Finite Group Databasebeta

← #29698056

Subgroups — #29698056

Order 15616.

1116 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839832C212yesmaximal
2C2×#16839784C211yesmaximal, characteristic, complemented
3C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
4C42×D244C211yesmaximal, characteristic
5#21446540C212yesmaximal, complemented
6#21446000C212yesmaximal, complemented
7#21445995C212yesmaximal, complemented
8#21445796C211yesmaximal, characteristic, complemented
9#21445796C211yesmaximal, characteristic, complemented
10#21445672C211yesmaximal, characteristic, complemented
11C61×#1842C211yesmaximal, characteristic, complemented
12C22×#11758189C2212yes—
13C22×C8×D61C2212yes—
14C2×C4×D244C411yescharacteristic
15C2×C4×D244C411yescharacteristic
16C2×C4×D244C2211yescharacteristic
17C2×C42×D61C2211yescharacteristic
18C2×C42×D61C2211yescharacteristic
19#16839832—28no—
20#16839784—24no—
21C2×C4⋊#92418C2211yescharacteristic
22C2×C4⋊#92418C2211yescharacteristic
23C2×#14519706C2212yes—
24C2×#14519706C2212yes—
25C4⋊C8×D61—24no—
26C2×#14519689C2212yes—
27C2×#14519689C2212yes—
28C2×#14519688C2212yes—
29C4×C4⋊Dic61C411yescharacteristic
30C4×C4⋊Dic61C411yescharacteristic
31C4×C4⋊Dic61C2211yescharacteristic
32C4×#11758198C411yescharacteristic
33C4×#11758198C411yescharacteristic
34C4×#11758198C411yescharacteristic
35C4×#11758198C411yescharacteristic
36C4×#11758198C2211yescharacteristic
37C4×#11758198C2211yescharacteristic
38#16839517C2212yes—
39#16839516C2212yes—
40#16839516C2212yes—
41C2×C4⋊C8×C61C2211yescharacteristic
42C2×C4⋊C8×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2212yes—
44C2×C4⋊C8×C61C2212yes—
45C43×C61C2211yescharacteristic
46C22×D244C2×C411yescharacteristic
47C22×C4×D61C2×C411yescharacteristic
48C22×C4×D61C2×C411yescharacteristic
49C22×C4×D61C2×C411yescharacteristic
50C22×C4×D61C2×C411yescharacteristic
51C22×C4×D61C2311yescharacteristic
52C22×C4×D61C2311yescharacteristic
53C2×#11758189—24no—
54C2×#11758189—24no—
55C2×#11758189—42no—
56C22×#92418C2312yes—
57C22×#92418C2312yes—
58C2×C8×D61—24no—
59C2×C8×D61—24no—
60C2×C8×D61—42no—
61C4×D244—24no—
62C4×D244—24no—
63C4×D244D414yes—
64C4×D244Q814yes—
65C2×C4⋊Dic61C2×C411yescharacteristic
66C2×C4⋊Dic61C2×C411yescharacteristic
67C2×C4⋊Dic61C2×C411yescharacteristic
68C2×#11758198C2×C411yescharacteristic
69C2×#11758198C2×C411yescharacteristic
70C2×#11758198C2×C411yescharacteristic
71C2×#11758198C2×C411yescharacteristic
72C2×#11758198C2×C411yescharacteristic
73C2×#11758198C2×C411yescharacteristic
74C2×C4×Dic61C2×C411yescharacteristic
75C2×C4×Dic61C2×C411yescharacteristic
76C2×C4×Dic61C2×C411yescharacteristic
77C2×C4×Dic61C2×C411yescharacteristic
78C2×C4×Dic61C2311yescharacteristic
79C2×C4×Dic61C2311yescharacteristic
80C42×D61—22no—
81C42×D61—22no—
82C42×D61—22no—
83C42×D61—22no—
84C4⋊#92418—22no—
85C4⋊#92418D414yescomplemented
86#14519706—24no—
87#14519706—24no—
88#14519689—24no—
89#14519689—24no—
90#14519688—24no—
