Finite Group Databasebeta

← #29689600

Subgroups — #29689600

Order 15616.

1170 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840290C211yesmaximal, characteristic
2C2×#16839814C211yesmaximal, characteristic, complemented
3C2×C4⋊C8×D61C211yesmaximal, characteristic, complemented
4#21446753C212yesmaximal, complemented
5#21446611C212yesmaximal, complemented
6#21446110C212yesmaximal, complemented
7#21446108C212yesmaximal, complemented
8#21445803C211yesmaximal, characteristic, complemented
9#21445800C211yesmaximal, characteristic, complemented
10#21445800C211yesmaximal, characteristic, complemented
11C61×#1798C211yesmaximal, characteristic, complemented
12C22×C8×D61C2212yes—
13#16840290—24no—
14#16840290—24no—
15C2×#14519772C411yescharacteristic
16C2×#14519770C411yescharacteristic
17C2×#14519767C411yescharacteristic
18C2×#14519763C411yescharacteristic
19C2×#14519762C411yescharacteristic
20C2×#14519759C411yescharacteristic
21C2×#14519755C411yescharacteristic
22C2×#14519750C411yescharacteristic
23C2×C4×Dic122C2211yescharacteristic
24C2×C4×D244C2211yescharacteristic
25C2×C4×C22⋊D61C2211yescharacteristic
26C2×C4×C22⋊D61C2211yescharacteristic
27C2×#14519734C2211yescharacteristic
28C2×C42×D61C2211yescharacteristic
29#16839814—24no—
30C2×C4⋊#92418C2211yescharacteristic
31C2×C4⋊#92418C2211yescharacteristic
32C2×#14519706C2212yes—
33C4⋊C8×D61—24no—
34C2×C22⋊#92418C2212yes—
35C2×#14519689C2212yes—
36C2×#14519689C2212yes—
37C2×#14519689C2212yes—
38C2×C8×Dic61C2212yes—
39#16839517C2212yes—
40#16839516C2212yes—
41#16839516C2212yes—
42C2×#117×C61C2211yescharacteristic
43C2×C4⋊C8×C61C2211yescharacteristic
44C2×C4⋊C8×C61C2211yescharacteristic
45C2×C22⋊C8×C61C2212yes—
46C61×#336C2212yes—
47C22×Dic122C2×C411yescharacteristic
48C22×D244C2×C411yescharacteristic
49C22×C22⋊D61C2×C411yescharacteristic
50C22×C22⋊D61C2×C411yescharacteristic
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C22×C4×D61C2311yescharacteristic
55C22×#92418C2312yes—
56C22×#92418C2312yes—
57C2×C8×D61—24no—
58C2×C8×D61—42no—
59C2×C8×D61—42no—
60#14519772—24no—
61#14519770—24no—
62#14519767—24no—
63#14519763—24no—
64#14519762—24no—
65#14519759—24no—
66#14519755—24no—
67#14519750—24no—
68C4×Dic122—22no—
69C4×Dic122—22no—
70C4×D244—22no—
71C4×D244—22no—
72C4×C22⋊D61—22no—
73C4×C22⋊D61—22no—
74C4×C22⋊D61—22no—
75C4×C22⋊D61—22no—
76#14519734—22no—
77#14519734—22no—
78C2×C4⋊Dic61C2×C411yescharacteristic
79C2×C4⋊Dic61C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×#11758196C2×C411yescharacteristic
83C2×#11758196C2×C411yescharacteristic
84C2×#11758196C2×C411yescharacteristic
85C2×#11758196C2×C411yescharacteristic
86C2×C22⋊Dic61C2×C411yescharacteristic
87C2×C22⋊Dic61C2×C411yescharacteristic
88C2×#11758198C2×C411yescharacteristic
89C2×#11758198C2×C411yescharacteristic
90C2×#11758198C2×C411yescharacteristic
91C2×#11758198C2×C411yescharacteristic
