Finite Group Databasebeta

← #29714509

Subgroups — #29714509

Order 15616.

1022 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840373C211yesmaximal, characteristic
2C2×#16840069C211yesmaximal, characteristic, complemented
3#21448495C211yesmaximal, characteristic
4#21447020C211yesmaximal, characteristic
5#21447013C211yesmaximal, characteristic
6#21446826C211yesmaximal, characteristic
7#21446826C211yesmaximal, characteristic
8#21446774C211yesmaximal, characteristic, complemented
9#21446743C211yesmaximal, characteristic, complemented
10#21446162C211yesmaximal, characteristic, complemented
11#21446162C211yesmaximal, characteristic, complemented
12#21446085C211yesmaximal, characteristic, complemented
13#21446071C211yesmaximal, characteristic, complemented
14#21445800C211yesmaximal, characteristic
15C2×C61×#462C211yesmaximal, characteristic, complemented
16#16840373—24no—
17#16840373—24no—
18C2×#14519772—21no—
19C2×#14519772—21no—
20C2×#14519769C2211yescharacteristic
21C2×#14519767—21no—
22C2×#14519767—21no—
23C2×#14519753C2211yescharacteristic
24C2×#14519753C2211yescharacteristic
25C2×#14519753C2211yescharacteristic
26C2×Q8×Dic61C2211yescharacteristic
27C2×Q8×Dic61C2211yescharacteristic
28C2×#14519734C2211yescharacteristic
29#16840069—24no—
30C2×#14519710C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×#14519701C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519701C2211yescharacteristic
39C2×#14519689C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41#16839669C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839657—21no—
44#16839653—21no—
45#16839652—21no—
46#16839646—21no—
47#16839619C2211yescharacteristic
48C4×C4⋊Dic61C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C22×Q16×C61C2211yescharacteristic
56C22×Q16×C61C2211yescharacteristic
57C2×C4⋊Q8×C61C2211yescharacteristic
58C2×C4×Q8×C61C2211yescharacteristic
59C61×#462—24no—
60C2×C4⋊C8×C61C2211yescharacteristic
61C2×#103×C61C2211yescharacteristic
62C2×#103×C61C2211yescharacteristic
63C22×C4×D61C2311yescharacteristic
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66#14519772—28no—
67#14519772—28no—
68#14519769—22no—
69#14519769D414yes—
70#14519767—28no—
71#14519767—28no—
72#14519753—22no—
73#14519753—22no—
74#14519753—22no—
75#14519753—22no—
76#14519753—22no—
77#14519753—22no—
78Q8×Dic61—22no—
79Q8×Dic61—22no—
80Q8×Dic61—22no—
81Q8×Dic61—22no—
82#14519734—22no—
83#14519734—22no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196D411yescharacteristic
97C2×#11758196D411yescharacteristic
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198D411yescharacteristic
103C2×#11758198D411yescharacteristic
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107C2×C4×Dic61C2311yescharacteristic
108#14519710—22no—
109#14519708—22no—
110C4⋊#92418—22no—
111#14519701—22no—
112#14519701—22no—
113#14519701—22no—
114#14519701—22no—
115#14519701—41no—
116#14519701—41no—
117#14519689—41no—