91#14519684C2×C411yescharacteristic
92#14519684C2×C411yescharacteristic
93#14519684C2×C411yescharacteristic
94#14519684C2×C411yescharacteristic
95#14519684C2×C411yescharacteristic
96#14519684C2×C411yescharacteristic
97C22×C8×C61C2312yes—
98C22×C8×C61C2312yes—
99C2×C42×C61C2×C411yescharacteristic
100C2×C42×C61C2×C411yescharacteristic
101C2×C42×C61C2×C411yescharacteristic
102C2×C42×C61C2×C411yescharacteristic
103C2×C42×C61C2311yescharacteristic
104C2×C42×C61C2311yescharacteristic
105C2×C42×C61C2311yescharacteristic
106C4⋊C8×C61—22no—
107C4⋊C8×C61—22no—
108C4⋊C8×C61—24no—
109C4⋊C8×C61—42no—
110C23×D61C22×C411yescharacteristic
111C23×D61C22×C411yescharacteristic
112C22×C4×C61C22×C411yescharacteristic
113C22×C4×C61C22×C411yescharacteristic
114C22×C4×C61C22×C411yescharacteristic
115C22×C4×C61C22×C411yescharacteristic
116C22×C4×C61C22×C411yescharacteristic
117C22×C4×C61C22×C411yescharacteristic
118C22×C4×C61C2411yescharacteristic
119C22×Dic61C22×C411yescharacteristic
120C22×Dic61C22×C411yescharacteristic
121C22×Dic61C22×C411yescharacteristic
122C22×Dic61C22×C411yescharacteristic
123C22×Dic61C22×C411yescharacteristic
124C22×Dic61C22×C411yescharacteristic
125C2×D244—22no—
126C2×D244—22no—
127C2×D244C4⋊C414yes—
128C2×C4×D61—22no—
129C2×C4×D61—22no—
130C2×C4×D61—21no—
131C2×C4×D61—21no—
132C2×C4×D61—21no—
133C2×C4×D61—21no—
134C2×C4×D61—21no—
135C2×C4×D61—21no—
136C2×C4×D61—21no—
137C2×C4×D61—21no—
138C2×C4×D61—22no—
139C2×C4×D61—22no—
140C2×C4×D61—22no—
141C2×C4×D61—22no—
142C2×C4×D61—22no—
143C2×C4×D61—22no—
144C2×C4×D61—22no—
145C2×C4×D61—41no—
146C2×C4×D61—41no—
147C2×C4×D61M1614yescomplemented
148C2×C4×D61C2×D412yes—
149C2×C4×D61C2×D412yes—
150C2×C4×D61C2×Q812yes—
151C2×C4×D61C2×Q812yes—
152C2×C8×C61—22no—
153C2×C8×C61—22no—
154C2×C8×C61—22no—
155C2×C8×C61—22no—
156C2×C8×C61—22no—
157C2×C8×C61—22no—
158C42×C61—22no—
159C42×C61—22no—
160C42×C61—22no—
161C42×C61—21no—
162C42×C61—21no—
163C42×C61—41no—
164C42×C61C2×D412yes—
165C42×C61C2×D412yes—
166C42×C61C2×Q812yes—
167C42×C61#4112yes—
168#11758198—22no—
169#11758198—22no—
170#11758198—22no—
171#11758198C4⋊C414yes—
172#11758198C4⋊C414yes—
173#11758198M1614yescomplemented
174C4⋊Dic61—22no—
175C4⋊Dic61—22no—
176C4⋊Dic61C4⋊C414yes—
177C4×Dic61—22no—
178C4×Dic61—22no—
179C4×Dic61—22no—
180C4×Dic61—22no—
181C4×Dic61—21no—
182C4×Dic61—21no—
183C4×Dic61—21no—
184C4×Dic61—21no—
185C2×#92418—22no—
186C2×#92418—22no—
187C2×#92418—22no—
188C2×#92418—22no—
189C2×#92418—22no—
190C2×#92418C2×D414yes—
191#11758189—48no—
192C8×D61—48no—
193C23×C61C23×C411yescharacteristic
194C22×D61—21no—
195C22×D61—21no—
196C22×D61—21no—
197C22×D61—21no—
198C22×D61—21no—
199C22×D61—21no—
200C22×D61—21no—
201C22×D61—21no—
202C22×D61—21no—
203C22×D61—21no—
204C22×D61C2×C4⋊C412yes—
205C22×D61C2×C4⋊C412yes—
206C22×D61C2×M1612yes—
207C22×D61C2×M1612yes—
208C2×C4×C61—21no—
209C2×C4×C61—21no—
210C2×C4×C61—21no—
211C2×C4×C61—21no—
212C2×C4×C61—21no—
