Finite Group Databasebeta

← #30617920

Subgroups — #30617920

Order 15616.

1434 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840779C211yesmaximal, characteristic, complemented
2C22×#14519844C211yesmaximal, characteristic, complemented
3#21461990C212yesmaximal, complemented
4#21461942C212yesmaximal, complemented
5#21461794C212yesmaximal, complemented
6#21461792C212yesmaximal, complemented
7#21461676C212yesmaximal, complemented
8#21461675C212yesmaximal, complemented
9C2×C4×#11758224C211yesmaximal, characteristic
10C23×#92420C2212yes—
11#16840779—24no—
12C2×#14519844—24no—
13C2×#14519844C2212yes—
14C2×#14519844C2212yes—
15C2×#14519844C2212yes—
16C2×#14519834C2214yes—
17C2×C22⋊#92420C2212yes—
18C2×C22⋊#92420C2214yes—
19C2×C4×#92420C2212yes—
20#16840628C2214yes—
21#16840628C2214yes—
22C22×#11758224C411yescharacteristic
23C4×#11758224—24no—
24C4×#11758224—24no—
25C2×C4×Dic122C411yescharacteristic
26C2×#14519742C2211yescharacteristic
27C2×D4×Dic61C2211yescharacteristic
28C2×C4×C22⋊D61C411yescharacteristic
29C2×C4×C22⋊D61C411yescharacteristic
30C2×#14519737C2211yescharacteristic
31C2×#14519737C2211yescharacteristic
32C2×#14519735C411yescharacteristic
33C2×#14519733C411yescharacteristic
34C2×#14519733C411yescharacteristic
35C22×C4×Dic61C2211yescharacteristic
36C22×C4×Dic61C2211yescharacteristic
37C2×C42×D61C2211yescharacteristic
38C2×C4×D4×C61C411yescharacteristic
39C22×#92420—24no—
40C22×#92420—44no—
41C22×#92420C2314yes—
42C22×#92420C2314yes—
43#14519844—24no—
44#14519844—24no—
45#14519844—24no—
46#14519834—44no—
47C22⋊#92420—24no—
48C22⋊#92420—44no—
49C4×#92420—24no—
50C2×#11758224—24no—
51C2×#11758224—24no—
52C2×#11758224C818yescomplemented
53C22×Dic122C2×C411yescharacteristic
54C22×C22⋊D61C2×C411yescharacteristic
55C22×C22⋊D61C2×C411yescharacteristic
56C23×Dic61C2×C411yescharacteristic
57C23×Dic61C2×C411yescharacteristic
58C23×Dic61C2311yescharacteristic
59C23×Dic61C2311yescharacteristic
60C22×C4×D61C2×C411yescharacteristic
61C22×C4×D61C2×C411yescharacteristic
62C22×C4×D61C2311yescharacteristic
63C4×Dic122—24no—
64#14519742—22no—
65#14519742—22no—
66D4×Dic61—22no—
67D4×Dic61—22no—
68C4×C22⋊D61—24no—
69C4×C22⋊D61—24no—
70#14519737—22no—
71#14519737—22no—
72#14519737—22no—
73#14519737—22no—
74#14519735—24no—
75#14519733—24no—
76#14519733—24no—
77C2×C4⋊Dic61C2×C411yescharacteristic
78C2×#11758196C2×C411yescharacteristic
79C2×#11758196C2×C411yescharacteristic
80C2×#11758196C2×C411yescharacteristic
81C2×#11758196C2×C411yescharacteristic
82C2×C22⋊Dic61C2×C411yescharacteristic
83C2×C22⋊Dic61C2×C411yescharacteristic
84C2×#11758198C2×C411yescharacteristic
85C2×#11758198C2×C411yescharacteristic
86C2×C4×Dic61—22no—
87C2×C4×Dic61—22no—
88C2×C4×Dic61C2×C411yescharacteristic
89C2×C4×Dic61C2×C411yescharacteristic
90C2×C4×Dic61C2×C411yescharacteristic
91C2×C4×Dic61C2×C411yescharacteristic