118#14519689—41no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×Q8×C61C2311yescharacteristic
125C22×Q8×C61C2311yescharacteristic
126C22×Q8×C61C2311yescharacteristic
127C2×Q16×C61—22no—
128C2×Q16×C61—22no—
129C2×Q16×C61—22no—
130C2×Q16×C61—22no—
131C2×Q16×C61—41no—
132C2×Q16×C61—41no—
133C22×C8×C61C2311yescharacteristic
134C22×C8×C61C2311yescharacteristic
135C4⋊Q8×C61—22no—
136C4⋊Q8×C61D414yes—
137C4×Q8×C61—22no—
138C4×Q8×C61—22no—
139C2×C4⋊C4×C61—21no—
140C2×C4⋊C4×C61—21no—
141C2×C4⋊C4×C61—21no—
142C2×C4⋊C4×C61C2311yescharacteristic
143C2×C42×C61—21no—
144C2×C42×C61C2311yescharacteristic
145C4⋊C8×C61—22no—
146#103×C61—22no—
147#103×C61—22no—
148C23×D61C2×D411yescharacteristic
149C2×Q8×C61—21no—
150C2×Q8×C61—21no—
151C2×Q8×C61—21no—
152C2×Q8×C61—21no—
153C2×Q8×C61—21no—
154C2×Q8×C61—21no—
155C2×Q8×C61—21no—
156C2×Q8×C61—21no—
157C2×Q8×C61—21no—
158C2×Q8×C61—21no—
159C2×Q8×C61—21no—
160C2×Q8×C61—21no—
161C2×Q8×C61—21no—
162C2×Q8×C61—21no—
163C2×Q8×C61—21no—
164C2×Q8×C61C2×D412yes—
165C2×Q8×C61C2×D412yes—
166C2×Q8×C61#4112yes—
167C22×C4×C61—21no—
168C22×C4×C61—21no—
169C22×C4×C61—21no—
170C22×C4×C61—21no—
171C22×C4×C61C2×D411yescharacteristic
172C22×C4×C61C2×D411yescharacteristic
173C22×C4×C61C2411yescharacteristic
174C22×Dic61—21no—
175C22×Dic61—21no—
176C22×Dic61—21no—
177C22×Dic61—21no—
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C2×C4×D61—21no—
182C2×C4×D61—21no—
183C2×C4×D61—21no—
184C2×C4×D61—21no—
185C2×C4×D61—21no—
186C2×C4×D61—21no—
187Q16×C61—44no—
188Q16×C61—44no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61—21no—
194C2×C8×C61—21no—
195C4⋊C4×C61—24no—
196C4⋊C4×C61—24no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—42no—
200C42×C61—42no—
201C42×C61C2×D412yes—
202C42×C61C2×D412yes—
203#11758198—24no—
204#11758198—24no—
205#11758198—24no—
206#11758198—22no—
207#11758198—22no—
208#11758198—81no—
209C4⋊Dic61—24no—
210C4⋊Dic61—24no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61C2×D412yes—
217C4⋊Dic61C2×D412yes—
218C4⋊Dic61#4112yes—
219#11758196—24no—
220#11758196—24no—
221#11758196—24no—
222#11758196—24no—
223#11758196—24no—
224#11758196—22no—
225#11758196—22no—
226#11758196—81no—
227C4×Dic61—24no—
228C4×Dic61—24no—
229C4×Dic61—21no—
230C4×Dic61—21no—
231C4×Dic61—81no—
232C2×#92418—21no—
233C2×#92418—21no—
234C2×#92418—21no—
235C2×#92418—21no—
236C2×#92418—21no—
237C2×#92418—21no—
238C23×C61C22×D411yescharacteristic
239C22×D61—21no—
240C22×D61—21no—
241C22×D61—21no—
242C22×D61—21no—
243C22×D61—21no—
244C22×D61—21no—
245C22×D61—21no—
246Q8×C61—22no—
247Q8×C61—22no—
248Q8×C61—22no—
249Q8×C61—22no—
250Q8×C61—22no—
251Q8×C61—22no—
252Q8×C61—41no—
253Q8×C61—41no—
254Q8×C61—41no—
255Q8×C61—41no—
256Q8×C61—41no—
257Q8×C61—41no—
258C2×C4×C61—21no—
259C2×C4×C61—21no—
260C2×C4×C61—21no—
261C2×C4×C61—21no—
262C2×C4×C61—22no—
263C2×C4×C61—22no—
264C2×C4×C61—22no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—41no—
270C2×C4×C61—41no—
271C2×C4×C61—41no—
272C2×C4×C61—41no—
273C2×C4×C61—41no—
