Finite Group Databasebeta

← #29729871

Subgroups — #29729871

Order 15616.

894 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840111C211yesmaximal, characteristic, complemented
2C2×#16839965C211yesmaximal, characteristic
3#21450770C211yesmaximal, characteristic
4#21448876C211yesmaximal, characteristic, complemented
5#21447231C211yesmaximal, characteristic, complemented
6#21447227C211yesmaximal, characteristic, complemented
7#21447135C211yesmaximal, characteristic
8#21446769C211yesmaximal, characteristic, complemented
9#21446534C211yesmaximal, characteristic, complemented
10#21446177C211yesmaximal, characteristic
11#21446176C211yesmaximal, characteristic
12#21445825C211yesmaximal, characteristic, complemented
13#21445822C211yesmaximal, characteristic, complemented
14#102×Dic61C211yesmaximal, characteristic
15C61×#2055C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×#14519776C2211yescharacteristic
18C2×#14519775C2211yescharacteristic
19C2×#14519754—21no—
20C2×#14519754—21no—
21C2×Q8×Dic61C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×D4×Dic61C2211yescharacteristic
24#16840111—24no—
25#16839965—24no—
26C2×#14519706C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×#14519701C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×C8×Dic61C2211yescharacteristic
35#16839698—21no—
36#16839698—21no—
37#16839667C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839659—21no—
40#16839649—21no—
41#16839644—21no—
42#16839632C2211yescharacteristic
43#16839631C2211yescharacteristic
44C4⋊C4×Dic61C2211yescharacteristic
45C4×#11758196C2211yescharacteristic
46#16839613—21no—
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839517C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×SD16×C61C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#390C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122—21no—
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C2×#11758200—22no—
65C2×#11758200—22no—
66C2×#11758200—41no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69#14519776—22no—
70#14519776—22no—
71#14519775—22no—
72#14519775D414yes—
73#14519754—44no—
74#14519754—44no—
75Q8×Dic61—22no—
76Q8×Dic61—41no—
77C4×Dic122—22no—
78C4×Dic122—41no—
79D4×Dic61—22no—
80D4×Dic61—22no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61C2311yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196D411yescharacteristic
94C2×#11758196D411yescharacteristic
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×C22⋊Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61C2311yescharacteristic
102C2×C4×Dic61C2311yescharacteristic
103#14519706—22no—
104#14519706—41no—
105#14519702—22no—
106#14519702—22no—
107#14519701—22no—
108#14519701—22no—
109#14519700—22no—
110#14519698—22no—
111C8×Dic61—22no—
112#14519684—21no—
113#14519684—21no—
114#14519684—21no—
115#14519684D411yescharacteristic
116#14519684D411yescharacteristic
117C22×Q8×C61C2311yescharacteristic
118C22×D4×C61C2311yescharacteristic
119C2×SD16×C61—22no—
120C2×SD16×C61—22no—
121C2×SD16×C61—41no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C22⋊C4×C61—21no—
127C2×C22⋊C4×C61—21no—
128C2×C42×C61C2311yescharacteristic
129C4⋊C8×C61—41no—
130#103×C61—22no—
131#102×C61—22no—
132#95×C61—21no—
133#95×C61—21no—
134C24×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61C2×D412yes—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61—21no—
144C2×D4×C61—21no—
145C2×D4×C61—21no—
146C2×D4×C61#4112yes—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—42no—
170C2×Dic122—42no—
171C2×Dic122—42no—
172C2×Dic122C2×D412yes—
173SD16×C61—44no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—42no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191#11758200—44no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—42no—
198C4⋊Dic61—42no—
199C4⋊Dic61—42no—
200C4⋊Dic61#4112yes—
201#11758196—24no—
202#11758196—24no—
203#11758196—22no—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—42no—
208#11758196SD1614yescomplemented
