Finite Group Databasebeta

← #29740153

Subgroups — #29740153

Order 15616.

826 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839986C211yesmaximal, characteristic
2C2×Q16×Dic61C211yesmaximal, characteristic, complemented
3#21449615C211yesmaximal, characteristic
4Q8×C4⋊Dic61C211yesmaximal, characteristic, complemented
5#21447346C211yesmaximal, characteristic
6#21447309C211yesmaximal, characteristic, complemented
7#21447227C211yesmaximal, characteristic, complemented
8#21447057C211yesmaximal, characteristic
9#21446503C211yesmaximal, characteristic
10#21446377C211yesmaximal, characteristic, complemented
11#21446174C211yesmaximal, characteristic
12#21446172C211yesmaximal, characteristic
13#21446075C211yesmaximal, characteristic, complemented
14#21446075C211yesmaximal, characteristic, complemented
15C61×#1878C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×#14519766C2211yescharacteristic
18C2×Q8×Dic61C2211yescharacteristic
19C2×Q8×Dic61C2211yescharacteristic
20C2×Q8×Dic61C2211yescharacteristic
21#16839986—24no—
22Q16×Dic61—24no—
23C2×#14519711C2211yescharacteristic
24C2×#14519711C2211yescharacteristic
25C2×#14519706C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519699C2211yescharacteristic
30C2×#14519699C2211yescharacteristic
31C2×#14519697C2211yescharacteristic
32C2×C8×Dic61C2211yescharacteristic
33#16839699—21no—
34#16839689—21no—
35#16839664C2211yescharacteristic
36#16839663C2211yescharacteristic
37#16839661—21no—
38#16839661—21no—
39#16839661C2211yescharacteristic
40#16839661C2211yescharacteristic
41#16839661C2211yescharacteristic
42#16839654—21no—
43#16839654—21no—
44#16839632—21no—
45#16839632C2211yescharacteristic
46C4×C4⋊Dic61—21no—
47C4×C4⋊Dic61C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C22×Q16×C61C2211yescharacteristic
55C2×C4×Q8×C61C2211yescharacteristic
56C2×#103×C61C2211yescharacteristic
57C2×#103×C61C2211yescharacteristic
58C61×#391C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C22×D244C2311yescharacteristic
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201—22no—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67#14519766—22no—
68#14519766—41no—
69Q8×Dic61—22no—
70Q8×Dic61—22no—
71Q8×Dic61—22no—
72Q8×Dic61—41no—
73Q8×Dic61—41no—
74Q8×Dic61—41no—
75C2×C4⋊Dic61—21no—
76C2×C4⋊Dic61—21no—
77C2×C4⋊Dic61—21no—
78C2×C4⋊Dic61—21no—
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—21no—
81C2×C4⋊Dic61—21no—
82C2×C4⋊Dic61—21no—
83C2×C4⋊Dic61D411yescharacteristic
84C2×C4⋊Dic61D411yescharacteristic
85C2×C4⋊Dic61C2311yescharacteristic
86C2×C4⋊Dic61C2311yescharacteristic
87C2×C4⋊Dic61C2311yescharacteristic
88C2×#11758198—21no—
89C2×#11758198—21no—
90C2×#11758198—21no—
91C2×#11758198—21no—
92C2×#11758198—21no—
93C2×#11758198—21no—
94C2×C4×Dic61—21no—
95C2×C4×Dic61—21no—
96C2×C4×Dic61—21no—
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99#14519711—22no—
100#14519711—41no—
101#14519706—22no—
102#14519701—22no—
103#14519701—22no—
104#14519701—22no—
105#14519699—22no—
106#14519699—41no—
107#14519697—22no—
108C8×Dic61—22no—
109#14519684—21no—
110#14519684—21no—
111#14519684—21no—
112#14519684D411yescharacteristic
113#14519684D411yescharacteristic
114C22×Q8×C61C2311yescharacteristic
115C22×Q8×C61C2311yescharacteristic
116C2×Q16×C61—22no—
117C2×Q16×C61—22no—
118C2×Q16×C61D414yes—
119C22×C8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C4×Q8×C61—22no—
122C4×Q8×C61—22no—
123C2×C4⋊C4×C61—21no—
