Finite Group Databasebeta

← #29736712

Subgroups — #29736712

Order 15616.

1010 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×#16840390C211yesmaximal, characteristic
2C2×#16840129C211yesmaximal, characteristic, complemented
3C2×#16840111C211yesmaximal, characteristic, complemented
4#21451589C211yesmaximal, characteristic
5#21446969C211yesmaximal, characteristic, complemented
6#21446968C211yesmaximal, characteristic, complemented
7#21446959C211yesmaximal, characteristic, complemented
8#21446958C211yesmaximal, characteristic, complemented
9#21446790C211yesmaximal, characteristic, complemented
10#21446782C211yesmaximal, characteristic, complemented
11#21446131C211yesmaximal, characteristic
12#21446127C211yesmaximal, characteristic
13#21446124C211yesmaximal, characteristic
14#21445587C211yesmaximal, characteristic
15C61×#1873C211yesmaximal, characteristic
16#16840390—24no—
17#16840390—24no—
18C2×#14519777—21no—
19C2×#14519776C2211yescharacteristic
20C2×#14519776C2211yescharacteristic
21C2×#14519775C2211yescharacteristic
22C2×#14519769—21no—
23C2×#14519768C2211yescharacteristic
24C2×#14519768C2211yescharacteristic
25C2×#14519756C2211yescharacteristic
26C2×#14519754—21no—
27C2×#14519754—21no—
28C2×#14519736C2211yescharacteristic
29C22×C4⋊Dic61C2211yescharacteristic
30#16840129—24no—
31#16840111—24no—
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×C22⋊#92418C2211yescharacteristic
39C2×#14519687C2211yescharacteristic
40C2×C8×Dic61C2211yescharacteristic
41#16839695—21no—
42#16839695—21no—
43#16839690—21no—
44#16839690—21no—
45#16839669C2211yescharacteristic
46#16839667C2211yescharacteristic
47#16839627C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55#16839516C2211yescharacteristic
56C2×C22⋊Q8×C61C2211yescharacteristic
57C2×#117×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C2×C22⋊C8×C61C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122—21no—
65C22×Dic122C2311yescharacteristic
66C23×Dic61—21no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519777—28no—
70#14519776—22no—
71#14519776—22no—
72#14519776—22no—
73#14519776D414yes—
74#14519775—22no—
75#14519775—22no—
76#14519769—28no—
77#14519768—22no—
78#14519768—22no—
79#14519768—22no—
80#14519768D414yes—
81#14519756—22no—
82#14519756—22no—
83#14519754—28no—
84#14519754—28no—
85#14519736—22no—
86#14519736—22no—
87C2×C4⋊Dic61—21no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61—21no—
90C2×C4⋊Dic61—22no—
91C2×C4⋊Dic61—22no—
92C2×C4⋊Dic61D411yescharacteristic
93C2×C4⋊Dic61D411yescharacteristic
94C2×C4⋊Dic61C2311yescharacteristic
95C2×C4⋊Dic61C2311yescharacteristic
96C2×C4⋊Dic61C2311yescharacteristic
97C2×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196—21no—
101C2×#11758196—21no—
102C2×#11758196—21no—
103C2×#11758196—21no—
104C2×C22⋊Dic61—21no—
105C2×C22⋊Dic61D411yescharacteristic
106C2×C22⋊Dic61D411yescharacteristic
107C2×C4×Dic61C2311yescharacteristic
108C2×C4×Dic61C2311yescharacteristic
109#14519702—22no—
110#14519702—22no—
111#14519701—22no—
112#14519701—22no—
113#14519700—22no—
114#14519700—22no—
115C22⋊#92418—22no—
116#14519687—22no—
117C8×Dic61—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124C22×Q8×C61C2311yescharacteristic
125C23×C4×C61C2311yescharacteristic
126C22×C8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22⋊Q8×C61—22no—
129C22⋊Q8×C61—22no—
130#117×C61—22no—
131#117×C61—22no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C22⋊C4×C61—21no—
138C2×C42×C61—21no—
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×C61—21no—
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×C61#4112yes—
159C22×C4×C61#4112yes—
160C22×C4×C61C2411yescharacteristic
161C22×Dic61—21no—
162C22×Dic61—21no—
163C22×Dic61—24no—
164C22×Dic61—21no—
165C22×Dic61—21no—
166C22×Dic61—81no—
167C22×Dic61—81no—
168C22×Dic61C2×D411yescharacteristic
169C22×Dic61C2×D411yescharacteristic
170C22×Dic61C2×D411yescharacteristic
171C22×Dic61C2×D411yescharacteristic
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122—21no—
175C2×Dic122—21no—
176C2×Dic122—21no—
177C2×Dic122—21no—
178C2×Dic122—24no—
179C2×Dic122—24no—
180C2×Dic122—81no—
181C2×Dic122—81no—
182C2×Dic122—81no—
183C2×Dic122—81no—
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—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C4⋊C4×C61—21no—
194C4⋊C4×C61—42no—
195C4⋊C4×C61C2×D412yes—
196C4⋊C4×C61C2×D412yes—
