Finite Group Databasebeta

← #29740618

Subgroups — #29740618

Order 15616.

874 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21449270C211yesmaximal, characteristic
2D4×C4⋊Dic61C211yesmaximal, characteristic
3#21447378C211yesmaximal, characteristic, complemented
4#21447252C211yesmaximal, characteristic, complemented
5#21447229C211yesmaximal, characteristic, complemented
6#21447016C211yesmaximal, characteristic, complemented
7#21446454C211yesmaximal, characteristic, complemented
8#21446405C211yesmaximal, characteristic, complemented
9#21446242C211yesmaximal, characteristic, complemented
10#21446124C211yesmaximal, characteristic, complemented
11#21446124C211yesmaximal, characteristic, complemented
12#21446014C211yesmaximal, characteristic, complemented
13#21445906C211yesmaximal, characteristic, complemented
14C2×#16839525C211yesmaximal, characteristic, complemented
15C61×#1869C211yesmaximal, characteristic
16C2×#14519756C2211yescharacteristic
17C2×#14519756C2211yescharacteristic
18C2×D4×Dic61—21no—
19C22×C4⋊Dic61C2211yescharacteristic
20C22×C4⋊Dic61C2211yescharacteristic
21C22×C4⋊Dic61C2211yescharacteristic
22C2×#14519709C2211yescharacteristic
23C2×#14519709C2211yescharacteristic
24C2×#14519702C2211yescharacteristic
25C2×#14519702C2211yescharacteristic
26C2×#14519702C2211yescharacteristic
27C2×#14519702C2211yescharacteristic
28#16839695—21no—
29#16839685—21no—
30#16839668C2211yescharacteristic
31#16839661—21no—
32#16839660C2211yescharacteristic
33#16839642—21no—
34#16839637—21no—
35#16839637—21no—
36#16839632C2211yescharacteristic
37#16839622C2211yescharacteristic
38#16839622C2211yescharacteristic
39#16839622C2211yescharacteristic
40#16839620—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52C23×C8×C61C2211yescharacteristic
53C2×C4×D4×C61C2211yescharacteristic
54C2×C22⋊C8×C61C2211yescharacteristic
55C2×C22⋊C8×C61C2211yescharacteristic
56C2×C22⋊C8×C61C2211yescharacteristic
57C61×#336C2211yescharacteristic
58C61×#336C2211yescharacteristic
59C22×Dic122C2311yescharacteristic
60C23×Dic61—21no—
61C23×Dic61—21no—
62C23×Dic61—21no—
63#14519756—22no—
64#14519756—22no—
65#14519756—22no—
66#14519756—41no—
67D4×Dic61—44no—
68C2×C4⋊Dic61—22no—
69C2×C4⋊Dic61—22no—
70C2×C4⋊Dic61—22no—
71C2×C4⋊Dic61—21no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61—21no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—41no—
77C2×C4⋊Dic61—41no—
78C2×C4⋊Dic61—41no—
79C2×C4⋊Dic61C2311yescharacteristic
80C2×C4⋊Dic61C2311yescharacteristic
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×C22⋊Dic61—21no—
90C2×C22⋊Dic61—21no—
91C2×C22⋊Dic61—21no—
92C2×C22⋊Dic61—21no—
93C2×C22⋊Dic61—21no—
94C2×C4×Dic61—21no—
95#14519709—22no—
96#14519709—41no—
97#14519702—22no—
98#14519702—41no—
99#14519702—41no—
100#14519702—41no—
101#14519684—21no—
102#14519684—21no—
103#14519684—21no—
104#14519684—21no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×D4×C61D411yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C23×C4×C61C2311yescharacteristic
111C22×C8×C61—22no—
112C22×C8×C61—41no—
113C22×C8×C61—41no—
114C22×C8×C61C2311yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C4×D4×C61—22no—
119C4×D4×C61—22no—
120C2×C4⋊C4×C61D411yescharacteristic
121C2×C22⋊C4×C61D411yescharacteristic
122C2×C22⋊C4×C61D411yescharacteristic
123C2×C42×C61C2311yescharacteristic
124C22⋊C8×C61—22no—
125C22⋊C8×C61—41no—
126C22⋊C8×C61—41no—
127C24×C61C2×D411yescharacteristic
128C24×C61C2×D411yescharacteristic
129C2×D4×C61—22no—
130C2×D4×C61—22no—
131C2×D4×C61SD1614yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61#4112yes—
147C22×C4×C61#4112yes—
148C22×C4×C61C2411yescharacteristic
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—42no—
162C22×Dic61—42no—
163C22×Dic61—81no—
164C22×Dic61—81no—
165C22×Dic61—81no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—44no—
185C4⋊C4×C61—22no—
186C22⋊C4×C61—22no—
187C22⋊C4×C61—22no—
188C42×C61—21no—
189C42×C61—21no—
190C22⋊Dic61—42no—
191C22⋊Dic61—42no—
192C22⋊Dic61—42no—
193C22⋊Dic61—42no—
194C22⋊Dic61—81no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—42no—
208C4⋊Dic61—42no—
209C4⋊Dic61—42no—
210C4⋊Dic61—42no—
211C4⋊Dic61—81no—
212C4⋊Dic61#4112yes—