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C4⋊C4×C61C2311yescharacteristic
126C2×C42×C61—21no—
127C2×C42×C61C2311yescharacteristic
128#103×C61—22no—
129#103×C61—22no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—21no—
133C2×Q8×C61—21no—
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×Q8×C61—21no—
141C2×Q8×C61C2×D412yes—
142C2×Q8×C61C2×D412yes—
143C2×Q8×C61#4112yes—
144C2×Q8×C61#4112yes—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C22×C4×C61C2411yescharacteristic
152C22×Dic61—21no—
153C22×Dic61—21no—
154C22×Dic61—21no—
155C22×Dic61C2×D411yescharacteristic
156C22×Dic61C2×D411yescharacteristic
157C22×Dic61C2×D411yescharacteristic
158C22×Dic61C2×D411yescharacteristic
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×D244—21no—
163C2×D244—21no—
164C2×D244—21no—
165Q16×C61—28no—
166C2×C8×C61—21no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D412yes—
172C4⋊C4×C61—24no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61#4112yes—
177C42×C61—24no—
178C42×C61—21no—
179C42×C61—21no—
180#11758201—44no—
181#11758198—42no—
182#11758198—42no—
183#11758198—42no—
184#11758198—42no—
185#11758198—81no—
186#11758198—81no—
187C4⋊Dic61—22no—
188C4⋊Dic61—22no—
189C4⋊Dic61—21no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—42no—
196C4⋊Dic61—42no—
197C4⋊Dic61—42no—
198C4⋊Dic61—42no—
199C4⋊Dic61—81no—
200C4⋊Dic61—81no—
201C4⋊Dic61—81no—
202C4⋊Dic61—81no—
203C4×Dic61—21no—
204C4×Dic61—21no—
205C4×Dic61—21no—
206C4×Dic61—21no—
207C4×Dic61—42no—
208C4×Dic61—42no—
209C4×Dic61—42no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C23×C61C22×D411yescharacteristic
217C22×D61—41no—
218C22×D61—41no—
219C22×D61—41no—
220C22×D61—41no—
221C22×D61—41no—
222C22×D61—41no—
223C22×D61—41no—
224Q8×C61—22no—
225Q8×C61—22no—
226Q8×C61—22no—
227Q8×C61—22no—
228Q8×C61—22no—
229Q8×C61—41no—
230Q8×C61—41no—
231Q8×C61#12314yes—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—22no—
236C2×C4×C61—21no—
237C2×C4×C61—21no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
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#13512yes—
250C2×C4×C61C22×D411yescharacteristic
251C2×C4×C61C2×#4111yescharacteristic
252C2×C4×C61C2×#4111yescharacteristic
253C2×C4×C61C2×#4111yescharacteristic
254C2×C4×C61H(5,𝔽2)11yescharacteristic
255C2×C4×C612−1+411yescharacteristic
256C2×Dic61—21no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—21no—
262C2×Dic61—21no—
263C2×Dic61—21no—
264C2×Dic61—21no—
265C2×Dic61—21no—
266C2×Dic61—21no—
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—
276C2×Dic61#13712yes—
277D244—41no—
278D244—41no—
279D244—41no—
280D244—41no—
281C8×C61—22no—
282C8×C61—41no—
283#92418—41no—
284#92418—41no—
285#26169—611nomaximal, Sylow
286C22×C61#52511yescharacteristic
287C22×C61D4211yescharacteristic
288C22×C61#54811yescharacteristic
289C22×C61C2×#13511yescharacteristic
290C22×C61C2×#13711yescharacteristic
291C22×C61#57611yescharacteristic
292C22×C61#57811yescharacteristic
293C2×D61—81no—
294C2×D61—81no—
295C2×D61—81no—
296C2×D61—81no—
297C2×D61—81no—
298C2×D61—81no—
299C2×D61—81no—
300C4×C61—22no—
301C4×C61—24no—
302C4×C61—41no—
303C4×C61—42no—
304C4×C61—42no—
305C4×C61—81no—
306C4×C61C2×#12311yescharacteristic
307C4×C61#53911yescharacteristic
308C4×C61#54711yescharacteristic
309C4×C61#55311yescharacteristic
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61—81no—
315Dic61—81no—
316Dic61—81no—
317C2×#464—611no—
318C2×C4×Q16—611no—
319#2448—611no—
