Finite Group Databasebeta

← #29702493

Subgroups — #29702493

Order 15616.

954 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839985C211yesmaximal, characteristic
2C2×#16839870C211yesmaximal, characteristic, complemented
3C2×C4×#11758190C211yesmaximal, characteristic
4#21448888C211yesmaximal, characteristic
5C4⋊C4×Dic122C211yesmaximal, characteristic, complemented
6#21446635C211yesmaximal, characteristic
7#21446583C211yesmaximal, characteristic, complemented
8#21446516C211yesmaximal, characteristic
9#21446495C211yesmaximal, characteristic, complemented
10#21446173C211yesmaximal, characteristic
11#21446159C211yesmaximal, characteristic, complemented
12#21446090C211yesmaximal, characteristic, complemented
13#21446083C211yesmaximal, characteristic
14#21445650C211yesmaximal, characteristic, complemented
15C61×#1771C211yesmaximal, characteristic, complemented
16C22×Dic244C2211yescharacteristic
17C22×#11758190C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×Dic122C2211yescharacteristic
21C2×C4×Dic122C2211yescharacteristic
22C2×C4×Dic122C2211yescharacteristic
23C2×C4×D244C2211yescharacteristic
24#16839985C418yescomplemented
25#16839870—24no—
26C4×#11758190—24no—
27C2×#14519709C2211yescharacteristic
28C2×#14519709C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519702C2211yescharacteristic
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519699C2211yescharacteristic
35C2×#14519699C2211yescharacteristic
36C2×#14519699C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839664C2211yescharacteristic
39#16839660C2211yescharacteristic
40#16839660C2211yescharacteristic
41#16839659—21no—
42#16839659—21no—
43#16839652—21no—
44#16839651—21no—
45#16839633—21no—
46#16839632C2211yescharacteristic
47#16839631—21no—
48C4⋊C4×Dic61—21no—
49C4×C4⋊Dic61C2211yescharacteristic
50#16839609—21no—
51#16839525C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839525C2211yescharacteristic
54#16839525C2211yescharacteristic
55C2×C4⋊C8×C61C2211yescharacteristic
56C2×#97×C61C2211yescharacteristic
57C2×#97×C61C2211yescharacteristic
58C2×C4×C8×C61C2211yescharacteristic
59C4×C4⋊C4×C61C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×D244C2311yescharacteristic
65C22×C4×D61—21no—
66C2×Dic244—22no—
67C2×Dic244—22no—
68C2×Dic244C2×C414yescomplemented
69C2×#11758190—22no—
70C2×#11758190—22no—
71C2×#11758190C2×C414yes—
72#14519765—22no—
73#14519765C2×C414yes—
74C4×Dic122—22no—
75C4×Dic122—22no—
76C4×Dic122—22no—
77C4×Dic122—22no—
78C4×Dic122—41no—
79C4×Dic122—41no—
80C4×Dic122—41no—
81C4×Dic122C2×C414yescomplemented
82C4×D244—22no—
83C4×D244—41no—
84C2×C4⋊Dic61—21no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196—21no—
96C2×#11758196—21no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61—21no—
106C2×C4×Dic61—21no—
107#14519709—22no—
108#14519709—22no—
109#14519702—22no—
110#14519702—22no—
111#14519702—41no—
112#14519702C2×C414yescomplemented
113#14519702D414yes—
114#14519699—22no—
115#14519699—41no—
116#14519699C2×C414yes—
117#14519684—21no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122C22×C8×C61C2311yescharacteristic
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C22×C8×C61C2311yescharacteristic
126C2×C4⋊C4×C61D411yescharacteristic
127C2×C4⋊C4×C61D411yescharacteristic
128C2×C42×C61C2311yescharacteristic
129C2×C42×C61C2311yescharacteristic
130C2×C42×C61C2311yescharacteristic
131C4⋊C8×C61C2×C414yescomplemented
132#97×C61—22no—
133#97×C61—22no—
134C4×C8×C61—22no—
135#95×C61D411yescharacteristic
136#95×C61D411yescharacteristic
137C23×D61—21no—
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×D411yescharacteristic
144C22×C4×C61C2411yescharacteristic
145C22×Dic61—21no—
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61—21no—
151C22×Dic61—21no—
152C2×D244—21no—
153C2×D244—21no—
154C2×D244—21no—
155C2×D244—21no—
156C2×D244—21no—
157C2×D244C22×C412yes—
158C2×C4×D61—42no—
159C2×C4×D61—42no—
160C2×C4×D61—81no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164C2×Dic122—21no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122C22×C412yes—
170C2×Dic122C22×C412yes—
171C2×Dic122C2×D412yes—
172C2×Dic122#4112yes—
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×C61—21no—
181C2×C8×C61—21no—
182C2×C8×C61C22×C412yes—
183C2×C8×C61C22×C412yes—
184C2×C8×C61C2×D412yes—
185C4⋊C4×C61—22no—
186C4⋊C4×C61—41no—
187C42×C61—21no—
188C42×C61—21no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61C22×C412yes—