92C2×C4×Dic61C2×C411yescharacteristic
93C2×C4×Dic61D414yes—
94C2×C4×Dic61D414yes—
95C2×C4×Dic61C2311yescharacteristic
96C2×C4×Dic61C2311yescharacteristic
97C2×C4×Dic61C2311yescharacteristic
98C2×C4×Dic61C2311yescharacteristic
99C42×D61—22no—
100C42×D61—22no—
101C22×D4×C61C2×C411yescharacteristic
102C23×C4×C61C2×C411yescharacteristic
103C23×C4×C61C2×C411yescharacteristic
104C4×D4×C61—24no—
105C2×C4⋊C4×C61C2×C411yescharacteristic
106C2×C22⋊C4×C61C2×C411yescharacteristic
107C2×C22⋊C4×C61C2×C411yescharacteristic
108C2×C42×C61C2×C411yescharacteristic
109C24×C61C22×C411yescharacteristic
110C24×C61C22×C411yescharacteristic
111C23×D61C22×C411yescharacteristic
112C2×D4×C61—22no—
113C2×D4×C61—22no—
114C2×D4×C61C2×C814yes—
115C22×C4×C61—22no—
116C22×C4×C61—42no—
117C22×C4×C61—42no—
118C22×C4×C61C22⋊C414yes—
119C22×C4×C61C22×C411yescharacteristic
120C22×C4×C61C22×C411yescharacteristic
121C22×C4×C61C22×C411yescharacteristic
122C22×C4×C61C22×C411yescharacteristic
123C22×C4×C61C22×C411yescharacteristic
124C2×C22⋊D61—22no—
125C2×C22⋊D61—22no—
126C2×C22⋊D61—22no—
127C2×C22⋊D61—22no—
128C2×C22⋊D61C2×C814yes—
129C2×C22⋊D61C2×C814yescomplemented
130C22×Dic61—22no—
131C22×Dic61—22no—
132C22×Dic61—22no—
133C22×Dic61—22no—
134C22×Dic61—21no—
135C22×Dic61—21no—
136C22×Dic61—22no—
137C22×Dic61—22no—
138C22×Dic61C22⋊C414yes—
139C22×Dic61C2×C814yes—
140C22×Dic61C2×C814yescomplemented
141C22×Dic61M1614yes—
142C22×Dic61C22×C411yescharacteristic
143C22×Dic61C22×C411yescharacteristic
144C22×Dic61C22×C411yescharacteristic
145C22×Dic61C22×C411yescharacteristic
146C22×Dic61C22×C411yescharacteristic
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C2×D412yes—
149C22×Dic61C2×D412yes—
150C22×Dic61C2411yescharacteristic
151#11758224—232no—
152C2×C4×D61—22no—
153C2×C4×D61—22no—
154C2×C4×D61—22no—
155C2×C4×D61—21no—
156C2×C4×D61—21no—
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61—42no—
160C2×C4×D61C2×C814yescomplemented
161C2×Dic122—22no—
162C2×Dic122—22no—
163C2×Dic122C2×C814yescomplemented
164C2×#92420—24no—
165C2×#92420—24no—
166C2×#92420—24no—
167C2×#92420—24no—
168C2×#92420—24no—
169C2×#92420—24no—
170C2×#92420—48no—
171C4⋊C4×C61—22no—
172C22⋊C4×C61—22no—
173C22⋊C4×C61—22no—
174C42×C61—22no—
175C22⋊Dic61—22no—
176C22⋊Dic61—22no—
177#11758198—22no—
178#11758198—22no—
179C4⋊Dic61—22no—
180#11758196—22no—
181#11758196—22no—
182#11758196—22no—
183#11758196—22no—
184C4×Dic61—22no—
185C4×Dic61—22no—
186C4×Dic61—22no—
187C4×Dic61—21no—
188C4×Dic61—21no—
189C4×Dic61—21no—
190C4×Dic61—21no—
191C4×Dic61C22⋊C414yes—
192C4×Dic61C22⋊C414yes—
193C4×Dic61C2×D412yes—
194C4×Dic61C2×D412yes—
195C4×Dic61C2×D412yes—
196C4×Dic61C2×D412yes—
197C23×C61—22no—
198C23×C61—22no—
199C23×C61—21no—
200C23×C61—22no—
201C23×C61—21no—
202C23×C61—22no—
203C23×C61C2×C22⋊C412yes—
204C23×C61C22×C812yes—
205C23×C61C22×C812yes—
206C23×C61C2×M1612yes—
207C23×C61C23×C411yescharacteristic
