Finite Group Databasebeta

← #29734404

Subgroups — #29734404

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840370C211yesmaximal, characteristic
2C2×#16840162C211yesmaximal, characteristic, complemented
3C2×#16840129C211yesmaximal, characteristic, complemented
4#21451654C211yesmaximal, characteristic
5#21447025C211yesmaximal, characteristic
6#21447020C211yesmaximal, characteristic
7#21447017C211yesmaximal, characteristic
8#21447014C211yesmaximal, characteristic
9#21446786C211yesmaximal, characteristic, complemented
10#21446619C211yesmaximal, characteristic, complemented
11#21446090C211yesmaximal, characteristic, complemented
12#21446073C211yesmaximal, characteristic, complemented
13#21446072C211yesmaximal, characteristic, complemented
14#21445584C211yesmaximal, characteristic
15C61×#1865C211yesmaximal, characteristic, complemented
16#16840370—24no—
17#16840370—24no—
18C2×#14519775C2211yescharacteristic
19C2×#14519772—21no—
20C2×#14519771—21no—
21C2×#14519769—21no—
22C2×#14519768C2211yescharacteristic
23C2×#14519768C2211yescharacteristic
24C2×#14519767—21no—
25C2×#14519757C2211yescharacteristic
26C2×#14519753C2211yescharacteristic
27C2×C4×Dic122C2211yescharacteristic
28C2×#14519734C2211yescharacteristic
29#16840162—24no—
30#16840129—24no—
31C2×#14519710C2211yescharacteristic
32C2×#14519710C2211yescharacteristic
33C2×#14519709C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519702C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519701C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39C2×#14519700C2211yescharacteristic
40C2×#14519689C2211yescharacteristic
41C2×#14519689C2211yescharacteristic
42C2×#14519687C2211yescharacteristic
43C2×C8×Dic61C2211yescharacteristic
44#16839696—21no—
45#16839693—21no—
46#16839686—21no—
47#16839684—21no—
48#16839660C2211yescharacteristic
49#16839631C2211yescharacteristic
50#16839619C2211yescharacteristic
51#16839619C2211yescharacteristic
52#16839525C2211yescharacteristic
53#16839524C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839517C2211yescharacteristic
56C2×C4×Q8×C61C2211yescharacteristic
57C2×#106×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C61×#389C2211yescharacteristic
61C61×#340C2211yescharacteristic
62C61×#336C2211yescharacteristic
63C22×Dic122C2311yescharacteristic
64C22×C4×D61C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519775—22no—
68#14519775D414yes—
69#14519772—28no—
70#14519771—28no—
71#14519769—28no—
72#14519768—22no—
73#14519768—22no—
74#14519768—22no—
75#14519768D414yes—
76#14519767—28no—
77#14519757—22no—
78#14519757—22no—
79#14519753—22no—
80#14519753—22no—
81C4×Dic122—22no—
82C4×Dic122—22no—
83#14519734—22no—
84#14519734—22no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61—21no—
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×#11758196—21no—
98C2×#11758196—21no—
99C2×#11758196—21no—
100C2×#11758196D411yescharacteristic
101C2×#11758196D411yescharacteristic
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198—21no—
106C2×#11758198D411yescharacteristic
107C2×#11758198D411yescharacteristic
108C2×C4×Dic61—21no—
109C2×C4×Dic61C2311yescharacteristic
110#14519710—22no—
111#14519710—22no—
112#14519709—22no—
113#14519702—22no—
114#14519702—22no—
115#14519701—22no—
116#14519701—22no—
117#14519700—22no—
118#14519700—22no—
119#14519689—22no—
120#14519689—22no—
121#14519687—22no—
122C8×Dic61—22no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126C22×Q8×C61C2311yescharacteristic
