Finite Group Databasebeta

← #29689675

Subgroups — #29689675

Order 15616.

1108 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459115C211yesmaximal, characteristic
2#21448217C211yesmaximal, characteristic, complemented
3#21446799C212yesmaximal, complemented
4#21446650C212yesmaximal, complemented
5#21446121C214yesmaximal, complemented
6#21445836C214yesmaximal, complemented
7C2×C61×#423C211yesmaximal, characteristic, complemented
8#16840319C411yescharacteristic, complemented
9#16840319C411yescharacteristic, complemented
10#16840319C2211yescharacteristic
11#16840312C411yescharacteristic, complemented
12#16840312C411yescharacteristic, complemented
13#16840312C2211yescharacteristic
14#16840296—21no—
15#16840296—21no—
16#16840296—21no—
17#16840296—21no—
18#16840296C412yescomplemented
19#16840296C412yescomplemented
20#16840273—21no—
21#16840273—21no—
22#16840273C2212yes—
23C4×#11758222—21no—
24C4×#11758222—21no—
25C4×#11758222C2212yes—
26C2×#14519703C2218yes—
27#16839630—21no—
28#16839628—21no—
29#16839625—21no—
30#16839622—21no—
31C4×C4⋊Dic61C2212yes—
32C4×C22⋊Dic61C2212yes—
33#16839608C2212yes—
34#16839523C2218yes—
35C22×M16×C61C2212yes—
36C22×C42×C61C2211yescharacteristic
37C61×#423—21no—
38C61×#423—21no—
39C61×#423C2212yes—
40C61×#423C2212yes—
41C2×C4⋊C8×C61—21no—
42C2×C4⋊C8×C61—21no—
43C2×#11758222—21no—
44C2×#11758222—21no—
45C2×#11758222C2312yes—
46#14519777C2×C411yescharacteristic
47#14519777C2×C411yescharacteristic
48#14519777C2×C411yescharacteristic
49#14519774C2×C411yescharacteristic
50#14519771—22no—
51#14519771—22no—
52#14519769—21no—
53#14519769—21no—
54#14519769C2×C412yes—
55#14519765—21no—
56#14519765—21no—
57#14519765C2×C412yes—
58#14519760—22no—
59#14519760—22no—
60#14519760—22no—
61#14519756—21no—
62#14519756—21no—
63#14519756—21no—
64#14519756—21no—
65#14519756C2×C412yes—
66#14519756C2×C412yes—
67#14519751—21no—
68#14519751—21no—
69#14519751—21no—
70#14519751—21no—
71#14519751C2×C412yes—
72#14519751C2×C412yes—
73#14519746—22no—
74C4×Dic122—21no—
75C4×Dic122—21no—
76C4×Dic122—21no—
77C4×Dic122—21no—
78C4×Dic122C2×C412yes—
79C4×Dic122C2×C412yes—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244—21no—
83C4×D244—21no—
84C4×D244C2×C412yes—
85C4×D244C2×C412yes—
86C4×C22⋊D61—22no—
87C4×C22⋊D61—22no—
88C4×C22⋊D61—22no—
89C4×C22⋊D61—22no—
90C4×C22⋊D61—22no—
91C4×C22⋊D61—22no—
92#14519736—21no—
93#14519736—21no—
94#14519736C2312yes—
95#14519734—22no—
96#14519734—22no—
97#14519734—22no—
98C2×C4⋊Dic61—22no—
99C2×C22⋊Dic61—22no—
100C2×C4×Dic61C2314yes—
101C42×D61—22no—
102#14519703—48no—
103#14519684—22no—
104#14519684—22no—
105#14519684—22no—
106#14519684—22no—
107C23×C4×C61D411yescharacteristic
108C23×C4×C61D411yescharacteristic
109C23×C4×C61C2311yescharacteristic
110C2×M16×C61—22no—
111C2×M16×C61—44no—
112C2×M16×C61C2314yes—
113C22×C8×C61—22no—
114C2×C42×C61C2×C411yescharacteristic
115C2×C42×C61C2×C411yescharacteristic
116C2×C42×C61C2×C411yescharacteristic
117C2×C42×C61C2×C411yescharacteristic
118C2×C42×C61D411yescharacteristic
119C2×C42×C61D411yescharacteristic
120C2×C42×C61D411yescharacteristic
121C2×C42×C61D411yescharacteristic
122C2×C42×C61D411yescharacteristic
123C2×C42×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C4⋊C8×C61—22no—
127C4⋊C8×C61—22no—
128C4⋊C8×C61—22no—
129C4⋊C8×C61—22no—
130C24×C61C2×D411yescharacteristic
131C22×C4×C61—22no—
132C22×C4×C61—22no—
133C22×C4×C61—42no—
134C22×C4×C61—42no—
135C22×C4×C61C22⋊C411yescharacteristic
136C22×C4×C61C22⋊C411yescharacteristic
