Finite Group Databasebeta

← #29696678

Subgroups — #29696678

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
2C2×#16839893C211yesmaximal, characteristic, complemented
3#21446791C211yesmaximal, characteristic, complemented
4#21446787C211yesmaximal, characteristic, complemented
5#21446783C211yesmaximal, characteristic, complemented
6#21446776C211yesmaximal, characteristic, complemented
7#21446776C211yesmaximal, characteristic, complemented
8#21446771C211yesmaximal, characteristic, complemented
9#21445826C211yesmaximal, characteristic
10#21445826C211yesmaximal, characteristic
11#21445816C211yesmaximal, characteristic
12#21445647C211yesmaximal, characteristic
13#21445647C211yesmaximal, characteristic
14C4×#14519697C211yesmaximal, characteristic
15C61×#2355C211yesmaximal, 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—
28#16839893—24no—
29C2×C4⋊#92418C2211yescharacteristic
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519697C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37C2×#14519688C2211yescharacteristic
38C2×C4×#92418C2211yescharacteristic
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#391—21no—
53C61×#387—21no—
54C61×#387—21no—
55C61×#384C2211yescharacteristic
56C61×#382C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C4×C4⋊C4×C61—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C4×C4⋊C4×C61C2211yescharacteristic
61C4×C4⋊C4×C61C2211yescharacteristic
62C22×Dic122—21no—
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C22×C4×D61—21no—
66C22×C4×D61—21no—
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C22×#92418C2311yescharacteristic
71#14519777—22no—
72#14519777C2×C414yes—
73#14519769—22no—
74#14519769—22no—
75#14519769C2×C414yes—
76#14519769C2×C414yes—
77#14519755—28no—
78#14519755—28no—
79#14519753—28no—
80C4×D244—22no—
81C4×D244—22no—
82C4×D244—22no—
83C4×D244C2×C414yes—
84C4×D244C2×C414yes—
85C4×D244C2×C414yes—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C4⋊C4×D61—28no—
95C2×#11758198—21no—
96C2×#11758198—21no—
97C2×#11758198—21no—
98C4⋊#92418—22no—
99C4⋊#92418—22no—
100C4⋊#92418—22no—
101#14519697—22no—
102#14519697—22no—
103#14519697—22no—
104#14519697Q814yescomplemented
105#14519688—22no—
106#14519688—22no—
107C4×#92418—22no—
108C22×Q8×C61D411yescharacteristic
109C4×Q8×C61C2×C414yes—
110C4×Q8×C61D414yes—
111C2×C4⋊C4×C61—21no—
112C2×C4⋊C4×C61—21no—
113C2×C4⋊C4×C61D411yescharacteristic
114C2×C4⋊C4×C61D411yescharacteristic
115C2×C4⋊C4×C61D411yescharacteristic
116C2×C4⋊C4×C61C2311yescharacteristic
117C2×C4⋊C4×C61C2311yescharacteristic
118C2×C4⋊C4×C61C2311yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C42×C61—21no—
121C2×C42×C61—21no—
122C2×C42×C61—21no—
123C2×C42×C61C2311yescharacteristic
124C2×C42×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#95×C61—21no—
127#95×C61—21no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132#95×C61—21no—
133C23×D61—21no—
134C2×Q8×C61—41no—
135C2×Q8×C61—41no—
136C2×Q8×C61C22⋊C414yes—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—21no—
140C22×C4×C61—21no—
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2×D411yescharacteristic
147C22×C4×C61C2411yescharacteristic
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—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—42no—
177C4⋊C4×C61—81no—
178C4⋊C4×C61C22⋊C414yes—
179C4⋊C4×C61C22⋊C414yes—
180C4⋊C4×C61C22⋊C414yes—
181C4⋊C4×C61C2×Q812yes—
182C42×C61—42no—
183C42×C61—81no—
184C42×C61—81no—