127C22×C8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C4×Q8×C61—22no—
130C4×Q8×C61—22no—
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61—21no—
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C42×C61—21no—
137C2×C42×C61C2311yescharacteristic
138#106×C61—22no—
139#103×C61—22no—
140#103×C61—22no—
141#95×C61—21no—
142C23×D61C2×D411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D412yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—21no—
165C2×C4×D61—21no—
166C2×C4×D61—21no—
167C2×C4×D61—21no—
168C2×C4×D61—21no—
169C2×Dic122—21no—
170C2×Dic122—21no—
171C2×Dic122—21no—
172C2×Dic122—21no—
173C2×Dic122—21no—
174C2×Dic122C2×D412yes—
175C2×C8×C61—21no—
176C2×C8×C61—21no—
177C2×C8×C61—21no—
178C2×C8×C61—21no—
179C2×C8×C61—21no—
180C2×C8×C61#4112yes—
181C4⋊C4×C61—24no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61—21no—
185C4⋊C4×C61—21no—
186C4⋊C4×C61—21no—
187C4⋊C4×C61—81no—
188C4⋊C4×C61C2×D412yes—
189C42×C61—24no—
190C42×C61—21no—
191C42×C61—21no—
192#11758198—24no—
193#11758198—24no—
194#11758198—22no—
195#11758198—22no—
196#11758198—81no—
197#11758198—81no—
198C4⋊Dic61—24no—
199C4⋊Dic61—21no—
200C4⋊Dic61—24no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—42no—
206C4⋊Dic61C2×D412yes—
207#11758196—24no—
208#11758196—24no—
209#11758196—24no—
210#11758196—24no—
211#11758196—24no—
212#11758196—22no—
213#11758196—22no—
214#11758196—42no—
215#11758196—81no—
216#11758196—81no—
217#11758196—81no—
218C4×Dic61—24no—
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D412yes—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418#4112yes—
227C23×C61C22×D411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235Q8×C61—22no—
236Q8×C61—22no—
237Q8×C61—41no—
238Q8×C61—41no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—22no—
242C2×C4×C61—22no—
243C2×C4×C61—21no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—21no—
247C2×C4×C61—41no—
248C2×C4×C61—41no—
249C2×C4×C61—41no—
250C2×C4×C61—41no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61C22×D411yescharacteristic
258C2×C4×C61C22×D411yescharacteristic
259C2×C4×C61C2×#4111yescharacteristic
260C2×C4×C61C2×#4111yescharacteristic
261C2×C4×C612−1+411yescharacteristic
262C2×C4×C612−1+411yescharacteristic
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
267C2×Dic61—22no—
268C2×Dic61—22no—
269C2×Dic61—21no—
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—41no—
273C2×Dic61—41no—
274C2×Dic61—41no—
275C2×Dic61—41no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61#12012yes—
283C2×Dic61#12012yes—
284C4×D61—41no—
285C4×D61—41no—
286C4×D61—41no—
287C4×D61—41no—
288Dic122—22no—
289Dic122—22no—
290Dic122—41no—
291Dic122—41no—
292C8×C61—22no—
293C8×C61—41no—
294#92418—22no—
295#92418—41no—
296#25683—611nomaximal, Sylow
297C22×C61C2×#12011yescharacteristic
298C22×C61#53211yescharacteristic
299C22×C61#53611yescharacteristic
300C22×C61#57711yescharacteristic
301C22×C61#57711yescharacteristic
302C22×C61#57811yescharacteristic
303C22×C61#57811yescharacteristic
304C2×D61—41no—
305C2×D61—41no—
306C2×D61—41no—
307C2×D61—41no—
308C2×D61—41no—
309C2×D61—41no—
310C2×D61—41no—
311C4×C61—42no—
312C4×C61—41no—
313C4×C61—41no—
314C4×C61—42no—
315C4×C61—81no—
316C4×C61—81no—
317C4×C61#54711yescharacteristic
318C4×C61#54711yescharacteristic
319C4×C61#54811yescharacteristic
320C4×C61#54811yescharacteristic
321Dic61—22no—