92C2×#11758198C2×C411yescharacteristic
93C2×#11758198C2×C411yescharacteristic
94C2×C4×Dic61C2×C411yescharacteristic
95C2×C4×Dic61C2×C411yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C42×D61D414yes—
99C42×D61D414yes—
100C4⋊#92418—22no—
101C4⋊#92418—22no—
102#14519706—24no—
103C22⋊#92418—42no—
104#14519689—24no—
105#14519689—42no—
106#14519689—42no—
107C8×Dic61—24no—
108C23×C4×C61C2311yescharacteristic
109C22×C8×C61C2312yes—
110C22×C8×C61C2312yes—
111#117×C61—22no—
112#117×C61—22no—
113C2×C4⋊C4×C61C2×C411yescharacteristic
114C2×C4⋊C4×C61C2×C411yescharacteristic
115C2×C22⋊C4×C61C2×C411yescharacteristic
116C2×C22⋊C4×C61C2×C411yescharacteristic
117C2×C42×C61C2311yescharacteristic
118C2×C42×C61C2311yescharacteristic
119C4⋊C8×C61—22no—
120C4⋊C8×C61—22no—
121C22⋊C8×C61—42no—
122C24×C61C22×C411yescharacteristic
123C23×D61C22×C411yescharacteristic
124C23×D61C22×C411yescharacteristic
125C22×C4×C61—21no—
126C22×C4×C61—21no—
127C22×C4×C61—21no—
128C22×C4×C61—21no—
129C22×C4×C61—21no—
130C22×C4×C61—21no—
131C22×C4×C61C22×C411yescharacteristic
132C22×C4×C61C22×C411yescharacteristic
133C22×C4×C61C22×C411yescharacteristic
134C22×C4×C61C22×C411yescharacteristic
135C22×C4×C61C22×C411yescharacteristic
136C22×C4×C61C2411yescharacteristic
137C2×C22⋊D61—22no—
138C2×C22⋊D61—24no—
139C2×C22⋊D61—22no—
140C2×C22⋊D61—24no—
141C22×Dic61C22×C411yescharacteristic
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C2×D244—22no—
148C2×D244—24no—
149C2×C4×D61—24no—
150C2×C4×D61—21no—
151C2×C4×D61—21no—
152C2×C4×D61—21no—
153C2×C4×D61—21no—
154C2×C4×D61—24no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—21no—
158C2×C4×D61—21no—
159C2×C4×D61—21no—
160C2×C4×D61—21no—
161C2×C4×D61C22⋊C414yes—
162C2×C4×D61C22⋊C414yes—
163C2×C4×D61C2×D412yes—
164C2×C4×D61C2×D412yes—
165C2×Dic122—22no—
166C2×Dic122—24no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C2×C8×C61—22no—
170C2×C8×C61—22no—
171C2×C8×C61—22no—
172C2×C8×C61—22no—
173C4⋊C4×C61—22no—
174C4⋊C4×C61—22no—
175C22⋊C4×C61—22no—
176C22⋊C4×C61—22no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61C2×D412yes—
180C42×C61C2×D412yes—
181C22⋊Dic61—22no—
182C22⋊Dic61—22no—
183#11758198—22no—
184#11758198—22no—
185#11758198—22no—
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189C4⋊Dic61—22no—
190C4⋊Dic61—22no—
191#11758196—22no—
192#11758196—22no—
193#11758196—22no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61C22⋊C414yes—
200C4×Dic61C22⋊C414yes—
201C4×Dic61C2×D412yes—
202C4×Dic61C2×D412yes—
203C2×#92418—22no—
204C2×#92418—22no—
205C2×#92418—22no—
206C2×#92418—22no—
207C2×#92418—22no—
208C2×#92418—22no—
209C8×D61—48no—
210C23×C61—21no—
211C23×C61—21no—
212C23×C61—21no—
213C23×C61—22no—
214C23×C61—22no—
215C23×C61C23×C411yescharacteristic
216C22×D61—21no—
217C22×D61—21no—
218C22×D61—22no—
219C22×D61—21no—
220C22×D61—21no—
221C22×D61—21no—