137C22×C4×C61C22⋊C411yescharacteristic
138C22×C4×C61C22⋊C411yescharacteristic
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22⋊C411yescharacteristic
142C22×C4×C61C22⋊C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C2×C22⋊D61—22no—
154C2×C22⋊D61—22no—
155C2×C22⋊D61—22no—
156C22×Dic61—24no—
157#11758222—24no—
158#11758222—24no—
159#11758222—44no—
160#11758222—44no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—22no—
165C2×C4×D61—22no—
166C2×C4×D61—22no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170M16×C61—28no—
171M16×C61—44no—
172C2×C8×C61—24no—
173C2×C8×C61—44no—
174C42×C61C22⋊C411yescharacteristic
175C42×C61C22⋊C411yescharacteristic
176C42×C61C22⋊C411yescharacteristic
177C42×C61C22⋊C411yescharacteristic
178C42×C61C22⋊C411yescharacteristic
179C42×C61C22⋊C411yescharacteristic
180C42×C61C22⋊C411yescharacteristic
181C42×C61C22⋊C411yescharacteristic
182C42×C61C22×C411yescharacteristic
183C42×C61C22×C411yescharacteristic
184C42×C61C22×C411yescharacteristic
185C42×C61C22×C411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2×D411yescharacteristic
188C42×C61C2×D411yescharacteristic
189C42×C61C2×D411yescharacteristic
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—44no—
194#11758198—22no—
195#11758198—22no—
196#11758198—22no—
197#11758198—22no—
198#11758198—22no—
199#11758198—22no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—44no—
205C4⋊Dic61C22×C412yes—
206C4⋊Dic61C22×C412yes—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196—22no—
211#11758196—22no—
212#11758196—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61—22no—
216C4×Dic61—24no—
217C4×Dic61—24no—
218C23×C61—22no—
219C23×C61—22no—
220C23×C61—41no—
221C23×C61C2×C22⋊C411yescharacteristic
222C23×C61C2×C22⋊C411yescharacteristic
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C2×C4×C61—22no—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—22no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C411yescharacteristic
247C2×C4×C61C2×C22⋊C411yescharacteristic
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C411yescharacteristic
250C2×C4×C61C2×C22⋊C411yescharacteristic
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C23×C411yescharacteristic
255C2×C4×C61C22×D411yescharacteristic
256C2×C4×C61H(5,𝔽2)12yes—
257C2×C4×C612−1+412yes—
258C22⋊D61—44no—
259C22⋊D61—44no—
260C22⋊D61—44no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—44no—
265C2×Dic61—44no—
266C2×Dic61—44no—
267D244—24no—
268D244—42no—
269D244—42no—
270C4×D61—44no—
271C4×D61—42no—
272C4×D61—42no—
273C4×D61—44no—
274Dic122—24no—
275Dic122—42no—
276Dic122—42no—
277C8×C61—44no—
278#21857—611nomaximal, Sylow
279C22×C61—21no—
280C22×C61—21no—
281C22×C61—21no—
282C22×C61—21no—
283C22×C61—21no—
284C22×C61—21no—
285C22×C61—21no—
286C22×C61—44no—
287C22×C61#42112yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C22×C61#53712yes—
290C2×D61—42no—
291C2×D61—42no—
292C2×D61—42no—
293C4×C61—21no—
294C4×C61—21no—
295C4×C61—21no—
296C4×C61—21no—
297C4×C61—21no—
298C4×C61—21no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61#42112yes—
303C4×C61#52012yes—
304C4×C61#53511yescharacteristic
305C4×C61#53611yescharacteristic
306Dic61—42no—
307Dic61—42no—
308Dic61—42no—
309Dic61—84no—
310#3444—611no—
311C2×#423D6111yescharacteristic, complemented
312#2283—611no—
313#1995—612no—
314#1963—612no—
315#1871—614no—
316#1806—614no—
317C2×C61—22no—
318C2×C61—22no—
319C2×C61—41no—
320C2×C61#231111yescharacteristic