322Dic61—22no—
323Dic61—42no—
324Dic61—41no—
325Dic61—42no—
326Dic61—81no—
327Dic61—81no—
328C2×#541—611no—
329C2×#499—611no—
330C2×#491—611no—
331#2629—611no—
332#2036—611no—
333#2036—611no—
334#2035—611no—
335#2035—611no—
336#1992—611no—
337#1956—611no—
338#1865—611no—
339#1865D6111yescharacteristic, complemented
340#1863—611no—
341#1863—611no—
342#1750—611no—
343C2×C61#243211yescharacteristic
344C2×C61#301311yescharacteristic
345C2×C61#301311yescharacteristic
346C2×C61#317411yescharacteristic
347C2×C61#317711yescharacteristic
348C2×C61#324211yescharacteristic
349C2×C61#324211yescharacteristic
350D61—81no—
351#541—1224no—
352#541—1224no—
353C2×C4⋊Q8—611no—
354C2×#126—1221no—
355C2×#126—1221no—
356C2×#125—611no—
357C2×#125—611no—
358C2×#125—1221no—
359C2×#125—1221no—
360C2×C22⋊Q8—611no—
361C2×C22⋊Q8—611no—
362C2×C4×Q8—611no—
363C2×C4×Q8C2×D6111yescharacteristic
364C2×#117—611no—
365#499—1224no—
366#491—1224no—
367C2×#107—611no—
368C2×#107—611no—
369C2×#106—611no—
370C2×#106C2×D6111yescharacteristic
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#103C2×D6111yescharacteristic
379C2×C22⋊C8—611no—
380C2×C22⋊C8—611no—
381C2×#97—611no—
382C2×C4×C8—611no—
383#398—1221no—
384#398—1221no—
385#396—1221no—
386#396—1221no—
387#389—611no—
388#389C2×D6111yescharacteristic
389#384—611no—
390#380—611no—
391#380—611no—
392#340—611no—
393#340—611no—
394#340—611no—
395#340C2×D6111yescharacteristic
396#336—611no—
397#336C2×D6111yescharacteristic
398C61#2568311yescharacteristic, complemented, Sylow
399C22×Q8—611no—
400C22×Q8C22×D6111yescharacteristic
401C23×C4—611no—
402C22×C8—611no—
403C22×C8—611no—
404C22×C8C22×D6111yescharacteristic
405C22×C8C22×D6111yescharacteristic
406C4⋊Q8—614no—
407C4⋊Q8—1222no—
408#126—1228no—
409#126—1228no—
410#125—614no—
411#125—1228no—
412#125—1222no—
413#125—1222no—
414#125—1222no—
415#125—1228no—
416C22⋊Q8—1222no—
417C22⋊Q8—1222no—
418C22⋊Q8—1222no—
419C22⋊Q8—1222no—
420C4×Q8—22no—
421C4×Q8—22no—
422C4×Q8—1222no—
423C4×Q8—1222no—
424#117—1222no—
425#117—1222no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—611no—
429C2×C4⋊C4—611no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—1221no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4—1221no—
437C2×C4⋊C4—1221no—
438C2×C4⋊C4—1221no—
439C2×C4⋊C4—1221no—
440C2×C4⋊C4—1221no—
441C2×C4⋊C4—1221no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4C22×D6111yescharacteristic
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C22⋊C4—611no—
449C2×C22⋊C4—611no—
450C2×C22⋊C4—1221no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C22⋊C4—1221no—
454C2×C42—21no—
455C2×C42—611no—
456C2×C42—1221no—
457C2×C42C22×D6111yescharacteristic
458#107—1222no—
459#107—1222no—
460#106—22no—
461#106—1222no—
462#103—22no—
463#103—22no—
464#103—1222no—
465#103—1222no—
466#103—1222no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470C22⋊C8—1222no—
471C22⋊C8—1222no—
472#97—1222no—
473C4×C8—1222no—
474#95—21no—
475#95—1221no—
476#95—1221no—
477#95—1221no—
478C24—611no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—612no—
485C2×Q8—1221no—
486C2×Q8—1221no—
487C2×Q8—1221no—
488C2×Q8—1221no—
489C2×Q8—1221no—
490C2×Q8D4×D6112yes—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—611no—
496C22×C4—611no—
497C22×C4—611no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4—1221no—
506C22×C4—1221no—
507C22×C4—1221no—
508C22×C4D4×D6111yescharacteristic
509C22×C4D4×D6111yescharacteristic