213C2×C4×C61—21no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—21no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61C2×C4⋊C412yes—
220C2×C4×C61C2×C4⋊C412yes—
221C2×C4×C61C4×D412yes—
222C2×C4×C61C4×D412yes—
223C2×C4×C61C2×M1612yes—
224C2×C4×C61C2×M1612yes—
225C2×C4×C61#13112yes—
226C2×C4×C61C22×D411yescharacteristic
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×Q811yescharacteristic
229C2×C4×C61C2×#4111yescharacteristic
230C2×C4×C61H(5,𝔽2)11yescharacteristic
231C2×C4×C612−1+411yescharacteristic
232C2×Dic61—21no—
233C2×Dic61—21no—
234C2×Dic61—21no—
235C2×Dic61—21no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—21no—
244C2×Dic61—21no—
245C2×Dic61—21no—
246C2×Dic61C2×C4⋊C412yes—
247C2×Dic61C2×C4⋊C412yes—
248C2×Dic61C2×M1612yes—
249C2×Dic61C2×M1612yes—
250D244—28no—
251C4×D61—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—28no—
260C4×D61—44no—
261C4×D61—44no—
262C4×D61—44no—
263C8×C61—42no—
264C8×C61—42no—
265#92418—24no—
266#92418—42no—
267#22648—611nomaximal, Sylow
268C22×C61C22×C4⋊C411yescharacteristic
269C22×C61C2×C4×D411yescharacteristic
270C22×C61#52011yescharacteristic
271C22×C61C22×M1611yescharacteristic
272C22×C61C22×M1611yescharacteristic
273C22×C61C2×#13111yescharacteristic
274C22×C61#56811yescharacteristic
275C2×D61—22no—
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C2×D61—22no—
280C2×D61—22no—
281C2×D61—22no—
282C2×D61—22no—
283C2×D61—22no—
284C2×D61—41no—
285C2×D61—41no—
286C2×D61—41no—
287C2×D61—41no—
288C2×D61#42314yescomplemented
289C4×C61—22no—
290C4×C61—22no—
291C4×C61—22no—
292C4×C61—22no—
293C4×C61—41no—
294C4×C61#43514yescomplemented
295C4×C61D4211yescharacteristic
296C4×C61#54711yescharacteristic
297C4×C61D4×Q811yescharacteristic
298C4×C61#55411yescharacteristic
299Dic61—22no—
300Dic61—22no—
301Dic61—22no—
302Dic61—22no—
303Dic61—41no—
304Dic61—41no—
305C2×#435—612no—
306C2×#423—611no—
307C22×C4⋊C8—611no—
308C42×D4—611no—
309#1940—612no—
310#1851—612no—
311#1850—612no—
312#1842D6111yescharacteristic, complemented
313#1797—611no—
314#1797—611no—
315#1767—611no—
316C2×C61D4×C4⋊C411yescharacteristic
317C2×C61C2×#42311yescharacteristic
318C2×C61#290611yescharacteristic
319C2×C61C2×#43511yescharacteristic
320C2×C61D4×M1611yescharacteristic
321C2×C61#298911yescharacteristic
322C2×C61#299011yescharacteristic
323D61—24no—
324D61—42no—
325C22×M16—612no—
326C23×C8—612no—
327C2×C4×D4—611no—
328C2×C4×D4—611no—
329C2×C4×D4—611no—
330C22×C42—611no—
331C22×C42—611no—
332#435—1228no—
333#423—1224no—
334C2×C4⋊C8—611no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—612no—
337C2×C4⋊C8—611no—
338C2×C4⋊C8—1224no—
339C2×C4⋊C8C2×D6111yescharacteristic
340C2×C4⋊C8C2×D6111yescharacteristic
341C2×C4⋊C8C2×D6112yes—
342C2×C4⋊C8C2×D6112yes—
343C2×C22⋊C8—612no—
344C2×C22⋊C8—612no—
345C2×#97—612no—
346C4×C4⋊C4—611no—
347C4×C4⋊C4—611no—
348C4×C4⋊C4—611no—
349C4×C22⋊C4—611no—
350C4×C22⋊C4—611no—
351C4×C22⋊C4—611no—
352C4×C22⋊C4—611no—
353C4×C22⋊C4—611no—
354C4×C22⋊C4—611no—
355C43C2×D6111yescharacteristic
356#336—612no—
