Finite Group Databasebeta

← #29713321

Subgroups — #29713321

Order 15616.

1072 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840314C211yesmaximal, characteristic
2C2×#16840103C211yesmaximal, characteristic
3#21448301C211yesmaximal, characteristic
4#21447150C211yesmaximal, characteristic, complemented
5#21447145C211yesmaximal, characteristic, complemented
6#21447063C211yesmaximal, characteristic, complemented
7#21447063C211yesmaximal, characteristic, complemented
8#21446132C211yesmaximal, characteristic
9#21446129C211yesmaximal, characteristic
10#21446085C211yesmaximal, characteristic, complemented
11#21446084C211yesmaximal, characteristic, complemented
12#21446071C211yesmaximal, characteristic, complemented
13#21446070C211yesmaximal, characteristic, complemented
14#21445885C211yesmaximal, characteristic
15C2×C61×#485C211yesmaximal, characteristic
16#16840314—24no—
17#16840314—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519767—21no—
21C2×#14519756C2211yescharacteristic
22C2×#14519754—21no—
23C2×Q8×Dic61C2211yescharacteristic
24C2×#14519742—21no—
25C2×#14519742C2211yescharacteristic
26C2×#14519736C2211yescharacteristic
27C2×#14519733—21no—
28C22×C4×Dic61C2211yescharacteristic
29#16840103—24no—
30C2×#14519710C2211yescharacteristic
31C2×#14519708C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×C22⋊#92418C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839658C2211yescharacteristic
41#16839658C2211yescharacteristic
42#16839658C2211yescharacteristic
43#16839634—21no—
44#16839634—21no—
45C4×#11758196—21no—
46C4×#11758196—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839516C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C22⋊Q8×C61C2211yescharacteristic
56C2×#117×C61C2211yescharacteristic
57C61×#485—24no—
58C2×#107×C61C2211yescharacteristic
59C2×#106×C61C2211yescharacteristic
60C2×#103×C61C2211yescharacteristic
61C2×#103×C61C2211yescharacteristic
62C2×C22⋊C8×C61C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122C2311yescharacteristic
65C23×Dic61D411yescharacteristic
66C23×Dic61D411yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—22no—
71#14519775—22no—
72#14519775—22no—
73#14519767—28no—
74#14519756—22no—
75#14519756D414yes—
76#14519754—28no—
77Q8×Dic61—22no—
78Q8×Dic61—22no—
79#14519742—28no—
80#14519742—22no—
81#14519742—22no—
82#14519736—22no—
83#14519736—22no—
84#14519733—28no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C4×Dic61—22no—
101C2×C4×Dic61—22no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61D411yescharacteristic
106C2×C4×Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519710—22no—
110#14519708—22no—
111#14519702—22no—
112#14519702—22no—
113#14519701—22no—
114#14519701—22no—
115#14519700—22no—
116#14519700—22no—
117C22⋊#92418—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×Q8×C61C2311yescharacteristic
123C23×C4×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C22⋊Q8×C61—22no—
127C22⋊Q8×C61D414yes—
128#117×C61—22no—
129#117×C61—22no—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C22⋊C4×C61—21no—
135C2×C22⋊C4×C61—21no—
136C2×C42×C61—21no—
137#107×C61—22no—
138#106×C61—22no—
139#103×C61—22no—
140#103×C61—22no—
141C22⋊C8×C61—22no—
142C24×C61C2×D411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61C2×D411yescharacteristic
158C22×C4×C61C2×D412yes—
159C22×C4×C61C2×D412yes—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—22no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—21no—
166C22×Dic61—21no—
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—22no—
170C22×Dic61—41no—
171C22×Dic61C2×D411yescharacteristic
172C22×Dic61C2×D411yescharacteristic
173C22×Dic61C2×D411yescharacteristic
174C22×Dic61C2×D411yescharacteristic
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—24no—
180C2×Dic122—21no—
181C2×Dic122—42no—
182C2×Dic122—42no—
183C2×Dic122C2×D412yes—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C2×C8×C61—21no—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C4⋊C4×C61—24no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61C2×D412yes—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—42no—
199C42×C61—42no—
200C22⋊Dic61—24no—
201C22⋊Dic61—24no—