208C22×D61—21no—
209C22×D61—22no—
210C22×D61—21no—
211C22×D61—22no—
212C22×D61C22×C812yes—
213D4×C61—28no—
214C2×C4×C61—21no—
215C2×C4×C61—21no—
216C2×C4×C61—22no—
217C2×C4×C61—22no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—22no—
221C2×C4×C61—22no—
222C2×C4×C61—44no—
223C2×C4×C61—44no—
224C2×C4×C61C2×C22⋊C412yes—
225C2×C4×C61C2×C22⋊C412yes—
226C2×C4×C61C22×C812yes—
227C22⋊D61—28no—
228C22⋊D61—28no—
229C2×Dic61—21no—
230C2×Dic61—22no—
231C2×Dic61—21no—
232C2×Dic61—22no—
233C2×Dic61—22no—
234C2×Dic61—24no—
235C2×Dic61—28no—
236C2×Dic61—21no—
237C2×Dic61—21no—
238C2×Dic61—22no—
239C2×Dic61—22no—
240C2×Dic61—41no—
241C2×Dic61—41no—
242C2×Dic61—42no—
243C2×Dic61C22⋊C8116yescomplemented
244C2×Dic61C2×C22⋊C412yes—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C412yes—
247C2×Dic61C22×C812yes—
248C2×Dic61C22×C812yes—
249C2×Dic61C22×C812yes—
250C2×Dic61C2×M1612yes—
251C2×Dic61C2×M1612yes—
252C2×Dic61C22×D411yescharacteristic
253C2×Dic61C22×D411yescharacteristic
254C2×Dic61H(5,𝔽2)12yes—
255C2×Dic612−1+412yes—
256C4×D61—28no—
257C4×D61—44no—
258C4×D61—41no—
259C4×D61—41no—
260C4×D61—42no—
261Dic122—28no—
262#92420—44no—
263#92420—44no—
264#21847—611nomaximal, Sylow
265C22×C61—22no—
266C22×C61—22no—
267C22×C61—24no—
268C22×C61—24no—
269C22×C61—22no—
270C22×C61—24no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—42no—
274C22×C61C2×C22⋊C814yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C22×C61#52012yes—
277C22×C61C23×C811yescharacteristic
278C22×C61C22×M1611yescharacteristic
279C22×C61#56812yes—
280C2×D61—22no—
281C2×D61—24no—
282C2×D61—41no—
283C2×D61—41no—
284C2×D61—42no—
285C4×C61—22no—
286C4×C61—22no—
287C4×C61—22no—
288C4×C61—41no—
289C4×C61—41no—
290Dic61—22no—
291Dic61—22no—
292Dic61—41no—
293Dic61—41no—
294Dic61C2×C22⋊C814yes—
295Dic61C2×C22⋊C814yes—
296Dic61#53511yescharacteristic
297Dic61#53611yescharacteristic
298Dic61#53712yes—
299C2×C4×#41—611no—
300C2×#431—611no—
301C22×C4⋊C8—611no—
302#1984—612no—
303#1954—612no—
304#1869—612no—
305#1869—612no—
306#1797—612no—
307#1797—612no—
308C2×C61—24no—
309C2×C61—42no—
310C2×C61#231111yescharacteristic
311C2×C61C22×C22⋊C811yescharacteristic
312C2×C61#287911yescharacteristic
313C2×C61#298812yes—
314C2×C61#298912yes—
315D61—42no—
316C22×#41—611no—
317C23×C8—612no—
318C4×#41—1224no—
319C4×#41—1224no—
320C2×C4×Q8—611no—
321C2×C4×Q8—611no—
322C2×C4×D4—611no—
323C2×C4×D4—611no—
324C2×C4×D4—611no—
325C2×C4×D4—611no—
326C2×C4×D4—611no—
327C2×C4×D4#692311yescharacteristic
328C2×#117—611no—
329C2×#117—611no—
330C2×#117—611no—
331C22×C42—611no—
332C22×C42—611no—
333C22×C42—611no—
334#431—1224no—
335C2×C4⋊C8—612no—
336C2×C4⋊C8—612no—
337C2×C4⋊C8—612no—
338C2×C4⋊C8—1224no—
339C2×C22⋊C8—612no—
340C2×C22⋊C8—614no—
341C2×C22⋊C8—614no—
342C2×C4×C8—612no—
343#336—614no—
344#336—614no—
345C61#2184711yescharacteristic, complemented, Sylow