321C2×C61C2×#42111yescharacteristic
322C2×C61C2×#42111yescharacteristic
323D61—82no—
324C22×M16C2×D6112yes—
325#533—611no—
326#533—611no—
327#533—611no—
328#532—611no—
329#532—611no—
330#532—611no—
331#529—612no—
332#529—612no—
333#529—1221no—
334#529—1221no—
335#529—1221no—
336#529—1221no—
337#525—612no—
338#525—1221no—
339#525—1221no—
340C4×#41—612no—
341C4×#41—1221no—
342C4×#41—1221no—
343C22×C42C2×D6111yescharacteristic
344#423—21no—
345#423—21no—
346#423C2×D6112yes—
347#423C2×D6112yes—
348C2×C4⋊C8—21no—
349C2×C4⋊C8—21no—
350C2×#104—618no—
351#383—1221no—
352#382—1221no—
353#381—1221no—
354#380—1221no—
355C4×C4⋊C4—612no—
356C4×C22⋊C4—612no—
357#376—612no—
358#339—618no—
359C61#2185711yescharacteristic, complemented, Sylow
360C2×#41—612no—
361C2×#41—1221no—
362C2×#41—1221no—
363C23×C4D24411yescharacteristic
364C23×C4D24411yescharacteristic
365C23×C4C22×D6111yescharacteristic
366C2×M16—22no—
367C2×M16—44no—
368C2×M16C22×D6114yes—
369C22×C8—22no—
370C4⋊Q8—611no—
371C4⋊Q8—611no—
372C4⋊Q8—611no—
373#127—611no—
374#126—1222no—
375#126—1222no—
376#125—612no—
377#125—1221no—
378#125—1221no—
379#124—612no—
380#124—1221no—
381#124—1221no—
382#123—1222no—
383#123—1222no—
384#123—1222no—
385C22⋊Q8—612no—
386C22⋊Q8—612no—
387C22⋊Q8—1221no—
388C22⋊Q8—1221no—
389C22⋊Q8—1221no—
390C22⋊Q8—1221no—
391#121—612no—
392#121—612no—
393#121—1221no—
394#121—1221no—
395#121—1221no—
396#121—1221no—
397#120—1222no—
398C4×Q8—612no—
399C4×Q8—612no—
400C4×Q8—1221no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1221no—
404C4×D4—612no—
405C4×D4—612no—
406C4×D4—1222no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1221no—
413C4×D4—1221no—
414C4×D4—1221no—
415C4×D4—1221no—
416#117—612no—
417#117—1221no—
418#117—1221no—
419#117—1222no—
420#117—1222no—
421#117—1222no—
422C2×C4⋊C4—1222no—
423C2×C22⋊C4—1222no—
424C2×C42—614no—
425C2×C42—1222no—
426C2×C42C4×D6111yescharacteristic
427C2×C42C4×D6111yescharacteristic
428C2×C42C4×D6111yescharacteristic
429C2×C42C4×D6111yescharacteristic
430C2×C42D24411yescharacteristic
431C2×C42D24411yescharacteristic
432C2×C42C22⋊D6111yescharacteristic
433C2×C42C22⋊D6111yescharacteristic
434C2×C42C22⋊D6111yescharacteristic
435C2×C42C22⋊D6111yescharacteristic
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C4⋊C8—22no—
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442#104—2448no—
443#95—1222no—
444#95—1222no—
445#95—1222no—
446#95—1222no—
447C24C2×D24411yescharacteristic
448#41—1224no—
449#41—1224no—
450#41—2444no—
451#41—2444no—
452C2×Q8—612no—
453C2×Q8—1221no—
454C2×Q8—1221no—
455C2×D4—612no—
456C2×D4—1222no—
457C2×D4—1222no—
458C2×D4—1222no—
459C2×D4—1221no—
460C2×D4—1221no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—42no—
464C22×C4—42no—
465C22×C4—1222no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4—1224no—
469C22×C4#1175819811yescharacteristic
470C22×C4#1175819811yescharacteristic
471C22×C4#1175819811yescharacteristic
472C22×C4#1175819811yescharacteristic
473C22×C4#1175819811yescharacteristic
474C22×C4#1175819811yescharacteristic
475C22×C4#1175819811yescharacteristic
476C22×C4#1175819811yescharacteristic
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4C2×C4×D6111yescharacteristic
479C22×C4C2×D24411yescharacteristic
480C22×C4C2×D24411yescharacteristic
481C22×C4C2×D24411yescharacteristic
482C22×C4C2×D24411yescharacteristic
483C22×C4C2×D24411yescharacteristic
484C22×C4C2×C22⋊D6111yescharacteristic
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C23×D6111yescharacteristic