192C42×C61#4112yes—
193#11758198—24no—
194#11758198—24no—
195#11758198—42no—
196#11758198—42no—
197#11758198—81no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—42no—
207C4⋊Dic61—81no—
208C4⋊Dic61C22×C412yes—
209C4⋊Dic61C2×D412yes—
210#11758196—24no—
211#11758196—42no—
212#11758196—42no—
213#11758196—42no—
214#11758196—81no—
215#11758196—81no—
216C4×Dic61—24no—
217C4×Dic61—42no—
218C4×Dic61—42no—
219C4×Dic61—42no—
220C4×Dic61—81no—
221Dic244—28no—
222#11758190—28no—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
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—21no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61C4×D412yes—
247C2×C4×C61C4×D412yes—
248C2×C4×C61#13712yes—
249C2×C4×C61C23×C411yescharacteristic
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61H(5,𝔽2)11yescharacteristic
254C2×C4×C612−1+411yescharacteristic
255C2×Dic61—22no—
256C2×Dic61—22no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—22no—
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—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276D244—22no—
277D244—22no—
278D244—41no—
279D244—41no—
280C4×D61—84no—
281Dic122—22no—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122C4×D414yes—
289C8×C61—22no—
290C8×C61—22no—
291C8×C61—22no—
292C8×C61—41no—
293#23013—611nomaximal, Sylow
294C22×C61C2×C4×D411yescharacteristic
295C22×C61D4211yescharacteristic
296C22×C61#54811yescharacteristic
297C22×C61C2×#13711yescharacteristic
298C22×C61#57511yescharacteristic
299C22×C61#57711yescharacteristic
300C22×C61#57711yescharacteristic
301C2×D61—42no—
302C2×D61—42no—
303C2×D61—42no—
304C2×D61—81no—
305C2×D61—81no—
306C2×D61—81no—
307C2×D61—81no—
308C4×C61—22no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—41no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61C2×C4×D411yescharacteristic
315C4×C61#52011yescharacteristic
316C4×C61#54711yescharacteristic
317C4×C61#55311yescharacteristic
318Dic61—24no—
319Dic61—42no—
320Dic61—42no—
321Dic61—42no—
322Dic61—81no—
323Dic61—81no—
324Dic61—81no—
325C2×#464—611no—
326C2×#441—611no—
327C2×C4×SD16—611no—
328#2367—611no—
329Q8×C4⋊C4—611no—
330#1959—611no—
331#1948—611no—
332#1936—611no—
333#1933—611no—
334#1880—611no—
335#1878—611no—
336#1865—611no—
337#1864—611no—
338#1771D6111yescharacteristic, complemented
339#1763—611no—
340C2×C61#236111yescharacteristic
341C2×C61C4×#13711yescharacteristic
342C2×C61#295211yescharacteristic
343C2×C61#326611yescharacteristic
344C2×C61#327111yescharacteristic
345C2×C61#336911yescharacteristic
346C2×C61#337111yescharacteristic
347D61—82no—
348C22×Q16—611no—
349C22×SD16—611no—
350C2×#124—611no—
351C2×C4×Q8—611no—
352C2×C4×Q8—611no—
353C2×C4×Q8—611no—
354C2×C4×Q8—611no—
355C2×C4×D4—611no—
356#464—618no—
357#441—1224no—
358C4×SD16—1224no—
359C2×#106—611no—
360C2×#106—611no—
361C2×C4⋊C8C2×D6111yescharacteristic
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#102—611no—
368C2×#102—611no—
369C2×#102—611no—
370C2×#97C2×D6111yescharacteristic
371C2×#97C2×D6111yescharacteristic
372C2×C4×C8C2×D6111yescharacteristic
373#391—611no—
374#390—611no—
375#389—611no—
376#389—611no—
377#389—1221no—
378#389—1221no—
379#388—1221no—
380#388—1221no—
381#385—1221no—
382#384—611no—
383#384—1221no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—1221no—
386C4×C4⋊C4C2×D6111yescharacteristic
387C4×C22⋊C4—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#336C2×D6111yescharacteristic
393#336C2×D6111yescharacteristic
394C61#2301311yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8—611no—
397C22×D4—611no—
398C23×C4—1221no—
399C2×Q16—614no—
400C2×Q16—1222no—
401C2×Q16—1222no—
402C2×SD16—614no—
403C2×SD16—1222no—
404C2×SD16—1222no—
405C22×C8C22×D6111yescharacteristic
406C22×C8C22×D6111yescharacteristic
407C22×C8C22×D6111yescharacteristic
408C22×C8C22×D6111yescharacteristic
409#124—614no—
410#124—1222no—
411C4×Q8—614no—
412C4×Q8—1222no—
413C4×Q8—1222no—
414C4×Q8—1222no—
415C4×Q8—1222no—
416C4×Q8—2441no—
417C4×Q8—2441no—
418C4×Q8—2441no—
419C4×D4—1222no—
420C4×D4—2441no—
421C2×C4⋊C4—611no—
422C2×C4⋊C4—611no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—611no—
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—1221no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4D24411yescharacteristic
435C2×C4⋊C4D24411yescharacteristic
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—1221no—
442C2×C42—1221no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448C2×C42C22×D6111yescharacteristic
