Finite Group Databasebeta

← #29696686

Subgroups — #29696686

Order 15616.

1066 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840472C211yesmaximal, characteristic
2#21448776C211yesmaximal, characteristic, complemented
3#21446792C211yesmaximal, characteristic, complemented
4#21446788C211yesmaximal, characteristic, complemented
5#21446784C211yesmaximal, characteristic, complemented
6#21446774C211yesmaximal, characteristic, complemented
7#21446774C211yesmaximal, characteristic, complemented
8#21446770C211yesmaximal, characteristic, complemented
9#21445830C211yesmaximal, characteristic
10#21445830C211yesmaximal, characteristic
11#21445820C211yesmaximal, characteristic
12#21445645C211yesmaximal, characteristic
13#21445645C211yesmaximal, characteristic
14C4×#14519699C211yesmaximal, characteristic
15C2×C61×#446C211yesmaximal, characteristic, complemented
16#16840472—24no—
17#16840472C418yescomplemented
18C2×#14519777C2211yescharacteristic
19C2×#14519769C2211yescharacteristic
20C2×#14519769C2211yescharacteristic
21C2×#14519755—21no—
22C2×#14519755—21no—
23C2×#14519753—21no—
24C2×C4×D244C2211yescharacteristic
25C2×C4×D244C2211yescharacteristic
26C2×C4×D244C2211yescharacteristic
27C2×C4⋊C4×D61—21no—
28C2×#14519699C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519699C2211yescharacteristic
32#16839669—21no—
33#16839648—21no—
34#16839648—21no—
35#16839632C2211yescharacteristic
36#16839628C2211yescharacteristic
37#16839628C2211yescharacteristic
38C4⋊C4×Dic61—21no—
39C4×C4⋊Dic61C2211yescharacteristic
40C4×C4⋊Dic61C2211yescharacteristic
41C4×C4⋊Dic61C2211yescharacteristic
42#16839525C2211yescharacteristic
43#16839525C2211yescharacteristic
44#16839525C2211yescharacteristic
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839525C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839525C2211yescharacteristic
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54C2×C4×Q8×C61C2211yescharacteristic
55C61×#446—24no—
56C2×C4⋊C8×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×C4⋊C8×C61C2211yescharacteristic
59C2×#97×C61C2211yescharacteristic
60C2×#97×C61C2211yescharacteristic
61C2×C4×C8×C61C2211yescharacteristic
62C22×Dic122—21no—
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67#14519777—22no—
68#14519777C2×C414yes—
69#14519769—22no—
70#14519769—22no—
71#14519769C2×C414yes—
72#14519769C2×C414yes—
73#14519755—28no—
74#14519755—28no—
75#14519753—28no—
76C4×D244—22no—
77C4×D244—22no—
78C4×D244—22no—
79C4×D244C2×C414yes—
80C4×D244C2×C414yes—
81C4×D244C2×C414yes—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61—21no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C4⋊C4×D61—28no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103#14519699—22no—
104#14519699—22no—
105#14519699—22no—
106#14519699Q814yescomplemented
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684—21no—
113#14519684—21no—
114C22×Q8×C61D411yescharacteristic
115C22×C8×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C22×C8×C61C2311yescharacteristic
119C4×Q8×C61C2×C414yes—
120C4×Q8×C61D414yes—
121C2×C4⋊C4×C61D411yescharacteristic
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126C2×C42×C61C2311yescharacteristic
127C4⋊C8×C61—22no—
128C4⋊C8×C61—22no—
129C4⋊C8×C61—22no—
130#97×C61—22no—
131#97×C61—22no—
132C4×C8×C61—22no—
133C23×D61—21no—
134C2×Q8×C61—41no—
135C2×Q8×C61—41no—
136C2×Q8×C61C22⋊C414yes—
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
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—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C2×D244—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244C22×C412yes—
155C2×D244C2×Q812yes—
156C2×D244#4112yes—
157C2×C4×D61—24no—
158C2×C4×D61—24no—
159C2×C4×D61—24no—
160C2×C4×D61—42no—
161C2×C4×D61—42no—
162C2×C4×D61—81no—
163C2×C4×D61—81no—
164C2×C4×D61—81no—
165C2×C4×D61—81no—
166C2×Dic122—24no—
167C2×Dic122—81no—
168C2×Dic122—81no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—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×C61C2×Q812yes—
181C4⋊C4×C61C22⋊C414yes—
182C4⋊C4×C61C22⋊C414yes—
183C4⋊C4×C61C22⋊C414yes—
184C42×C61C22×C412yes—
185C42×C61C22×C412yes—
186C42×C61C22×C412yes—
187C42×C61C2×D412yes—
188C42×C61C2×D412yes—
189C42×C61C2×D412yes—
190#11758198—24no—
191#11758198—24no—
192#11758198—24no—
193C4⋊Dic61—24no—
194C4⋊Dic61—21no—