346C2×#41—618no—
347C2×#41—1224no—
348C2×#41—1224no—
349C22×Q8—611no—
350C22×D4—611no—
351C22×D4—611no—
352C22×D4C2×#692311yescharacteristic
353C23×C4—611no—
354C23×C4—611no—
355C23×C4—611no—
356C23×C4—611no—
357C23×C4—611no—
358C23×C4—611no—
359C23×C4—611no—
360C23×C4C2×#692311yescharacteristic
361C23×C4C2×#692311yescharacteristic
362C22×C8—614no—
363C22×C8—614no—
364C22×C8—1224no—
365C22×C8—2444no—
366C4×Q8—1224no—
367C4×Q8—1222no—
368C4×Q8—1222no—
369C4×D4—24no—
370C4×D4—1222no—
371C4×D4—1222no—
372C4×D4—1224no—
373C4×D4—1222no—
374C4×D4—1222no—
375C4×D4—1222no—
376C4×D4—1222no—
377C4×D4—1224no—
378#117—1224no—
379#117—1224no—
380#117—1224no—
381C2×C4⋊C4—611no—
382C2×C4⋊C4—611no—
383C2×C4⋊C4—611no—
384C2×C4⋊C4—611no—
385C2×C4⋊C4—611no—
386C2×C4⋊C4C2×#692311yescharacteristic
387C2×C22⋊C4—611no—
388C2×C22⋊C4—611no—
389C2×C22⋊C4—611no—
390C2×C22⋊C4—611no—
391C2×C22⋊C4C2×#692311yescharacteristic
392C2×C22⋊C4C2×#692311yescharacteristic
393C2×C42—614no—
394C2×C42—614no—
395C2×C42—611no—
396C2×C42—611no—
397C2×C42—611no—
398C2×C42—611no—
399C2×C42—611no—
400C2×C42—611no—
401C2×C42—611no—
402C2×C42—611no—
403C2×C42—611no—
404C2×C42—1222no—
405C2×C42—1222no—
406C2×C42—1222no—
407C2×C42—1222no—
408C2×C42C2×#692311yescharacteristic
409C4⋊C8—1224no—
410C4⋊C8—1224no—
411C4⋊C8—1224no—
412C22⋊C8—1224no—
413C22⋊C8—2444no—
414C22⋊C8—2444no—
415C4×C8—1224no—
416C24—611no—
417C24C22×#692311yescharacteristic
418C24C22×#692311yescharacteristic
419#41—12232no—
420C2×Q8—614no—
421C2×Q8—1222no—
422C2×Q8—1222no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—614no—
426C2×D4—614no—
427C2×D4—1222no—
428C2×D4—1222no—
429C2×D4—1222no—
430C2×D4—1222no—
431C2×D4D61⋊C814yes—
432C22×C4—22no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—614no—
436C22×C4—614no—
437C22×C4—614no—
438C22×C4—611no—
439C22×C4—611no—
440C22×C4—611no—
441C22×C4—612no—
442C22×C4—612no—
443C22×C4—614no—
444C22×C4—611no—
445C22×C4—611no—
446C22×C4—614no—
447C22×C4—611no—
448C22×C4—611no—
449C22×C4—1222no—
450C22×C4—1222no—
451C22×C4—1222no—
452C22×C4—1221no—
453C22×C4—1221no—
454C22×C4—1222no—
455C22×C4—1222no—
456C22×C4—1222no—
457C22×C4—1222no—
458C22×C4—1222no—
459C22×C4—1222no—
460C22×C4—1221no—
461C22×C4—1221no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4—2442no—
465C22×C4C22⋊#692314yes—
466C22×C4C22×#692311yescharacteristic
467C22×C4C22×#692311yescharacteristic
468C22×C4C22×#692311yescharacteristic
469C22×C4C22×#692311yescharacteristic
470C22×C4C22×#692311yescharacteristic
471C2×C8—1224no—
472C2×C8—1224no—
473C2×C8—1224no—
474C2×C8—1224no—
475C2×C8—1224no—
476C2×C8—1224no—
477C2×C8—2448no—
478C4⋊C4—22no—
479C4⋊C4—1222no—
480C4⋊C4—1222no—
481C4⋊C4—1222no—
482C4⋊C4—1222no—
483C4⋊C4—1222no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—1222no—
487C22⋊C4—1222no—
488C22⋊C4—1222no—
489C22⋊C4—1222no—
490C42—22no—