197C22⋊C4×C61—24no—
198C22⋊C4×C61—42no—
199C42×C61—24no—
200C22⋊Dic61—22no—
201C22⋊Dic61—22no—
202C22⋊Dic61—42no—
203C4⋊Dic61—24no—
204C4⋊Dic61—24no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—24no—
208C4⋊Dic61—22no—
209C4⋊Dic61—22no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61C2×D412yes—
213C4⋊Dic61C2×D412yes—
214#11758196—24no—
215#11758196—24no—
216#11758196—24no—
217#11758196—24no—
218#11758196—42no—
219#11758196—42no—
220#11758196—42no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61C2×D412yes—
224C4×Dic61C2×D412yes—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C23×C61—21no—
232C23×C61—21no—
233C23×C61—21no—
234C23×C61—21no—
235C23×C61—21no—
236C23×C61—21no—
237C23×C61#12412yes—
238C23×C61C22×D411yescharacteristic
239Q8×C61—41no—
240Q8×C61—41no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—22no—
250C2×C4×C61—22no—
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—41no—
261C2×C4×C61#12412yes—
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×D411yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C612−1+411yescharacteristic
267C2×C4×C612−1+411yescharacteristic
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—21no—
279C2×Dic61—21no—
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—
289C2×Dic61—41no—
290C2×Dic61—84no—
291C2×Dic61#12012yes—
292C2×Dic61#12012yes—
293Dic122—41no—
294Dic122—41no—
295Dic122—41no—
296Dic122—41no—
297Dic122—84no—
298Dic122—84no—
299C8×C61—41no—
300C8×C61—41no—
301#92418—41no—
302#92418—41no—
303#25914—611nomaximal, Sylow
304C22×C61—22no—
305C22×C61—22no—
306C22×C61—22no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61—41no—
310C22×C61—41no—
311C22×C61C2×#12011yescharacteristic
312C22×C61C2×#12411yescharacteristic
313C22×C61#53611yescharacteristic
314C22×C61#57711yescharacteristic
315C22×C61#57711yescharacteristic
316C22×C61#57811yescharacteristic
317C22×C61#57811yescharacteristic
318C4×C61—22no—
319C4×C61—42no—
320C4×C61—42no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61#54811yescharacteristic
324C4×C61#54811yescharacteristic
325C4×C61#54811yescharacteristic
326C4×C61#54811yescharacteristic
327Dic61—22no—
328Dic61—22no—
329Dic61—41no—
330Dic61—42no—
331Dic61—42no—
332Dic61—41no—
333Dic61—81no—
334Dic61—81no—
335C2×#544—611no—
336C2×#491—611no—
337C2×#487—611no—
338#2621—611no—
339#2026—611no—
340#2026—611no—
341#2024—611no—
342#2024—611no—
343#1993—611no—
344#1991—611no—
345#1873—611no—
346#1873D6111yescharacteristic
347#1872—611no—
348#1872—611no—
349#1750—611no—
350C2×C61—42no—
351C2×C61#243611yescharacteristic
352C2×C61#300911yescharacteristic
353C2×C61#301111yescharacteristic
354C2×C61#313711yescharacteristic
355C2×C61#313711yescharacteristic
356C2×C61#324211yescharacteristic
357C2×C61#324211yescharacteristic
358#544—1224no—
359#544—1224no—
360C2×C4⋊Q8—611no—
361C2×C4⋊Q8—611no—
362C2×C4⋊Q8—611no—
363C2×C4⋊Q8—1221no—
364C2×#125—611no—
365C2×#125—611no—
366C2×#125—1221no—
367C2×C22⋊Q8—611no—
368C2×C22⋊Q8—1221no—
369C2×C22⋊Q8—1221no—
370C2×C22⋊Q8C2×D6111yescharacteristic
371C2×#117—611no—
372C2×#117C2×D6111yescharacteristic
373C22×C4⋊C4—611no—
374#491—1224no—
375#487—1224no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103C2×D6111yescharacteristic
383C2×#103C2×D6111yescharacteristic
384C2×C22⋊C8—611no—
385C2×C22⋊C8C2×D6111yescharacteristic
386C2×#97—611no—
387C2×C4×C8—611no—
388#398—1221no—
389#398—1221no—
390#397—1221no—
391#397—1221no—
392#391—611no—
393#391—611no—
394#382—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#340—611no—
400#340—611no—
401#340C2×D6111yescharacteristic
402#340C2×D6111yescharacteristic
403#336—611no—
404#336—611no—
405C61#2591411yescharacteristic, complemented, Sylow
406C22×Q8—611no—
407C22×Q8—1221no—
408C22×Q8—1221no—
409C22×Q8C22×D6111yescharacteristic
410C23×C4—1221no—
411C23×C4C22×D6111yescharacteristic
412C22×C8—611no—
413C22×C8—611no—
414C22×C8C22×D6111yescharacteristic
415C22×C8C22×D6111yescharacteristic
416C4⋊Q8—614no—
417C4⋊Q8—1228no—
418C4⋊Q8—1222no—
419C4⋊Q8—1222no—
420C4⋊Q8—1222no—
421C4⋊Q8—1222no—
422C4⋊Q8—1222no—
423#125—614no—
424#125—1228no—
425#125—1222no—
426#125—1222no—
427#125—1222no—
428C22⋊Q8—22no—
429C22⋊Q8—22no—
430C22⋊Q8—1228no—
431C22⋊Q8—1228no—
432C22⋊Q8—1222no—
433C22⋊Q8—1222no—
434#117—22no—
435#117—22no—
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—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1222no—
