Finite Group Databasebeta

← #29696715

Subgroups — #29696715

Order 15616.

1004 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758222C211yesmaximal, characteristic
2#21448297C211yesmaximal, characteristic, complemented
3#21446799C212yesmaximal, complemented
4#21446795C211yesmaximal, characteristic, complemented
5#21446763C211yesmaximal, characteristic, complemented
6#21446763C212yesmaximal, complemented
7#21445836C212yesmaximal
8#21445836C211yesmaximal, characteristic
9#21445632C212yesmaximal
10C4×#14519703C211yesmaximal, characteristic
11Q8×M16×C61C211yesmaximal, characteristic, complemented
12Q8×Dic122C411yescharacteristic, complemented
13#16840490C411yescharacteristic, complemented
14#16840490C412yescomplemented
15#16840472C411yescharacteristic, complemented
16#16840472C412yescomplemented
17#16840467—22no—
18#16840467—21no—
19Q8×D244C411yescharacteristic, complemented
20Q8×C22⋊D61—21no—
21#16840319C2211yescharacteristic
22#16840319C2212yes—
23#16840316—21no—
24#16840316—22no—
25C4×#11758222C2212yes—
26C4×#11758222C2211yescharacteristic
27C4×Q8×D61—21no—
28C2×#14519703C2212yes—
29C2×#14519703C2211yescharacteristic
30C2×#14519703C2211yescharacteristic
31#16839669—21no—
32#16839661—21no—
33#16839648—22no—
34C4×C4⋊Dic61C2211yescharacteristic
35C4×C4⋊Dic61C2212yes—
36#16839608C2212yes—
37C42×Dic61C2211yescharacteristic
38#16839523C2212yes—
39#16839523C2212yes—
40#16839523C2211yescharacteristic
41#16839523C2212yes—
42#16839523C2212yes—
43#16839523C2212yes—
44#16839523C2211yescharacteristic
45C2×C4×Q8×C61C2211yescharacteristic
46C61×#446—21no—
47C61×#446—22no—
48C8×Q8×C61—21no—
49C61×#423C2211yescharacteristic
50C61×#423C2212yes—
51C4×M16×C61C2212yes—
52C4×M16×C61C2211yescharacteristic
53C2×#11758222C2311yescharacteristic
54C2×Q8×D61—21no—
55#14519777C2×C411yescharacteristic
56#14519777C2×C412yes—
57#14519777C2×C411yescharacteristic
58#14519777C2×C412yes—
59#14519776—22no—
60#14519776—21no—
61#14519775—21no—
62#14519775—22no—
63#14519769C2×C412yes—
64#14519769C2×C412yes—
65#14519769C2×C412yes—
66#14519767—22no—
67#14519767—22no—
68#14519767—22no—
69#14519757—22no—
70#14519757—21no—
71#14519756—21no—
72#14519756—22no—
73#14519756—21no—
74#14519756—22no—
75#14519755—22no—
76#14519755—22no—
77#14519755—22no—
78#14519753—21no—
79#14519753—22no—
80Q8×Dic61—21no—
81C4×Dic122—21no—
82C4×Dic122—22no—
83C4×Dic122—22no—
84C4×Dic122—21no—
85C4×Dic122C2×C412yes—
86C4×Dic122C2×C411yescharacteristic
87C4×Dic122C2×C412yes—
88C4×Dic122C2×C411yescharacteristic
89#14519742—22no—
90#14519742—21no—
91C4×D244C2×C412yes—
92C4×D244C2×C411yescharacteristic
93C4×D244C2×C412yes—
94C4×D244C2×C411yescharacteristic
95C4×C22⋊D61—21no—
96C4×C22⋊D61—22no—
97C4×C22⋊D61—21no—
98C4×C22⋊D61—22no—
99#14519736C2312yes—
100#14519736C2311yescharacteristic
101#14519734—22no—
102#14519734—21no—
103C2×C4⋊Dic61—22no—
104C2×C4⋊Dic61—21no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—21no—
107C2×C4×Dic61—21no—
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110C2×C4×Dic61C2312yes—
