Finite Group Databasebeta

← #29734340

Subgroups — #29734340

Order 15616.

1038 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×#16840316C211yesmaximal, characteristic
2C2×#16840170C211yesmaximal, characteristic, complemented
3C2×#16840129C211yesmaximal, characteristic, complemented
4#21454128C211yesmaximal, characteristic
5#21447024C211yesmaximal, characteristic
6#21447020C211yesmaximal, characteristic
7#21447018C211yesmaximal, characteristic
8#21447014C211yesmaximal, characteristic
9#21446790C211yesmaximal, characteristic, complemented
10#21446637C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446087C211yesmaximal, characteristic, complemented
13#21446072C211yesmaximal, characteristic, complemented
14#21445580C211yesmaximal, characteristic
15C61×#1863C211yesmaximal, characteristic, complemented
16#16840316—24no—
17#16840316—24no—
18C2×#14519776C2211yescharacteristic
19C2×#14519775C2211yescharacteristic
20C2×#14519768C2211yescharacteristic
21C2×#14519767—21no—
22C2×#14519757C2211yescharacteristic
23C2×#14519755—21no—
24C2×#14519753C2211yescharacteristic
25C2×C4×Dic122—21no—
26C2×C4×Dic122C2211yescharacteristic
27C2×#14519734—21no—
28C2×C42×D61C2211yescharacteristic
29#16840170—24no—
30#16840129—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519709C2211yescharacteristic
33C2×#14519708C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×#14519687C2211yescharacteristic
44#16839660C2211yescharacteristic
45#16839648—21no—
46#16839644—21no—
47#16839636—21no—
48#16839633—21no—
49#16839627C2211yescharacteristic
50#16839619C2211yescharacteristic
51#16839619C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839517C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#107×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122C2311yescharacteristic
65C22×C4×D61D411yescharacteristic
66C22×C4×D61D411yescharacteristic
67C22×C4×D61C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70#14519776—22no—
71#14519776D414yes—
72#14519775—22no—
73#14519775D414yes—
74#14519768—22no—
75#14519768—22no—
76#14519767—28no—
77#14519757—22no—
78#14519757—22no—
79#14519755—28no—
80#14519753—22no—
81#14519753—22no—
82C4×Dic122—28no—
83C4×Dic122—22no—
84C4×Dic122—22no—
85#14519734—28no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758196—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61D411yescharacteristic
104C2×C4×Dic61D411yescharacteristic
105C2×C4×Dic61C2311yescharacteristic
106C42×D61—22no—
107C42×D61—22no—
108#14519710—22no—
109#14519709—22no—
110#14519708—22no—
111#14519702—22no—
112#14519702—22no—
113#14519701—22no—
114#14519701—22no—
115#14519700—22no—
116#14519700—22no—
117#14519689—22no—
118#14519689—22no—
119#14519687—22no—
120#14519687—22no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C4×Q8×C61—22no—
130C4×Q8×C61—22no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138#107×C61—22no—
139#103×C61—22no—
140#103×C61—22no—
141#95×C61—21no—
142C23×D61C2×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×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—22no—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—22no—
173C2×C4×D61—41no—
174C2×C4×D61—41no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—21no—
179C2×Dic122—21no—
180C2×Dic122—24no—
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#4112yes—
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×C61—42no—
197C4⋊C4×C61C2×D412yes—
198C42×C61—24no—
199C42×C61—21no—
200C42×C61—21no—
201#11758198—24no—
202#11758198—24no—
203#11758198—42no—
204#11758198—42no—
205C4⋊Dic61—24no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—42no—
212C4⋊Dic61C2×D412yes—
213#11758196—24no—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—42no—
219#11758196—42no—
220C4×Dic61—24no—
221C4×Dic61—22no—
222C4×Dic61—22no—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D412yes—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418#4112yes—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61—21no—