222C22×D61—22no—
223C22×D61—22no—
224C22×D61C2×C22⋊C412yes—
225C22×D61#13114yes—
226C2×C4×C61—22no—
227C2×C4×C61—22no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—22no—
234C2×C4×C61—22no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61C2×C22⋊C412yes—
240C2×C4×C61C2×C22⋊C412yes—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C2×C4×C61H(5,𝔽2)11yescharacteristic
245C2×C4×C612−1+411yescharacteristic
246C2×C4×C612−1+411yescharacteristic
247C22⋊D61—44no—
248C22⋊D61—44no—
249C2×Dic61—21no—
250C2×Dic61—21no—
251C2×Dic61—22no—
252C2×Dic61—21no—
253C2×Dic61—21no—
254C2×Dic61—21no—
255C2×Dic61—21no—
256C2×Dic61—21no—
257C2×Dic61—22no—
258C2×Dic61—22no—
259C2×Dic61C2×C22⋊C412yes—
260C2×Dic61C2×C22⋊C412yes—
261C2×Dic61C2×C22⋊C412yes—
262C2×Dic61#13114yes—
263D244—44no—
264C4×D61—28no—
265C4×D61—22no—
266C4×D61—22no—
267C4×D61—22no—
268C4×D61—22no—
269C4×D61—28no—
270C4×D61—41no—
271C4×D61—41no—
272C4×D61—41no—
273C4×D61—41no—
274Dic122—44no—
275C8×C61—42no—
276C8×C61—42no—
277#92418—42no—
278#92418—42no—
279#21848—611nomaximal, Sylow
280C22×C61—41no—
281C22×C61—41no—
282C22×C61—41no—
283C22×C61—42no—
284C22×C61—42no—
285C22×C61C22×C22⋊C411yescharacteristic
286C22×C61#52011yescharacteristic
287C22×C61#52011yescharacteristic
288C22×C61C2×#13112yes—
289C22×C61C2×#13112yes—
290C2×D61—22no—
291C2×D61—22no—
292C2×D61—24no—
293C2×D61—24no—
294C2×D61—41no—
295C2×D61—41no—
296C2×D61—41no—
297C2×D61—42no—
298C2×D61—42no—
299C2×D61#40814yes—
300C4×C61—22no—
301C4×C61—22no—
302C4×C61—41no—
303C4×C61—41no—
304C4×C61—41no—
305C4×C61#53511yescharacteristic
306C4×C61#53611yescharacteristic
307C4×C61#53711yescharacteristic
308C4×C61#53711yescharacteristic
309Dic61—22no—
310Dic61—22no—
311Dic61—41no—
312Dic61—41no—
313Dic61—41no—
314Dic61#40814yes—
315C2×#529—611no—
316C2×#431—611no—
317C22×C4⋊C8—611no—
318#1984—612no—
319#1954—612no—
320#1869—612no—
321#1869—612no—
322#1798—611no—
323#1798—611no—
324#1798—611no—
325#1798D6111yescharacteristic, complemented
326C2×C61—81no—
327C2×C61#231111yescharacteristic
328C2×C61C2×#40811yescharacteristic
329C2×C61C2×#40811yescharacteristic
330C2×C61#299412yes—
331C2×C61#299412yes—
332D61—24no—
333D61—81no—
334C23×C8—612no—
335#529—1224no—
336#529—1224no—
337C2×#126—611no—
338C2×#126—611no—
339C2×#125—611no—
340C2×#124—611no—
341C2×#123—611no—
342C2×#123—611no—
343C2×C22⋊Q8—611no—
344C2×#121—611no—
345C2×C4×Q8—611no—
346C2×C4×D4—611no—
347C2×C4×D4—611no—
348C2×C4×D4—611no—
349C2×#117—611no—
350C2×#117C2×D6111yescharacteristic
351C22×C42—611no—
352#431—1224no—
353C2×C4⋊C8—612no—
354C2×C4⋊C8—611no—
355C2×C4⋊C8—611no—
356C2×C4⋊C8—1224no—
357C2×C4⋊C8C2×D6111yescharacteristic
358C2×C4⋊C8C2×D6111yescharacteristic
359C2×C22⋊C8—612no—
360C2×C22⋊C8—612no—
361C2×C22⋊C8—612no—
362C2×C22⋊C8—612no—
363C2×C22⋊C8C2×D6112yes—
364C2×C4×C8—612no—
365#336—612no—
366#336—612no—
367#336—612no—