111C2×C4×Dic61C2311yescharacteristic
112C42×D61—21no—
113C42×D61—22no—
114#14519703—24no—
115#14519703—22no—
116#14519703Q814yescomplemented
117#14519684—22no—
118#14519684—22no—
119#14519684—21no—
120#14519684—22no—
121C22×Q8×C61D411yescharacteristic
122C2×M16×C61C2312yes—
123C2×M16×C61C2311yescharacteristic
124C2×M16×C61C2311yescharacteristic
125C4×Q8×C61C2×C411yescharacteristic
126C4×Q8×C61C2×C412yes—
127C4×Q8×C61C2×C411yescharacteristic
128C4×Q8×C61D411yescharacteristic
129C4×Q8×C61D411yescharacteristic
130C4×Q8×C61D412yes—
131C2×C4⋊C4×C61D412yes—
132C2×C4⋊C4×C61D411yescharacteristic
133C2×C42×C61C2311yescharacteristic
134C2×C42×C61C2312yes—
135C4⋊C8×C61—21no—
136C4⋊C8×C61—22no—
137C4⋊C8×C61—22no—
138C4⋊C8×C61—21no—
139#97×C61—22no—
140#97×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61—22no—
143C2×Q8×C61—42no—
144C2×Q8×C61C22⋊C411yescharacteristic
145C2×Q8×C61C22⋊C411yescharacteristic
146C2×Q8×C61C22⋊C412yes—
147C22×C4×C61C2×D411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D412yes—
150C22×C4×C61C2×D412yes—
151C22×C4×C61C2411yescharacteristic
152C2×C22⋊D61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—22no—
156Q8×D61—44no—
157#11758222—21no—
158#11758222—21no—
159#11758222C2×Q812yes—
160#11758222#4112yes—
161C2×D244C22×C411yescharacteristic
162C2×C4×D61—21no—
163C2×C4×D61—22no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—22no—
167C2×Dic122—22no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122—21no—
171C2×Dic122C22×C411yescharacteristic
172M16×C61—21no—
173M16×C61—22no—
174M16×C61—21no—
175M16×C61—22no—
176M16×C61—21no—
177M16×C61C2×Q812yescomplemented
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61—22no—
181C4⋊C4×C61C22⋊C412yes—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22⋊C412yes—
184C4⋊C4×C61C22⋊C412yes—
185C4⋊C4×C61C22⋊C412yes—
186C4⋊C4×C61C22⋊C412yes—
187C4⋊C4×C61C22⋊C411yescharacteristic
188C42×C61C22×C411yescharacteristic
189C42×C61C22×C412yes—
190C42×C61C22×C412yes—
191C42×C61C22×C411yescharacteristic
192C42×C61C2×D411yescharacteristic
193C42×C61C2×D412yes—
194C42×C61C2×D412yes—
195C42×C61C2×D411yescharacteristic
196C22⋊Dic61—22no—
197C22⋊Dic61—21no—
198#11758198—21no—
199#11758198—22no—
200#11758198—22no—
201#11758198—21no—
202C4⋊Dic61—22no—
203C4⋊Dic61—21no—
204C4⋊Dic61—22no—
205C4⋊Dic61—21no—
206C4⋊Dic61—44no—
207C4⋊Dic61—42no—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C22×C411yescharacteristic
210C4⋊Dic61C22×C412yes—
211C4⋊Dic61C22×C411yescharacteristic
212#11758196—22no—
213#11758196—22no—
214#11758196—22no—
215#11758196—21no—
216#11758196—22no—
217#11758196—22no—
218#11758196—21no—
219#11758196—22no—
220#11758196—22no—
221#11758196—22no—
222C4×Dic61—21no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61—21no—
226C4×Dic61—21no—
227C4×Dic61—22no—
228C4×Dic61—21no—
229C4×Dic61—21no—
230C4×Dic61—22no—
231C4×Dic61—21no—
232C4×Dic61—22no—
233C4×Dic61—42no—
234C4×Dic61—42no—
235C4×Dic61C2×Q812yescomplemented
236C23×C61C22×D411yescharacteristic
237C22×D61—21no—