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61C2×D412yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C23×C61—22no—
223C23×C61—41no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230Q8×C61—22no—
231Q8×C61—22no—
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—22no—
235D4×C61—22no—
236D4×C61—41no—
237D4×C61—41no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—22no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—41no—
245C2×C4×C61—41no—
246C2×C4×C61—41no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61C2×SD1612yes—
253C2×C4×C61C22×D411yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61C2×#4111yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C61H(5,𝔽2)11yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—22no—
264C2×Dic61—21no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—21no—
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—84no—
279C2×Dic61#12112yes—
280C2×Dic61#12112yes—
281C2×Dic61C2×SD1612yes—
282C2×Dic61C2×SD1612yes—
283C2×Dic61AGL(1,ℤ/8)12yes—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122—84no—
289C8×C61—41no—
290C8×C61—41no—
291#92418—41no—
292#92418—41no—
293#25320—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×#12111yescharacteristic
302C22×C61#54611yescharacteristic
303C22×C61#54811yescharacteristic
304C22×C61C22×SD1611yescharacteristic
305C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
306C22×C61#57811yescharacteristic
307C22×C61#57811yescharacteristic
308C4×C61—22no—
309C4×C61—41no—
310C4×C61—42no—
311C4×C61—81no—
312C4×C61—81no—
313C4×C61#52911yescharacteristic
314C4×C61#54211yescharacteristic
315C4×C61#54611yescharacteristic
316C4×C61#54711yescharacteristic
317Dic61—22no—
318Dic61—22no—
319Dic61—41no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323Dic61—81no—
324Dic61#46014yes—
325C2×#487—611no—
326C2×#461—611no—
327#2542—611no—
328#2365—611no—
329#2072—611no—
330#2072—611no—
331#2055—611no—
332#2055D6111yescharacteristic
333#1988—611no—
334#1939—611no—
335#1880—611no—
336#1880—611no—
337#1803—611no—
338#1803—611no—
339C4×#102—611no—
340C2×C61—82no—
341C2×C61#242711yescharacteristic
342C2×C61C2×#46011yescharacteristic
343C2×C61#304411yescharacteristic
344C2×C61#329811yescharacteristic
345C2×C61#333011yescharacteristic
346C2×C61#335911yescharacteristic
347C2×C61#336711yescharacteristic
348C22×SD16—611no—
349C22×SD16C2×D6111yescharacteristic
350C2×C4⋊Q8—611no—
351C2×C4⋊Q8—611no—
352C2×C22⋊Q8—1221no—
353C2×C22⋊Q8—1221no—
354C2×C4×Q8—611no—
355C2×C4×Q8—611no—
356C2×C4×D4—611no—
357#487—1224no—
358#461—1224no—
359C2×C4⋊C8—611no—
360C2×C4⋊C8—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#103C2×D6111yescharacteristic
368C2×#102—611no—
369C2×#102C2×D6111yescharacteristic
370C2×C4×C8—611no—
371#399—1221no—
372#399—1221no—
373#391—611no—
374#391C2×D6111yescharacteristic
375#390—611no—
376#390C2×D6111yescharacteristic
377#389—1221no—
378#387—1221no—
379#387—1221no—
380#384—611no—
381#384—611no—
382C4×C4⋊C4—611no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—1221no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340—611no—
389#340—611no—
390#340C2×D6111yescharacteristic
391#336—611no—
392#336—611no—
393C61#2532011yescharacteristic, complemented, Sylow
394C22×Q8—611no—
395C22×Q8—1221no—
396C22×Q8C22×D6111yescharacteristic
397C22×D4C22×D6111yescharacteristic
398C23×C4—1221no—
399C2×SD16—22no—
400C2×SD16—22no—
401C2×SD16—41no—
402C2×SD16—1222no—
403C2×SD16—1222no—
404C2×SD16—2441no—
405C22×C8—611no—
406C22×C8—611no—
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409C4⋊Q8—614no—
410C4⋊Q8—1222no—
411C4⋊Q8—1222no—
412C4⋊Q8—1222no—
413C22⋊Q8—2444no—
414C22⋊Q8—2444no—
415C4×Q8—1222no—
416C4×Q8—1222no—
417C4×Q8—2441no—
418C4×Q8—2441no—
419C4×D4—1222no—
420C4×D4—1222no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4—1221no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4—1221no—
432C2×C4⋊C4—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C4⋊C4C22×D6111yescharacteristic