213C4⋊Dic61#4112yes—
214#11758196—42no—
215#11758196—42no—
216#11758196—81no—
217C4×Dic61—42no—
218C23×C61—21no—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61#12312yes—
229C23×C61C2×SD1612yes—
230C23×C61C2×SD1612yes—
231C23×C61#13712yes—
232C23×C61C22×D411yescharacteristic
233D4×C61—28no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—22no—
241C2×C4×C61—22no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61#12312yes—
253C2×C4×C61C2×SD1612yes—
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61C2×#4111yescharacteristic
258C2×C4×C61H(5,𝔽2)11yescharacteristic
259C2×C4×C61H(5,𝔽2)11yescharacteristic
260C2×Dic61—22no—
261C2×Dic61—22no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61—41no—
267C2×Dic61—41no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
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—84no—
285C2×Dic61—84no—
286C2×Dic61—84no—
287Dic122—41no—
288Dic122—41no—
289Dic122—41no—
290Dic122—41no—
291C8×C61—41no—
292C8×C61—41no—
293C8×C61—41no—
294C8×C61—41no—
295#26200—611nomaximal, Sylow
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61—22no—
301C22×C61—22no—
302C22×C61—22no—
303C22×C61—22no—
304C22×C61—22no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61#48314yes—
310C22×C61C2×#12311yescharacteristic
311C22×C61#54611yescharacteristic
312C22×C61#54611yescharacteristic
313C22×C61C22×SD1611yescharacteristic
314C22×C61C2×#13711yescharacteristic
315C22×C61#57611yescharacteristic
316C22×C61#57811yescharacteristic
317C4×C61—22no—
318C4×C61—22no—
319C4×C61—41no—
320C4×C61—41no—
321C4×C61#52911yescharacteristic
322C4×C61#53811yescharacteristic
323C4×C61#55311yescharacteristic
324C4×C61#55311yescharacteristic
325Dic61—42no—
326Dic61—42no—
327Dic61—81no—
328Dic61—81no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332Dic61—81no—
333#2417—611no—
334D4×C4⋊C4—611no—
335#2099—611no—
336#2077—611no—
337#2072—611no—
338#2035—611no—
339#1927—611no—
340#1920—611no—
341#1892—611no—
342#1872—611no—
343#1872—611no—
344#1869D6111yescharacteristic
345#1853—611no—
346#1823—611no—
347C2×#340—611no—
348C2×C61—24no—
349C2×C61—42no—
350C2×C61#244411yescharacteristic
351C2×C61C2×#48311yescharacteristic
352C2×C61#309811yescharacteristic
353C2×C61#329511yescharacteristic
354C2×C61#329811yescharacteristic
355C2×C61#331211yescharacteristic
356C2×C61#331411yescharacteristic
357C23×C8C2×D6111yescharacteristic
358C2×C22⋊Q8—611no—
359C2×C22⋊Q8—611no—
360C2×C4×D4—1221no—
361C2×C4×D4C2×D6111yescharacteristic
362C22×C4⋊C4—611no—
363C22×C4⋊C4—611no—
364C22×C4⋊C4—611no—
365C2×#106—611no—
366C2×#106—611no—
367C2×#103—611no—
368C2×#103—611no—
369C2×#103—611no—
370C2×#103—611no—
371C2×C22⋊C8C2×D6111yescharacteristic
372C2×C22⋊C8C2×D6111yescharacteristic
373C2×C22⋊C8C2×D6111yescharacteristic
374#398—1221no—
375#396—1221no—
376#391—611no—
377#389—611no—
378#389—1221no—
379#386—1221no—
380#385—1221no—
381#385—1221no—
382#384—611no—
383#380—611no—
384#380—611no—
385#380—611no—
386#380—1221no—
387C4×C4⋊C4—611no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#336C2×D6111yescharacteristic
399#336C2×D6111yescharacteristic
400C61#2620011yescharacteristic, complemented, Sylow
401C22×Q8—611no—
402C22×D4D24411yescharacteristic
403C23×C4—1221no—
404C23×C4—1221no—
405C23×C4—1221no—
406C23×C4C22×D6111yescharacteristic
407C23×C4C22×D6111yescharacteristic
408C22×C8—22no—
409C22×C8—41no—
410C22×C8—41no—
411C22×C8C22×D6111yescharacteristic
412C22×C8C22×D6111yescharacteristic
413C22×C8C22×D6111yescharacteristic
414C22×C8C22×D6111yescharacteristic
415C22⋊Q8—1222no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—2441no—
419C4×D4—22no—
420C4×D4—22no—
421C4×D4—2444no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—611no—
427C2×C4⋊C4—611no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—1222no—
430C2×C4⋊C4—1222no—
431C2×C4⋊C4—1222no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—2441no—
441C2×C4⋊C4—2441no—
442C2×C4⋊C4—2441no—
443C2×C4⋊C4D24411yescharacteristic
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4—1221no—
448C2×C22⋊C4—1221no—
449C2×C22⋊C4D24411yescharacteristic
450C2×C22⋊C4D24411yescharacteristic
