Finite Group Databasebeta

← #29723882

Subgroups — #29723882

Order 15616.

932 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839916C211yesmaximal, characteristic
2#21449316C211yesmaximal, characteristic
3Q8×#11758198C211yesmaximal, characteristic
4#21447364C211yesmaximal, characteristic, complemented
5#21447243C211yesmaximal, characteristic
6#21447140C211yesmaximal, characteristic, complemented
7#21447013C211yesmaximal, characteristic
8#21446495C211yesmaximal, characteristic, complemented
9#21446425C211yesmaximal, characteristic
10#21446162C211yesmaximal, characteristic, complemented
11#21446160C211yesmaximal, characteristic, complemented
12#21446087C211yesmaximal, characteristic, complemented
13#21445832C211yesmaximal, characteristic, complemented
14#21445752C211yesmaximal, characteristic
15C2×C4×Q16×C61C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758202C2211yescharacteristic
18C2×#14519753C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20#16839916—24no—
21C2×#14519708C2211yescharacteristic
22C2×#14519706C2211yescharacteristic
23C2×#14519702C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519700C2211yescharacteristic
29C2×#14519689C2211yescharacteristic
30C2×#14519689C2211yescharacteristic
31#16839700—21no—
32#16839688—21no—
33#16839669C2211yescharacteristic
34#16839668—21no—
35#16839668C2211yescharacteristic
36#16839667—21no—
37#16839667C2211yescharacteristic
38#16839660C2211yescharacteristic
39#16839653—21no—
40#16839646—21no—
41#16839627C2211yescharacteristic
42#16839619—21no—
43#16839619C2211yescharacteristic
44#16839619C2211yescharacteristic
45C4×#11758198—21no—
46C4×#11758198C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53C22×Q16×C61C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C4×Q16×C61—24no—
57C2×#107×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×C4×C8×C61C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C2×Q8×D61—22no—
64C2×Q8×D61—22no—
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—41no—
68C2×Q8×D61—41no—
69C22×C4×D61—21no—
70C22×C4×D61C2311yescharacteristic
71C2×#11758202—22no—
72C2×#11758202—22no—
73C2×#11758202—41no—
74C22×#92418C2311yescharacteristic
75C22×#92418C2311yescharacteristic
76#14519753—22no—
77#14519753—41no—
78Q8×Dic61—22no—
79Q8×Dic61—41no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×#11758198D411yescharacteristic
90C2×#11758198D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61C2311yescharacteristic
93#14519708—41no—
94#14519706—41no—
95#14519702—22no—
96#14519702—22no—
97#14519701—22no—
98#14519701—22no—
99#14519701—41no—
100#14519700—41no—
101#14519689—22no—
102#14519689—41no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114#14519684D411yescharacteristic
115#14519684D411yescharacteristic
116C22×Q8×C61C2311yescharacteristic
117C22×Q8×C61C2311yescharacteristic
118C2×Q16×C61—22no—
119C2×Q16×C61—22no—
120C2×Q16×C61D414yes—
121C22×C8×C61C2311yescharacteristic
122C22×C8×C61C2311yescharacteristic
123C4×Q8×C61—22no—
124C4×Q8×C61—22no—
125C4×Q8×C61—41no—
126C4×Q8×C61—41no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61—21no—
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—22no—
135#103×C61—22no—
136#103×C61—22no—
137C4×C8×C61—22no—
138C23×D61C2×D411yescharacteristic
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61—21no—
142C2×Q8×C61—21no—
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61C2×D412yes—
148C2×Q8×C61C2×D412yes—
149C2×Q8×C61#4112yes—
150C2×Q8×C61#4112yes—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61C2×D411yescharacteristic
157C22×C4×C61C2411yescharacteristic
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165Q8×D61—44no—
166Q8×D61—44no—
167Q8×D61—44no—
168Q8×D61—44no—
169C2×C4×D61—24no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—21no—
173C2×C4×D61—21no—
174C2×C4×D61—21no—
175C2×C4×D61—21no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—24no—
183C2×Dic122—21no—
184C2×Dic122—42no—
185C2×Dic122—42no—
186C2×Dic122#4112yes—
