Finite Group Databasebeta

← #29688012

Subgroups — #29688012

Order 15616.

818 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21448561C211yesmaximal, characteristic, complemented
2#21448532C211yesmaximal, characteristic
3#21446789C211yesmaximal, characteristic, complemented
4#21446636C211yesmaximal, characteristic
5#21446518C211yesmaximal, characteristic
6#21446511C211yesmaximal, characteristic, complemented
7#21446504C211yesmaximal, characteristic
8#21446497C211yesmaximal, characteristic, complemented
9#21445947C211yesmaximal, characteristic, complemented
10#21445651C211yesmaximal, characteristic, complemented
11#21445646C211yesmaximal, characteristic
12#103×Dic61C211yesmaximal, characteristic, complemented
13#102×Dic61C211yesmaximal, characteristic
14#21445505C211yesmaximal, characteristic, complemented
15C2×C61×#488C211yesmaximal, characteristic
16C2×Q8×Dic61C2211yescharacteristic
17C2×D4×Dic61C2211yescharacteristic
18C2×#14519701C2211yescharacteristic
19C2×#14519701C2211yescharacteristic
20C2×#14519698C2211yescharacteristic
21C2×#14519698C2211yescharacteristic
22C2×#14519688C2211yescharacteristic
23C2×#14519687C2211yescharacteristic
24C2×#14519687C2211yescharacteristic
25C2×C8×Dic61C2211yescharacteristic
26C2×C8×Dic61C2211yescharacteristic
27#16839669C2211yescharacteristic
28#16839665C2211yescharacteristic
29#16839661—21no—
30#16839661—21no—
31#16839661C2211yescharacteristic
32#16839661C2211yescharacteristic
33#16839655—21no—
34#16839655—21no—
35#16839648—21no—
36#16839648—21no—
37#16839637—21no—
38#16839637—21no—
39#16839632C2211yescharacteristic
40#16839632C2211yescharacteristic
41C4⋊C4×Dic61C2211yescharacteristic
42C4⋊C4×Dic61C2211yescharacteristic
43#16839608C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52C2×#125×C61C2211yescharacteristic
53C2×#124×C61C2211yescharacteristic
54C61×#488C418yescomplemented
55C2×#103×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C2×C4×C8×C61C2211yescharacteristic
60C23×Dic61—21no—
61C22×#92418C2311yescharacteristic
62C22×#92418C2311yescharacteristic
63Q8×Dic61—22no—
64Q8×Dic61—22no—
65D4×Dic61—22no—
66D4×Dic61—22no—
67C2×C4⋊Dic61—21no—
68C2×C4⋊Dic61—21no—
69C2×C4⋊Dic61—21no—
70C2×C4⋊Dic61—21no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C4×Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61C2311yescharacteristic
85C2×C4×Dic61C2311yescharacteristic
86#14519701—22no—
87#14519701—22no—
88#14519698—22no—
89#14519698—22no—
90#14519688—22no—
91#14519687—22no—
92#14519687—22no—
93C8×Dic61—22no—
94C8×Dic61—22no—
95#14519684—21no—
96#14519684—21no—
97#14519684—21no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101#14519684D411yescharacteristic
102#14519684D411yescharacteristic
103#14519684D411yescharacteristic
104#14519684D411yescharacteristic
105C22×Q8×C61C2311yescharacteristic
106C22×D4×C61C2311yescharacteristic
107C22×C8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#125×C61—22no—
110#125×C61C2×C414yescomplemented
111#124×C61—22no—
112#124×C61C2×C414yes—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61C2311yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C22⋊C4×C61—21no—
118C2×C22⋊C4×C61—21no—
119C2×C42×C61C2311yescharacteristic
120#103×C61C2×C414yescomplemented
121#103×C61C2×C414yescomplemented
122#102×C61C2×C414yes—