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⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4—1221no—
455C2×C4⋊C4—1221no—
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C4⋊C4C22×D6111yescharacteristic
458C2×C4⋊C4C22×D6111yescharacteristic
459C2×C22⋊C4—21no—
460C2×C22⋊C4—21no—
461C2×C22⋊C4—611no—
462C2×C22⋊C4—611no—
463C2×C22⋊C4—1221no—
464C2×C42—21no—
465C2×C42—611no—
466C2×C42—611no—
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—22no—
476C22⋊C8—1222no—
477#97—1222no—
478C4×C8—1222no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24D4×D6111yescharacteristic
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—1224no—
493C2×Q8—1224no—
494C2×Q8—1221no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—4881no—
501C2×Q8—4881no—
502C2×Q8—4881no—
503C2×Q8—4881no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—611no—
513C22×C4—611no—
514C22×C4—611no—
515C22×C4—611no—
516C22×C4—1224no—
517C22×C4—1221no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—4881no—
522C22×C4—4881no—
523C22×C4#1175822212yes—
524C22×C4D4×D6111yescharacteristic
525C22×C4#1175822412yes—
526C22×C4C23×D6111yescharacteristic
527C2×C8—21no—
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—1221no—
534C2×C8—1221no—
535C2×C8—1221no—
536C2×C8—1221no—
537C2×C8—1221no—
538C2×C8—1221no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—42no—
544C4⋊C4—612no—
545C4⋊C4—612no—
546C4⋊C4—1222no—
547C4⋊C4—1222no—
548C4⋊C4—1224no—
549C4⋊C4—1224no—
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—1221no—
559C4⋊C4—2442no—
560C4⋊C4—2442no—
561C4⋊C4—2442no—
562C4⋊C4D4×D6112yes—
563C4⋊C4D4×D6112yes—
564C22⋊C4—24no—
565C22⋊C4—42no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—2442no—
569C42—24no—
570C42—612no—
571C42—612no—
572C42—1221no—
573C42—1221no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23#1451976312yes—
581C23C2×D4×D6111yescharacteristic
582Q8—41no—
583Q8—41no—
584Q8—41no—
585Q8—41no—
586Q8—4884no—
587Q8—4884no—
588Q8—2441no—
589Q8—2441no—
590Q8—2441no—
591Q8—2441no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
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—41no—
609C2×C4—41no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—1221no—
613C2×C4—1221no—
614C2×C4—1221no—
615C2×C4—1221no—
616C2×C4—1221no—
617C2×C4—1221no—
618C2×C4—1222no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—4884no—
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#1451976312yes—
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×#1175822411yescharacteristic
639C2×C4C2×#1175822211yescharacteristic
640C2×C4#1451982411yescharacteristic
641C2×C4#1451982411yescharacteristic
642C8—41no—
643C8—41no—
644C8—2441no—
645C8—2441no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22#120×D6111yescharacteristic
654C22C2×#1451976311yescharacteristic
655C22#1684033411yescharacteristic
656C22#1684056911yescharacteristic
657C22#1684057111yescharacteristic
658C22#1684057611yescharacteristic
659C22#1684057811yescharacteristic
660C4—22no—
661C4—42no—
662C4—42no—
663C4—81no—
664C4—81no—
665C4—1222no—
666C4—1222no—
667C4—4881no—
668C4—4881no—
669C4—2441no—
670C4—2441no—
671C4—2442no—
672C4—2442no—
673C4#1684041811yescharacteristic
674C4#1684041811yescharacteristic
675C4#1684042111yescharacteristic
676C4#1684042111yescharacteristic
677C2—42no—
678C2#2144947511yescharacteristic
679C2#2145509311yescharacteristic
680C2#2145511111yescharacteristic
681C2#2145618011yescharacteristic
682C2#2145618811yescharacteristic
683C2#2145708511yescharacteristic
684C2#2145709311yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1873C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1873D61
hypocenterC61#25914
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1873

Complements

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

NH ≅ G/Ntype
2. C2×#16840129C2semidirect
3. C2×#16840111C2semidirect
5. #21446969C2semidirect
6. #21446968C2semidirect
7. #21446959C2semidirect
8. #21446958C2semidirect
9. #21446790C2semidirect
10. #21446782C2semidirect
405. C61#25914semidirect

Maximal subgroups

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

subgroupindex
C2×#168403902
C2×#168401292
C2×#168401112
#214515892
#214469692
#214469682
#214469592
#214469582
#214467902
#214467822
#214461312
#214461272
#214461242
#214455872
C61×#18732
#2591461