Finite Group Databasebeta

← #29702491

Subgroups — #29702491

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×#16839984C211yesmaximal, characteristic
2C2×#16839867C211yesmaximal, characteristic, complemented
3C2×C4×#11758201C211yesmaximal, characteristic
4#21448888C211yesmaximal, characteristic
5#21446633C211yesmaximal, characteristic
6#21446580C211yesmaximal, characteristic, complemented
7#21446515C211yesmaximal, characteristic
8#21446494C211yesmaximal, characteristic, complemented
9#21446174C211yesmaximal, characteristic
10#21446160C211yesmaximal, characteristic, complemented
11#21446086C211yesmaximal, characteristic, complemented
12#21446079C211yesmaximal, characteristic
13#21445694C211yesmaximal, characteristic, complemented
14#21445649C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
16C22×#11758202C2211yescharacteristic
17C22×#11758201C2211yescharacteristic
18C2×#14519765C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×C4×D244C2211yescharacteristic
21#16839984C418yescomplemented
22#16839867—24no—
23C4×#11758201—24no—
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519700C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519697C2211yescharacteristic
33C2×#14519697C2211yescharacteristic
34C2×#14519697C2211yescharacteristic
35C2×#14519688C2211yescharacteristic
36C2×#14519688C2211yescharacteristic
37C2×C4×#92418C2211yescharacteristic
38#16839668C2211yescharacteristic
39#16839664C2211yescharacteristic
40#16839652—21no—
41#16839651—21no—
42#16839633—21no—
43C4×C4⋊Dic61C2211yescharacteristic
44#16839609—21no—
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839517C2211yescharacteristic
50#16839517C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C2×C4×Q8×C61C2211yescharacteristic
53C2×C4×Q8×C61C2211yescharacteristic
54C61×#389—21no—
55C61×#389—21no—
56C61×#389C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#384—21no—
59C61×#384C2211yescharacteristic
60C4×C4⋊C4×C61—21no—
61C4×C4⋊C4×C61C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×D244C2311yescharacteristic
64C22×C4×D61—21no—
65C2×#11758202—22no—
66C2×#11758202—22no—
67C2×#11758202C2×C414yescomplemented
68C2×#11758201—22no—
69C2×#11758201—22no—
70C2×#11758201C2×C414yes—
71C22×#92418C2311yescharacteristic
72C22×#92418C2311yescharacteristic
73C22×#92418C2311yescharacteristic
74C22×#92418C2311yescharacteristic
75#14519765—22no—
76#14519765C2×C414yes—
77C4×Dic122—22no—
78C4×Dic122—41no—
79C4×D244—22no—
80C4×D244—41no—
81C2×C4⋊Dic61C2311yescharacteristic
82C2×C4⋊Dic61C2311yescharacteristic
83C2×#11758196—21no—
84C2×#11758198—21no—
85C2×#11758198—21no—
86C2×#11758198—21no—
87C2×#11758198—21no—
88C2×#11758198—21no—
89C2×C4×Dic61—21no—
90C2×C4×Dic61—21no—
91#14519708—22no—
92#14519708—22no—
93C4⋊#92418C2×C414yescomplemented
94#14519701—22no—
95#14519701D414yes—
96#14519700—22no—
97#14519700—41no—
98#14519700C2×C414yescomplemented
99#14519697—22no—
100#14519697—41no—
101#14519697C2×C414yes—
102#14519688—22no—
103#14519688—22no—
104C4×#92418—22no—
105#14519684—21no—
106#14519684—21no—
107#14519684—21no—
108C22×Q8×C61C2311yescharacteristic
109C4×Q8×C61—22no—
110C4×Q8×C61—22no—
111C4×Q8×C61—22no—
112C4×Q8×C61—41no—
113C4×Q8×C61—41no—
114C4×Q8×C61C2×C414yescomplemented
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—21no—
118C2×C4⋊C4×C61—21no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—21no—
121C2×C4⋊C4×C61—21no—
122C2×C4⋊C4×C61D411yescharacteristic
123C2×C4⋊C4×C61D411yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C4⋊C4×C61C2311yescharacteristic
127C2×C42×C61—21no—
128C2×C42×C61—21no—
129C2×C42×C61—21no—
130C2×C42×C61C2311yescharacteristic
131C2×C42×C61C2311yescharacteristic
132C2×C42×C61C2311yescharacteristic
133#95×C61—21no—
134#95×C61—21no—
135#95×C61D411yescharacteristic
136#95×C61D411yescharacteristic
137C23×D61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C22×C412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2411yescharacteristic
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244C22×C412yes—
164C2×C4×D61—42no—
165C2×C4×D61—42no—
166C2×C4×D61—81no—
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122C22×C412yes—
173C4⋊C4×C61—24no—