187Q16×C61—28no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D412yes—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—42no—
199C4⋊C4×C61—42no—
200C42×C61—21no—
201C42×C61—21no—
202C42×C61—42no—
203C42×C61—42no—
204#11758202—44no—
205#11758198—22no—
206#11758198—22no—
207#11758198—42no—
208#11758198—42no—
209#11758198—42no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—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×#92418—21no—
228C23×C61C22×D411yescharacteristic
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61#13512yes—
236Q8×C61—22no—
237Q8×C61—22no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243Q8×C61#12314yes—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61#13712yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61H(5,𝔽2)11yescharacteristic
267C2×C4×C61H(5,𝔽2)11yescharacteristic
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C4×D61—48no—
290C4×D61—41no—
291C4×D61—41no—
292C4×D61—41no—
293C4×D61—41no—
294Dic122—22no—
295Dic122—22no—
296Dic122—48no—
297Dic122—41no—
298Dic122—41no—
299C8×C61—22no—
300C8×C61—41no—
301#92418—41no—
302#92418—41no—
303#24902—611nomaximal, Sylow
304C22×C61#52511yescharacteristic
305C22×C61D4211yescharacteristic
306C22×C61#54611yescharacteristic
307C22×C61C2×#13511yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C22×C61#57811yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C4×C61—24no—
319C4×C61—22no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12311yescharacteristic
325C4×C61#53811yescharacteristic
326C4×C61#54611yescharacteristic
327C4×C61#55111yescharacteristic
328Dic61—41no—
329Dic61—42no—
330Dic61—41no—
331Dic61—41no—
332Dic61—81no—
333Dic61—81no—
334Dic61—81no—
335C2×#451—611no—
336C2×C4×Q16D6111yescharacteristic, complemented
337#2421—611no—
338Q8×C22⋊C4—611no—
339#2097—611no—
340#2075—611no—
341#2056—611no—
342#2035—611no—
343#1933—611no—
344#1923—611no—
345#1878—611no—
346#1878—611no—
347#1865—611no—
348#1805—611no—
349#1787—611no—
350C2×C61#241811yescharacteristic
351C2×C61#310811yescharacteristic
352C2×C61#312711yescharacteristic
353C2×C61#327011yescharacteristic
354C2×C61#328511yescharacteristic
355C2×C61#330011yescharacteristic
356C2×C61#332811yescharacteristic
357D61—42no—
358C23×Q8—611no—
359C22×Q16—611no—
360C22×Q16C2×D6111yescharacteristic
361C2×C22⋊Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8C2×D6111yescharacteristic
364C2×C4×Q8C2×D6111yescharacteristic
365#451—1224no—
366C4×Q16—24no—
367C2×#107C2×D6111yescharacteristic
368C2×#106—611no—
369C2×C4⋊C8—611no—
370C2×#103—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103C2×D6111yescharacteristic
377C2×#103C2×D6111yescharacteristic
378C2×C22⋊C8—611no—
379C2×C22⋊C8—611no—
380C2×C4×C8C2×D6111yescharacteristic
381#400—1221no—
382#397—1221no—
383#391—611no—
384#391—611no—
385#391—611no—
386#391—1221no—
387#391—1221no—
388#389—611no—
389#388—1221no—
390#387—1221no—
391#382—611no—
392#380—611no—
393#380—611no—
394#380—1221no—
395C4×C22⋊C4—611no—
396C4×C22⋊C4—1221no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#336—611no—
402#336—611no—
403C61#2490211yescharacteristic, complemented, Sylow
404C22×Q8—611no—
405C22×Q8—1221no—
406C22×Q8—1222no—
407C22×Q8—1222no—
408C22×Q8—1222no—
409C22×Q8—1222no—
410C22×Q8—2441no—
411C22×Q8—2441no—
412C22×Q8C22×D6111yescharacteristic
413C22×Q8C22×D6111yescharacteristic
414C23×C4—611no—
415C23×C4—1221no—
416C2×Q16—22no—
417C2×Q16—22no—
418C2×Q16—1222no—
419C2×Q16—1222no—
420C2×Q16—2441no—
421C2×Q16C22⋊D6114yes—
422C22×C8—611no—
423C22×C8—611no—
424C22×C8C22×D6111yescharacteristic
425C22×C8C22×D6111yescharacteristic
426C22⋊Q8—1222no—
427C22⋊Q8—2441no—
428C4×Q8—22no—
429C4×Q8—22no—
430C4×Q8—41no—
431C4×Q8—41no—
432C4×Q8—1222no—
433C4×Q8—2441no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4C22×D6111yescharacteristic
443C2×C4⋊C4C22×D6111yescharacteristic
444C2×C22⋊C4—611no—
445C2×C22⋊C4—611no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C42—21no—
450C2×C42—21no—