123#102×C61C2×C414yes—
124C4×C8×C61C2×C414yescomplemented
125C24×C61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61—21no—
130C2×Q8×C61—21no—
131C2×Q8×C61C22×C412yes—
132C2×D4×C61—21no—
133C2×D4×C61—21no—
134C2×D4×C61—21no—
135C2×D4×C61—21no—
136C2×D4×C61—21no—
137C2×D4×C61C22×C412yes—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2411yescharacteristic
144C22×Dic61—21no—
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—42no—
149C22×Dic61—81no—
150C22×Dic61—81no—
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C22×Dic61C2×D411yescharacteristic
155C2×C8×C61—21no—
156C2×C8×C61—21no—
157C2×C8×C61—21no—
158C2×C8×C61—21no—
159C2×C8×C61C22×C412yes—
160C2×C8×C61C22×C412yes—
161C4⋊C4×C61—24no—
162C4⋊C4×C61—24no—
163C4⋊C4×C61—21no—
164C4⋊C4×C61—21no—
165C4⋊C4×C61C22×C412yes—
166C4⋊C4×C61C22×C412yes—
167C22⋊C4×C61—24no—
168C22⋊C4×C61—24no—
169C42×C61C22×C412yes—
170C42×C61#4112yes—
171C22⋊Dic61—42no—
172C22⋊Dic61—81no—
173C22⋊Dic61—81no—
174C4⋊Dic61—21no—
175C4⋊Dic61—21no—
176C4⋊Dic61—21no—
177C4⋊Dic61—21no—
178C4⋊Dic61—21no—
179C4⋊Dic61—21no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—42no—
183C4⋊Dic61—81no—
184C4⋊Dic61—81no—
185C4⋊Dic61—81no—
186C4⋊Dic61—81no—
187C4⋊Dic61—81no—
188C4⋊Dic61—81no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61—42no—
192C4×Dic61—81no—
193C4×Dic61—81no—
194C4×Dic61#4112yes—
195C4×Dic61#4112yes—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418#4112yes—
201C2×#92418#4112yes—
202C23×C61—22no—
203C23×C61—41no—
204C23×C61—41no—
205C23×C61—41no—
206C23×C61—41no—
207C23×C61—41no—
208C23×C61—41no—
209C23×C61C22×D411yescharacteristic
210Q8×C61—22no—
211Q8×C61—22no—
212Q8×C61—41no—
213Q8×C61—41no—
214D4×C61—22no—
215D4×C61—22no—
216D4×C61—41no—
217D4×C61—41no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—22no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61C4×D412yes—
232C2×C4×C61C4×D412yes—
233C2×C4×C61C23×C411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C2×C4×C61C2×#4111yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C2×C4×C61C2×#4111yescharacteristic
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61—21no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—41no—
250C2×Dic61—41no—
251C2×Dic61—41no—
252C2×Dic61—41no—
253C2×Dic61—41no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—84no—
262C2×Dic61#12412yes—
263C2×Dic61#12412yes—
264C8×C61—22no—
265C8×C61—22no—
266#92418—22no—
267#92418—22no—
268#21643—611nomaximal, Sylow
269C22×C61—42no—
270C22×C61—42no—
271C22×C61—42no—
272C22×C61—81no—
273C22×C61—81no—
274C22×C61—81no—
275C22×C61—81no—
276C22×C61C2×C4×D411yescharacteristic
277C22×C61C2×#12411yescharacteristic
278C22×C61#53211yescharacteristic
279C22×C61#57611yescharacteristic
280C22×C61#57711yescharacteristic
281C22×C61#57711yescharacteristic
282C22×C61#57811yescharacteristic
283C4×C61—22no—
284C4×C61—22no—
285C4×C61—42no—
286C4×C61—42no—
287C4×C61—42no—
288C4×C61C4×#4111yescharacteristic
289C4×C61C4×#4111yescharacteristic
290C4×C61#52911yescharacteristic
291C4×C61#52911yescharacteristic
292Dic61—22no—
293Dic61—22no—
294Dic61—41no—
295Dic61—41no—
296Dic61—81no—
297Dic61—81no—
298Dic61—81no—
299Dic61—81no—
300C2×#488D6111yescharacteristic
301#2330—611no—
302#2326—611no—
303#1992—611no—
304#1959—611no—