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—42no—
205C4⋊Dic61—81no—
206C4⋊Dic61C22×C412yes—
207C4⋊Dic61C22×C412yes—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C2×Q812yes—
210#11758196—24no—
211#11758196—24no—
212#11758196—24no—
213#11758196—24no—
214C4×Dic61—42no—
215C4×Dic61—81no—
216C4×Dic61—81no—
217C23×C61C22×D411yescharacteristic
218C22×D61—22no—
219C22×D61—22no—
220C22×D61—22no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224C22×D61—41no—
225Q8×C61—44no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61C2×C22⋊C412yes—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C2×C22⋊C412yes—
242C2×C4×C61C2×C22⋊C412yes—
243C2×C4×C61C2×C22⋊C412yes—
244C2×C4×C61C23×C411yescharacteristic
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×Q811yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C2×C4×C612−1+411yescharacteristic
249C2×C4×C612−1+411yescharacteristic
250C2×Dic61—22no—
251C2×Dic61—22no—
252C2×Dic61—22no—
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
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—
271D244—22no—
272D244—22no—
273D244—22no—
274D244C4×Q814yes—
275C4×D61—48no—
276C4×D61—84no—
277C4×D61—84no—
278Dic122—84no—
279C8×C61—22no—
280C8×C61—41no—
281C8×C61—41no—
282C8×C61—41no—
283#22526—611nomaximal, Sylow
284C22×C61C22×C22⋊C411yescharacteristic
285C22×C61#54811yescharacteristic
286C22×C61D4×Q811yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57511yescharacteristic
289C22×C61#57611yescharacteristic
290C22×C61#57711yescharacteristic
291C2×D61—24no—
292C2×D61—42no—
293C2×D61—42no—
294C2×D61—42no—
295C2×D61—42no—
296C2×D61—42no—
297C2×D61—42no—
298C4×C61—22no—
299C4×C61—22no—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—22no—
303C4×C61—22no—
304C4×C61C2×C4×Q811yescharacteristic
305C4×C61#51911yescharacteristic
306C4×C61#54811yescharacteristic
307C4×C61D4×Q811yescharacteristic
308Dic61—42no—
309Dic61—42no—
310Dic61—42no—
311Dic61—42no—
312Dic61—81no—
313Dic61—81no—
314Dic61—81no—
315C2×#554—611no—
316C2×#446D6111yescharacteristic, complemented
317#2355—611no—
318#1993—611no—
319#1992—611no—
320#1991—611no—
321#1989—611no—
322#1989—611no—
323#1988—611no—
324#1804—611no—
325#1804—611no—
326#1802—611no—
327#1762—611no—
328#1762—611no—
329C4×#102—611no—
330C2×C61Q8×C22⋊C411yescharacteristic
331C2×C61#289611yescharacteristic
332C2×C61#289711yescharacteristic
333C2×C61#337011yescharacteristic
334C2×C61#337611yescharacteristic
335C2×C61#338811yescharacteristic
336C2×C61#338911yescharacteristic
337D61—44no—
338#554—618no—
339#554—1224no—
340C2×C4⋊Q8—611no—
341C2×#125—611no—
342C2×#125—611no—
343C2×C22⋊Q8—1221no—
344C2×C22⋊Q8—1221no—
345C2×C22⋊Q8—1221no—
346C2×C4×Q8C2×D6111yescharacteristic
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C22×C4⋊C4—1221no—
351#446—24no—
352C2×C4⋊C8C2×D6111yescharacteristic
353C2×C4⋊C8C2×D6111yescharacteristic
354C2×C4⋊C8C2×D6111yescharacteristic
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#97C2×D6111yescharacteristic
360C2×#97C2×D6111yescharacteristic
361C2×C4×C8C2×D6111yescharacteristic
362#391—1221no—
363#387—1221no—
364#387—1221no—
365#384—611no—
366#382—611no—
367#382—611no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—611no—
370C4×C4⋊C4—611no—
371C4×C4⋊C4—1221no—
372#340—611no—
373#340—611no—
374#340—611no—
375#340—611no—
376#340—611no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340—611no—
381#340—611no—
382#340—611no—
383#340—611no—
384C61#2252611yescharacteristic, complemented, Sylow
385C22×Q8—1221no—
386C22×Q8D24411yescharacteristic
387C22×D4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4—1221no—
391C22×C8C22×D6111yescharacteristic
392C22×C8C22×D6111yescharacteristic
393C22×C8C22×D6111yescharacteristic
394C22×C8C22×D6111yescharacteristic
395C4⋊Q8—614no—
396C4⋊Q8—1222no—
397#125—614no—
398#125—614no—
399#125—1222no—
400#125—1222no—
401C22⋊Q8—1228no—
402C22⋊Q8—1228no—
403C22⋊Q8—1228no—
404C4×Q8C4×D6114yes—
405C4×Q8C22⋊D6114yes—
406C4×D4—614no—
407C4×D4—614no—
408C4×D4—614no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1228no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C4⋊C4D24411yescharacteristic
428C2×C4⋊C4D24411yescharacteristic
429C2×C4⋊C4D24411yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—1221no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—22no—
440C4⋊C8—22no—