238Q8×C61—22no—
239Q8×C61—22no—
240Q8×C61—42no—
241C2×C4×C61—22no—
242C2×C4×C61—41no—
243C2×C4×C61—42no—
244C2×C4×C61—42no—
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C2×C22⋊C412yes—
247C2×C4×C61C2×C22⋊C412yes—
248C2×C4×C61C2×C22⋊C411yescharacteristic
249C2×C4×C61C2×C22⋊C412yes—
250C2×C4×C61C2×C22⋊C412yes—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C23×C411yescharacteristic
254C2×C4×C61C22×D411yescharacteristic
255C2×C4×C61C22×Q811yescharacteristic
256C2×C4×C61C2×#4111yescharacteristic
257C2×C4×C612−1+411yescharacteristic
258C2×C4×C612−1+411yescharacteristic
259C22⋊D61—22no—
260C22⋊D61—22no—
261C2×Dic61—21no—
262C2×Dic61—22no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—41no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275D244—21no—
276D244C4×Q812yes—
277C4×D61—22no—
278C4×D61—22no—
279C4×D61—44no—
280C4×D61—42no—
281C4×D61—81no—
282C4×D61—82no—
283Dic122—21no—
284Dic122—44no—
285Dic122—42no—
286Dic122—81no—
287Dic122—82no—
288Dic122C4×Q812yes—
289C8×C61—22no—
290C8×C61—41no—
291C8×C61—42no—
292#22529—611nomaximal, Sylow
293C22×C61—21no—
294C22×C61—21no—
295C22×C61C22×C22⋊C411yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61D4×Q811yescharacteristic
298C2×D61—22no—
299C2×D61—41no—
300C2×D61—41no—
301C4×C61—22no—
302C4×C61—21no—
303C4×C61—22no—
304C4×C61—21no—
305C4×C61—22no—
306C4×C61—21no—
307C4×C61—22no—
308C4×C61#42112yes—
309C4×C61C2×C4×Q811yescharacteristic
310C4×C61#51911yescharacteristic
311C4×C61#54811yescharacteristic
312C4×C61D4×Q811yescharacteristic
313Dic61—22no—
314Dic61—42no—
315Dic61—42no—
316Dic61—41no—
317Dic61—42no—
318Dic61—42no—
319Dic61—41no—
320Dic61—82no—
321Q8×#41—611no—
322Q8×M16D6111yescharacteristic, complemented
323#2295—611no—
324#1995—612no—
325#1994—611no—
326#1986—611no—
327#1986—612no—
328#1806—611no—
329#1806—612no—
330#1759—612no—
331C4×#104—611no—
332C2×C61—21no—
333C2×C61—21no—
334C2×C61Q8×C22⋊C411yescharacteristic
335C2×C61C2×#42111yescharacteristic
336C2×C61#290111yescharacteristic
337D61—42no—
338Q82—611no—
339#557—612no—
340#557—611no—
341#554—611no—
342#554—612no—
343#554—1222no—
344#554—1221no—
345D4×Q8—611no—
346D4×Q8—1221no—
347#533—611no—
348#533—612no—
349#533—1221no—
350#533—1222no—
351C4×#41—611no—
352C4×#41—612no—
353C2×C4×Q8—1221no—
354C2×C4×Q8C2×D6111yescharacteristic
355#446—21no—
356#446—22no—
357C8×Q8—21no—
358#423C2×D6111yescharacteristic
359#423C2×D6112yes—
360C2×#104—611no—
361C2×#104—611no—
362C2×#104—612no—
363C4×M16C2×D6111yescharacteristic
364C4×M16C2×D6112yes—
365#391—1221no—
366#389—1221no—
367#387—1222no—
368C4×C4⋊C4—611no—
369C4×C4⋊C4—612no—
370#376—612no—
371C43—611no—
372#339—612no—
373#339—612no—
374#339—612no—
375#339—611no—
376#339—612no—
377#339—612no—
378#339—611no—
379C61#2252911yescharacteristic, complemented, Sylow
380C2×#41—611no—
381C22×Q8—1221no—
382C22×Q8D24411yescharacteristic
383C2×M16C22×D6112yes—
384C2×M16C22×D6111yescharacteristic
385C2×M16C22×D6111yescharacteristic
386C4⋊Q8—611no—
387C4⋊Q8—612no—
388C4⋊Q8—611no—
389C4⋊Q8—612no—