357#336—612no—
358#336—612no—
359C61#2264811yescharacteristic, complemented, Sylow
360C22×D4—611no—
361C23×C4—611no—
362C23×C4—611no—
363C23×C4—611no—
364C23×C4—611no—
365C23×C4—611no—
366C23×C4—611no—
367C2×M16—1224no—
368C2×M16—1224no—
369C2×M16—2442no—
370C22×C8—612no—
371C22×C8—612no—
372C22×C8—1224no—
373C22×C8—1224no—
374C22×C8—2442no—
375C22×C8C22×D6112yes—
376C22×C8C22×D6112yes—
377C4×D4—614no—
378C4×D4—614no—
379C4×D4—1224no—
380C4×D4—1224no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C22⋊C4—611no—
385C2×C22⋊C4—611no—
386C2×C22⋊C4—611no—
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C42—611no—
391C2×C42—611no—
392C2×C42—611no—
393C2×C42—611no—
394C2×C42—611no—
395C2×C42—611no—
396C2×C42—1222no—
397C2×C42—1222no—
398C2×C42—1222no—
399C2×C42—1222no—
400C2×C42C4×D6111yescharacteristic
401C2×C42C4×D6111yescharacteristic
402C2×C42C4×D6111yescharacteristic
403C2×C42C4×D6111yescharacteristic
404C2×C42C22×D6111yescharacteristic
405C2×C42C22×D6111yescharacteristic
406C2×C42C22×D6111yescharacteristic
407C4⋊C8—22no—
408C4⋊C8—22no—
409C4⋊C8—24no—
410C4⋊C8—42no—
411C4⋊C8—614no—
412C4⋊C8—1224no—
413C4⋊C8—1224no—
414C4⋊C8—1222no—
415C22⋊C8—1224no—
416C22⋊C8—1224no—
417#97—1224no—
418#95—611no—
419#95—611no—
420#95—611no—
421#95—611no—
422#95—611no—
423#95—611no—
424C24—611no—
425C24—611no—
426C2×D4—614no—
427C2×D4—1222no—
428C2×D4—1222no—
429C22×C4—611no—
430C22×C4—611no—
431C22×C4—611no—
432C22×C4—611no—
433C22×C4—612no—
434C22×C4—612no—
435C22×C4—612no—
436C22×C4—612no—
437C22×C4—614no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—1222no—
441C22×C4—1222no—
442C22×C4—1221no—
443C22×C4—1221no—
444C22×C4—1221no—
445C22×C4—1221no—
446C22×C4—1221no—
447C22×C4—1221no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1222no—
453C22×C4—1222no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—2441no—
458C22×C4—2441no—
459C22×C4C2×C4×D6111yescharacteristic
460C22×C4C2×C4×D6111yescharacteristic
461C22×C4C2×C4×D6111yescharacteristic
462C22×C4C2×C4×D6111yescharacteristic
463C22×C4C2×C4×D6111yescharacteristic
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4C23×D6111yescharacteristic
466M16—2448no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—614no—
474C2×C8—1222no—
475C2×C8—1222no—
476C2×C8—1222no—
477C2×C8—1222no—
478C2×C8—1222no—
479C2×C8—2448no—
480C4⋊C4—614no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C22⋊C4—614no—
484C22⋊C4—614no—
485C22⋊C4—614no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C42—21no—
490C42—21no—
491C42—22no—
492C42—22no—
493C42—22no—
494C42—41no—
495C42—1221no—
496C42—1221no—
497C42—1221no—
498C42—1221no—
499C42—1222no—
500C42—1222no—
501C42—1222no—
502C42—1222no—
503C42D4×D6112yes—
504C42D4×D6112yes—
505C42Q8×D6112yes—
506C42#1175822612yes—
507C23—612no—
508C23—612no—
509C23—612no—
510C23—612no—
511C23—1221no—
512C23—1221no—
513C23—1221no—
514C23—1221no—
515C23—1221no—
516C23—1221no—
517C23—1221no—
518C23—1221no—
519C23—1221no—