185C42×C61C22×C412yes—
186C42×C61C22×C412yes—
187C42×C61C22×C412yes—
188C42×C61C2×D412yes—
189C42×C61C2×D412yes—
190C42×C61C2×D412yes—
191#11758198—24no—
192#11758198—24no—
193#11758198—24no—
194C4⋊Dic61—24no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61C22×C412yes—
199C4⋊Dic61C22×C412yes—
200C4⋊Dic61C22×C412yes—
201#11758196—24no—
202#11758196—24no—
203#11758196—24no—
204#11758196—24no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418C2×Q812yes—
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×C61—22no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
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×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C412yes—
252C2×C4×C61C2×C22⋊C412yes—
253C2×C4×C61C2×C22⋊C412yes—
254C2×C4×C61C2×C22⋊C412yes—
255C2×C4×C61C2×C22⋊C412yes—
256C2×C4×C61C23×C411yescharacteristic
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×Q811yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C612−1+411yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
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—
279#92418—22no—
280#92418—41no—
281#92418—41no—
282#92418—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×C61—42no—
305C4×C61—81no—
306C4×C61—81no—
307C4×C61—81no—
308C4×C61C2×C4×Q811yescharacteristic
309C4×C61#51911yescharacteristic
310C4×C61#54811yescharacteristic
311C4×C61D4×Q811yescharacteristic
312Dic61—42no—
313Dic61—42no—
314Dic61—42no—
315C2×#554—611no—
316C2×#446—611no—
317#2355D6111yescharacteristic, complemented
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—1224no—
352C2×C4⋊C8—611no—
353C2×C4⋊C8—611no—
354C2×C4⋊C8—611no—
355C2×#102—611no—
356C2×#102—611no—
357C2×#102—611no—
358C2×#102—611no—
359C2×#97—611no—
360C2×#97—611no—
361C2×C4×C8—611no—
362#391—21no—
363#387—21no—
364#387—21no—
365#384C2×D6111yescharacteristic
366#382C2×D6111yescharacteristic
367#382C2×D6111yescharacteristic
368C4×C4⋊C4—21no—
369C4×C4⋊C4C2×D6111yescharacteristic
370C4×C4⋊C4C2×D6111yescharacteristic
371C4×C4⋊C4C2×D6111yescharacteristic
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×Q8C22⋊D6111yescharacteristic
387C22×D4—611no—
388C23×C4—1221no—
389C23×C4—1221no—
390C23×C4—1221no—
391C22×C8—611no—
392C22×C8—611no—
393C22×C8—611no—
394C22×C8—611no—
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×Q8D24414yes—
406C4×D4—614no—
407C4×D4—614no—
408C4×D4—614no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C2×C4⋊C4—21no—
413C2×C4⋊C4—21no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—611no—
416C2×C4⋊C4—611no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1228no—
423C2×C4⋊C4C22⋊D6111yescharacteristic
424C2×C4⋊C4C22⋊D6111yescharacteristic
425C2×C4⋊C4C22⋊D6111yescharacteristic
426C2×C4⋊C4C22×D6111yescharacteristic
427C2×C4⋊C4C22×D6111yescharacteristic
428C2×C4⋊C4C22×D6111yescharacteristic
429C2×C4⋊C4C22×D6111yescharacteristic
430C2×C22⋊C4—1221no—
431C2×C22⋊C4—1221no—
432C2×C22⋊C4—1221no—
433C2×C42—21no—
434C2×C42—21no—
435C2×C42—21no—
436C2×C42C22×D6111yescharacteristic
437C2×C42C22×D6111yescharacteristic
438C2×C42C22×D6111yescharacteristic
439C4⋊C8—1222no—
440C4⋊C8—1222no—
441C4⋊C8—1222no—
442#102—614no—
443#102—1222no—
444#102—1222no—
445#102—1222no—
446#97—1222no—
447#97—1222no—
448C4×C8—1222no—
449#95—21no—
450#95—21no—
451#95—21no—
452#95—21no—
453#95—21no—
454#95—21no—
455#95—21no—
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—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—1221no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1224no—