390C4⋊Q8—1221no—
391C4⋊Q8—1222no—
392C4⋊Q8—1222no—
393C4⋊Q8—1221no—
394#125—612no—
395#125—612no—
396#125—612no—
397#125—1222no—
398#125—1222no—
399#125—1222no—
400C22⋊Q8—1221no—
401C22⋊Q8—1222no—
402C22⋊Q8—1221no—
403C22⋊Q8—1222no—
404C22⋊Q8—1221no—
405C22⋊Q8—1222no—
406C22⋊Q8—1222no—
407C22⋊Q8—1221no—
408C22⋊Q8—1222no—
409C22⋊Q8—1222no—
410C22⋊Q8—1222no—
411C4×Q8—611no—
412C4×Q8—612no—
413C4×Q8—611no—
414C4×Q8—612no—
415C4×Q8—1221no—
416C4×Q8—1222no—
417C4×Q8—1221no—
418C4×Q8—1221no—
419C4×Q8—1222no—
420C4×Q8—1222no—
421C4×Q8—1221no—
422C4×Q8C4×D6111yescharacteristic
423C4×Q8C4×D6111yescharacteristic
424C4×Q8C4×D6112yes—
425C4×Q8C22⋊D6111yescharacteristic
426C4×Q8C22⋊D6111yescharacteristic
427C4×Q8C22⋊D6112yes—
428C4×D4—611no—
429C4×D4—612no—
430C4×D4—611no—
431C4×D4—612no—
432C4×D4—1221no—
433C4×D4—1222no—
434C4×D4—1221no—
435C4×D4—1222no—
436#117—611no—
437#117—612no—
438#117—1222no—
439#117—1221no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1222no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4D24412yes—
445C2×C4⋊C4D24411yescharacteristic
446C2×C42—611no—
447C2×C42—611no—
448C2×C42—612no—
449C2×C42—1221no—
450C2×C42—1222no—
451C2×C42—1221no—
452C2×C42—1221no—
453C2×C42C22×D6111yescharacteristic
454C2×C42C22×D6112yes—
455C4⋊C8—21no—
456C4⋊C8—22no—
457C4⋊C8—22no—
458C4⋊C8—21no—
459#104—614no—
460#104—1222no—
461#104—1224no—
462#97—22no—
463#97—21no—
464C4×C8—21no—
465C4×C8—22no—
466#95—1222no—
467#95—1222no—
468#95—1221no—
469#95—1222no—
470#41—612no—
471#41—612no—
472#41—1221no—
473#41—1221no—
474C2×Q8—42no—
475C2×Q8—611no—
476C2×Q8—1222no—
477C2×Q8—1221no—
478C2×Q8—1222no—
479C2×Q8—1221no—
480C2×Q8—2444no—
481C2×Q8#1175819811yescharacteristic
482C2×Q8#1175819811yescharacteristic
483C2×Q8#1175819812yes—
484C2×D4—611no—
485C2×D4—1221no—
486C22×C4—1221no—
487C22×C4—1221no—
488C22×C4—1221no—
489C22×C4—1221no—
490C22×C4—1222no—
491C22×C4—1222no—
492C22×C4—1221no—
493C22×C4—1222no—
494C22×C4C2×D24411yescharacteristic
495C22×C4C2×D24412yes—
496C22×C4C2×D24411yescharacteristic
497C22×C4C2×D24412yes—
498C22×C4C23×D6111yescharacteristic
499M16—21no—
500M16—21no—
501M16—22no—
502M16—22no—
503M16—21no—
504M16Q8×D6112yescomplemented
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—22no—
508C4⋊C4—611no—
509C4⋊C4—612no—
510C4⋊C4—611no—
511C4⋊C4—612no—
512C4⋊C4—1222no—
513C4⋊C4—1222no—
514C4⋊C4—1222no—
515C4⋊C4—1222no—
516C4⋊C4—1221no—
517C4⋊C4—1221no—
518C4⋊C4—1222no—
519C4⋊C4—1222no—
520C4⋊C4—1221no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1222no—
525C4⋊C4—1221no—
526C4⋊C4—2442no—
527C4⋊C4—2444no—
528C4⋊C4#1175819812yes—
529C4⋊C4#1175819811yescharacteristic
530C4⋊C4#1175819812yes—
531C4⋊C4#1175819812yes—
532C4⋊C4#1175819812yes—
533C4⋊C4#1175819812yes—
534C4⋊C4#1175819811yescharacteristic
535C22⋊C4—1221no—
536C22⋊C4—1222no—
537C22⋊C4—1221no—
538C22⋊C4—1222no—
539C22⋊C4—1222no—
540C22⋊C4—1221no—
541C42—612no—
542C42—1221no—
543C42—1222no—
544C42—1221no—
545C42—1221no—
546C42—1222no—
547C42—1222no—
548C42—1221no—
549C42—1222no—
550C42—1221no—
