Finite Group Databasebeta

← #29742876

Subgroups — #29742876

Order 15616.

812 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840068C211yesmaximal, characteristic, complemented
2C2×#16839802C211yesmaximal, characteristic
3#21449533C211yesmaximal, characteristic
4#21448824C211yesmaximal, characteristic
5#21447354C211yesmaximal, characteristic, complemented
6#21447311C211yesmaximal, characteristic, complemented
7#21447201C211yesmaximal, characteristic, complemented
8#21447155C211yesmaximal, characteristic, complemented
9#21446574C211yesmaximal, characteristic, complemented
10#21446395C211yesmaximal, characteristic, complemented
11#21446250C211yesmaximal, characteristic
12#21446129C211yesmaximal, characteristic
13#21446074C211yesmaximal, characteristic, complemented
14#21446001C211yesmaximal, characteristic
15C61×#1852C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519768C2211yescharacteristic
18C2×#14519756C2211yescharacteristic
19C2×#14519742C2211yescharacteristic
20C2×#14519736C2211yescharacteristic
21#16840068—24no—
22#16839802—24no—
23C2×#14519710C2211yescharacteristic
24C2×#14519710C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519706C2211yescharacteristic
28C2×#14519706C2211yescharacteristic
29C2×#14519702C2211yescharacteristic
30C2×#14519702C2211yescharacteristic
31C2×#14519700C2211yescharacteristic
32C2×#14519700C2211yescharacteristic
33#16839698—21no—
34#16839685—21no—
35#16839667C2211yescharacteristic
36#16839661C2211yescharacteristic
37#16839659—21no—
38#16839658C2211yescharacteristic
39#16839658C2211yescharacteristic
40#16839657—21no—
41#16839649—21no—
42#16839643—21no—
43#16839635—21no—
44#16839631C2211yescharacteristic
45C4⋊C4×Dic61C2211yescharacteristic
46C4×C22⋊Dic61—21no—
47C4×C22⋊Dic61C2211yescharacteristic
48#16839525C2211yescharacteristic
49#16839525C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839524C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×C4⋊C4×C61C2211yescharacteristic
55C2×#107×C61C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×C22⋊C8×C61C2211yescharacteristic
58C61×#380C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#340C2211yescharacteristic
61C22×Dic122C2311yescharacteristic
62C2×#11758194—22no—
63C2×#11758194—22no—
64C2×#11758194—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519768—22no—
68#14519768—41no—
69#14519756—22no—
70#14519756Q814yes—
71#14519742—22no—
72#14519742—41no—
73#14519736—22no—
74#14519736—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C4⋊Dic61C2311yescharacteristic
79C2×C4⋊Dic61C2311yescharacteristic
80C2×#11758196—21no—
81C2×#11758196—21no—
82C2×#11758196—21no—
83C2×#11758196—21no—
84C2×#11758196—21no—
85C2×#11758196—21no—
86C2×#11758196—21no—
87C2×C22⋊Dic61—21no—
88C2×C22⋊Dic61—21no—
89C2×C22⋊Dic61D411yescharacteristic
90C2×C22⋊Dic61D411yescharacteristic
91C2×C4×Dic61—21no—
92C2×C4×Dic61—21no—
93C2×C4×Dic61—21no—
94C2×C4×Dic61C2311yescharacteristic
95C2×C4×Dic61C2311yescharacteristic
96#14519710—22no—
97#14519710—22no—
98#14519708—22no—
99#14519708D414yes—
100#14519706—22no—
101#14519706—22no—
102#14519702—22no—
103#14519702—41no—
104#14519700—22no—
105#14519700—22no—
106#14519684—21no—
107#14519684—21no—
108#14519684—21no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C23×C4×C61—21no—