174C4⋊C4×C61—22no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—41no—
181C4⋊C4×C61—42no—
182C4⋊C4×C61—42no—
183C4⋊C4×C61—42no—
184C4⋊C4×C61—81no—
185C4⋊C4×C61—81no—
186C4⋊C4×C61—81no—
187C4⋊C4×C61C22×C412yes—
188C42×C61—24no—
189C42×C61—21no—
190C42×C61—21no—
191C42×C61—21no—
192C42×C61—21no—
193C42×C61—42no—
194C42×C61—42no—
195C42×C61C22×C412yes—
196C42×C61#4112yes—
197#11758202—28no—
198#11758201—28no—
199#11758198—24no—
200#11758198—24no—
201#11758198—42no—
202#11758198—42no—
203#11758198—81no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61C2×D412yes—
208#11758196—42no—
209C4×Dic61—42no—
210C4×Dic61—81no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C2×#92418C22×C412yes—
222C2×#92418C2×D412yes—
223C23×C61C22×D411yescharacteristic
224C22×D61—22no—
225C22×D61—41no—
226C22×D61—41no—
227C22×D61—41no—
228C22×D61—41no—
229C22×D61—41no—
230C22×D61—41no—
231Q8×C61—22no—
232Q8×C61—22no—
233Q8×C61—22no—
234Q8×C61C4×D414yes—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—21no—
244C2×C4×C61—22no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61C4×D412yes—
263C2×C4×C61C4×D412yes—
264C2×C4×C61#13712yes—
265C2×C4×C61C23×C411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61H(5,𝔽2)11yescharacteristic
270C2×C4×C612−1+411yescharacteristic
271C2×Dic61—22no—
272C2×Dic61—22no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280D244—22no—
281D244—22no—
282D244—41no—
283D244—41no—
284C4×D61—84no—
285Dic122—22no—
286Dic122—22no—
287Dic122—41no—
288Dic122—41no—
289#92418—22no—
290#92418—22no—
291#92418—22no—
292#92418—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—24no—
309C4×C61—22no—
310C4×C61—22no—
311C4×C61—22no—
312C4×C61—42no—
313C4×C61—41no—
314C4×C61—41no—
315C4×C61—41no—
316C4×C61—81no—
317C4×C61—81no—
318C4×C61C2×C4×D411yescharacteristic
319C4×C61#52011yescharacteristic
320C4×C61#54711yescharacteristic
321C4×C61#55311yescharacteristic
322Dic61—42no—
323Dic61—42no—
324Dic61—81no—
325C2×#464—611no—
326C2×#441—611no—
327C2×C4×SD16—611no—
328#2367—611no—
329Q8×C4⋊C4D6111yescharacteristic, complemented
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#1771—611no—
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×Q8C2×D6111yescharacteristic
353C2×C4×Q8C2×D6111yescharacteristic
354C2×C4×Q8C2×D6111yescharacteristic
355C2×C4×D4—611no—
356#464—618no—
357#441—1224no—
358C4×SD16—1224no—
359C2×#106—611no—
360C2×#106—611no—
361C2×C4⋊C8—611no—
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×#97—611no—
371C2×#97—611no—
372C2×C4×C8—611no—
373#391—611no—
374#390—611no—
375#389—21no—
376#389—21no—
377#389C2×D6111yescharacteristic
378#389C2×D6111yescharacteristic
379#388—1221no—
380#388—1221no—
381#385—1221no—
382#384—21no—
383#384C2×D6111yescharacteristic
384C4×C4⋊C4—21no—
385C4×C4⋊C4—611no—
386C4×C4⋊C4C2×D6111yescharacteristic
387C4×C22⋊C4—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340—611no—
392#336—611no—
393#336—611no—
394C61#2301311yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8C22×D6111yescharacteristic
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×C8—611no—
406C22×C8—611no—
407C22×C8—611no—
408C22×C8—611no—
409#124—614no—
410#124—1222no—
411C4×Q8—22no—
412C4×Q8—22no—
413C4×Q8—22no—
414C4×Q8—41no—
415C4×Q8—41no—
416C4×Q8—1222no—
417C4×Q8—2441no—
418C4×Q8C4×D6114yescomplemented
419C4×D4—1222no—
420C4×D4—2441no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—21no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—1221no—
431C2×C4⋊C4C22⋊D6111yescharacteristic
432C2×C4⋊C4C22⋊D6111yescharacteristic
433C2×C4⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—1221no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—21no—
442C2×C42—21no—
443C2×C42—21no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42C22×D6111yescharacteristic
447C2×C42C22×D6111yescharacteristic
448C2×C42C22×D6111yescharacteristic
449#106—1222no—
450#106—1222no—
451C4⋊C8—614no—
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—1222no—
461#97—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—21no—
465#95—1221no—
466#95—1221no—
467#95—1221no—
468#95C22⋊D6111yescharacteristic
469#95C22⋊D6111yescharacteristic
470C24—1221no—