368#336C2×D6112yes—
369C61#2184811yescharacteristic, complemented, Sylow
370C22×Q8—611no—
371C22×D4—611no—
372C22×D4—611no—
373C22×D4—611no—
374C23×C4—611no—
375C23×C4—611no—
376C23×C4—611no—
377C23×C4—611no—
378C23×C4C22×D6111yescharacteristic
379C22×C8—612no—
380C22×C8—612no—
381C22×C8—1224no—
382C22×C8—2442no—
383C22×C8—2442no—
384C22×C8C22×D6112yes—
385C22×C8C22×D6112yes—
386#126—1224no—
387#126—1224no—
388#125—1224no—
389#124—1224no—
390#123—1224no—
391#123—1224no—
392C22⋊Q8—1224no—
393#121—1224no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396C4×D4—1222no—
397C4×D4—1222no—
398C4×D4—1222no—
399C4×D4—1222no—
400C4×D4—1222no—
401C4×D4—1222no—
402#117—22no—
403#117—22no—
404#117—1222no—
405#117—1222no—
406C2×C4⋊C4—611no—
407C2×C4⋊C4—611no—
408C2×C4⋊C4—611no—
409C2×C4⋊C4—611no—
410C2×C4⋊C4—611no—
411C2×C4⋊C4—611no—
412C2×C4⋊C4—611no—
413C2×C4⋊C4—611no—
414C2×C4⋊C4C4×D6111yescharacteristic
415C2×C4⋊C4C4×D6111yescharacteristic
416C2×C22⋊C4—611no—
417C2×C22⋊C4—611no—
418C2×C22⋊C4—611no—
419C2×C22⋊C4—611no—
420C2×C22⋊C4—611no—
421C2×C22⋊C4—611no—
422C2×C22⋊C4—611no—
423C2×C22⋊C4—611no—
424C2×C22⋊C4C4×D6111yescharacteristic
425C2×C22⋊C4C4×D6111yescharacteristic
426C2×C42—611no—
427C2×C42—611no—
428C2×C42—611no—
429C2×C42—614no—
430C2×C42—614no—
431C2×C42—611no—
432C2×C42C22×D6111yescharacteristic
433C2×C42C22×D6111yescharacteristic
434C4⋊C8—22no—
435C4⋊C8—22no—
436C4⋊C8—1224no—
437C4⋊C8—1222no—
438C4⋊C8—1222no—
439C22⋊C8—42no—
440C22⋊C8—1224no—
441C22⋊C8—2442no—
442C22⋊C8—2442no—
443C22⋊C8—2442no—
444C4×C8—1224no—
445C24—611no—
446C24—611no—
447C24C2×C4×D6111yescharacteristic
448C2×Q8—1222no—
449C2×Q8—1224no—
450C2×D4—1222no—
451C2×D4—1224no—
452C2×D4—1222no—
453C2×D4—1224no—
454C2×D4—1222no—
455C2×D4—1224no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—612no—
466C22×C4—612no—
467C22×C4—614no—
468C22×C4—614no—
469C22×C4—611no—
470C22×C4—611no—
471C22×C4—611no—
472C22×C4—1224no—
473C22×C4—1224no—
474C22×C4—1221no—
475C22×C4—1221no—
476C22×C4—1221no—
477C22×C4—1221no—
478C22×C4—1221no—
479C22×C4—1221no—
480C22×C4—1221no—
481C22×C4—1221no—
482C22×C4—1221no—
483C22×C4—1221no—
484C22×C4C2×C4×D6111yescharacteristic
485C22×C4C2×C4×D6111yescharacteristic
486C22×C4C2×C4×D6111yescharacteristic
487C22×C4C2×C4×D6111yescharacteristic
488C22×C4C2×C4×D6111yescharacteristic
489C22×C4C23×D6111yescharacteristic
490C2×C8—22no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—22no—
495C2×C8—22no—
496C2×C8—1222no—
497C2×C8—1222no—
498C2×C8—1222no—
499C2×C8—1222no—
500C2×C8—1222no—
501C2×C8—1222no—
502C2×C8—2448no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C22⋊C4—22no—
514C22⋊C4—22no—
515C22⋊C4—1222no—
516C22⋊C4—1222no—
517C22⋊C4—1222no—
518C22⋊C4—1222no—
519C22⋊C4—1222no—
520C22⋊C4—1222no—
521C22⋊C4—1222no—
522C22⋊C4—1222no—
523C42—21no—
524C42—21no—
525C42—612no—
526C42—612no—
527C42—614no—
