—1222no—
420C22×C8—2441no—
421C22×C8C22×D6111yescharacteristic
422C22×C8C22×D6111yescharacteristic
423C4⋊Q8—1222no—
424C4⋊Q8—2441no—
425C22⋊Q8—2444no—
426C4×Q8—22no—
427C4×Q8—41no—
428C4×Q8—1222no—
429C4×Q8—2441no—
430C2×C4⋊C4—21no—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—21no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—611no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—1222no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—2441no—
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C4⋊C4C22×D6111yescharacteristic
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C22⋊C4—611no—
447C2×C22⋊C4—611no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4—1221no—
450C2×C42—21no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#106—1222no—
455#106—1222no—
456#106—2441no—
457#106—2441no—
458C4⋊C8—41no—
459C4⋊C8—1222no—
460#103—22no—
461#103—22no—
462#103—614no—
463#103—1222no—
464#103—1222no—
465#103—1222no—
466#103—2441no—
467#103—2441no—
468C22⋊C8—1222no—
469#95—611no—
470#95—611no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479C24—611no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—612no—
484C2×Q8—1224no—
485C2×Q8—1221no—
486C2×Q8—1228no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8—1221no—
491C2×Q8—4881no—
492C2×Q8—2442no—
493C2×Q8—2442no—
494C2×Q8—2442no—
495C2×Q8—2442no—
496C2×Q8—2444no—
497C2×Q8—2444no—
498C2×Q8—2444no—
499C2×Q8#1175822212yes—
500C2×Q8D4×D6112yes—
501C2×Q8Q8×D6112yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—612no—
509C22×C4—612no—
510C22×C4—611no—
511C22×C4—1224no—
512C22×C4—1221no—
513C22×C4—1221no—
514C22×C4—1221no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—4881no—
521C22×C4—2442no—
522C22×C4—2442no—
523C22×C4—2442no—
524C22×C4—2442no—
525C22×C4D4×D6111yescharacteristic
526C22×C4D4×D6111yescharacteristic
527C22×C4C23×D6111yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—612no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—2444no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—42no—
548C4⋊C4—42no—
549C4⋊C4—81no—
550C4⋊C4—612no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C22⋊C4—614no—
562C22⋊C4—1222no—
563C22⋊C4—2442no—
564C22⋊C4—2442no—
565C42—21no—
566C42—21no—
567C42—42no—
568C42—1221no—
569C42—1221no—
570C42—2442no—
571C23—612no—
572C23—612no—
573C23—612no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—1222no—
583Q8—1222no—
584Q8—4884no—
585Q8—2448no—
586Q8—2441no—
587Q8—2441no—
588Q8#1451975514yescomplemented
589C2×C4—22no—
590C2×C4—21no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—612no—
607C2×C4—612no—
608C2×C4—1221no—
609C2×C4—1221no—
610C2×C4—1221no—
611C2×C4—1221no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1222no—
621C2×C4—4884no—
622C2×C4—2448no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4—2441no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4SD16×D6112yes—
634C2×C4C2×D4×D6111yescharacteristic
635C2×C4C2×Q8×D6111yescharacteristic
636C2×C4#41×D6111yescharacteristic
637C2×C4C2×#1175822211yescharacteristic
638C2×C4#1451982211yescharacteristic
639C2×C4#1451982311yescharacteristic
640C8—41no—
641C8—41no—
642C8—1222no—
643C8—2441no—
644C22—614no—
645C22—1222no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—1222no—
651C22C22⋊Q8×D6111yescharacteristic
652C22#1684039611yescharacteristic
653C22#1684040211yescharacteristic
654C22C2×SD16×D6111yescharacteristic
655C22#137×D6111yescharacteristic
656C22#1684057211yescharacteristic
657C22#1684057511yescharacteristic
658C4—24no—
659C4—22no—
660C4—41no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—1222no—
665C4—1222no—
666C4—4881no—
667C4—4881no—
668C4—2442no—
669C4—2442no—
670C4—2441no—
671C4C2×#1451975511yescharacteristic
672C4#1684038711yescharacteristic
673C4#1684040111yescharacteristic
674C4#1684043611yescharacteristic
675C2—1224no—
676C2#2144948711yescharacteristic
677C2#475×D6111yescharacteristic
678C2#2145565611yescharacteristic
679C2#2145739511yescharacteristic
680C2#2145751011yescharacteristic
681C2#2145768011yescharacteristic
682C2#2145793111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1803C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1803D61
hypocenterC61#25955
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
1. C2×#16840045C2semidirect
5. #21447363C2semidirect
6. #21447201C2semidirect
7. #21447199C2semidirect
8. #21446959C2semidirect
9. #21446087C2semidirect
10. #21446085C2semidirect
15. C61×#1803C2semidirect
69. C2×Q8×D61Q8semidirect
240. Q8×C61C22⋊Q8semidirect
316. C2×D61#475semidirect
347. #1803D61semidirect
402. C61#25955semidirect
588. Q8#14519755semidirect

Maximal subgroups

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

subgroupindex
C2×#168400452
C2×#103×D612
#214495242
Q8×#117581982
#214473632
#214472012
#214471992
#214469592
#214460872
#214460852
#214460132
#214459952
#214458432
#214457522
C61×#18032
#2595561