451C2×C42—611no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454#107—22no—
455#106—2441no—
456C4⋊C8—2441no—
457#103—22no—
458#103—22no—
459#103—1222no—
460#103—1222no—
461#103—1222no—
462#103—1222no—
463#103—2441no—
464#103—2441no—
465C22⋊C8—1222no—
466C22⋊C8—2441no—
467C4×C8—22no—
468#95—611no—
469#95—611no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473#95—1221no—
474#95—1221no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481C24—611no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—612no—
491C2×Q8—1224no—
492C2×Q8—1221no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—2442no—
498C2×Q8—2442no—
499C2×Q8—2444no—
500C2×Q8—2444no—
501C2×Q8—2444no—
502C2×Q8—2444no—
503C2×Q8#1175822212yes—
504C2×Q8#1175822212yes—
505C2×Q8C2×C22⋊D6112yes—
506C2×Q8C2×C22⋊D6112yes—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—1224no—
515C22×C4—1221no—
516C22×C4—1221no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1221no—
524C22×C4—1221no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4D4×D6111yescharacteristic
528C22×C4D4×D6111yescharacteristic
529C22×C4C23×D6111yescharacteristic
530Q16—28no—
531Q16—2444no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8C2×C22⋊D6112yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—42no—
550C4⋊C4—1221no—
551C4⋊C4—1221no—
552C4⋊C4—1221no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—2442no—
557C4⋊C4—2442no—
558C4⋊C4—2442no—
559C22⋊C4—1222no—
560C22⋊C4—1222no—
561C22⋊C4—2442no—
562C22⋊C4—2442no—
563C22⋊C4—2442no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—42no—
568C42—1221no—
569C42—1221no—
570C42—2442no—
571C23—612no—
572C23—1221no—
573C23—1221no—
574C23—1221no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23C2×D4×D6111yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—41no—
585Q8—41no—
586Q8—1222no—
587Q8—1222no—
588Q8—2448no—
589Q8—2441no—
590Q8—2441no—
591Q8#1451976014yes—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—1221no—
610C2×C4—1222no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—2448no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
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×C4—2441no—
634C2×C4—2441no—
635C2×C4#1451980912yes—
636C2×C4C22×C22⋊D6111yescharacteristic
637C2×C4#41×D6111yescharacteristic
638C2×C4C2×#1175822211yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982211yescharacteristic
641C2×C4#1451982311yescharacteristic
642C8—22no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22#1684027111yescharacteristic
654C22D4×C22⋊D6111yescharacteristic
655C22#1684040811yescharacteristic
656C22#135×D6111yescharacteristic
657C22C2×#1451980911yescharacteristic
658C22#1684057511yescharacteristic
659C22#1684057611yescharacteristic
660C4—24no—
661C4—22no—
662C4—42no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—4881no—
667C4—4881no—
668C4—4881no—
669C4—2441no—
670C4—2441no—
671C4—2442no—
672C4—2441no—
673C4C2×#1451976011yescharacteristic
674C4#1684034511yescharacteristic
675C4#1684040011yescharacteristic
676C4#1684044111yescharacteristic
677C2—2442no—
678C2#2144927511yescharacteristic
679C2#2145590311yescharacteristic
680C2#2145611611yescharacteristic
681C2#2145738911yescharacteristic
682C2#2145755411yescharacteristic
683C2#2145771111yescharacteristic
684C2#2145810411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C4×Q16×C61C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×C4×Q16D61
hypocenterC61#24902
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
4. #21447364C2semidirect
6. #21447140C2semidirect
8. #21446495C2semidirect
10. #21446162C2semidirect
11. #21446160C2semidirect
12. #21446087C2semidirect
13. #21445832C2semidirect
15. C2×C4×Q16×C61C2semidirect
336. C2×C4×Q16D61semidirect
403. C61#24902semidirect

Maximal subgroups

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

subgroupindex
C2×#168399162
#214493162
Q8×#117581982
#214473642
#214472432
#214471402
#214470132
#214464952
#214464252
#214461622
#214461602
#214460872
#214458322
#214457522
C2×C4×Q16×C612
#2490261