Finite Group Databasebeta

← #29702494

Subgroups — #29702494

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×#16839983C211yesmaximal, characteristic
2C2×#16839868C211yesmaximal, characteristic, complemented
3C2×C4×#11758200C211yesmaximal, characteristic
4#21448892C211yesmaximal, characteristic
5#21448733C211yesmaximal, characteristic, complemented
6#21446634C211yesmaximal, characteristic
7#21446579C211yesmaximal, characteristic, complemented
8#21446518C211yesmaximal, characteristic
9#21446494C211yesmaximal, characteristic, complemented
10#21446171C211yesmaximal, characteristic
11#21446159C211yesmaximal, characteristic, complemented
12#21446088C211yesmaximal, characteristic, complemented
13#21445690C211yesmaximal, characteristic, complemented
14#21445648C211yesmaximal, characteristic, complemented
15C61×#1864C211yesmaximal, characteristic
16C22×Dic244C2211yescharacteristic
17C22×#11758200C2211yescharacteristic
18C2×#14519764C2211yescharacteristic
19C2×C4×Dic122C2211yescharacteristic
20C2×#14519742C2211yescharacteristic
21C2×#14519742C2211yescharacteristic
22C2×#14519742C2211yescharacteristic
23#16839983C418yescomplemented
24#16839868—24no—
25C4×#11758200—24no—
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519702C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519700C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33C2×#14519698C2211yescharacteristic
34C2×#14519687C2211yescharacteristic
35C2×#14519687C2211yescharacteristic
36C2×C4×#92418C2211yescharacteristic
37#16839668C2211yescharacteristic
38#16839665C2211yescharacteristic
39#16839661C2211yescharacteristic
40#16839659—21no—
41#16839657—21no—
42#16839656—21no—
43#16839650—21no—
44#16839635—21no—
45#16839631—21no—
46#16839631C2211yescharacteristic
47C4⋊C4×Dic61C2211yescharacteristic
48C4×#11758196C2211yescharacteristic
49#16839613—21no—
50C4×C22⋊Dic61—21no—
51#16839525C2211yescharacteristic
52#16839524C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839516C2211yescharacteristic
55C2×C4×D4×C61C2211yescharacteristic
56C2×#106×C61C2211yescharacteristic
57C2×#102×C61C2211yescharacteristic
58C2×#102×C61C2211yescharacteristic
59C61×#389C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C61×#336C2211yescharacteristic
62C22×Dic122C2311yescharacteristic
63C22×Dic122C2311yescharacteristic
64C2×Dic244—22no—
65C2×Dic244—22no—
66C2×Dic244C2×C414yescomplemented
67C2×#11758200—22no—
68C2×#11758200—22no—
69C2×#11758200C2×C414yes—
70C22×#92418C2311yescharacteristic
71C22×#92418C2311yescharacteristic
72#14519764—22no—
73#14519764C2×C414yes—
74C4×Dic122—22no—
75C4×Dic122—41no—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—41no—
80#14519742—41no—
81#14519742C2×C414yescomplemented
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61C2311yescharacteristic
84C2×C4⋊Dic61C2311yescharacteristic
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×#11758196—21no—
88C2×#11758196—21no—
89C2×#11758196—21no—
90C2×#11758196—21no—
91C2×#11758196D411yescharacteristic
92C2×#11758196D411yescharacteristic
93C2×C22⋊Dic61—21no—
94C2×C22⋊Dic61—21no—
95C2×C22⋊Dic61—21no—
96C2×C22⋊Dic61—21no—
97C2×C4×Dic61—21no—
98C2×C4×Dic61—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61C2311yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104#14519708—22no—
105#14519706C2×C414yescomplemented
106#14519702—22no—
107#14519702—41no—
108#14519700—22no—