510C22×C4C23×D6111yescharacteristic
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—612no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8#1175822212yes—
523C4⋊C4—24no—
524C4⋊C4—21no—
525C4⋊C4—21no—
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—81no—
530C4⋊C4—612no—
531C4⋊C4—1222no—
532C4⋊C4—1222no—
533C4⋊C4—1224no—
534C4⋊C4—1224no—
535C4⋊C4—1224no—
536C4⋊C4—1224no—
537C4⋊C4—1224no—
538C4⋊C4—1224no—
539C4⋊C4—1224no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—1221no—
544C4⋊C4—1221no—
545C4⋊C4—4881no—
546C4⋊C4—4881no—
547C4⋊C4—4881no—
548C4⋊C4—2442no—
549C4⋊C4—2442no—
550C4⋊C4D4×D6112yes—
551C22⋊C4—1222no—
552C22⋊C4—1222no—
553C22⋊C4—1224no—
554C22⋊C4—1224no—
555C22⋊C4—4881no—
556C22⋊C4—4881no—
557C42—24no—
558C42—21no—
559C42—21no—
560C42—612no—
561C42—612no—
562C42—1224no—
563C23—1221no—
564C23—1221no—
565C23—1221no—
566C23—1221no—
567C23—1221no—
568C23—1221no—
569C23—1221no—
570C23C2×D4×D6111yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—41no—
574Q8—41no—
575Q8—1222no—
576Q8—1222no—
577Q8—2441no—
578Q8—2441no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—612no—
598C2×C4—612no—
599C2×C4—1221no—
600C2×C4—1221no—
601C2×C4—1221no—
602C2×C4—1221no—
603C2×C4—1222no—
604C2×C4—1222no—
605C2×C4—1221no—
606C2×C4—1221no—
607C2×C4—1221no—
608C2×C4—2441no—
609C2×C4—2441no—
610C2×C4—2441no—
611C2×C4—2441no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4C2×D4×D6111yescharacteristic
623C2×C4C2×D4×D6111yescharacteristic
624C2×C4#41×D6111yescharacteristic
625C2×C4C2×#1175822211yescharacteristic
626C2×C4#1451982411yescharacteristic
627C2×C4#1451982511yescharacteristic
628C8—22no—
629C8—41no—
630C8—1222no—
631C8—2441no—
632C22—2441no—
633C22—2441no—
634C22—2441no—
635C22—2441no—
636C22—2441no—
637C22—2441no—
638C22—2441no—
639C22#120×D6111yescharacteristic
640C22#1684030911yescharacteristic
641C22#1684033511yescharacteristic
642C22#1684056811yescharacteristic
643C22#1684056911yescharacteristic
644C22#1684057811yescharacteristic
645C22#1684057811yescharacteristic
646C4—41no—
647C4—41no—
648C4—42no—
649C4—42no—
650C4—81no—
651C4—81no—
652C4—1222no—
653C4—1222no—
654C4—4881no—
655C4—4881no—
656C4—2442no—
657C4—2441no—
658C4—2442no—
659C4#1684041011yescharacteristic
660C4#1684041311yescharacteristic
661C4#1684042111yescharacteristic
662C4#1684042211yescharacteristic
663C2—4881no—
664C2#2144944511yescharacteristic
665C2#2145512911yescharacteristic
666C2#2145513811yescharacteristic
667C2#2145644311yescharacteristic
668C2#2145647611yescharacteristic
669C2#2145708411yescharacteristic
670C2#2145709411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1865C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#1865D61
hypocenterC61#25683
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1865

Complements

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

NH ≅ G/Ntype
2. C2×#16840162C2semidirect
3. C2×#16840129C2semidirect
9. #21446786C2semidirect
10. #21446619C2semidirect
11. #21446090C2semidirect
12. #21446073C2semidirect
13. #21446072C2semidirect
15. C61×#1865C2semidirect
339. #1865D61semidirect
398. C61#25683semidirect

Maximal subgroups

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

subgroupindex
C2×#168403702
C2×#168401622
C2×#168401292
#214516542
#214470252
#214470202
#214470172
#214470142
#214467862
#214466192
#214460902
#214460732
#214460722
#214455842
C61×#18652
#2568361