237C22×D61—21no—
238C22×D61#13712yes—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
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×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61#13712yes—
261C2×C4×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C2×#4111yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C612−1+411yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—22no—
273C2×Dic61—22no—
274C2×Dic61—21no—
275C2×Dic61—21no—
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×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61#12012yes—
287C2×Dic61#12012yes—
288C4×D61—28no—
289C4×D61—44no—
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—22no—
302#92418—41no—
303#25677—611nomaximal, Sylow
304C22×C61C2×#12011yescharacteristic
305C22×C61#53211yescharacteristic
306C22×C61#53611yescharacteristic
307C22×C61C2×#13711yescharacteristic
308C22×C61C2×#13711yescharacteristic
309C22×C61#57511yescharacteristic
310C22×C61#57511yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—41no—
315C2×D61—41no—
316C2×D61—41no—
317C2×D61—41no—
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61#54711yescharacteristic
325C4×C61#54711yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61#54811yescharacteristic
328Dic61—22no—
329Dic61—22no—
330Dic61—42no—
331Dic61—41no—
332Dic61—42no—
333Dic61—81no—
334Dic61—81no—
335C2×#533—611no—
336C2×#501—611no—
337C2×#491—611no—
338#2851—611no—
339#2036—611no—
340#2036—611no—
341#2035—611no—
342#2035—611no—
343#1993—611no—
344#1960—611no—
345#1865—611no—
346#1865—611no—
347#1863—611no—
348#1863D6111yescharacteristic, complemented
349#1749—611no—
350C2×C61#243211yescharacteristic
351C2×C61#300411yescharacteristic
352C2×C61#300511yescharacteristic
353C2×C61#317211yescharacteristic
354C2×C61#318111yescharacteristic
355C2×C61#323611yescharacteristic
356C2×C61#323711yescharacteristic
357D61—42no—
358#533—1224no—
359#533—1224no—
360C2×C4⋊Q8—611no—
361C2×C4⋊Q8—611no—
362C2×#125—611no—
363C2×#125—1221no—
364C2×C22⋊Q8—611no—
365C2×C22⋊Q8—611no—
366C2×C22⋊Q8—1221no—
367C2×C4×Q8—611no—
368C2×C4×Q8—1221no—
369C2×C4×Q8C2×D6111yescharacteristic
370C2×#117—1221no—
371C22×C42—611no—
372#501—1224no—
373#491—1224no—
374C2×#107—611no—
375C2×#107C2×D6111yescharacteristic
376C2×#106—611no—
377C2×#106—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—611no—
383C2×#103—611no—
384C2×#103C2×D6111yescharacteristic
385C2×#103C2×D6111yescharacteristic
386C2×C22⋊C8—611no—
387C2×C22⋊C8—611no—
388C2×#97—611no—
389C2×#97—611no—
390#389—611no—
391#389C2×D6111yescharacteristic
392#387—1221no—
393#387—1221no—
394#385—1221no—
395#385—1221no—
396#382—611no—
397#380—611no—
398#380—611no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340C2×D6111yescharacteristic
403#336—611no—
404#336C2×D6111yescharacteristic
405C61#2567711yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—1221no—
408C22×Q8C22×D6111yescharacteristic
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—611no—
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—614no—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420#125—1222no—
421#125—1222no—
422#125—1228no—
423C22⋊Q8—1222no—
424C22⋊Q8—1228no—
425C22⋊Q8—1222no—
426C22⋊Q8—1222no—
427C22⋊Q8—1222no—
428C4×Q8—22no—
429C4×Q8—22no—
430C4×Q8—1222no—
431C4×Q8—1222no—
432C4×Q8—1228no—
433#117—1228no—
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⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C4⋊C4C22×D6111yescharacteristic
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C42—21no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—611no—
459C2×C42—1222no—
460C2×C42—1222no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#107—22no—
464#107—1222no—
465#106—1222no—
466#106—1222no—
467#103—22no—
468#103—22no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#103—1222no—
473#103—1222no—
474#103—1222no—