109#14519700C2×C414yescomplemented
110#14519700D414yes—
111#14519698—22no—
112#14519687—22no—
113#14519687—22no—
114C4×#92418—22no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119#14519684D411yescharacteristic
120#14519684D411yescharacteristic
121C22×D4×C61C2311yescharacteristic
122C23×C4×C61—21no—
123C22×C8×C61C2311yescharacteristic
124C22×C8×C61C2311yescharacteristic
125C4×D4×C61—22no—
126C4×D4×C61—41no—
127C2×C4⋊C4×C61—21no—
128C2×C4⋊C4×C61C2311yescharacteristic
129C2×C4⋊C4×C61C2311yescharacteristic
130C2×C4⋊C4×C61C2311yescharacteristic
131C2×C22⋊C4×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#106×C61—22no—
134#102×C61—41no—
135#102×C61C2×C414yes—
136#95×C61—21no—
137C24×C61—21no—
138C2×D4×C61—21no—
139C2×D4×C61—21no—
140C2×D4×C61—21no—
141C2×D4×C61—21no—
142C2×D4×C61—21no—
143C2×D4×C61C22×C412yes—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—42no—
148C22×C4×C61—42no—
149C22×C4×C61—81no—
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2×D411yescharacteristic
152C22×C4×C61C2411yescharacteristic
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61—21no—
156C22×Dic61—21no—
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C22×Dic61C2×D411yescharacteristic
160C22×Dic61C2×D411yescharacteristic
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×C61C22×C412yes—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—21no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—81no—
184C4⋊C4×C61C22×C412yes—
185C4⋊C4×C61C2×D412yes—
186C22⋊C4×C61—42no—
187C42×C61—21no—
188C42×C61—21no—
189C22⋊Dic61—24no—
190C22⋊Dic61—24no—
191C22⋊Dic61—42no—
192C22⋊Dic61—81no—
193#11758200—28no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—81no—
199#11758196—22no—
200#11758196—24no—
201#11758196—41no—
202#11758196—42no—
203#11758196—42no—
204#11758196—42no—
205#11758196—42no—
206#11758196—81no—
207C4×Dic61—21no—
208C4×Dic61—21no—
209C4×Dic61—24no—
210C4×Dic61—42no—
211C4×Dic61—42no—
212C4×Dic61—42no—
213C4×Dic61—81no—
214C4×Dic61C22×C412yes—
215C4×Dic61#4112yes—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—21no—
219C2×#92418—21no—
220C2×#92418C22×C412yes—
221C2×#92418C2×D412yes—
222Dic244—28no—
223C23×C61—22no—
224C23×C61—41no—
225C23×C61—41no—
226C23×C61—41no—
227C23×C61—41no—
228C23×C61—41no—
229C23×C61—41no—
230C23×C61C22×D411yescharacteristic
231D4×C61—22no—
232D4×C61—22no—
233D4×C61—41no—
234D4×C61—41no—
235C2×C4×C61—21no—
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—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×C61#13712yes—
251C2×C4×C61C23×C411yescharacteristic
252C2×C4×C61C22×D411yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61C2×#4111yescharacteristic
255C2×C4×C61H(5,𝔽2)11yescharacteristic
256C2×C4×C612−1+411yescharacteristic
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61—22no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—22no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—41no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61C4×D412yes—
280C2×Dic61C4×D412yes—
281Dic122—22no—
282Dic122—22no—
283Dic122—22no—
284Dic122—22no—
285Dic122—22no—
286Dic122—41no—
287Dic122—41no—
288Dic122C4×D414yes—
289C8×C61—22no—
290C8×C61—41no—
291#92418—22no—
292#92418—22no—
293#23013—611nomaximal, Sylow
294C22×C61—42no—
295C22×C61—42no—
296C22×C61—42no—
297C22×C61—81no—
298C22×C61—81no—
299C22×C61—81no—
300C22×C61—81no—