477C22×C4—1224no—
478C22×C4—1224no—
479C22×C4—4881no—
480C22×C4—4881no—
481C22×C4—4881no—
482C22×C4—4881no—
483C22×C4—2442no—
484C22×C4—2442no—
485C22×C4C2×C22⋊D6111yescharacteristic
486C22×C4C2×C22⋊D6111yescharacteristic
487C22×C4C2×C22⋊D6111yescharacteristic
488C22×C4C2×C22⋊D6111yescharacteristic
489C22×C4C2×C22⋊D6111yescharacteristic
490C22×C4C2×C22⋊D6111yescharacteristic
491C22×C4C23×D6111yescharacteristic
492C2×C8—612no—
493C2×C8—1221no—
494C2×C8—1221no—
495C2×C8—1221no—
496C2×C8—1221no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8—1221no—
504C4⋊C4—21no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—81no—
513C4⋊C4—612no—
514C4⋊C4—612no—
515C4⋊C4—612no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4—1224no—
520C4⋊C4—1224no—
521C4⋊C4—1224no—
522C4⋊C4—1224no—
523C4⋊C4—1224no—
524C4⋊C4#1175819814yes—
525C4⋊C4#1175819814yes—
526C4⋊C4#1175819814yes—
527C4⋊C4Q8×D6112yes—
528C22⋊C4—1224no—
529C22⋊C4—1224no—
530C22⋊C4—1224no—
531C42—42no—
532C42—81no—
533C42—81no—
534C42C2×C4×D6112yes—
535C42C2×C4×D6112yes—
536C42C2×C4×D6112yes—
537C42C2×D24412yes—
538C42C2×D24412yes—
539C42C2×D24412yes—
540C23—1222no—
541C23—1222no—
542C23—1222no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23C22×C22⋊D6111yescharacteristic
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—22no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—1222no—
579C2×C4—1222no—
580C2×C4—1222no—
581C2×C4—4884no—
582C2×C4—4884no—
583C2×C4—2448no—
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×D24411yescharacteristic
598C2×C4C2×Q8×D6111yescharacteristic
599C2×C4#41×D6111yescharacteristic
600C2×C4#1451982411yescharacteristic
601C2×C4#1451982511yescharacteristic
602C8—1222no—
603C8—2441no—
604C8—2441no—
605C8—2441no—
606C22—1224no—
607C22—2442no—
608C22—2442no—
609C22—2442no—
610C22—2442no—
611C22—2442no—
612C22—2442no—
613C22C22×#1175819811yescharacteristic
614C22#1684041911yescharacteristic
615C22Q8×C22⋊D6111yescharacteristic
616C22#1684056011yescharacteristic
617C22#1684056011yescharacteristic
618C22#1684056511yescharacteristic
619C22#1684057011yescharacteristic
620C4—22no—
621C4—22no—
622C4—22no—
623C4—22no—
624C4—22no—
625C4—22no—
626C4—42no—
627C4—81no—
628C4—81no—
629C4—81no—
630C4—2442no—
631C4—2442no—
632C4—2442no—
633C4C4×Q8×D6111yescharacteristic
634C4#1684023811yescharacteristic
635C4#1684042411yescharacteristic
636C4Q8×D24411yescharacteristic
637C2—2444no—
638C2Q8×#1175819811yescharacteristic
639C2#2145433611yescharacteristic
640C2#2145434211yescharacteristic
641C2#2145856611yescharacteristic
642C2#2145864011yescharacteristic
643C2#2145875811yescharacteristic
644C2#2145877311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#2355C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#2355D61
hypocenterC61#22526
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2355

Complements

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

NH ≅ G/Ntype
2. C2×#16839893C2semidirect
3. #21446791C2semidirect
4. #21446787C2semidirect
5. #21446783C2semidirect
6. #21446776C2semidirect
7. #21446776C2semidirect
8. #21446771C2semidirect
15. C61×#2355C2semidirect
17. #16840472C4semidirect
104. #14519697Q8semidirect
317. #2355D61semidirect
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
C2×#168398932
#214467912
#214467872
#214467832
#214467762
#214467712
#214458262
#214458162
#214456472
C4×#145196972
C61×#23552
#2252661