202C4⋊Dic61—24no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61C2×D412yes—
207C4⋊Dic61#4112yes—
208C4⋊Dic61#4112yes—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214#11758196—42no—
215#11758196—42no—
216#11758196—42no—
217#11758196—42no—
218C4×Dic61—24no—
219C4×Dic61—24no—
220C4×Dic61—24no—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61—21no—
224C4×Dic61—21no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61—21no—
238C23×C61—21no—
239C23×C61#12012yes—
240C23×C61C22×D411yescharacteristic
241Q8×C61—22no—
242Q8×C61—22no—
243Q8×C61—41no—
244Q8×C61—41no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
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—41no—
262C2×C4×C61—41no—
263C2×C4×C61#12012yes—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C612−1+411yescharacteristic
269C2×C4×C612−1+411yescharacteristic
270C2×Dic61—28no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—22no—
279C2×Dic61—22no—
280C2×Dic61—22no—
281C2×Dic61—22no—
282C2×Dic61—21no—
283C2×Dic61—21no—
284C2×Dic61—44no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61—41no—
289C2×Dic61—41no—
290C2×Dic61—41no—
291C2×Dic61—41no—
292C2×Dic61—41no—
293C2×Dic61#13512yes—
294C2×Dic61#13712yes—
295C2×Dic61#13712yes—
296Dic122—22no—
297Dic122—22no—
298Dic122—48no—
299Dic122—41no—
300Dic122—41no—
301C8×C61—41no—
302C8×C61—41no—
303#92418—41no—
304#92418—41no—
305#24065—611nomaximal, Sylow
306C22×C61—22no—
307C22×C61—22no—
308C22×C61—22no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61—41no—
312C22×C61—41no—
313C22×C61C2×#12011yescharacteristic
314C22×C61#52511yescharacteristic
315C22×C61#53611yescharacteristic
316C22×C61C2×#13511yescharacteristic
317C22×C61C2×#13711yescharacteristic
318C22×C61C2×#13711yescharacteristic
319C22×C61#57511yescharacteristic
320C4×C61—22no—
321C4×C61—42no—
322C4×C61—42no—
323C4×C61—81no—
324C4×C61—81no—
325C4×C61#54711yescharacteristic
326C4×C61#54711yescharacteristic
327C4×C61#54711yescharacteristic
328C4×C61#54711yescharacteristic
329Dic61—22no—
330Dic61—22no—
331Dic61—22no—
332Dic61—42no—
333Dic61—42no—
334Dic61—42no—
335Dic61—42no—
336Dic61—41no—
337C2×#533—611no—
338C2×#485—611no—
339C2×#485D6111yescharacteristic
340#2296—611no—
341#2058—611no—
342#2057—611no—
343#2043—611no—
344#2043—611no—
345#1873—611no—
346#1873—611no—
347#1865—611no—
348#1865—611no—
349#1863—611no—
350#1863—611no—
351#1817—611no—
352C2×C61—42no—
353C2×C61#240611yescharacteristic
354C2×C61#300211yescharacteristic
355C2×C61#300411yescharacteristic
356C2×C61#310711yescharacteristic
357C2×C61#311611yescharacteristic
358C2×C61#323111yescharacteristic
359C2×C61#323711yescharacteristic
360#533—1224no—
361#533—1224no—
362C2×C4⋊Q8—611no—
363C2×C4⋊Q8—611no—
364C2×#125—1221no—
365C2×C22⋊Q8—611no—
366C2×C22⋊Q8—1221no—
367C2×C22⋊Q8C2×D6111yescharacteristic
368C2×C4×Q8—611no—
369C2×C4×Q8—611no—
370C2×C4×Q8—1221no—
371C2×#117—611no—
372C2×#117—1221no—
373C2×#117C2×D6111yescharacteristic
374C22×C42—611no—
375#485—24no—
376#485—1224no—
377C2×#107—611no—
378C2×#107C2×D6111yescharacteristic
379C2×#106—611no—
380C2×#106C2×D6111yescharacteristic
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103C2×D6111yescharacteristic
388C2×#103C2×D6111yescharacteristic
389C2×C22⋊C8—611no—
390C2×C22⋊C8C2×D6111yescharacteristic
391#389—611no—
392#389—611no—
393#389—611no—
394#389—611no—
395#385—1221no—
396#385—1221no—
397C4×C4⋊C4—1221no—
398C4×C4⋊C4—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#336—611no—
404#336—611no—
405#336—611no—
406#336—611no—
407C61#2406511yescharacteristic, complemented, Sylow
408C22×Q8—611no—
409C22×Q8—1221no—
410C22×Q8C22×D6111yescharacteristic
411C23×C4—611no—
412C23×C4—611no—
413C23×C4C22×D6111yescharacteristic
414C22×C8—611no—
415C22×C8—611no—
416C22×C8C22×D6111yescharacteristic
417C22×C8C22×D6111yescharacteristic
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420C4⋊Q8—1222no—
421C4⋊Q8—1222no—
422#125—1228no—
423C22⋊Q8—22no—
424C22⋊Q8—614no—
425C22⋊Q8—1228no—
426C22⋊Q8—1222no—
427C22⋊Q8C22⋊D6114yes—
428C4×Q8—1222no—
429C4×Q8—1228no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1222no—
433#117—22no—
434#117—22no—
435#117—1228no—
436#117—1222no—
437#117—1222no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—21no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1221no—