475C22⋊C8—1222no—
476C22⋊C8—1222no—
477#97—1222no—
478#97—1222no—
479#95—21no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24—611no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—612no—
492C2×Q8—1224no—
493C2×Q8—1221no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—2442no—
499C2×Q8—2442no—
500C2×Q8D4×D6112yes—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—611no—
506C22×C4—611no—
507C22×C4—611no—
508C22×C4—1222no—
509C22×C4—1222no—
510C22×C4—1222no—
511C22×C4—1222no—
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—1221no—
521C22×C4—1221no—
522C22×C4—2441no—
523C22×C4—2441no—
524C22×C4D4×D6111yescharacteristic
525C22×C4D4×D6111yescharacteristic
526C22×C4C23×D6111yescharacteristic
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—612no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8#1175822212yes—
539C4⋊C4—24no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—42no—
546C4⋊C4—612no—
547C4⋊C4—1224no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1224no—
553C4⋊C4—1221no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—2442no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4D4×D6112yes—
562C22⋊C4—1224no—
563C22⋊C4—1224no—
564C22⋊C4—2442no—
565C22⋊C4—2442no—
566C42—24no—
567C42—21no—
568C42—21no—
569C42—612no—
570C42—612no—
571C42—1222no—
572C42—1222no—
573C42—1224no—
574C23—612no—
575C23—1221no—
576C23—1221no—
577C23—1221no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23C2×D4×D6111yescharacteristic
582Q8—22no—
583Q8—22no—
584Q8—41no—
585Q8—41no—
586Q8—1222no—
587Q8—1222no—
588Q8—2448no—
589Q8—2441no—
590Q8—2441no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—21no—
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—41no—
607C2×C4—41no—
608C2×C4—612no—
609C2×C4—612no—
610C2×C4—1228no—
611C2×C4—1221no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1222no—
617C2×C4—1222no—
618C2×C4—1221no—
619C2×C4—1221no—
620C2×C4—2444no—
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×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4#41×D6111yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982411yescharacteristic
641C2×C4#1451982511yescharacteristic
642C8—22no—
643C8—41no—
644C8—1222no—
645C8—2441no—
646C22—1222no—
647C22—1222no—
648C22—1222no—
649C22—2441no—
650C22—2441no—
651C22—2441no—
652C22—2441no—
653C22#120×D6111yescharacteristic
654C22#1684030911yescharacteristic
655C22#1684033511yescharacteristic
656C22#137×D6111yescharacteristic
657C22C2×#1451980911yescharacteristic
658C22#1684055811yescharacteristic
659C22#1684055911yescharacteristic
660C4—22no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—81no—
666C4—1222no—
667C4—1222no—
668C4—4881no—
669C4—4881no—
670C4—2442no—
671C4—2441no—
672C4—2442no—
673C4#1684041011yescharacteristic
674C4#1684041311yescharacteristic
675C4#1684042111yescharacteristic
676C4#1684042211yescharacteristic
677C2—2442no—
678C2#2144944511yescharacteristic
679C2#2145505411yescharacteristic
680C2#2145505711yescharacteristic
681C2#2145642311yescharacteristic
682C2#2145652411yescharacteristic
683C2#2145702311yescharacteristic
684C2#2145703111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1863C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1863D61
hypocenterC61#25677
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1863

Complements

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

NH ≅ G/Ntype
2. C2×#16840170C2semidirect
3. C2×#16840129C2semidirect
9. #21446790C2semidirect
10. #21446637C2semidirect
11. #21446090C2semidirect
12. #21446087C2semidirect
13. #21446072C2semidirect
15. C61×#1863C2semidirect
348. #1863D61semidirect
405. C61#25677semidirect

Maximal subgroups

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

subgroupindex
C2×#168403162
C2×#168401702
C2×#168401292
#214541282
#214470242
#214470202
#214470182
#214470142
#214467902
#214466372
#214460902
#214460872
#214460722
#214455802
C61×#18632
#2567761