115C23×C4×C61C2311yescharacteristic
116C22×C8×C61C2311yescharacteristic
117C22×C8×C61C2311yescharacteristic
118C2×C4⋊C4×C61—22no—
119C2×C4⋊C4×C61—21no—
120C2×C4⋊C4×C61—41no—
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C22⋊C4×C61—21no—
126#107×C61—22no—
127#107×C61—41no—
128C22⋊C8×C61—22no—
129#95×C61—21no—
130C24×C61C2×D411yescharacteristic
131C22×C4×C61—21no—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61—21no—
137C22×C4×C61—21no—
138C22×C4×C61—21no—
139C22×C4×C61—42no—
140C22×C4×C61—42no—
141C22×C4×C61—81no—
142C22×C4×C61C2×D411yescharacteristic
143C22×C4×C61C2×Q812yes—
144C22×C4×C61#4112yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61—21no—
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153C22×Dic61C2×D411yescharacteristic
154C2×Dic122—21no—
155C2×Dic122—21no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122C2×Q812yes—
160C2×C8×C61—21no—
161C2×C8×C61—21no—
162C2×C8×C61—21no—
163C2×C8×C61—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—42no—
174C4⋊C4×C61C2×D412yes—
175C22⋊C4×C61—81no—
176C22⋊Dic61—24no—
177C22⋊Dic61—22no—
178C22⋊Dic61—41no—
179C22⋊Dic61—42no—
180C4⋊Dic61—21no—
181C4⋊Dic61—21no—
182C4⋊Dic61—21no—
183C4⋊Dic61—21no—
184C4⋊Dic61—42no—
185C4⋊Dic61—81no—
186C4⋊Dic61C2×D412yes—
187C4⋊Dic61C2×Q812yes—
188#11758196—24no—
189#11758196—42no—
190#11758196—42no—
191#11758196—42no—
192#11758196—42no—
193#11758196—81no—
194#11758196—81no—
195C4×Dic61—21no—
196C4×Dic61—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—42no—
200C4×Dic61—42no—
201C4×Dic61—81no—
202#11758194—28no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207C2×#92418C2×D412yes—
208C2×#92418#4112yes—
209C23×C61—21no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—21no—
214C23×C61—21no—
215C23×C61C22⋊Q812yes—
216C23×C61C22×D411yescharacteristic
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—22no—
223C2×C4×C61—22no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—41no—
234C2×C4×C61—41no—
235C2×C4×C61—84no—
236C2×C4×C61C22⋊Q812yes—
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×Q811yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C2×C4×C61C2×#4111yescharacteristic
241C2×C4×C61H(5,𝔽2)11yescharacteristic
242C2×C4×C612−1+411yescharacteristic
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—22no—
246C2×Dic61—22no—
247C2×Dic61—21no—
248C2×Dic61—21no—
249C2×Dic61—22no—
250C2×Dic61—21no—
251C2×Dic61—21no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262C2×Dic61—41no—
263C2×Dic61—41no—
264C2×Dic61—41no—
265C2×Dic61—41no—
266C2×Dic61#13712yes—
267Dic122—22no—
268Dic122—22no—
269Dic122—41no—
270Dic122—41no—
271C8×C61—41no—
272C8×C61—41no—
273#92418—22no—
274#92418—22no—
275#26362—611nomaximal, Sylow
276C22×C61—22no—
277C22×C61—22no—
278C22×C61—22no—
279C22×C61—41no—
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61C2×C22⋊Q811yescharacteristic
284C22×C61D4211yescharacteristic
285C22×C61#54711yescharacteristic
286C22×C61C2×#13711yescharacteristic
287C22×C61#57511yescharacteristic
288C22×C61#57611yescharacteristic
289C22×C61#57811yescharacteristic
290C4×C61—22no—
291C4×C61—42no—
292C4×C61—81no—
293C4×C61—81no—
294C4×C61—81no—
295C4×C61#52511yescharacteristic
296C4×C61#53911yescharacteristic
297C4×C61D4×Q811yescharacteristic
298C4×C61#55411yescharacteristic
299Dic61—22no—