274C2×C4×C61#12012yes—
275C2×C4×C61#12012yes—
276C2×C4×C61C22×D411yescharacteristic
277C2×C4×C61C22×D411yescharacteristic
278C2×C4×C61C2×#4111yescharacteristic
279C2×C4×C61C2×#4111yescharacteristic
280C2×C4×C61H(5,𝔽2)11yescharacteristic
281C2×C4×C612−1+411yescharacteristic
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—22no—
285C2×Dic61—22no—
286C2×Dic61—21no—
287C2×Dic61—21no—
288C2×Dic61—22no—
289C2×Dic61—21no—
290C2×Dic61—21no—
291C2×Dic61—21no—
292C2×Dic61—21no—
293C2×Dic61—21no—
294C2×Dic61—41no—
295C2×Dic61—41no—
296C2×Dic61—41no—
297C2×Dic61—41no—
298C2×Dic61—41no—
299C2×Dic61—41no—
300C2×Dic61—41no—
301C2×Dic61—41no—
302C2×Dic61—41no—
303C4×D61—41no—
304C4×D61—41no—
305C4×D61—41no—
306C4×D61—41no—
307C8×C61—41no—
308C8×C61—41no—
309#92418—41no—
310#92418—41no—
311#24165—611nomaximal, Sylow
312C22×C61C2×#12011yescharacteristic
313C22×C61#52511yescharacteristic
314C22×C61#53711yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C22×C61#57811yescharacteristic
318C22×C61#57811yescharacteristic
319C2×D61—41no—
320C2×D61—41no—
321C2×D61—41no—
322C2×D61—41no—
323C2×D61—41no—
324C2×D61—41no—
325C2×D61—41no—
326C4×C61—22no—
327C4×C61—22no—
328C4×C61—42no—
329C4×C61—42no—
330C4×C61—42no—
331C4×C61—81no—
332C4×C61#54611yescharacteristic
333C4×C61#54711yescharacteristic
334C4×C61#54811yescharacteristic
335C4×C61#55011yescharacteristic
336Dic61—42no—
337Dic61—41no—
338Dic61—41no—
339Dic61—42no—
340Dic61—41no—
341Dic61—42no—
342Dic61—81no—
343C2×#541—611no—
344C2×#479—611no—
345C2×#462D6111yescharacteristic, complemented
346#2323—611no—
347#2036—611no—
348#2035—611no—
349#2001—611no—
350#2001—611no—
351#1989—611no—
352#1981—611no—
353#1878—611no—
354#1878—611no—
355#1865—611no—
356#1863—611no—
357#1798—611no—
358C2×C61#240711yescharacteristic
359C2×C61#301111yescharacteristic
360C2×C61#301311yescharacteristic
361C2×C61#312011yescharacteristic
362C2×C61#312711yescharacteristic
363C2×C61#325811yescharacteristic
364C2×C61#326111yescharacteristic
365D61—81no—
366C22×Q16C2×D6111yescharacteristic
367C22×Q16C2×D6111yescharacteristic
368#541—1224no—
369#541—1224no—
370C2×C4⋊Q8C2×D6111yescharacteristic
371C2×#126—1221no—
372C2×#126—1221no—
373C2×#125—611no—
374C2×#125—1221no—
375C2×#125—1221no—
376C2×C22⋊Q8—611no—
377C2×C22⋊Q8—611no—
378C2×C22⋊Q8—611no—
379C2×C4×Q8—611no—
380C2×C4×Q8—611no—
381C2×C4×Q8C2×D6111yescharacteristic
382C2×#117—611no—
383#479—1224no—
384#462—24no—
385C2×#107—611no—
386C2×#106—611no—
387C2×C4⋊C8—611no—
388C2×C4⋊C8C2×D6111yescharacteristic
389C2×#103—611no—
390C2×#103—611no—
391C2×#103—611no—
392C2×#103—611no—
393C2×#103—611no—
394C2×#103—611no—
395C2×#103C2×D6111yescharacteristic
396C2×#103C2×D6111yescharacteristic
397C2×C22⋊C8—611no—
398C2×C22⋊C8—611no—
399#391—611no—
400#389—611no—
401#389—1221no—
402#388—1221no—
403#388—1221no—
404#387—1221no—
405#380—611no—
406C4×C4⋊C4—611no—
407#340—611no—
408#340—611no—
409#340—611no—
410#340—611no—
411#336—611no—
412#336—611no—