459C2×C42—21no—
460C2×C42—611no—
461C2×C42—611no—
462C2×C42—611no—
463C2×C42—611no—
464C2×C42—1221no—
465C2×C42—1221no—
466C2×C42—1221no—
467C2×C42—1222no—
468C2×C42—1222no—
469#107—22no—
470#107—1222no—
471#106—22no—
472#106—1222no—
473#103—22no—
474#103—22no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—1222no—
479#103—1222no—
480#103—1222no—
481C22⋊C8—22no—
482C22⋊C8—1222no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487C24D4×D6111yescharacteristic
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—612no—
494C2×Q8—1224no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—2442no—
501C2×Q8—2442no—
502C2×Q8C2×C22⋊D6112yes—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—611no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—1222no—
516C22×C4—1222no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—1222no—
524C22×C4—2441no—
525C22×C4D4×D6112yes—
526C22×C4D4×D6111yescharacteristic
527C22×C4C2×C22⋊D6112yes—
528C22×C4C23×D6111yescharacteristic
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C2×C8—1221no—
540C2×C8—1221no—
541C4⋊C4—21no—
542C4⋊C4—24no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—612no—
548C4⋊C4—612no—
549C4⋊C4—612no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1224no—
554C4⋊C4—1224no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1224no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4—2442no—
563C4⋊C4C2×C22⋊D6112yes—
564C22⋊C4—24no—
565C22⋊C4—42no—
566C22⋊C4—1224no—
567C22⋊C4—1224no—
568C42—42no—
569C42—1224no—
570C42—1224no—
571C42—1224no—
572C42—1221no—
573C42—1221no—
574C42—1221no—
575C42—1221no—
576C42—1222no—
577C42—1222no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—21no—
582C23—21no—
583C23—21no—
584C23#1451974512yes—
585C23C2×D4×D6111yescharacteristic
586Q8—22no—
587Q8—22no—
588Q8—41no—
589Q8—41no—
590Q8—1222no—
591Q8—1222no—
592Q8—2448no—
593Q8—2441no—
594Q8—2441no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—612no—
614C2×C4—612no—
615C2×C4—612no—
616C2×C4—1228no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—1222no—
625C2×C4—1222no—
626C2×C4—1222no—
627C2×C4—1222no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—2444no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4#1451974512yes—
640C2×C4C2×D4×D6111yescharacteristic
641C2×C4C22×C22⋊D6111yescharacteristic
642C2×C4#41×D6111yescharacteristic
643C2×C4#41×D6111yescharacteristic
644C2×C4#1451982411yescharacteristic
645C2×C4#1451982411yescharacteristic
646C8—41no—
647C8—41no—
648C8—2441no—
649C8—2441no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22C2×#1451974511yescharacteristic
658C22#1684027011yescharacteristic
659C22#1684033411yescharacteristic
660C22#135×D6111yescharacteristic
661C22#137×D6111yescharacteristic
662C22#137×D6111yescharacteristic
663C22#1684055911yescharacteristic
664C4—22no—
665C4—42no—
666C4—42no—
667C4—81no—
668C4—81no—
669C4—1222no—
670C4—1222no—
671C4—1222no—
672C4—2442no—
673C4—2442no—
674C4—2441no—
675C4—2442no—
676C4—2442no—
677C4#1684041011yescharacteristic
678C4#1684041011yescharacteristic
679C4#1684041111yescharacteristic
680C4#1684041111yescharacteristic
681C2—42no—
682C2#2144915111yescharacteristic
683C2#2145503311yescharacteristic
684C2#2145505111yescharacteristic
685C2#2145588511yescharacteristic
686C2#2145597611yescharacteristic
687C2#2145697611yescharacteristic
688C2#2145703011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#485C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenterC2×#485D61
hypocenterC61#24065
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#485

Complements

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

NH ≅ G/Ntype
4. #21447150C2semidirect
5. #21447145C2semidirect
6. #21447063C2semidirect
7. #21447063C2semidirect
10. #21446085C2semidirect
11. #21446084C2semidirect
12. #21446071C2semidirect
13. #21446070C2semidirect
407. C61#24065semidirect

Maximal subgroups

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

subgroupindex
C2×#168403142
C2×#168401032
#214483012
#214471502
#214471452
#214470632
#214461322
#214461292
#214460852
#214460842
#214460712
#214460702
#214458852
C2×C61×#4852
#2406561