320Q8×C4⋊C4—611no—
321#2093—611no—
322#2086—611no—
323#2072—611no—
324#2042—611no—
325#1934—611no—
326#1916—611no—
327#1880—611no—
328#1880—611no—
329#1878D6111yescharacteristic, complemented
330#1863—611no—
331#1863—611no—
332C2×C61#244311yescharacteristic
333C2×C61#310811yescharacteristic
334C2×C61#312411yescharacteristic
335C2×C61#327011yescharacteristic
336C2×C61#328711yescharacteristic
337C2×C61#336411yescharacteristic
338C2×C61#337511yescharacteristic
339D61—161no—
340C22×Q16C2×D6111yescharacteristic
341C22×SD16—611no—
342C2×#124—611no—
343C2×C4×Q8—611no—
344C2×C4×Q8—611no—
345C2×C4×Q8—611no—
346C2×C4×Q8C2×D6111yescharacteristic
347#464—1224no—
348C4×Q16—1224no—
349C2×#107—611no—
350C2×#107—611no—
351C2×C4⋊C8—611no—
352C2×#103—611no—
353C2×#103—611no—
354C2×#103—611no—
355C2×#103C2×D6111yescharacteristic
356C2×#103C2×D6111yescharacteristic
357C2×#102—611no—
358C2×#102—611no—
359C2×#102—611no—
360C2×C4×C8—611no—
361#399—1221no—
362#397—1221no—
363#391C2×D6111yescharacteristic
364#390—611no—
365#390—611no—
366#389—611no—
367#389—611no—
368#389—611no—
369#389—1221no—
370#389—1221no—
371#388—1221no—
372#388—1221no—
373#384—611no—
374#384—1221no—
375C4×C4⋊C4—611no—
376C4×C4⋊C4—1221no—
377#340—611no—
378#340—611no—
379#340—611no—
380#340C2×D6111yescharacteristic
381#336—611no—
382#336—611no—
383#336—611no—
384#336C2×D6111yescharacteristic
385C61#2616911yescharacteristic, complemented, Sylow
386C22×Q8C22×D6111yescharacteristic
387C22×Q8C22×D6111yescharacteristic
388C22×D4—611no—
389C2×Q16—22no—
390C2×Q16—22no—
391C2×Q16D24414yes—
392C2×SD16—1222no—
393C2×SD16—1222no—
394C2×SD16—1222no—
395C22×C8—611no—
396C22×C8—611no—
397C22×C8C22×D6111yescharacteristic
398C22×C8C22×D6111yescharacteristic
399#124—1222no—
400#124—2441no—
401C4×Q8—22no—
402C4×Q8—22no—
403C4×Q8—1222no—
404C4×Q8—1222no—
405C4×Q8—1222no—
406C4×Q8—2441no—
407C4×Q8—2441no—
408C4×Q8—2441no—
409C2×C4⋊C4—21no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4—611no—
415C2×C4⋊C4—1221no—
416C2×C4⋊C4—1221no—
417C2×C4⋊C4—1221no—
418C2×C4⋊C4—1221no—
419C2×C4⋊C4—1221no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1221no—
423C2×C4⋊C4C22×D6111yescharacteristic
424C2×C4⋊C4C22×D6111yescharacteristic
425C2×C22⋊C4—1221no—
426C2×C22⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C22⋊C4—1221no—
431C2×C42—21no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—1221no—
435C2×C42—1221no—
436C2×C42—1221no—
437C2×C42C22×D6111yescharacteristic
438#107—1222no—
439#107—2441no—
440C4⋊C8—1222no—
441#103—22no—
442#103—22no—
443#103—1222no—
444#103—1222no—
445#103—1222no—
446#102—1222no—
447#102—1222no—
448#102—2441no—
449C4×C8—1222no—
450#95—21no—
451#95—21no—
452#95—611no—
453#95—611no—
454#95—1221no—
455#95—1221no—
456#95—1221no—
457C24—1221no—
458C2×Q8—21no—
459C2×Q8—21no—
460C2×Q8—21no—
461C2×Q8—21no—
462C2×Q8—21no—
463C2×Q8—21no—
464C2×Q8—21no—
465C2×Q8—21no—
466C2×Q8C2×D24412yes—
467C2×Q8C2×D24412yes—
468C2×Q8#1175822412yes—
469C2×Q8#1175822412yes—
470C2×D4—1221no—
471C2×D4—1221no—
472C2×D4—1221no—
473C2×D4—1221no—
474C2×D4—1221no—
475C2×D4—1221no—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—1221no—
485C22×C4—1221no—
486C22×C4—1221no—
487C22×C4D4×D6111yescharacteristic
488C22×C4D4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490Q16—28no—
491SD16—2444no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—1221no—
498C2×C8—1221no—
499C2×C8—1221no—
500C2×C8—1221no—
501C2×C8—1221no—