413C61#2416511yescharacteristic, complemented, Sylow
414C22×Q8C22×D6111yescharacteristic
415C22×Q8C22×D6111yescharacteristic
416C22×Q8C22×D6111yescharacteristic
417C23×C4—611no—
418C2×Q16—22no—
419C2×Q16—22no—
420C2×Q16—22no—
421C2×Q16—22no—
422C2×Q16—41no—
423C2×Q16—41no—
424C22×C8—611no—
425C22×C8—611no—
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428C4⋊Q8—22no—
429C4⋊Q8C22⋊D6114yes—
430#126—1228no—
431#126—1228no—
432#125—614no—
433#125—1228no—
434#125—1228no—
435#125—1222no—
436C22⋊Q8—1222no—
437C22⋊Q8—1222no—
438C22⋊Q8—1222no—
439C22⋊Q8—1222no—
440C22⋊Q8—1222no—
441C22⋊Q8—1222no—
442C4×Q8—22no—
443C4×Q8—22no—
444C4×Q8—1222no—
445C4×Q8—1222no—
446C4×Q8—1222no—
447C4×Q8—1222no—
448#117—1222no—
449#117—1222no—
450C2×C4⋊C4—21no—
451C2×C4⋊C4—21no—
452C2×C4⋊C4—21no—
453C2×C4⋊C4—611no—
454C2×C4⋊C4—611no—
455C2×C4⋊C4—611no—
456C2×C4⋊C4—611no—
457C2×C4⋊C4—611no—
458C2×C4⋊C4—611no—
459C2×C4⋊C4—1221no—
460C2×C4⋊C4—1221no—
461C2×C4⋊C4—1221no—
462C2×C4⋊C4—1221no—
463C2×C4⋊C4—1221no—
464C2×C4⋊C4—1221no—
465C2×C4⋊C4—1221no—
466C2×C4⋊C4—1221no—
467C2×C4⋊C4C22×D6111yescharacteristic
468C2×C22⋊C4—611no—
469C2×C22⋊C4—611no—
470C2×C22⋊C4—1221no—
471C2×C22⋊C4—1221no—
472C2×C22⋊C4—1221no—
473C2×C22⋊C4—1221no—
474C2×C42—21no—
475C2×C42—611no—
476C2×C42—1221no—
477C2×C42—1221no—
478C2×C42—1221no—
479C2×C42C22×D6111yescharacteristic
480#107—1222no—
481#106—1222no—
482C4⋊C8—22no—
483C4⋊C8—1222no—
484#103—22no—
485#103—22no—
486#103—1222no—
487#103—1222no—
488#103—1222no—
489#103—1222no—
490#103—2441no—
491#103—2441no—
492C22⋊C8—2441no—
493C22⋊C8—2441no—
494#95—1221no—
495#95—1221no—
496#95—1221no—
497#95—1221no—
498#95—1221no—
499C24—611no—
500C2×Q8—21no—
501C2×Q8—21no—
502C2×Q8—21no—
503C2×Q8—21no—
504C2×Q8—21no—
505C2×Q8—21no—
506C2×Q8—21no—
507C2×Q8—21no—
508C2×Q8—21no—
509C2×Q8—21no—
510C2×Q8—21no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—21no—
514C2×Q8—21no—
515C2×Q8#1175822412yes—
516C2×Q8C2×C22⋊D6112yes—
517C2×Q8C2×C22⋊D6112yes—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—21no—
521C22×C4—21no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—611no—
525C22×C4—1221no—
526C22×C4—1221no—
527C22×C4—1221no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1221no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—1221no—
534C22×C4—1221no—
535C22×C4D4×D6111yescharacteristic
536C22×C4D4×D6111yescharacteristic
537C22×C4C23×D6111yescharacteristic
538Q16—44no—
539Q16—44no—
540C2×C8—21no—
541C2×C8—21no—
542C2×C8—21no—
543C2×C8—21no—
544C2×C8—21no—
545C2×C8—21no—
546C2×C8—1221no—
547C2×C8—1221no—
548C2×C8—1221no—
549C2×C8—1221no—
550C2×C8—1221no—
551C2×C8—1221no—
552C4⋊C4—24no—
553C4⋊C4—24no—
554C4⋊C4—21no—
555C4⋊C4—21no—
556C4⋊C4—42no—
557C4⋊C4—612no—
558C4⋊C4—612no—
559C4⋊C4—612no—
560C4⋊C4—1224no—
561C4⋊C4—1224no—
562C4⋊C4—1224no—
563C4⋊C4—1224no—
564C4⋊C4—1224no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1222no—
569C4⋊C4—1222no—
570C4⋊C4—1224no—
571C4⋊C4—1224no—
572C4⋊C4—1221no—