520C23—1221no—
521C23C22×C4×D6111yescharacteristic
522D4—1228no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—612no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—612no—
538C2×C4—1221no—
539C2×C4—1221no—
540C2×C4—1221no—
541C2×C4—1221no—
542C2×C4—1221no—
543C2×C4—1221no—
544C2×C4—1221no—
545C2×C4—1221no—
546C2×C4—1222no—
547C2×C4—1222no—
548C2×C4—1228no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1221no—
559C2×C4—1221no—
560C2×C4—1221no—
561C2×C4—2444no—
562C2×C4—2444no—
563C2×C4—2444no—
564C2×C4C4⋊C4×D6112yes—
565C2×C4C4⋊C4×D6112yes—
566C2×C4#1451973912yes—
567C2×C4#1451973912yes—
568C2×C4M16×D6112yes—
569C2×C4M16×D6112yes—
570C2×C4#1451978312yes—
571C2×C4C2×D4×D6111yescharacteristic
572C2×C4C2×D4×D6111yescharacteristic
573C2×C4C2×Q8×D6111yescharacteristic
574C2×C4C2×#1175822611yescharacteristic
575C2×C4#1451982211yescharacteristic
576C2×C4#1451982511yescharacteristic
577C8—42no—
578C8—42no—
579C8—1224no—
580C8—2442no—
581C22—614no—
582C22—1222no—
583C22—1222no—
584C22—1222no—
585C22—1222no—
586C22—1222no—
587C22—1222no—
588C22—1222no—
589C22—1222no—
590C22—1222no—
591C22—2441no—
592C22—2441no—
593C22—2441no—
594C22—2441no—
595C22C2×C4⋊C4×D6111yescharacteristic
596C22C2×#1451973911yescharacteristic
597C22#1684024311yescharacteristic
598C22C2×M16×D6111yescharacteristic
599C22C2×M16×D6111yescharacteristic
600C22C2×#1451978311yescharacteristic
601C22#1684052811yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—22no—
606C4—41no—
607C4—1222no—
608C4—1222no—
609C4—1222no—
610C4—1222no—
611C4—2441no—
612C4—2441no—
613C4#1683983214yescomplemented
614C4#1684039711yescharacteristic
615C4#1684041511yescharacteristic
616C4#1684042711yescharacteristic
617C4#1684047311yescharacteristic
618C2—1224no—
619C2—2442no—
620C2#2144871911yescharacteristic
621C2#423×D6111yescharacteristic
622C2#2145438411yescharacteristic
623C2C2×#1683983211yescharacteristic
624C2#2145455411yescharacteristic
625C2#2145493311yescharacteristic
626C2#2145494111yescharacteristic
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1842C2
prime radicalC22×D244C2×C4
cosocleC22×C4×C61C24
hypercenter#1842D61
hypocenterC61#22648
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1842

Complements

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

NH ≅ G/Ntype
2. C2×#16839784C2semidirect
3. C2×C4⋊C8×D61C2semidirect
5. #21446540C2semidirect
6. #21446000C2semidirect
7. #21445995C2semidirect
8. #21445796C2semidirect
9. #21445796C2semidirect
10. #21445672C2semidirect
11. C61×#1842C2semidirect
85. C4⋊#92418D4semidirect
147. C2×C4×D61M16semidirect
173. #11758198M16semidirect
288. C2×D61#423semidirect
294. C4×C61#435semidirect
312. #1842D61semidirect
359. C61#22648semidirect
613. C4#16839832semidirect

Maximal subgroups

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

subgroupindex
C2×#168398322
C2×#168397842
C2×C4⋊C8×D612
C42×D2442
#214465402
#214460002
#214459952
#214457962
#214456722
C61×#18422
#2264861