300Dic61—42no—
301Dic61—41no—
302Dic61—42no—
303Dic61—41no—
304Dic61—41no—
305Dic61—42no—
306Dic61—81no—
307C2×#479—611no—
308C2×#428—611no—
309#2441—611no—
310#2360—611no—
311#2095—611no—
312#2086—611no—
313#2066—611no—
314#2059—611no—
315#1947—611no—
316#1919—611no—
317#1894—611no—
318#1873—611no—
319#1863—611no—
320#1852D6111yescharacteristic
321#1851—611no—
322C2×C61—42no—
323C2×C61#244911yescharacteristic
324C2×C61#307811yescharacteristic
325C2×C61#308311yescharacteristic
326C2×C61#326811yescharacteristic
327C2×C61#327611yescharacteristic
328C2×C61#334611yescharacteristic
329C2×C61#335211yescharacteristic
330C22×M16—611no—
331C2×#125—611no—
332C2×C22⋊Q8—611no—
333C2×C4×Q8—611no—
334C2×#117—611no—
335C22×C4⋊C4C2×D6111yescharacteristic
336#479—1224no—
337#428—1224no—
338C2×#107—611no—
339C2×#107—611no—
340C2×#107C2×D6111yescharacteristic
341C2×#107C2×D6111yescharacteristic
342C2×#106—611no—
343C2×#106—611no—
344C2×C4⋊C8—611no—
345C2×C4⋊C8—611no—
346C2×#103—611no—
347C2×#103—611no—
348C2×#103—611no—
349C2×#103—611no—
350C2×C22⋊C8C2×D6111yescharacteristic
351#399—1221no—
352#396—1221no—
353#391—611no—
354#389—611no—
355#389—611no—
356#389—611no—
357#389—1221no—
358#389—1221no—
359#387—1221no—
360#387—1221no—
361#385—1221no—
362#384—611no—
363#380C2×D6111yescharacteristic
364C4×C4⋊C4—611no—
365C4×C22⋊C4—611no—
366C4×C22⋊C4—1221no—
367#340—611no—
368#340—611no—
369#340—611no—
370#340—611no—
371#340C2×D6111yescharacteristic
372#340C2×D6111yescharacteristic
373#336—611no—
374#336—611no—
375C61#2636211yescharacteristic, complemented, Sylow
376C22×Q8—611no—
377C23×C4—21no—
378C23×C4C22×D6111yescharacteristic
379C2×M16—1222no—
380C2×M16—1222no—
381C2×M16—1222no—
382C22×C8—611no—
383C22×C8—611no—
384C22×C8C22×D6111yescharacteristic
385C22×C8C22×D6111yescharacteristic
386#125—1222no—
387#125—2441no—
388C22⋊Q8—614no—
389C22⋊Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—2441no—
392#117—1222no—
393#117—2441no—
394C2×C4⋊C4—22no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—41no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4—611no—
400C2×C4⋊C4—1221no—
401C2×C4⋊C4—1221no—
402C2×C4⋊C4—1221no—
403C2×C4⋊C4—1221no—
404C2×C4⋊C4—1221no—
405C2×C4⋊C4—1221no—
406C2×C4⋊C4—1221no—
407C2×C4⋊C4—1221no—
408C2×C4⋊C4—1221no—
409C2×C4⋊C4C22×D6111yescharacteristic
410C2×C4⋊C4C22×D6111yescharacteristic
411C2×C4⋊C4C22×D6111yescharacteristic
412C2×C4⋊C4C22×D6111yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—611no—
415C2×C22⋊C4—611no—
416C2×C22⋊C4—1221no—
417C2×C22⋊C4—1221no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—1221no—
421C2×C42—1221no—
422C2×C42—1221no—
423#107—22no—
424#107—41no—
425#107—1222no—
426#107—1222no—
427#106—614no—
428#106—1222no—
429C4⋊C8—1222no—
430C4⋊C8—1222no—
431#103—1222no—
432#103—1222no—
433#103—1222no—
434#103—2441no—
435C22⋊C8—22no—
436#95—21no—
437#95—611no—
438#95—611no—
439#95—1221no—
440#95—1221no—
441#95—1221no—
442#95—1221no—
443#95—1221no—
444#95—1221no—
445C24D4×D6111yescharacteristic
446C2×Q8—612no—
447C2×Q8—1221no—
448C2×Q8—1221no—
449C2×Q8—1221no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—81no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—1221no—
468C22×C4—1221no—
469C22×C4—1221no—
470C22×C4—1221no—
471C22×C4#1175822212yes—
472C22×C4D4×D6111yescharacteristic