471C2×Q8—21no—
472C2×Q8—21no—
473C2×Q8—21no—
474C2×Q8—612no—
475C2×Q8—1221no—
476C2×Q8—1221no—
477C2×Q8—1221no—
478C2×Q8—1221no—
479C2×Q8—1221no—
480C2×Q8C2×C4×D6112yes—
481C2×Q8D4×D6112yes—
482C2×Q8#1175822612yes—
483C2×D4—612no—
484C2×D4—1221no—
485C2×D4—1221no—
486C2×D4—1221no—
487C2×D4—1221no—
488C2×D4—1221no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—1221no—
494C22×C4—1221no—
495C22×C4—1221no—
496C22×C4—4881no—
497C22×C4—2442no—
498C22×C4—2442no—
499C22×C4C2×C22⋊D6111yescharacteristic
500C22×C4C2×C22⋊D6111yescharacteristic
501C22×C4C2×C22⋊D6111yescharacteristic
502C22×C4C2×C22⋊D6111yescharacteristic
503C22×C4C2×C22⋊D6111yescharacteristic
504C22×C4C2×C22⋊D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506Q16—1228no—
507SD16—1228no—
508C2×C8—612no—
509C2×C8—612no—
510C2×C8—612no—
511C2×C8—1221no—
512C2×C8—1221no—
513C2×C8—1221no—
514C2×C8—1221no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C4⋊C4—22no—
521C4⋊C4—21no—
522C4⋊C4—24no—
523C4⋊C4—21no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—41no—
528C4⋊C4—42no—
529C4⋊C4—42no—
530C4⋊C4—42no—
531C4⋊C4—81no—
532C4⋊C4—81no—
533C4⋊C4—81no—
534C4⋊C4—612no—
535C4⋊C4—1221no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—2442no—
539C4⋊C4C2×C4×D6112yes—
540C22⋊C4—1224no—
541C22⋊C4—1224no—
542C22⋊C4—4881no—
543C22⋊C4—2442no—
544C22⋊C4—2442no—
545C42—21no—
546C42—21no—
547C42—21no—
548C42—21no—
549C42—24no—
550C42—42no—
551C42—42no—
552C42—4881no—
553C42—2442no—
554C42C2×C4×D6112yes—
555C42#1175822212yes—
556C23—1222no—
557C23—2441no—
558C23—2441no—
559C23—2441no—
560C23—2441no—
561C23—2441no—
562C23—2441no—
563C23C22×C22⋊D6111yescharacteristic
564Q8—22no—
565Q8—22no—
566Q8—22no—
567Q8—1222no—
568Q8—1222no—
569Q8—2441no—
570Q8—2441no—
571Q8#1451973914yes—
572D4—1222no—
573D4—1222no—
574D4—2441no—
575D4—2441no—
576C2×C4—22no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—4884no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4C4×C22⋊D6112yes—
614C2×C4C4×C22⋊D6112yes—
615C2×C4#1451980812yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4C2×#1175822211yescharacteristic
619C2×C4C2×#1175822611yescharacteristic
620C2×C4#1451982211yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—1222no—
623C8—1222no—
624C8—1222no—
625C8—2441no—
626C22—4881no—
627C22—4881no—
628C22—4881no—
629C22—4881no—
630C22—2442no—
631C22—2442no—
632C22—2442no—
633C22C2×C4×C22⋊D6111yescharacteristic
634C22D4×C22⋊D6111yescharacteristic
635C22#1684042311yescharacteristic
636C22C2×#1451980811yescharacteristic
637C22#1684056011yescharacteristic
638C22#1684057011yescharacteristic
639C22#1684057311yescharacteristic
640C4—22no—
641C4—22no—
642C4—24no—
643C4—22no—
644C4—42no—
645C4—41no—
646C4—41no—
647C4—41no—
648C4—81no—
649C4—81no—
650C4—4881no—
651C4—2442no—
652C4—2442no—
653C4C2×#1451973911yescharacteristic
654C4#1684024311yescharacteristic
655C4#1684041311yescharacteristic
656C4#1684046011yescharacteristic
657C2—4882no—
658C2#2144883511yescharacteristic
659C2C4×#1451980811yescharacteristic
660C2#2145470011yescharacteristic
661C2#2145733011yescharacteristic
662C2#2145740011yescharacteristic
663C2#2145855911yescharacteristic
664C2#2145858711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingQ8×C4⋊C4×C61C2
socleC23×C61C22×D4
prime radicalC22×D244C23
cosocleC22×C4×C61C24
hypercenterQ8×C4⋊C4D61
hypocenterC61#23013
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ Q8×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#16839867C2semidirect
6. #21446580C2semidirect
8. #21446494C2semidirect
10. #21446160C2semidirect
11. #21446086C2semidirect
13. #21445694C2semidirect
14. #21445649C2semidirect
15. Q8×C4⋊C4×C61C2semidirect
21. #16839984C4semidirect
67. C2×#11758202C2×C4semidirect
93. C4⋊#92418C2×C4semidirect
98. #14519700C2×C4semidirect
114. C4×Q8×C61C2×C4semidirect
329. Q8×C4⋊C4D61semidirect
394. C61#23013semidirect
418. C4×Q8C4×D61semidirect

Maximal subgroups

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

subgroupindex
C2×#168399842
C2×#168398672
C2×C4×#117582012
#214488882
#214466332
#214465802
#214465152
#214464942
#214461742
#214461602
#214460862
#214460792
#214456942
#214456492
Q8×C4⋊C4×C612
#2301361