441C4⋊C8—22no—
442#102—614no—
443#102—1222no—
444#102—1222no—
445#102—1222no—
446#97—22no—
447#97—22no—
448C4×C8—22no—
449#95—1221no—
450#95—1221no—
451#95—1221no—
452#95—1221no—
453#95—1221no—
454#95—1221no—
455#95—1221no—
456C24—1221no—
457C2×Q8—41no—
458C2×Q8—41no—
459C2×Q8—1224no—
460C2×Q8—4881no—
461C2×Q8—4881no—
462C2×Q8#1175819814yes—
463C2×D4—612no—
464C2×D4—612no—
465C2×D4—612no—
466C2×D4—1221no—
467C2×D4—1221no—
468C2×D4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4—1221no—
472C22×C4—1221no—
473C22×C4—1224no—
474C22×C4—1224no—
475C22×C4—1224no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—4881no—
480C22×C4—4881no—
481C22×C4—4881no—
482C22×C4—4881no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×D24411yescharacteristic
486C22×C4C2×D24411yescharacteristic
487C22×C4C2×D24411yescharacteristic
488C22×C4C2×D24411yescharacteristic
489C22×C4C2×D24411yescharacteristic
490C22×C4C2×D24411yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—21no—
501C2×C8—21no—
502C2×C8—21no—
503C2×C8Q8×D6112yes—
504C4⋊C4—612no—
505C4⋊C4—612no—
506C4⋊C4—612no—
507C4⋊C4—612no—
508C4⋊C4—1221no—
509C4⋊C4—1221no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1224no—
513C4⋊C4—1224no—
514C4⋊C4—1224no—
515C4⋊C4—1224no—
516C4⋊C4—1224no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1221no—
520C4⋊C4—1221no—
521C4⋊C4—1221no—
522C4⋊C4—1221no—
523C4⋊C4—4881no—
524C4⋊C4—2442no—
525C4⋊C4#1175819814yes—
526C4⋊C4#1175819814yes—
527C4⋊C4#1175819814yes—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C42—4881no—
532C42—4881no—
533C42—2442no—
534C42C2×C4×D6112yes—
535C42C2×C4×D6112yes—
536C42C2×C4×D6112yes—
537C42C2×C22⋊D6112yes—
538C42C2×C22⋊D6112yes—
539C42C2×C22⋊D6112yes—
540C23—1222no—
541C23—1222no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23C22×D24411yescharacteristic
548Q8—44no—
549Q8—4884no—
550D4—614no—
551D4—1222no—
552D4—1222no—
553D4—1222no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—1222no—
567C2×C4—1222no—
568C2×C4—1222no—
569C2×C4—1222no—
570C2×C4—4884no—
571C2×C4—4884no—
572C2×C4—2448no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4—2441no—
580C2×C4—2441no—
581C2×C4—2441no—
582C2×C4—2441no—
583C2×C4—2441no—
584C2×C4—2441no—
585C2×C4—2441no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C2×#1175819812yes—
591C2×C4C2×#1175819812yes—
592C2×C4C2×#1175819812yes—
593C2×C4C2×#1175819812yes—
594C2×C4C2×#1175819812yes—
595C2×C4C2×#1175819812yes—
596C2×C4C22×C4×D6111yescharacteristic
597C2×C4C22×C22⋊D6111yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#1451982411yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—22no—
603C8—41no—
604C8—41no—
605C8—41no—
606C22—1224no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22C22×#1175819811yescharacteristic
614C22#1684042411yescharacteristic
615C22Q8×D24411yescharacteristic
616C22#1684056211yescharacteristic
617C22#1684056211yescharacteristic
618C22#1684056711yescharacteristic
619C22#1684057411yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—4881no—
627C4—4881no—
628C4—4881no—
629C4—2442no—
630C4—2442no—
631C4—2442no—
632C4—2442no—
633C4C4×Q8×D6111yescharacteristic
634C4#1684023811yescharacteristic
635C4#1684041911yescharacteristic
636C4Q8×C22⋊D6111yescharacteristic
637C2—2444no—
638C2Q8×#1175819811yescharacteristic
639C2#2145433911yescharacteristic
640C2#2145434611yescharacteristic
641C2#2145857411yescharacteristic
642C2#2145864811yescharacteristic
643C2#2145876611yescharacteristic
644C2#2145877911yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC2×C61×#446C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterC2×#446D61
hypocenterC61#22526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#446

Complements

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

NH ≅ G/Ntype
2. #21448776C2semidirect
3. #21446792C2semidirect
4. #21446788C2semidirect
5. #21446784C2semidirect
6. #21446774C2semidirect
7. #21446774C2semidirect
8. #21446770C2semidirect
15. C2×C61×#446C2semidirect
17. #16840472C4semidirect
106. #14519699Q8semidirect
316. C2×#446D61semidirect
384. C61#22526semidirect

Maximal subgroups

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

subgroupindex
C2×#168404722
#214487762
#214467922
#214467882
#214467842
#214467742
#214467702
#214458302
#214458202
#214456452
C4×#145196992
C2×C61×#4462
#2252661