Finite Group Databasebeta

← #29699355

Subgroups — #29699355

Order 15616.

838 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839860C211yesmaximal, characteristic
2#21448790C211yesmaximal, characteristic
3#21446515C211yesmaximal, characteristic
4#21446501C211yesmaximal, characteristic
5#21446408C211yesmaximal, characteristic, complemented
6#21446387C211yesmaximal, characteristic, complemented
7#21446174C211yesmaximal, characteristic
8#21446174C211yesmaximal, characteristic
9#21445954C211yesmaximal, characteristic, complemented
10#21445834C211yesmaximal, characteristic, complemented
11#21445816C211yesmaximal, characteristic
12#21445694C211yesmaximal, characteristic, complemented
13#21445649C211yesmaximal, characteristic, complemented
14C4×#14519701C211yesmaximal, characteristic, complemented
15Q8×C4⋊C4×C61C211yesmaximal, characteristic, complemented
16C22×#11758201C2211yescharacteristic
17C2×C4×D244C2211yescharacteristic
18#16839860C418yescomplemented
19C2×#14519710C2211yescharacteristic
20C2×C4⋊#92418C2211yescharacteristic
21C2×#14519701C2211yescharacteristic
22C2×#14519701C2211yescharacteristic
23C2×#14519701C2211yescharacteristic
24C2×#14519697C2211yescharacteristic
25C2×#14519697C2211yescharacteristic
26C2×#14519697C2211yescharacteristic
27C2×#14519688C2211yescharacteristic
28C2×#14519688C2211yescharacteristic
29C2×C4×#92418C2211yescharacteristic
30#16839664C2211yescharacteristic
31#16839664C2211yescharacteristic
32#16839652—21no—
33#16839652—21no—
34#16839628C2211yescharacteristic
35#16839624—21no—
36C4×C4⋊Dic61C2211yescharacteristic
37C4×C4⋊Dic61C2211yescharacteristic
38#16839609—21no—
39#16839524C2211yescharacteristic
40#16839524C2211yescharacteristic
41#16839524C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839517C2211yescharacteristic
48#16839517C2211yescharacteristic
49C2×C4×Q8×C61C2211yescharacteristic
50C2×C4×Q8×C61C2211yescharacteristic
51C2×C4×Q8×C61C2211yescharacteristic
52C61×#389—21no—
53C61×#389—21no—
54C61×#389C2211yescharacteristic
55C61×#389C2211yescharacteristic
56C61×#384—21no—
57C61×#384C2211yescharacteristic
58C4×C4⋊C4×C61—21no—
59C4×C4⋊C4×C61C2211yescharacteristic
60C22×D244C2311yescharacteristic
61C22×C4×D61—21no—
62C2×#11758201—22no—
63C2×#11758201—22no—
64C2×#11758201C2×C414yes—
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68C22×#92418C2311yescharacteristic
69C4×D244—22no—
70C4×D244C2×C414yes—
71C2×C4⋊Dic61C2311yescharacteristic
72C2×C4⋊Dic61C2311yescharacteristic
73C2×C4⋊Dic61C2311yescharacteristic
74C2×#11758198—21no—
75C2×#11758198—21no—
76C2×#11758198—21no—
77C2×#11758198—21no—
78C2×#11758198—21no—
79C2×C4×Dic61—21no—
80C2×C4×Dic61—21no—
81#14519710C2×C414yes—
82C4⋊#92418—22no—
83#14519701—22no—
84#14519701—22no—
85#14519701C2×C414yes—
86#14519697—41no—
87#14519697—41no—
88#14519697C2×C414yes—
89#14519688—41no—
90#14519688C2×C414yes—
91C4×#92418—41no—
92#14519684—21no—
93#14519684—21no—
94#14519684—21no—
95#14519684—21no—
96C22×Q8×C61C2311yescharacteristic
97C4×Q8×C61—22no—
98C4×Q8×C61—22no—
99C4×Q8×C61—22no—
100C4×Q8×C61—41no—
101C4×Q8×C61—41no—
102C4×Q8×C61C2×C414yes—
103C2×C4⋊C4×C61—21no—
104C2×C4⋊C4×C61—21no—
105C2×C4⋊C4×C61—21no—
106C2×C4⋊C4×C61—21no—
107C2×C4⋊C4×C61—21no—
108C2×C4⋊C4×C61—21no—
109C2×C4⋊C4×C61—21no—
110C2×C4⋊C4×C61D411yescharacteristic
111C2×C4⋊C4×C61D411yescharacteristic
112C2×C4⋊C4×C61C2311yescharacteristic
113C2×C4⋊C4×C61C2311yescharacteristic
