Finite Group Databasebeta

← #29698300

Subgroups — #29698300

Order 15616.

1018 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839862C211yesmaximal, characteristic
2C2×#16839859C211yesmaximal, characteristic
3C2×#16839853C211yesmaximal, characteristic
4#21448733C211yesmaximal, characteristic, complemented
5D4×#11758196C211yesmaximal, characteristic
6#21446561C211yesmaximal, characteristic, complemented
7#21446559C211yesmaximal, characteristic, complemented
8#21446076C211yesmaximal, characteristic, complemented
9#21446070C211yesmaximal, characteristic, complemented
10#21446068C211yesmaximal, characteristic
11#21445977C211yesmaximal, characteristic, complemented
12#21445823C211yesmaximal, characteristic, complemented
13#21445817C211yesmaximal, characteristic
14#21445685C211yesmaximal, characteristic, complemented
15C61×#1862C211yesmaximal, characteristic
16C22×#11758200C2211yescharacteristic
17C2×C4×Dic122C2211yescharacteristic
18C2×#14519742C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×D4×Dic61C2211yescharacteristic
21C2×D4×Dic61C2211yescharacteristic
22C22×#11758196—21no—
23#16839862—24no—
24#16839859—24no—
25#16839853C418yescomplemented
26C2×#14519711C2211yescharacteristic
27C2×#14519710C2211yescharacteristic
28C2×#14519710C2211yescharacteristic
29C2×#14519708C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519700C2211yescharacteristic
34C2×#14519698C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519687C2211yescharacteristic
37C2×C8×Dic61C2211yescharacteristic
38#16839661C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839659—21no—
41#16839658C2211yescharacteristic
42#16839657—21no—
43#16839635—21no—
44#16839634—21no—
45#16839631—21no—
46#16839631C2211yescharacteristic
47#16839631C2211yescharacteristic
48#16839620—21no—
49C4×#11758196C2211yescharacteristic
50#16839613—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C23×Dic61—21no—
64C23×Dic61—21no—
65C2×#11758200C2×C414yes—
66C2×#11758200D414yes—
67C2×#11758200Q814yes—
68C22×#92418C2311yescharacteristic
69C22×#92418C2311yescharacteristic
70C4×Dic122—22no—
71C4×Dic122—41no—
72#14519742—22no—
73#14519742—22no—
74#14519742—41no—
75#14519742C2×C414yescomplemented
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—41no—
79D4×Dic61C2×C414yes—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×C4⋊Dic61C2311yescharacteristic
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—44no—
94C2×#11758196D411yescharacteristic
95C2×#11758196D411yescharacteristic
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#14519711—41no—
104#14519710—22no—
105#14519710—22no—
106#14519708C2×C414yescomplemented
107#14519706—22no—
108#14519702—22no—
109#14519702C2×C414yescomplemented
110#14519700—22no—
111#14519698—22no—
112#14519688—22no—
113#14519687—22no—
114C8×Dic61C2×C414yescomplemented
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684D411yescharacteristic
119#14519684D411yescharacteristic
120C22×D4×C61C2311yescharacteristic
121C23×C4×C61—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C4×D4×C61—22no—
125C4×D4×C61—41no—
126C2×C4⋊C4×C61—21no—
127C2×C4⋊C4×C61C2311yescharacteristic
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C22⋊C4×C61—21no—
131C2×C42×C61C2311yescharacteristic
132#107×C61—41no—
133#102×C61—22no—
134#102×C61C2×C414yes—
135#95×C61—21no—
136C24×C61—21no—
137C2×D4×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61C22×C412yes—
141C2×D4×C61C2×D412yes—
142C2×D4×C61C2×Q812yes—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—42no—
147C22×C4×C61—42no—
148C22×C4×C61—81no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—24no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—42no—
158C22×Dic61—42no—
159C22×Dic61—42no—
160C22×Dic61—42no—
161C22×Dic61—81no—
162C22×Dic61C2×D411yescharacteristic
163C22×Dic61C2×D411yescharacteristic
164C22×Dic61C2×D411yescharacteristic
165C22×Dic61C2×D411yescharacteristic
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C2×D412yes—
171C2×Dic122C2×Q812yes—
172C2×C8×C61—21no—
173C2×C8×C61—21no—
174C2×C8×C61—21no—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61C22×C412yes—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61C22×C412yes—
185C22⋊C4×C61—42no—
186C42×C61—21no—
187C42×C61—21no—
188C22⋊Dic61—24no—
189C22⋊Dic61—42no—
190#11758200C4⋊C4116yescomplemented
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—81no—
197C4⋊Dic61C22×C412yes—
198#11758196—24no—