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#106—1222no—
454#106—2441no—
455#103—1222no—
456#103—2441no—
457#103—2441no—
458#103—2441no—
459C22⋊C8—22no—
460C22⋊C8—41no—
461C22⋊C8—41no—
462#95—1221no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469C24C2×D24411yescharacteristic
470C24C2×D24411yescharacteristic
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×Q8—1221no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×D4—22no—
478C2×D4—22no—
479C2×D4#1175819014yes—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—4881no—
499C22×C4—4881no—
500C22×C4—4881no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2442no—
505C22×C4—2442no—
506C22×C4—2442no—
507C22×C4C2×D24411yescharacteristic
508C22×C4C2×D24411yescharacteristic
509C22×C4C2×D24411yescharacteristic
510C22×C4C2×D24411yescharacteristic
511C22×C4#1175822412yes—
512C22×C4#1175822412yes—
513C22×C4C23×D6111yescharacteristic
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8—21no—
518C2×C8—21no—
519C2×C8—21no—
520C2×C8—21no—
521C2×C8—21no—
522C2×C8—21no—
523C2×C8—21no—
524C2×C8—21no—
525C2×C8—21no—
526C2×C8—44no—
527C4⋊C4—22no—
528C4⋊C4—612no—
529C4⋊C4—612no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—1221no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—4881no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4—2442no—
546C4⋊C4—2442no—
547C4⋊C4—2442no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C22⋊C4—22no—
551C22⋊C4—22no—
552C22⋊C4—4881no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C22⋊C4—2442no—
556C22⋊C4—2442no—
557C42—21no—
558C42—21no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23#1451976112yes—
571C23C2×#1175819012yes—
572C23C2×#1175819012yes—
573C23#1451981012yes—
574C23C22×D24411yescharacteristic
575Q8—2441no—
576Q8—2441no—
577Q8—2441no—
578Q8—2441no—
579D4—28no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—4884no—
601C2×C4—4884no—
602C2×C4—4884no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4—2441no—
624C2×C4—2441no—
625C2×C4#1451976112yes—
626C2×C4C2×#1175819012yes—
627C2×C4#41×D6111yescharacteristic
628C2×C4#41×D6111yescharacteristic
629C2×C4C2×#1175822411yescharacteristic
630C2×C4C2×#1175822411yescharacteristic
631C2×C4#1451982211yescharacteristic
632C2×C4#1451982211yescharacteristic
633C8—41no—
634C8—41no—
635C8—41no—
636C8—41no—
637C22—22no—
638C22—22no—
639C22—22no—
640C22—22no—
641C22—22no—
642C22—22no—
643C22—22no—
644C22—22no—
645C22—22no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22—41no—
650C22#1684009314yes—
651C22C2×#1451976111yescharacteristic
652C22#1684040711yescharacteristic
653C22#1684040711yescharacteristic
654C22C22×#1175819011yescharacteristic
655C22C2×#1451981011yescharacteristic
656C22#1684056711yescharacteristic
657C22#1684057911yescharacteristic
658C4—22no—
659C4—22no—
660C4—41no—
661C4—41no—
662C4—4881no—
663C4—4881no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—4881no—
668C4—2442no—
669C4—2442no—
670C4#1684028911yescharacteristic
671C4#1684034711yescharacteristic
672C4#1684045511yescharacteristic
673C4#1684045511yescharacteristic
674C2—24no—
675C2—42no—
676C2#2144956711yescharacteristic
677C2C2×#1684009311yescharacteristic
678C2#2145579911yescharacteristic
679C2#2145764411yescharacteristic
680C2#2145768511yescharacteristic
681C2#2145787211yescharacteristic
682C2#2145790211yescharacteristic
subgroupG/H
centerC23C22×D244
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1869C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#1869D61
hypocenterC61#26200
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #21447378C2semidirect
4. #21447252C2semidirect
5. #21447229C2semidirect
6. #21447016C2semidirect
7. #21446454C2semidirect
8. #21446405C2semidirect
9. #21446242C2semidirect
10. #21446124C2semidirect
11. #21446124C2semidirect
12. #21446014C2semidirect
13. #21445906C2semidirect
14. C2×#16839525C2semidirect
400. C61#26200semidirect

Maximal subgroups

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

subgroupindex
#214492702
D4×C4⋊Dic612
#214473782
#214472522
#214472292
#214470162
#214464542
#214464052
#214462422
#214461242
#214460142
#214459062
C2×#168395252
C61×#18692
#2620061