114C2×C4⋊C4×C61C2311yescharacteristic
115C2×C42×C61—21no—
116C2×C42×C61—21no—
117C2×C42×C61—21no—
118C2×C42×C61C2311yescharacteristic
119C2×C42×C61C2311yescharacteristic
120C2×C42×C61C2311yescharacteristic
121#95×C61—21no—
122#95×C61—21no—
123#95×C61D411yescharacteristic
124#95×C61D411yescharacteristic
125C23×D61—21no—
126C2×Q8×C61—21no—
127C2×Q8×C61—21no—
128C2×Q8×C61—21no—
129C2×Q8×C61C22×C412yes—
130C2×Q8×C61#4112yes—
131C2×Q8×C61#4112yes—
132C22×C4×C61—21no—
133C22×C4×C61—21no—
134C22×C4×C61—21no—
135C22×C4×C61—21no—
136C22×C4×C61C2×D411yescharacteristic
137C22×C4×C61C2×D411yescharacteristic
138C22×C4×C61C2×D411yescharacteristic
139C22×C4×C61C2×D411yescharacteristic
140C22×C4×C61C2×D411yescharacteristic
141C22×C4×C61C2×D411yescharacteristic
142C22×C4×C61C2411yescharacteristic
143C22×Dic61—21no—
144C22×Dic61—21no—
145C22×Dic61—21no—
146C2×D244—21no—
147C2×D244—21no—
148C2×D244—21no—
149C2×D244—21no—
150C2×D244—21no—
151C2×D244C22×C412yes—
152C2×C4×D61—24no—
153C2×C4×D61—81no—
154C2×C4×D61—81no—
155C4⋊C4×C61—24no—
156C4⋊C4×C61—22no—
157C4⋊C4×C61—21no—
158C4⋊C4×C61—21no—
159C4⋊C4×C61—21no—
160C4⋊C4×C61—21no—
161C4⋊C4×C61—21no—
162C4⋊C4×C61—41no—
163C4⋊C4×C61—42no—
164C4⋊C4×C61—42no—
165C4⋊C4×C61—42no—
166C4⋊C4×C61—81no—
167C4⋊C4×C61—81no—
168C4⋊C4×C61—81no—
169C4⋊C4×C61C22×C412yes—
170C42×C61—24no—
171C42×C61—21no—
172C42×C61—21no—
173C42×C61—21no—
174C42×C61—21no—
175C42×C61—42no—
176C42×C61—42no—
177C42×C61C22×C412yes—
178C42×C61#4112yes—
179#11758201—28no—
180#11758198—24no—
181#11758198—42no—
182#11758198—42no—
183#11758198—81no—
184#11758198—81no—
185C4⋊Dic61—21no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61C22×C412yes—
191C4×Dic61—81no—
192C4×Dic61—81no—
193C2×#92418—21no—
194C2×#92418—21no—
195C2×#92418—21no—
196C2×#92418—21no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418C22×C412yes—
204C2×#92418C22×C412yes—
205C23×C61C22×D411yescharacteristic
206C22×D61—22no—
207C22×D61—41no—
208C22×D61—41no—
209C22×D61—41no—
210C22×D61—41no—
211C22×D61—41no—
212C22×D61—41no—
213Q8×C61—22no—
214Q8×C61—22no—
215Q8×C61—22no—
216Q8×C61#11714yes—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—22no—
220C2×C4×C61—22no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—22no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—41no—
241C2×C4×C61—41no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61C4×D412yes—
245C2×C4×C61C4×D412yes—
246C2×C4×C61#13712yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×C4×C61C2×#4111yescharacteristic
249C2×C4×C61C2×#4111yescharacteristic
250C2×C4×C61C2×#4111yescharacteristic
251C2×C4×C61H(5,𝔽2)11yescharacteristic
252C2×C4×C612−1+411yescharacteristic
253C2×Dic61—22no—
254C2×Dic61—41no—
255C2×Dic61—41no—
256C2×Dic61—41no—
257C2×Dic61—41no—
258C2×Dic61—41no—
259C2×Dic61—41no—
260C2×Dic61—41no—
261C2×Dic61—41no—
262D244—22no—
263D244—22no—
264D244—41no—
265D244—41no—
266C4×D61—84no—
267#92418—22no—
268#92418—22no—
269#92418—41no—
270#92418—41no—
271#22775—611nomaximal, Sylow
272C22×C61C2×C4×D411yescharacteristic
273C22×C61#54611yescharacteristic
274C22×C61#54811yescharacteristic
275C22×C61C2×#13711yescharacteristic
276C22×C61#57511yescharacteristic
277C22×C61#57611yescharacteristic
278C22×C61#57711yescharacteristic
279C2×D61—42no—
280C2×D61—42no—
281C2×D61—42no—
282C2×D61—81no—
283C2×D61—81no—
284C2×D61—81no—
285C2×D61—81no—
286C4×C61—24no—
287C4×C61—22no—