305#1936—611no—
306#1935—611no—
307#1934—611no—
308#1933—611no—
309#1834—611no—
310#1763—611no—
311#1762—611no—
312C4×#103—611no—
313C4×#102—611no—
314#1726—611no—
315C2×C61—82no—
316C2×C61C4×#12411yescharacteristic
317C2×C61#295111yescharacteristic
318C2×C61#295211yescharacteristic
319C2×C61#313611yescharacteristic
320C2×C61#313711yescharacteristic
321C2×C61#317811yescharacteristic
322C2×C61#317911yescharacteristic
323C2×#125C2×D6111yescharacteristic
324C2×#124C2×D6111yescharacteristic
325C2×C4×Q8—611no—
326C2×C4×D4—611no—
327#488Dic6118yescomplemented
328C2×#103—611no—
329C2×#103—611no—
330C2×#103C2×D6111yescharacteristic
331C2×#103C2×D6111yescharacteristic
332C2×#102—611no—
333C2×#102—611no—
334C2×#102C2×D6111yescharacteristic
335C2×#102C2×D6111yescharacteristic
336C2×#97—611no—
337C2×#97—611no—
338C2×#97—611no—
339C2×C4×C8—611no—
340C2×C4×C8—611no—
341C2×C4×C8C2×D6111yescharacteristic
342#391—611no—
343#390—611no—
344#389—611no—
345#389—611no—
346#389—1221no—
347#389—1221no—
348#388—1221no—
349#388—1221no—
350#387—1221no—
351#387—1221no—
352#385—1221no—
353#385—1221no—
354#384—611no—
355#384—611no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358#376—611no—
359#340—611no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367C61#2164311yescharacteristic, complemented, Sylow
368C22×Q8C22×D6111yescharacteristic
369C22×D4C22×D6111yescharacteristic
370C23×C4—1221no—
371C22×C8—611no—
372C22×C8—611no—
373C22×C8C22×D6111yescharacteristic
374C22×C8C22×D6111yescharacteristic
375#125—22no—
376#125C2×Dic6114yescomplemented
377#124—22no—
378#124C2×Dic6114yes—
379C4×Q8—1222no—
380C4×Q8—1222no—
381C4×D4—1222no—
382C4×D4—1222no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4—611no—
387C2×C4⋊C4—611no—
388C2×C4⋊C4—611no—
389C2×C4⋊C4—1221no—
390C2×C4⋊C4—1221no—
391C2×C4⋊C4—1221no—
392C2×C4⋊C4—1221no—
393C2×C4⋊C4—1221no—
394C2×C4⋊C4—1221no—
395C2×C4⋊C4—1221no—
396C2×C4⋊C4C22×D6111yescharacteristic
397C2×C4⋊C4C22×D6111yescharacteristic
398C2×C22⋊C4—21no—
399C2×C22⋊C4—21no—
400C2×C22⋊C4—1221no—
401C2×C22⋊C4—1221no—
402C2×C22⋊C4—1221no—
403C2×C42—611no—
404C2×C42—611no—
405C2×C42—1221no—
406C2×C42—1221no—
407C2×C42—1221no—
408C2×C42C22×D6111yescharacteristic
409#103—1222no—
410#103—1222no—
411#103C2×Dic6114yescomplemented
412#103C2×Dic6114yescomplemented
413#102—1222no—
414#102—1222no—
415#102C2×Dic6114yes—
416#102C2×Dic6114yes—
417#97—1222no—
418#97—1222no—
419#97—1222no—
420C4×C8—1222no—
421C4×C8—1222no—
422C4×C8C2×Dic6114yescomplemented
423#95—611no—
424#95—611no—
425#95—611no—
426#95—611no—
427#95—1221no—
428#95—1221no—
429#95—1221no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433C24—21no—
434C2×Q8—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8C22×Dic6112yes—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4C22×Dic6112yes—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—611no—
450C22×C4—611no—
451C22×C4—611no—
452C22×C4—611no—
453C22×C4—1221no—
454C22×C4—1221no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—4881no—
458C22×C4—4881no—
459C22×C4—2442no—
460C22×C4D4×D6111yescharacteristic
461C22×C4D4×D6111yescharacteristic
462C22×C4C23×D6111yescharacteristic
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C22×Dic6112yes—