491C42—612no—
492C42—612no—
493C42—612no—
494C42—612no—
495C42—614no—
496C42—614no—
497C42—1222no—
498C42—1222no—
499C42—1221no—
500C42—1221no—
501C42—1221no—
502C42—1221no—
503C42—1222no—
504C23—22no—
505C23—22no—
506C23—21no—
507C23—22no—
508C23—21no—
509C23—22no—
510C23—612no—
511C23—1221no—
512C23—1222no—
513C23—1221no—
514C23—1222no—
515C23C2×C22⋊#692312yes—
516C23C2×D61⋊C812yes—
517C23C2×D61⋊C812yes—
518C23D61⋊M1612yes—
519C23C23×#692311yescharacteristic
520Q8—1228no—
521D4—28no—
522D4—1228no—
523D4—1228no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—44no—
533C2×C4—44no—
534C2×C4—6116no—
535C2×C4—612no—
536C2×C4—612no—
537C2×C4—611no—
538C2×C4—611no—
539C2×C4—612no—
540C2×C4—612no—
541C2×C4—612no—
542C2×C4—612no—
543C2×C4—612no—
544C2×C4—612no—
545C2×C4—612no—
546C2×C4—612no—
547C2×C4—1228no—
548C2×C4—1222no—
549C2×C4—1222no—
550C2×C4—1224no—
551C2×C4—1228no—
552C2×C4—1221no—
553C2×C4—1222no—
554C2×C4—1221no—
555C2×C4—1221no—
556C2×C4—1221no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—2444no—
560C2×C4—2441no—
561C2×C4—2441no—
562C2×C4—2442no—
563C2×C4—2441no—
564C2×C4—2441no—
565C2×C4—2442no—
566C2×C4C2×C22⋊#692312yes—
567C2×C4C2×C22⋊#692312yes—
568C2×C4C2×D61⋊C812yes—
569C8—2444no—
570C8—2444no—
571C22—22no—
572C22—22no—
573C22—24no—
574C22—24no—
575C22—22no—
576C22—24no—
577C22—41no—
578C22—41no—
579C22—42no—
580C22—1222no—
581C22—1224no—
582C22—2441no—
583C22—2441no—
584C22—2442no—
585C22D61⋊(C22⋊C8)14yes—
586C22C22×C22⋊#692311yescharacteristic
587C22#1684083112yes—
588C22C22×D61⋊C811yescharacteristic
589C22C2×D61⋊M1611yescharacteristic
590C22#1684084012yes—
591C4—22no—
592C4—22no—
593C4—22no—
594C4—41no—
595C4—41no—
596C4—611no—
597C4—611no—
598C4—612no—
599C4—614no—
600C4—614no—
601C4—1222no—
602C4—1222no—
603C4—2441no—
604C4—2441no—
605C2—24no—
606C2—42no—
607C2—2442no—
608C2#2146229411yescharacteristic
609C2C2×D61⋊(C22⋊C8)11yescharacteristic
610C2#2146240311yescharacteristic
611C2#2146273512yes—
612C2#2146273912yes—
subgroupG/H
center, FrattiniC23C23×#6923
commutatorC22×C61C23×C8
FittingC2×C4×D4×C61C4
socleC23×C61C23×C4
prime radicalC22×#11758224C4
cosocleC22×Dic61C24
hypercenterC2×C4×D4#6923
hypocenterC61#21847
LCSG ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16840779C2semidirect
2. C22×#14519844C2semidirect
3. #21461990C2semidirect
4. #21461942C2semidirect
5. #21461794C2semidirect
6. #21461792C2semidirect
7. #21461676C2semidirect
8. #21461675C2semidirect
52. C2×#11758224C8semidirect
129. C2×C22⋊D61C2×C8semidirect
140. C22×Dic61C2×C8semidirect
160. C2×C4×D61C2×C8semidirect
163. C2×Dic122C2×C8semidirect
243. C2×Dic61C22⋊C8semidirect
345. C61#21847semidirect

Maximal subgroups

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

subgroupindex
C2×#168407792
C22×#145198442
#214619902
#214619422
#214617942
#214617922
#214616762
#214616752
C2×C4×#117582242
#2184761