288C4×C61—22no—
289C4×C61—22no—
290C4×C61—42no—
291C4×C61—41no—
292C4×C61—41no—
293C4×C61—41no—
294C4×C61—81no—
295C4×C61—81no—
296C4×C61C2×#11711yescharacteristic
297C4×C61#52011yescharacteristic
298C4×C61#55211yescharacteristic
299C4×C61#55311yescharacteristic
300Dic61—42no—
301Dic61—81no—
302Dic61—81no—
303C2×#440—611no—
304#2357—611no—
305Q8×C4⋊C4D6111yescharacteristic, complemented
306#1936—611no—
307#1934—611no—
308#1921—611no—
309#1918—611no—
310#1880—611no—
311#1880—611no—
312#1836—611no—
313#1805—611no—
314#1802—611no—
315#1771—611no—
316#1763—611no—
317C4×#103—611no—
318C2×C61#235211yescharacteristic
319C2×C61C4×#13711yescharacteristic
320C2×C61#294611yescharacteristic
321C2×C61#330911yescharacteristic
322C2×C61#332011yescharacteristic
323C2×C61#337111yescharacteristic
324C2×C61#337511yescharacteristic
325D61—82no—
326C22×SD16—611no—
327C2×C4×Q8C2×D6111yescharacteristic
328C2×C4×Q8C2×D6111yescharacteristic
329C2×C4×Q8C2×D6111yescharacteristic
330C2×C4×D4—611no—
331#440—618no—
332C2×#107—611no—
333C2×C4⋊C8—611no—
334C2×#103—611no—
335C2×#103—611no—
336C2×#103—611no—
337C2×#102—611no—
338C2×#102—611no—
339C2×#102—611no—
340C2×#97—611no—
341C2×#97—611no—
342C2×C4×C8—611no—
343#390—611no—
344#390—611no—
345#389—21no—
346#389—21no—
347#389C2×D6111yescharacteristic
348#389C2×D6111yescharacteristic
349#388—1221no—
350#388—1221no—
351#384—21no—
352#384C2×D6111yescharacteristic
353#382—611no—
354#381—1221no—
355C4×C4⋊C4—21no—
356C4×C4⋊C4—611no—
357C4×C4⋊C4—611no—
358C4×C4⋊C4C2×D6111yescharacteristic
359C4×C22⋊C4—1221no—
360#340—611no—
361#340—611no—
362#340—611no—
363#340—611no—
364#340—611no—
365#340—611no—
366#340—611no—
367#340—611no—
368#336—611no—
369#336—611no—
370C61#2277511yescharacteristic, complemented, Sylow
371C22×Q8C22×D6111yescharacteristic
372C22×D4—611no—
373C23×C4—1221no—
374C2×SD16—614no—
375C2×SD16—1222no—
376C2×SD16—1222no—
377C22×C8—611no—
378C22×C8—611no—
379C22×C8—611no—
380C22×C8—611no—
381C4×Q8—22no—
382C4×Q8—22no—
383C4×Q8—22no—
384C4×Q8—41no—
385C4×Q8—41no—
386C4×Q8C4×D6114yes—
387C4×D4—614no—
388C4×D4—1222no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—611no—
397C2×C4⋊C4—611no—
398C2×C4⋊C4—611no—
399C2×C4⋊C4C22⋊D6111yescharacteristic
400C2×C4⋊C4C22⋊D6111yescharacteristic
401C2×C4⋊C4C22×D6111yescharacteristic
402C2×C4⋊C4C22×D6111yescharacteristic
403C2×C4⋊C4C22×D6111yescharacteristic
404C2×C22⋊C4—1221no—
405C2×C22⋊C4—1221no—
406C2×C22⋊C4—1221no—
407C2×C22⋊C4—1221no—
408C2×C22⋊C4—1221no—
409C2×C42—21no—
410C2×C42—21no—
411C2×C42—21no—
412C2×C42—1221no—
413C2×C42—1221no—
414C2×C42C22×D6111yescharacteristic
415C2×C42C22×D6111yescharacteristic
416C2×C42C22×D6111yescharacteristic
417#107—614no—
418C4⋊C8—1222no—
419#103—614no—
420#103—1222no—
421#103—1222no—
422#102—614no—
423#102—2441no—
424#102—2441no—
425#97—614no—
426#97—2441no—
427C4×C8—2441no—
428#95—21no—
429#95—21no—
430#95—1221no—
431#95—1221no—
432#95—1221no—
433#95—1221no—
434#95C22⋊D6111yescharacteristic
435#95C22⋊D6111yescharacteristic
436C24—1221no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8C2×C4×D6112yes—
441C2×Q8#1175822412yes—
442C2×Q8#1175822612yes—
443C2×D4—612no—
444C2×D4—1221no—
445C2×D4—1221no—
446C2×D4—1221no—
447C2×D4—1221no—
448C2×D4—1221no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—1221no—
454C22×C4—1224no—
455C22×C4—1221no—
456C22×C4—1221no—
457C22×C4—4881no—
458C22×C4—4881no—