199#11758196—22no—
200#11758196—22no—
201#11758196—42no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206#11758196—42no—
207#11758196—81no—
208#11758196—81no—
209C4×Dic61—21no—
210C4×Dic61—21no—
211C4×Dic61—24no—
212C4×Dic61—42no—
213C4×Dic61—42no—
214C4×Dic61C22×C412yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418C22×C412yes—
220C2×#92418C2×D412yes—
221C2×#92418C2×Q812yes—
222C23×C61—22no—
223C23×C61—22no—
224C23×C61—22no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61C22×D411yescharacteristic
230D4×C61—22no—
231D4×C61—22no—
232D4×C61—22no—
233D4×C61C2×C4⋊C414yes—
234C2×C4×C61—22no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
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—84no—
250C2×C4×C61C23×C411yescharacteristic
251C2×C4×C61C22×D411yescharacteristic
252C2×C4×C61C22×Q811yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—48no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—84no—
278C2×Dic61C4×D412yes—
279C2×Dic61C4×D412yes—
280C2×Dic61AGL(1,ℤ/8)12yes—
281C2×Dic61#13712yes—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122C2×C4⋊C414yescomplemented
286C8×C61—22no—
287C8×C61—41no—
288#92418—41no—
289#92418C2×C4⋊C414yescomplemented
290#22671—611nomaximal, Sylow
291C22×C61—24no—
292C22×C61—42no—
293C22×C61—42no—
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—42no—
298C22×C61C2×C4×D411yescharacteristic
299C22×C61D4211yescharacteristic
300C22×C61D4×Q811yescharacteristic
301C22×C61C2×AGL(1,ℤ/8)11yescharacteristic
302C22×C61C2×#13711yescharacteristic
303C22×C61#57611yescharacteristic
304C22×C61#57811yescharacteristic
305C4×C61—42no—
306C4×C61—41no—
307C4×C61—41no—
308C4×C61—81no—
309C4×C61—81no—
310C4×C61C22×C4⋊C411yescharacteristic
311C4×C61#52011yescharacteristic
312C4×C61#54711yescharacteristic
313C4×C61#55411yescharacteristic
314Dic61—24no—
315Dic61—22no—
316Dic61—42no—
317Dic61—41no—
318Dic61—41no—
319Dic61—81no—
320Dic61—81no—
321Dic61#44014yescomplemented
322C2×#440—611no—
323C2×#440—611no—
324C2×C4×SD16—611no—
325Q8×C4⋊C4—611no—
326D4×C4⋊C4—611no—
327#1944—611no—
328#1944—611no—
329#1863—611no—
330#1863—611no—
331#1862—611no—
332#1862D6111yescharacteristic
333#1845—611no—
334#1803—611no—
335#1802—611no—
336#1770—611no—
337C2×C61—44no—
338C2×C61D4×C4⋊C411yescharacteristic
339C2×C61C2×#44011yescharacteristic
340C2×C61#295111yescharacteristic
341C2×C61#327411yescharacteristic
342C2×C61#327611yescharacteristic
343C2×C61#338511yescharacteristic
344C2×C61#338611yescharacteristic
345C22×SD16—611no—
346C2×C4×Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×Q8—611no—
349C2×C4×D4—611no—
350C2×C4×D4—611no—
351C2×C4×D4C2×D6111yescharacteristic
352C22×C4⋊C4—1221no—
353#440—1224no—
354#440—1224no—
355C4×SD16—618no—
356C2×#107—611no—
357C2×#107—611no—
358C2×#107—611no—
359C2×#107C2×D6111yescharacteristic
360C2×#106—611no—
361C2×C4⋊C8—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#102—611no—
366C2×#102C2×D6111yescharacteristic
367C2×#102C2×D6111yescharacteristic
368C2×#97—611no—
369C2×#97—611no—
370C2×C4×C8—611no—
371#389—611no—
372#389—611no—
373#389—611no—
374#389—1221no—
375#389—1221no—
376#389C2×D6111yescharacteristic
377#385—1221no—
378#385—1221no—
379#384—611no—
380#384—611no—
381#384—1221no—
382#380—1221no—
383C4×C4⋊C4—611no—
384C4×C4⋊C4—1221no—
385#340—611no—
386#340—611no—
387#340—611no—
388#340C2×D6111yescharacteristic
389#336—611no—
390#336C2×D6111yescharacteristic
391C61#2267111yescharacteristic, complemented, Sylow
392C22×Q8—611no—
393C22×D4C22×D6111yescharacteristic
394C23×C4—21no—
395C23×C4—1221no—
396C23×C4—1221no—
397C2×SD16—614no—
398C2×SD16—614no—
399C2×SD16—614no—
400C22×C8—611no—
401C22×C8—611no—
402C22×C8C22×D6111yescharacteristic
403C22×C8C22×D6111yescharacteristic
404C4×Q8—614no—
405C4×Q8—1222no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—2441no—
409C4×Q8—2441no—
410C4×D4—22no—
411C4×D4—41no—
412C4×D4—614no—
413C4×D4—1222no—
414C4×D4—1222no—
415C4×D4—2441no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—611no—
418C2×C4⋊C4—611no—
419C2×C4⋊C4—611no—
420C2×C4⋊C4—611no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4—1221no—
424C2×C4⋊C4—1221no—
425C2×C4⋊C4—1221no—
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—2444no—
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42C22×D6111yescharacteristic
445#107—41no—
446#107—1222no—
447#107—1222no—
448#107—2441no—
449#106—614no—
450C4⋊C8—1222no—