301C22×C61C2×C4×D411yescharacteristic
302C22×C61D4211yescharacteristic
303C22×C61#54811yescharacteristic
304C22×C61C2×#13711yescharacteristic
305C22×C61#57511yescharacteristic
306C22×C61#57711yescharacteristic
307C22×C61#57711yescharacteristic
308C4×C61—22no—
309C4×C61—42no—
310C4×C61—42no—
311C4×C61—41no—
312C4×C61—81no—
313C4×C61C2×C4×D411yescharacteristic
314C4×C61#52011yescharacteristic
315C4×C61#54711yescharacteristic
316C4×C61#55311yescharacteristic
317Dic61—24no—
318Dic61—22no—
319Dic61—22no—
320Dic61—42no—
321Dic61—41no—
322Dic61—41no—
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#1864D6111yescharacteristic
338#1771—611no—
339#1763—611no—
340C2×C61—82no—
341C2×C61#236111yescharacteristic
342C2×C61C4×#13711yescharacteristic
343C2×C61#295211yescharacteristic
344C2×C61#326611yescharacteristic
345C2×C61#327111yescharacteristic
346C2×C61#336911yescharacteristic
347C2×C61#337111yescharacteristic
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×D4C2×D6111yescharacteristic
356#464—618no—
357#441—1224no—
358C4×SD16—1224no—
359C2×#106—611no—
360C2×#106C2×D6111yescharacteristic
361C2×C4⋊C8—611no—
362C2×#103—611no—
363C2×#103—611no—
364C2×#103—611no—
365C2×#103—611no—
366C2×#103—611no—
367C2×#102—611no—
368C2×#102C2×D6111yescharacteristic
369C2×#102C2×D6111yescharacteristic
370C2×#97—611no—
371C2×#97—611no—
372C2×C4×C8—611no—
373#391—611no—
374#390—611no—
375#389—611no—
376#389—1221no—
377#389—1221no—
378#389C2×D6111yescharacteristic
379#388—1221no—
380#388—1221no—
381#385—1221no—
382#384—611no—
383#384—1221no—
384C4×C4⋊C4—611no—
385C4×C4⋊C4—611no—
386C4×C4⋊C4—1221no—
387C4×C22⋊C4—1221no—
388#340—611no—
389#340—611no—
390#340—611no—
391#340C2×D6111yescharacteristic
392#336—611no—
393#336C2×D6111yescharacteristic
394C61#2301311yescharacteristic, complemented, Sylow
395C22×Q8—611no—
396C22×Q8—611no—
397C22×D4C22×D6111yescharacteristic
398C23×C4—21no—
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×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—22no—
420C4×D4—41no—
421C2×C4⋊C4—21no—
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⋊C4C22×D6111yescharacteristic
434C2×C4⋊C4C22×D6111yescharacteristic
435C2×C4⋊C4C22×D6111yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—1221no—
438C2×C22⋊C4—1221no—
439C2×C22⋊C4—1221no—
440C2×C22⋊C4—1221no—
441C2×C42—611no—
442C2×C42—611no—
443C2×C42—1221no—
444C2×C42—1221no—
445C2×C42—1221no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42C22×D6111yescharacteristic
449#106—22no—
450#106—1222no—
451C4⋊C8—614no—
452#103—614no—
453#103—614no—
454#103—1222no—
455#103—1222no—
456#103—2441no—
457#102—41no—
458#102—1222no—
459#102C4×D6114yes—
460#97—1222no—
461#97—1222no—
462C4×C8—1222no—
463#95—21no—
464#95—611no—
465#95—611no—
466#95—1221no—
467#95—1221no—
468#95—1221no—
469#95—1221no—
470C24—21no—
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—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4C2×C4×D6112yes—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—42no—
493C22×C4—42no—
494C22×C4—81no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—611no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4D4×D6111yescharacteristic
504C22×C4D4×D6111yescharacteristic
505C22×C4C23×D6111yescharacteristic
506Q16—1228no—
507SD16—1228no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—612no—