459C22×C4C2×C22⋊D6111yescharacteristic
460C22×C4C2×C22⋊D6111yescharacteristic
461C22×C4C2×C22⋊D6111yescharacteristic
462C22×C4C2×C22⋊D6111yescharacteristic
463C22×C4C2×C22⋊D6111yescharacteristic
464C22×C4C2×C22⋊D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466SD16—1228no—
467C2×C8—612no—
468C2×C8—612no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—1221no—
476C2×C8—1221no—
477C2×C8—1221no—
478C2×C8—1221no—
479C4⋊C4—22no—
480C4⋊C4—21no—
481C4⋊C4—24no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—41no—
487C4⋊C4—42no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—81no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—612no—
494C4⋊C4—1221no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4C2×C4×D6112yes—
500C22⋊C4—1224no—
501C22⋊C4—4881no—
502C22⋊C4—4881no—
503C22⋊C4—2442no—
504C22⋊C4—2442no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—21no—
509C42—24no—
510C42—42no—
511C42—42no—
512C42—4881no—
513C42—4881no—
514C42C2×C4×D6112yes—
515C42#1175822212yes—
516C23—1222no—
517C23—2441no—
518C23—2441no—
519C23—2441no—
520C23—2441no—
521C23—2441no—
522C23—2441no—
523C23C22×C22⋊D6111yescharacteristic
524Q8—22no—
525Q8—22no—
526Q8—22no—
527Q8#1451973514yes—
528D4—1222no—
529D4—1222no—
530D4—2441no—
531D4—2441no—
532C2×C4—22no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—1222no—
560C2×C4—4884no—
561C2×C4—2441no—
562C2×C4—2441no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4C4×C22⋊D6112yes—
570C2×C4C4×C22⋊D6112yes—
571C2×C4#1451980812yes—
572C2×C4C22×C4×D6111yescharacteristic
573C2×C4C2×#1175822411yescharacteristic
574C2×C4C2×#1175822211yescharacteristic
575C2×C4C2×#1175822611yescharacteristic
576C2×C4#1451982211yescharacteristic
577C2×C4#1451982511yescharacteristic
578C8—1222no—
579C8—1222no—
580C8—2441no—
581C8—2441no—
582C22—4881no—
583C22—4881no—
584C22—4881no—
585C22—4881no—
586C22—2442no—
587C22—2442no—
588C22—2442no—
589C22C2×C4×C22⋊D6111yescharacteristic
590C22#1684040611yescharacteristic
591C22#1684042311yescharacteristic
592C22C2×#1451980811yescharacteristic
593C22#1684056011yescharacteristic
594C22#1684056511yescharacteristic
595C22#1684057011yescharacteristic
596C4—24no—
597C4—22no—
598C4—22no—
599C4—22no—
600C4—42no—
601C4—41no—
602C4—41no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—4881no—
607C4—4881no—
608C4—2442no—
609C4C2×#1451973511yescharacteristic
610C4#1684024311yescharacteristic
611C4#1684044711yescharacteristic
612C4#1684046011yescharacteristic
613C2—4882no—
614C2#2144874411yescharacteristic
615C2C4×#1451980811yescharacteristic
616C2#2145464711yescharacteristic
617C2#2145782511yescharacteristic
618C2#2145799011yescharacteristic
619C2#2145858711yescharacteristic
620C2#2145863311yescharacteristic
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#22775
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
5. #21446408C2semidirect
6. #21446387C2semidirect
9. #21445954C2semidirect
10. #21445834C2semidirect
12. #21445694C2semidirect
13. #21445649C2semidirect
14. C4×#14519701C2semidirect
15. Q8×C4⋊C4×C61C2semidirect
18. #16839860C4semidirect
305. Q8×C4⋊C4D61semidirect
370. C61#22775semidirect

Maximal subgroups

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

subgroupindex
C2×#168398602
#214487902
#214465152
#214465012
#214464082
#214463872
#214461742
#214459542
#214458342
#214458162
#214456942
#214456492
C4×#145197012
Q8×C4⋊C4×C612
#2277561