528C42—614no—
529C42—1221no—
530C42—1221no—
531C42D4×D6112yes—
532C42D4×D6112yes—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—22no—
537C23—22no—
538C23—612no—
539C23—614no—
540C23—1221no—
541C23—1221no—
542C23—1222no—
543C23—1221no—
544C23—1221no—
545C23—1221no—
546C23—1222no—
547C23—1222no—
548C23C22×C4×D6111yescharacteristic
549Q8—2444no—
550D4—2444no—
551D4—2444no—
552D4—2444no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—612no—
567C2×C4—614no—
568C2×C4—612no—
569C2×C4—612no—
570C2×C4—1228no—
571C2×C4—1228no—
572C2×C4—1221no—
573C2×C4—1221no—
574C2×C4—1222no—
575C2×C4—1222no—
576C2×C4—1222no—
577C2×C4—1222no—
578C2×C4—1222no—
579C2×C4—1221no—
580C2×C4—1221no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1221no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—2441no—
587C2×C4—2441no—
588C2×C4—2441no—
589C2×C4—2441no—
590C2×C4C22⋊C4×D6112yes—
591C2×C4C22⋊C4×D6112yes—
592C2×C4C2×D4×D6111yescharacteristic
593C2×C4C2×D4×D6111yescharacteristic
594C2×C4#1451982211yescharacteristic
595C2×C4#1451982311yescharacteristic
596C2×C4#1451982411yescharacteristic
597C2×C4#1451982511yescharacteristic
598C8—42no—
599C8—42no—
600C8—2442no—
601C8—2442no—
602C22—42no—
603C22—42no—
604C22—41no—
605C22—41no—
606C22—41no—
607C22—614no—
608C22—1224no—
609C22—1224no—
610C22—1222no—
611C22—1222no—
612C22—2442no—
613C22—2442no—
614C22—2441no—
615C22—2441no—
616C22—2441no—
617C22C2×C22⋊C4×D6111yescharacteristic
618C22#1684024211yescharacteristic
619C22#1684024311yescharacteristic
620C22#131×D6112yes—
621C22#131×D6112yes—
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—41no—
627C4—614no—
628C4—1222no—
629C4—1222no—
630C4—2441no—
631C4—2441no—
632C4—2441no—
633C4#1684032811yescharacteristic
634C4#1684033511yescharacteristic
635C4#1684034111yescharacteristic
636C4#1684034211yescharacteristic
637C2—81no—
638C2—1224no—
639C2—4881no—
640C2#2144840811yescharacteristic
641C2#408×D6111yescharacteristic
642C2#408×D6111yescharacteristic
643C2#2145497312yes—
644C2#2145497712yes—
subgroupG/H
center, FrattiniC22×C4C23×D61
commutator, socleC23×C61C23×C4
FittingC61×#1798C2
prime radicalC2×#14519750C4
cosocleC22×C4×C61C24
hypercenter#1798D61
hypocenterC61#21848
LCSG ▷ C23×C61 ▷ C61
UCSC1 ◁ C22×C4 ◁ #1798

Complements

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

NH ≅ G/Ntype
2. C2×#16839814C2semidirect
3. C2×C4⋊C8×D61C2semidirect
4. #21446753C2semidirect
5. #21446611C2semidirect
6. #21446110C2semidirect
7. #21446108C2semidirect
8. #21445803C2semidirect
9. #21445800C2semidirect
10. #21445800C2semidirect
11. C61×#1798C2semidirect
325. #1798D61semidirect
369. C61#21848semidirect

Maximal subgroups

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

subgroupindex
C2×#168402902
C2×#168398142
C2×C4⋊C8×D612
#214467532
#214466112
#214461102
#214461082
#214458032
#214458002
C61×#17982
#2184861