514C2×C8—612no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8C2×C4×D6112yes—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—81no—
525C4⋊C4—1224no—
526C4⋊C4—1222no—
527C4⋊C4—1221no—
528C4⋊C4—1221no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—4881no—
532C4⋊C4—4881no—
533C4⋊C4—2441no—
534C4⋊C4—2442no—
535C4⋊C4—2442no—
536C4⋊C4—2442no—
537C4⋊C4—2442no—
538C4⋊C4C2×C4×D6112yes—
539C4⋊C4D4×D6112yes—
540C22⋊C4—42no—
541C22⋊C4—1224no—
542C22⋊C4—1224no—
543C22⋊C4—4881no—
544C22⋊C4—2442no—
545C42—21no—
546C42—21no—
547C42—612no—
548C42—612no—
549C42—1224no—
550C42—1221no—
551C42—1221no—
552C42—4881no—
553C42—2442no—
554C42—2442no—
555C42—2442no—
556C23—22no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23C2×D4×D6111yescharacteristic
564Q8—614no—
565Q8—1222no—
566Q8—1222no—
567Q8—1222no—
568Q8—1222no—
569Q8—1222no—
570Q8—2441no—
571Q8—2441no—
572D4—22no—
573D4—22no—
574D4—41no—
575D4—41no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—22no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—84no—
591C2×C4—612no—
592C2×C4—612no—
593C2×C4—1222no—
594C2×C4—1221no—
595C2×C4—1221no—
596C2×C4—1221no—
597C2×C4—1221no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1221no—
604C2×C4—1221no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4#1451980612yes—
616C2×C4C22×C4×D6111yescharacteristic
617C2×C4C2×D4×D6111yescharacteristic
618C2×C4#41×D6111yescharacteristic
619C2×C4#41×D6111yescharacteristic
620C2×C4#1451982211yescharacteristic
621C2×C4#1451982511yescharacteristic
622C8—22no—
623C8—41no—
624C8—1222no—
625C8—1222no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—81no—
630C22—81no—
631C22—81no—
632C22—81no—
633C22C4×D4×D6111yescharacteristic
634C22#1684039411yescharacteristic
635C22#1684042211yescharacteristic
636C22C2×#1451980611yescharacteristic
637C22#1684055911yescharacteristic
638C22#1684057211yescharacteristic
639C22#1684057211yescharacteristic
640C4—22no—
641C4—42no—
642C4—42no—
643C4—41no—
644C4—81no—
645C4—1222no—
646C4—1222no—
647C4—1224no—
648C4—4881no—
649C4—4881no—
650C4—2442no—
651C4—2441no—
652C4—2441no—
653C4C4×D4×D6111yescharacteristic
654C4#1684024311yescharacteristic
655C4#1684041411yescharacteristic
656C4#1684045711yescharacteristic
657C2—82no—
658C2#2144883811yescharacteristic
659C2C4×#1451980611yescharacteristic
660C2#2145470111yescharacteristic
661C2#2145733311yescharacteristic
662C2#2145739711yescharacteristic
663C2#2145855811yescharacteristic
664C2#2145858511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1864C2
socleC23×C61C22×D4
prime radicalC22×D4×C61C23
cosocleC22×C4×C61C24
hypercenter#1864D61
hypocenterC61#23013
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1864

Complements

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

NH ≅ G/Ntype
2. C2×#16839868C2semidirect
5. #21448733C2semidirect
7. #21446579C2semidirect
9. #21446494C2semidirect
11. #21446159C2semidirect
12. #21446088C2semidirect
13. #21445690C2semidirect
14. #21445648C2semidirect
23. #16839983C4semidirect
66. C2×Dic244C2×C4semidirect
81. #14519742C2×C4semidirect
105. #14519706C2×C4semidirect
109. #14519700C2×C4semidirect
394. C61#23013semidirect

Maximal subgroups

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

subgroupindex
C2×#168399832
C2×#168398682
C2×C4×#117582002
#214488922
#214487332
#214466342
#214465792
#214465182
#214464942
#214461712
#214461592
#214460882
#214456902
#214456482
C61×#18642
#2301361