474C2×C8C22×Dic6112yes—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—24no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C4⋊C4—1221no—
486C4⋊C4—1221no—
487C4⋊C4—4881no—
488C4⋊C4—4881no—
489C4⋊C4—4881no—
490C4⋊C4—4881no—
491C4⋊C4—4881no—
492C4⋊C4—4881no—
493C4⋊C4—2442no—
494C4⋊C4C22×Dic6112yes—
495C4⋊C4C22×Dic6112yes—
496C22⋊C4—24no—
497C22⋊C4—24no—
498C22⋊C4—4881no—
499C22⋊C4—4881no—
500C22⋊C4—2442no—
501C42—612no—
502C42—612no—
503C42—1221no—
504C42—1221no—
505C42—4881no—
506C42—4881no—
507C42—2442no—
508C42#1175822412yes—
509C42C22×Dic6112yes—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23C2×D4×D6111yescharacteristic
518Q8—22no—
519Q8—22no—
520Q8—41no—
521Q8—41no—
522D4—22no—
523D4—22no—
524D4—41no—
525D4—41no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1221no—
547C2×C4—1221no—
548C2×C4—1221no—
549C2×C4—1221no—
550C2×C4—1221no—
551C2×C4—4884no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4—2441no—
556C2×C4—2441no—
557C2×C4—2441no—
558C2×C4—2441no—
559C2×C4—2441no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4D4×Dic6112yes—
565C2×C4D4×Dic6112yes—
566C2×C4C23×Dic6111yescharacteristic
567C2×C4#41×D6111yescharacteristic
568C2×C4#41×D6111yescharacteristic
569C2×C4#41×D6111yescharacteristic
570C2×C4#41×D6111yescharacteristic
571C2×C4C2×#1175822411yescharacteristic
572C8—22no—
573C8—22no—
574C8—1222no—
575C8—1222no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22—81no—
580C22—81no—
581C22—81no—
582C22—81no—
583C22C2×D4×Dic6111yescharacteristic
584C22#124×D6111yescharacteristic
585C22#1684031111yescharacteristic
586C22#1684056411yescharacteristic
587C22#1684056911yescharacteristic
588C22#1684056911yescharacteristic
589C22#1684057611yescharacteristic
590C4—22no—
591C4—22no—
592C4—42no—
593C4—42no—
594C4—42no—
595C4—1222no—
596C4—1222no—
597C4—4881no—
598C4—4881no—
599C4—4881no—
600C4—4881no—
601C4—2441no—
602C4—2441no—
603C4#41×Dic6111yescharacteristic
604C4#41×Dic6111yescharacteristic
605C4#1684029111yescharacteristic
606C4#1684029111yescharacteristic
607C2—82no—
608C2#124×Dic6111yescharacteristic
609C2#2145469511yescharacteristic
610C2#2145470311yescharacteristic
611C2#2145617211yescharacteristic
612C2#2145618311yescharacteristic
613C2#2145648611yescharacteristic
614C2#2145650111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#488C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenterC2×#488D61
hypocenterC61#21643
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#488

Complements

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

NH ≅ G/Ntype
1. #21448561C2semidirect
3. #21446789C2semidirect
6. #21446511C2semidirect
8. #21446497C2semidirect
9. #21445947C2semidirect
10. #21445651C2semidirect
12. #103×Dic61C2semidirect
14. #21445505C2semidirect
54. C61×#488C4semidirect
110. #125×C61C2×C4semidirect
120. #103×C61C2×C4semidirect
121. #103×C61C2×C4semidirect
124. C4×C8×C61C2×C4semidirect
327. #488Dic61semidirect
367. C61#21643semidirect
376. #125C2×Dic61semidirect
411. #103C2×Dic61semidirect
412. #103C2×Dic61semidirect
422. C4×C8C2×Dic61semidirect

Maximal subgroups

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

subgroupindex
#214485612
#214485322
#214467892
#214466362
#214465182
#214465112
#214465042
#214464972
#214459472
#214456512
#214456462
#103×Dic612
#102×Dic612
#214455052
C2×C61×#4882
#2164361