437C2×C22⋊C4—21no—
438C2×C22⋊C4—21no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C22⋊C4—1221no—
442C2×C42—611no—
443C2×C42—611no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42C22×D6111yescharacteristic
448C4⋊C8—41no—
449C4⋊C8—1222no—
450C4⋊C8—2441no—
451#103—22no—
452#103—1222no—
453#103—1222no—
454#103—1222no—
455#103—1222no—
456#103—1222no—
457#102—22no—
458#102—1222no—
459C4×C8—1222no—
460#95—21no—
461#95—21no—
462#95—611no—
463#95—611no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467C24—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—612no—
474C2×Q8—1221no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—2442no—
480C2×Q8—2442no—
481C2×Q8—2442no—
482C2×Q8D4×D6112yes—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4#1175822412yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—611no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—2442no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506SD16—44no—
507SD16—2444no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—614no—
525C4⋊C4—612no—
526C4⋊C4—1222no—
527C4⋊C4—1224no—
528C4⋊C4—1224no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—1221no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C4⋊C4—2442no—
541C22⋊C4—42no—
542C22⋊C4—42no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C22⋊C4—2442no—
546C42—21no—
547C42—21no—
548C42—612no—
549C42—612no—
550C42—1221no—
551C42—1221no—
552C42—2442no—
553C42—2442no—
554C42—2442no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23C2×D4×D6111yescharacteristic
563Q8—22no—
564Q8—22no—
565Q8—41no—
566Q8—41no—
567Q8—1222no—
568Q8—1222no—
569Q8—4884no—
570Q8—2441no—
571Q8—2441no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—612no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—612no—
594C2×C4—612no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1222no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—4884no—
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×C4SD16×D6112yes—
616C2×C4C2×D4×D6111yescharacteristic
617C2×C4#41×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4C2×#1175822411yescharacteristic
620C2×C4#1451982211yescharacteristic
621C2×C4#1451982411yescharacteristic
622C8—41no—
623C8—41no—
624C8—2441no—
625C8—2441no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22#121×D6111yescharacteristic
634C22#1684040111yescharacteristic
635C22#1684041811yescharacteristic
636C22C2×SD16×D6111yescharacteristic
637C22AGL(1,ℤ/8)×D6111yescharacteristic
638C22#1684057611yescharacteristic
639C22#1684057811yescharacteristic
640C4—22no—
641C4—42no—
642C4—41no—
643C4—81no—
644C4—81no—
645C4—614no—
646C4—1222no—
647C4—1222no—
648C4—4881no—
649C4—4881no—
650C4—2442no—
651C4—2441no—
652C4—2442no—
653C4#1684028911yescharacteristic
654C4#1684037611yescharacteristic
655C4#1684039911yescharacteristic
656C4#1684041011yescharacteristic
657C2—82no—
658C2#2144938911yescharacteristic
659C2#460×D6111yescharacteristic
660C2#2145534711yescharacteristic
661C2#2145767911yescharacteristic
662C2#2145811911yescharacteristic
663C2#2145845611yescharacteristic
664C2#2145853211yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2055C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
hypercenter#2055D61
hypocenterC61#25320
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2055

Complements

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

NH ≅ G/Ntype
1. C2×#16840111C2semidirect
4. #21448876C2semidirect
5. #21447231C2semidirect
6. #21447227C2semidirect
8. #21446769C2semidirect
9. #21446534C2semidirect
12. #21445825C2semidirect
13. #21445822C2semidirect
208. #11758196SD16semidirect
393. C61#25320semidirect

Maximal subgroups

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

subgroupindex
C2×#168401112
C2×#168399652
#214507702
#214488762
#214472312
#214472272
#214471352
#214467692
#214465342
#214461772
#214461762
#214458252
#214458222
#102×Dic612
C61×#20552
#2532061