573C4⋊C4—1221no—
574C4⋊C4—4881no—
575C22⋊C4—1224no—
576C22⋊C4—1224no—
577C22⋊C4—1224no—
578C22⋊C4—1222no—
579C22⋊C4—1222no—
580C22⋊C4—4881no—
581C42—42no—
582C42—1224no—
583C42—1224no—
584C42—1221no—
585C42—1221no—
586C42—4881no—
587C42D4×D6112yes—
588C42C2×C22⋊D6112yes—
589C23—1221no—
590C23—1221no—
591C23—1221no—
592C23—1221no—
593C23—1221no—
594C23—1221no—
595C23—1221no—
596C23C2×D4×D6111yescharacteristic
597Q8—22no—
598Q8—22no—
599Q8—22no—
600Q8—22no—
601Q8—22no—
602Q8—22no—
603Q8—41no—
604Q8—41no—
605Q8—41no—
606Q8—41no—
607Q8—41no—
608Q8—41no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—22no—
614C2×C4—22no—
615C2×C4—22no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—1221no—
626C2×C4—1221no—
627C2×C4—1222no—
628C2×C4—1222no—
629C2×C4—1221no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—1221no—
634C2×C4—1221no—
635C2×C4—1221no—
636C2×C4—1221no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4—2441no—
646C2×C4—2441no—
647C2×C4—2441no—
648C2×C4—2441no—
649C2×C4—2441no—
650C2×C4#1451974512yes—
651C2×C4#1451974512yes—
652C2×C4C2×D4×D6111yescharacteristic
653C2×C4C22×C22⋊D6111yescharacteristic
654C2×C4#41×D6111yescharacteristic
655C2×C4C2×#1175822411yescharacteristic
656C2×C4#1451982211yescharacteristic
657C2×C4#1451982411yescharacteristic
658C8—41no—
659C8—41no—
660C8—2441no—
661C8—2441no—
662C22—2441no—
663C22—2441no—
664C22—2441no—
665C22—2441no—
666C22—2441no—
667C22—2441no—
668C22—2441no—
669C22C2×#1451974511yescharacteristic
670C22#1684027211yescharacteristic
671C22#1684034111yescharacteristic
672C22#1684057111yescharacteristic
673C22#1684057511yescharacteristic
674C22#1684057511yescharacteristic
675C22#1684057811yescharacteristic
676C4—22no—
677C4—22no—
678C4—42no—
679C4—42no—
680C4—42no—
681C4—81no—
682C4—4881no—
683C4—2441no—
684C4—2442no—
685C4—2441no—
686C4—2442no—
687C4—2442no—
688C4—2441no—
689C4#1684040611yescharacteristic
690C4#1684041111yescharacteristic
691C4#1684041811yescharacteristic
692C4#1684043011yescharacteristic
693C2—4881no—
694C2#2144916011yescharacteristic
695C2#2145511011yescharacteristic
696C2#2145513311yescharacteristic
697C2#2145603211yescharacteristic
698C2#2145611711yescharacteristic
699C2#2145724311yescharacteristic
700C2#2145727111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#462C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenterC2×#462D61
hypocenterC61#24165
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
2. C2×#16840069C2semidirect
8. #21446774C2semidirect
9. #21446743C2semidirect
10. #21446162C2semidirect
11. #21446162C2semidirect
12. #21446085C2semidirect
13. #21446071C2semidirect
15. C2×C61×#462C2semidirect
345. C2×#462D61semidirect
413. C61#24165semidirect

Maximal subgroups

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

subgroupindex
C2×#168403732
C2×#168400692
#214484952
#214470202
#214470132
#214468262
#214467742
#214467432
#214461622
#214460852
#214460712
#214458002
C2×C61×#4622
#2416561