451#103—614no—
452#103—1222no—
453#103—1222no—
454#102—22no—
455#102—1222no—
456#102C4×D6114yes—
457#97—1222no—
458#97—1222no—
459C4×C8—614no—
460#95—21no—
461#95—611no—
462#95—611no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466C24—21no—
467C2×Q8—612no—
468C2×Q8—612no—
469C2×Q8—612no—
470C2×Q8—1221no—
471C2×Q8—1221no—
472C2×Q8—1221no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4C2×C4×D6112yes—
477C2×D4D4×D6112yes—
478C2×D4Q8×D6112yes—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—81no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—1224no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—4881no—
495C22×C4—2442no—
496C22×C4—2442no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4D4×D6111yescharacteristic
500C22×C4D4×D6111yescharacteristic
501C22×C4C23×D6111yescharacteristic
502SD16—6116no—
503C2×C8—21no—
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—612no—
509C2×C8—612no—
510C2×C8—612no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8C2×C4×D6112yes—
515C4⋊C4—21no—
516C4⋊C4—21no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—81no—
521C4⋊C4—612no—
522C4⋊C4—1222no—
523C4⋊C4—1222no—
524C4⋊C4—1224no—
525C4⋊C4—1221no—
526C4⋊C4—1221no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—4881no—
531C4⋊C4—4881no—
532C4⋊C4—4881no—
533C4⋊C4—2442no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4C2×C4×D6112yes—
540C22⋊C4—42no—
541C22⋊C4—1224no—
542C22⋊C4—2442no—
543C42—21no—
544C42—21no—
545C42—612no—
546C42—612no—
547C42—1224no—
548C42—1221no—
549C42—1221no—
550C42—2442no—
551C42—2442no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23C2×D4×D6111yescharacteristic
560Q8—614no—
561Q8—1222no—
562Q8—1222no—
563Q8—1222no—
564D4—22no—
565D4—22no—
566D4—22no—
567D4C4⋊C4×D6114yes—
568C2×C4—22no—
569C2×C4—21no—
570C2×C4—21no—
571C2×C4—21no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—84no—
584C2×C4—612no—
585C2×C4—612no—
586C2×C4—612no—
587C2×C4—612no—
588C2×C4—1222no—
589C2×C4—1221no—
590C2×C4—1221no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1221no—
599C2×C4—1221no—
600C2×C4—4884no—
601C2×C4—2448no—
602C2×C4—2441no—
603C2×C4—2441no—
604C2×C4—2441no—
605C2×C4—2441no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4C22×C4×D6111yescharacteristic
611C2×C4C2×D4×D6111yescharacteristic
612C2×C4C2×Q8×D6111yescharacteristic
613C2×C4#41×D6111yescharacteristic
614C2×C4#1451982211yescharacteristic
615C2×C4#1451982411yescharacteristic
616C8—22no—
617C8—41no—
618C8—614no—
619C8—2441no—
620C22—24no—
621C22—42no—
622C22—42no—
623C22—42no—
624C22—42no—
625C22—42no—
626C22—42no—
627C22C4×D4×D6111yescharacteristic
628C22#1684039411yescharacteristic
629C22#1684042511yescharacteristic
630C22AGL(1,ℤ/8)×D6111yescharacteristic
631C22#137×D6111yescharacteristic
632C22#1684056311yescharacteristic
633C22#1684057611yescharacteristic
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—614no—
640C4—1224no—
641C4—1222no—
642C4—4881no—
643C4—4881no—
644C4—2442no—
645C4—2441no—
646C4—2441no—
647C4C2×C4⋊C4×D6111yescharacteristic
648C4#1684024111yescharacteristic
649C4#1684041011yescharacteristic
650C4#1684046611yescharacteristic
651C2—44no—
652C2#2144871711yescharacteristic
653C2#440×D6111yescharacteristic
654C2#2145468911yescharacteristic
655C2#2145743711yescharacteristic
656C2#2145746311yescharacteristic
657C2#2145872411yescharacteristic
658C2#2145873511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1862C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1862D61
hypocenterC61#22671
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1862

Complements

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

NH ≅ G/Ntype
4. #21448733C2semidirect
6. #21446561C2semidirect
7. #21446559C2semidirect
8. #21446076C2semidirect
9. #21446070C2semidirect
11. #21445977C2semidirect
12. #21445823C2semidirect
14. #21445685C2semidirect
25. #16839853C4semidirect
75. #14519742C2×C4semidirect
106. #14519708C2×C4semidirect
109. #14519702C2×C4semidirect
114. C8×Dic61C2×C4semidirect
190. #11758200C4⋊C4semidirect
285. Dic122C2×C4⋊C4semidirect
289. #92418C2×C4⋊C4semidirect
321. Dic61#440semidirect
391. C61#22671semidirect

Maximal subgroups

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

subgroupindex
C2×#168398622
C2×#168398592
C2×#168398532
#214487332
D4×#117581962
#214465612
#214465592
#214460762
#214460702
#214460682
#214459772
#214458232
#214458172
#214456852
C61×#18622
#2267161