449#106—1222no—
450#106—1222no—
451C4⋊C8C4×D6114yescomplemented
452#103—614no—
453#103—614no—
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#102—614no—
458#102—1222no—
459#102—2441no—
460#97—22no—
461#97—22no—
462C4×C8—22no—
463#95—1221no—
464#95—1221no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95D24411yescharacteristic
469#95D24411yescharacteristic
470C24—1221no—
471C2×Q8—612no—
472C2×Q8—612no—
473C2×Q8—612no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8—1221no—
481C2×Q8—1221no—
482C2×Q8—1221no—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C22×C4—1221no—
490C22×C4—1221no—
491C22×C4—1221no—
492C22×C4—1221no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—4881no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4C2×D24411yescharacteristic
500C22×C4C2×D24411yescharacteristic
501C22×C4C2×D24411yescharacteristic
502C22×C4C2×D24411yescharacteristic
503C22×C4C2×D24411yescharacteristic
504C22×C4C2×D24411yescharacteristic
505C22×C4C23×D6111yescharacteristic
506Q16—1228no—
507SD16—1228no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—21no—
517C2×C8C2×C4×D6112yes—
518C2×C8C2×C4×D6112yes—
519C2×C8D4×D6112yes—
520C4⋊C4—22no—
521C4⋊C4—41no—
522C4⋊C4—612no—
523C4⋊C4—612no—
524C4⋊C4—1221no—
525C4⋊C4—1221no—
526C4⋊C4—1224no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—1221no—
533C4⋊C4—4881no—
534C4⋊C4—4881no—
535C4⋊C4—4881no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4—2442no—
539C4⋊C4—2442no—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—4881no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—1224no—
550C42—4881no—
551C42—2442no—
552C42—2442no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42#1175822212yes—
556C23—1222no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23C22×D24411yescharacteristic
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568Q8—1222no—
569Q8—1222no—
570Q8—2441no—
571Q8—2441no—
572D4—1222no—
573D4—1222no—
574D4—2441no—
575D4—2441no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—1222no—
592C2×C4—1222no—
593C2×C4—1222no—
594C2×C4—1222no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—4884no—
598C2×C4—2441no—
599C2×C4—2441no—
600C2×C4—2441no—
601C2×C4—2441no—
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×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4C4×D24412yes—
614C2×C4C4×D24412yes—
615C2×C4#1451981012yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4C2×#1175822211yescharacteristic
620C2×C4#1451982311yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—22no—
623C8—22no—
624C8—22no—
625C8—41no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22C2×C4×D24411yescharacteristic
634C22D4×D24411yescharacteristic
635C22#1684042411yescharacteristic
636C22C2×#1451981011yescharacteristic
637C22#1684056211yescharacteristic
638C22#1684057411yescharacteristic
639C22#1684057411yescharacteristic
640C4—22no—
641C4—22no—
642C4—22no—
643C4—41no—
644C4—41no—
645C4—41no—
646C4—1224no—
647C4—4881no—
648C4—4881no—
649C4—4881no—
650C4—2442no—
651C4—2442no—
652C4—2442no—
653C4C4×D4×D6111yescharacteristic
654C4#1684024411yescharacteristic
655C4#1684041311yescharacteristic
656C4#1684045611yescharacteristic
657C2—4882no—
658C2#2144883711yescharacteristic
659C2C4×#1451981011yescharacteristic
660C2#2145470411yescharacteristic
661C2#2145733411yescharacteristic
662C2#2145740911yescharacteristic
663C2#2145856111yescharacteristic
664C2#2145858911yescharacteristic
subgroupG/H
centerC23C22×D244
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1771C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenter#1771D61
hypocenterC61#23013
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1771

Complements

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

NH ≅ G/Ntype
2. C2×#16839870C2semidirect
5. C4⋊C4×Dic122C2semidirect
7. #21446583C2semidirect
9. #21446495C2semidirect
11. #21446159C2semidirect
12. #21446090C2semidirect
14. #21445650C2semidirect
15. C61×#1771C2semidirect
24. #16839985C4semidirect
68. C2×Dic244C2×C4semidirect
81. C4×Dic122C2×C4semidirect
112. #14519702C2×C4semidirect
131. C4⋊C8×C61C2×C4semidirect
338. #1771D61semidirect
394. C61#23013semidirect
451. C4⋊C8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399852
C2×#168398702
C2×C4×#117581902
#214488882
C4⋊C4×Dic1222
#214466352
#214465832
#214465162
#214464952
#214461732
#214461592
#214460902
#214460832
#214456502
C61×#17712
#2301361