551C42—1222no—
552C42—1221no—
553C42—2442no—
554C42—2442no—
555C42C2×C4×D6111yescharacteristic
556C42C2×C4×D6112yes—
557C42C2×C4×D6112yes—
558C42C2×C4×D6111yescharacteristic
559C42C2×C22⋊D6111yescharacteristic
560C42C2×C22⋊D6112yes—
561C42C2×C22⋊D6112yes—
562C42C2×C22⋊D6111yescharacteristic
563C23—1221no—
564C23C22×D24411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—42no—
568Q8—612no—
569Q8—1221no—
570Q8—4881no—
571Q8—4882no—
572Q8—2444no—
573Q8—2442no—
574D4—612no—
575D4—1222no—
576D4—1222no—
577D4—1221no—
578C2×C4—22no—
579C2×C4—41no—
580C2×C4—42no—
581C2×C4—42no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1221no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1222no—
591C2×C4—4881no—
592C2×C4—4882no—
593C2×C4—2444no—
594C2×C4—2442no—
595C2×C4—2442no—
596C2×C4—2442no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2442no—
602C2×C4C2×#1175819811yescharacteristic
603C2×C4C2×#1175819812yes—
604C2×C4C2×#1175819812yes—
605C2×C4C2×#1175819811yescharacteristic
606C2×C4C2×#1175819812yes—
607C2×C4C2×#1175819812yes—
608C2×C4C2×#1175819811yescharacteristic
609C2×C4C2×#1175819811yescharacteristic
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C22×C22⋊D6111yescharacteristic
612C2×C4C2×Q8×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4#1451982411yescharacteristic
615C2×C4#1451982511yescharacteristic
616C8—22no—
617C8—41no—
618C8—42no—
619C22—21no—
620C22—21no—
621C22—1222no—
622C22—2441no—
623C22—2441no—
624C22C22×#1175819811yescharacteristic
625C22#1684042411yescharacteristic
626C22Q8×D24411yescharacteristic
627C4—21no—
628C4—22no—
629C4—22no—
630C4—21no—
631C4—22no—
632C4—21no—
633C4—22no—
634C4—1222no—
635C4—4882no—
636C4—2442no—
637C4—2442no—
638C4—2442no—
639C4—2441no—
640C4—2442no—
641C4—2441no—
642C4#1683977812yes—
643C4C4×Q8×D6111yescharacteristic
644C4#1684023811yescharacteristic
645C4#1684041911yescharacteristic
646C4Q8×C22⋊D6111yescharacteristic
647C2—21no—
648C2—21no—
649C2—2442no—
650C2Q8×#1175819811yescharacteristic
651C2C2×#1683977811yescharacteristic
652C2#2145436411yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingQ8×M16×C61C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222C23
cosocleC22×C4×C61C24
hypercenterQ8×M16D61
hypocenterC61#22529
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C4×Q8 ◁ Q8×M16

Complements

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

NH ≅ G/Ntype
2. #21448297C2semidirect
3. #21446799C2semidirect
4. #21446795C2semidirect
5. #21446763C2semidirect
6. #21446763C2semidirect
11. Q8×M16×C61C2semidirect
12. Q8×Dic122C4semidirect
13. #16840490C4semidirect
14. #16840490C4semidirect
15. #16840472C4semidirect
16. #16840472C4semidirect
19. Q8×D244C4semidirect
116. #14519703Q8semidirect
177. M16×C61C2×Q8semidirect
235. C4×Dic61C2×Q8semidirect
322. Q8×M16D61semidirect
379. C61#22529semidirect
504. M16Q8×D61semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582222
#214482972
#214467992
#214467952
#214467632
#214458362
#214456322
C4×#145197032
Q8×M16×C612
#2252961