487M16—28no—
488M16—44no—
489C2×C8—24no—
490C2×C8—44no—
491C4⋊C4—612no—
492C4⋊C4—612no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—2444no—
504C22⋊C4—1222no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C22⋊C4—1222no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—2444no—
514C42—1222no—
515C42—1222no—
516C42—1222no—
517C42—1224no—
518C42—1224no—
519C42#1175819811yescharacteristic
520C42#1175819811yescharacteristic
521C42#1175819811yescharacteristic
522C42#1175819811yescharacteristic
523C42#1175819811yescharacteristic
524C42#1175819811yescharacteristic
525C42#1175819811yescharacteristic
526C42#1175819811yescharacteristic
527C42C2×C4×D6111yescharacteristic
528C42C2×C4×D6111yescharacteristic
529C42C2×C4×D6111yescharacteristic
530C42C2×C4×D6111yescharacteristic
531C42C2×C22⋊D6111yescharacteristic
532C42C2×C22⋊D6111yescharacteristic
533C42C2×C22⋊D6111yescharacteristic
534C42C2×C22⋊D6111yescharacteristic
535C23—22no—
536C23—22no—
537C23—41no—
538C23—1222no—
539C23C2×#1175819811yescharacteristic
540C23C2×#1175819811yescharacteristic
541C23C22×D24411yescharacteristic
542Q8—1224no—
543Q8—2442no—
544Q8—2442no—
545D4—1224no—
546D4—2442no—
547D4—2442no—
548D4—2444no—
549D4—2444no—
550D4—2444no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—1222no—
571C2×C4—1222no—
572C2×C4—1222no—
573C2×C4—2444no—
574C2×C4—2444no—
575C2×C4—2444no—
576C2×C4—2444no—
577C2×C4—2444no—
578C2×C4—2442no—
579C2×C4—2442no—
580C2×C4C2×#1175819811yescharacteristic
581C2×C4C2×#1175819811yescharacteristic
582C2×C4C2×#1175819811yescharacteristic
583C2×C4C2×#1175819811yescharacteristic
584C2×C4C2×#1175819811yescharacteristic
585C2×C4C2×#1175819811yescharacteristic
586C2×C4C2×#1175819811yescharacteristic
587C2×C4C2×#1175819811yescharacteristic
588C2×C4C2×#1175819811yescharacteristic
589C2×C4C2×#1175819811yescharacteristic
590C2×C4C22×C4×D6111yescharacteristic
591C2×C4C22×C22⋊D6111yescharacteristic
592C2×C4#1451982212yes—
593C2×C4#1451982512yes—
594C8—44no—
595C22—21no—
596C22—21no—
597C22—21no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—44no—
603C22—2442no—
604C22—2442no—
605C22—2442no—
606C22#1683977812yes—
607C22C22×#1175819811yescharacteristic
608C22#1684034312yes—
609C4—21no—
610C4—21no—
611C4—21no—
612C4—21no—
613C4—21no—
614C4—21no—
615C4—21no—
616C4—21no—
617C4—21no—
618C4—4884no—
619C4—2442no—
620C4—2442no—
621C4—2442no—
622C4#1683977812yes—
623C4#1684024312yes—
624C4#1684032911yescharacteristic
625C4#1684033611yescharacteristic
626C2—22no—
627C2—22no—
628C2—41no—
629C2—4882no—
630C2#2144841211yescharacteristic
631C2C2×#1683977811yescharacteristic
632C2C2×#1683977811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#423C2
socleC22×C61C22×C22⋊C4
prime radical#21459115C2
cosocleC22×C4×C61C24
hypercenterC2×#423D61
hypocenterC61#21857
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C2×#423

Complements

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

NH ≅ G/Ntype
2. #21448217C2semidirect
3. #21446799C2semidirect
4. #21446650C2semidirect
5. #21446121C2semidirect
6. #21445836C2semidirect
7. C2×C61×#423C2semidirect
8. #16840319C4semidirect
9. #16840319C4semidirect
11. #16840312C4semidirect
12. #16840312C4semidirect
18. #16840296C4semidirect
19. #16840296C4semidirect
311. C2×#423D61semidirect
359. C61#21857semidirect

Maximal subgroups

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

subgroupindex
#214591152
#214482172
#214467992
#214466502
#214461212
#214458362
C2×C61×#4232
#2185761