473C22×C4Q8×D6112yes—
474C22×C4C23×D6111yescharacteristic
475M16—1228no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—612no—
483C2×C8—612no—
484C2×C8—1221no—
485C2×C8—1221no—
486C2×C8—1221no—
487C2×C8—1221no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—42no—
496C4⋊C4—612no—
497C4⋊C4—612no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4—1224no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—4881no—
504C4⋊C4—4881no—
505C4⋊C4—4881no—
506C4⋊C4—2442no—
507C4⋊C4—2442no—
508C4⋊C4—2442no—
509C4⋊C4—2442no—
510C4⋊C4—2442no—
511C4⋊C4D4×D6112yes—
512C22⋊C4—81no—
513C22⋊C4—1222no—
514C22⋊C4—1224no—
515C22⋊C4—2441no—
516C22⋊C4—2442no—
517C42—1221no—
518C42—1221no—
519C42—1221no—
520C42—1221no—
521C42—4881no—
522C42—2442no—
523C42—2442no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23#1451975512yes—
531C23C2×D4×D6111yescharacteristic
532Q8—1222no—
533Q8—1222no—
534Q8—2441no—
535Q8—2441no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4—612no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1221no—
561C2×C4—1221no—
562C2×C4—1221no—
563C2×C4—1221no—
564C2×C4—1221no—
565C2×C4—1221no—
566C2×C4—1221no—
567C2×C4—1221no—
568C2×C4—1222no—
569C2×C4—1221no—
570C2×C4—2441no—
571C2×C4—2441no—
572C2×C4—2441no—
573C2×C4—2441no—
574C2×C4—2441no—
575C2×C4—2441no—
576C2×C4—2441no—
577C2×C4—2441no—
578C2×C4—2441no—
579C2×C4#1451975512yes—
580C2×C4C2×D4×D6111yescharacteristic
581C2×C4C2×Q8×D6111yescharacteristic
582C2×C4#41×D6111yescharacteristic
583C2×C4C2×#1175822211yescharacteristic
584C2×C4#1451982211yescharacteristic
585C2×C4#1451982411yescharacteristic
586C8—41no—
587C8—41no—
588C8—1222no—
589C8—1222no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22C2×#1451975511yescharacteristic
598C22#1684039411yescharacteristic
599C22#1684041011yescharacteristic
600C22#137×D6111yescharacteristic
601C22#1684055811yescharacteristic
602C22#1684056311yescharacteristic
603C22#1684057511yescharacteristic
604C4—22no—
605C4—42no—
606C4—81no—
607C4—81no—
608C4—81no—
609C4—1222no—
610C4—4881no—
611C4—2441no—
612C4—2442no—
613C4—2442no—
614C4—2442no—
615C4—2441no—
616C4—2441no—
617C4#1684027111yescharacteristic
618C4#1684035411yescharacteristic
619C4#1684042511yescharacteristic
620C4#1684046611yescharacteristic
621C2—42no—
622C2#2144963011yescharacteristic
623C2#2145562311yescharacteristic
624C2#2145567511yescharacteristic
625C2#2145735711yescharacteristic
626C2#2145745811yescharacteristic
627C2#2145830711yescharacteristic
628C2#2145839411yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1852C2
socleC23×C61C22×D4
prime radicalC24×C61C2×D4
hypercenter#1852D61
hypocenterC61#26362
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1852

Complements

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

NH ≅ G/Ntype
1. C2×#16840068C2semidirect
5. #21447354C2semidirect
6. #21447311C2semidirect
7. #21447201C2semidirect
8. #21447155C2semidirect
9. #21446574C2semidirect
10. #21446395C2semidirect
13. #21446074C2semidirect
375. C61#26362semidirect

Maximal subgroups

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

subgroupindex
C2×#168400682
C2×#168398022
#214495332
#214488242
#214473542
#214473112
#214472012
#214471552
#214465742
#214463952
#214462502
#214461292
#214460742
#214460012
C61×#18522
#2636261