502C2×C8—1221no—
503C2×C8C2×D24412yes—
504C4⋊C4—24no—
505C4⋊C4—21no—
506C4⋊C4—21no—
507C4⋊C4—21no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1221no—
511C4⋊C4—1221no—
512C4⋊C4—1221no—
513C4⋊C4—1221no—
514C4⋊C4—1221no—
515C4⋊C4—1221no—
516C4⋊C4—4881no—
517C4⋊C4—4881no—
518C4⋊C4—4881no—
519C4⋊C4—4881no—
520C4⋊C4—2442no—
521C4⋊C4—2442no—
522C4⋊C4—2442no—
523C4⋊C4—2442no—
524C4⋊C4#1175822412yes—
525C22⋊C4—4881no—
526C22⋊C4—4881no—
527C22⋊C4—2442no—
528C22⋊C4—2442no—
529C22⋊C4—2442no—
530C22⋊C4—2442no—
531C42—24no—
532C42—21no—
533C42—21no—
534C42—1221no—
535C42—1221no—
536C42—1221no—
537C42—1221no—
538C42—2442no—
539C42—2442no—
540C42—2442no—
541C23—2441no—
542C23—2441no—
543C23—2441no—
544C23—2441no—
545C23—2441no—
546C23—2441no—
547C23—2441no—
548C23C2×D4×D6111yescharacteristic
549Q8—22no—
550Q8—22no—
551Q8—22no—
552Q8—22no—
553Q8—22no—
554Q8—41no—
555Q8—41no—
556Q8#1451976114yes—
557D4—2441no—
558D4—2441no—
559D4—2441no—
560D4—2441no—
561C2×C4—21no—
562C2×C4—22no—
563C2×C4—21no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—22no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—22no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—612no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1221no—
585C2×C4—1221no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—1221no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4—2441no—
597C2×C4—2441no—
598C2×C4—2441no—
599C2×C4#1451980012yes—
600C2×C4C22×D24411yescharacteristic
601C2×C4C2×#1175822411yescharacteristic
602C2×C4C2×#1175822411yescharacteristic
603C2×C4C2×#1175822411yescharacteristic
604C2×C4#1451982311yescharacteristic
605C2×C4#1451982411yescharacteristic
606C8—22no—
607C8—41no—
608C8—2441no—
609C8—2441no—
610C22—4881no—
611C22—4881no—
612C22—4881no—
613C22—4881no—
614C22—4881no—
615C22—4881no—
616C22—4881no—
617C22#1684026811yescharacteristic
618C22D4×D24411yescharacteristic
619C22#1684041811yescharacteristic
620C22C2×#1451980011yescharacteristic
621C22#137×D6111yescharacteristic
622C22#1684056411yescharacteristic
623C22#1684057811yescharacteristic
624C4—22no—
625C4—24no—
626C4—42no—
627C4—41no—
628C4—42no—
629C4—81no—
630C4—1222no—
631C4—4881no—
632C4—4881no—
633C4—4881no—
634C4—2441no—
635C4—2441no—
636C4—2441no—
637C4C2×#1451976111yescharacteristic
638C4#1684035611yescharacteristic
639C4#1684041611yescharacteristic
640C4#1684046411yescharacteristic
641C2—9761no—
642C2#2144955211yescharacteristic
643C2#2145590011yescharacteristic
644C2#2145608211yescharacteristic
645C2#2145739111yescharacteristic
646C2#2145757811yescharacteristic
647C2#2145850811yescharacteristic
648C2#2145862811yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#1878C2
socleC23×C61C22×D4
prime radicalC22×D244C23
hypercenter#1878D61
hypocenterC61#26169
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1878

Complements

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

NH ≅ G/Ntype
2. C2×Q16×Dic61C2semidirect
4. Q8×C4⋊Dic61C2semidirect
6. #21447309C2semidirect
7. #21447227C2semidirect
10. #21446377C2semidirect
13. #21446075C2semidirect
14. #21446075C2semidirect
15. C61×#1878C2semidirect
329. #1878D61semidirect
385. C61#26169semidirect

Maximal subgroups

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

subgroupindex
C2×#168399862
C2×Q16×Dic612
#214496152
Q8×C4⋊Dic612
#214473462
#214473092
#214472272
#214470572
#214465032
#214463772
#214461742
#214461722
#214460752
C61×#18782
#2616961