Finite Group Databasebeta

← #30610465

Subgroups — #30610465

Order 15616.

1292 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21462495C211yesmaximal, characteristic, complemented
2C2×#16840765C211yesmaximal, characteristic, complemented
3#21461154C214yesmaximal, complemented
4#21461147C212yesmaximal, complemented
5#21461144C212yesmaximal, complemented
6#21461114C214yesmaximal, complemented
7C2×#41×Dic61C211yesmaximal, characteristic
8C22×#11758217—21no—
9#16840783—21no—
10#16840782—21no—
11#16840779C2212yes—
12#16840778C2212yescomplemented
13#16840765—21no—
14#16840765—21no—
15#16840765—21no—
16#16840765C2212yes—
17#16840765C2212yescomplemented
18C2×#14519844—21no—
19C2×C4⋊#92420C2212yescomplemented
20#16840732—21no—
21C4×#11758217C2212yescomplemented
22Q8⋊#92420C2218yescomplemented
23D4⋊#92420C2218yescomplemented
24#41×Dic61—21no—
25#41×Dic61—21no—
26#41×Dic61—21no—
27#41×Dic61—21no—
28#41×Dic61C412yes—
29#41×Dic61C412yes—
30#41×Dic61C2212yes—
31#41×Dic61C2212yes—
32C2×Q8×Dic61C411yescharacteristic
33C2×Q8×Dic61C2211yescharacteristic
34C2×D4×Dic61—21no—
35C2×D4×Dic61—21no—
36C2×D4×Dic61C411yescharacteristic
37C2×D4×Dic61C2211yescharacteristic
38C2×#14519736—21no—
39C2×#14519736C411yescharacteristic
40C22×C4×Dic61—21no—
41C22×C4×Dic61C2211yescharacteristic
42C22×#41×C61C411yescharacteristic
43C2×#11758217—22no—
44C2×#11758217—22no—
45C2×#11758217—22no—
46C2×#11758217—44no—
47C22×#92420—22no—
48#14519844—22no—
49#14519844—22no—
50#14519844—22no—
51C4⋊#92420—22no—
52C4⋊#92420—22no—
53C4⋊#92420C2314yes—
54C22⋊#92420—22no—
55C22⋊#92420—22no—
56#14519832—22no—
57C4×#92420C2314yes—
58C23×Dic61—21no—
59C23×Dic61—21no—
60C23×Dic61D411yescharacteristic
61C23×Dic61D411yescharacteristic
62Q8×Dic61—21no—
63Q8×Dic61—21no—
64Q8×Dic61—21no—
65Q8×Dic61—21no—
66Q8×Dic61C2×C412yes—
67Q8×Dic61C2×C412yes—
68Q8×Dic61C2×C412yes—
69Q8×Dic61C2312yes—
70D4×Dic61—22no—
71D4×Dic61—22no—
72D4×Dic61—22no—
73D4×Dic61—22no—
74D4×Dic61—22no—
75D4×Dic61—22no—
76D4×Dic61—22no—
77D4×Dic61—22no—
78D4×Dic61—21no—
79D4×Dic61—21no—
80D4×Dic61—21no—
81D4×Dic61—21no—
82D4×Dic61C2×C412yes—
83D4×Dic61C2×C412yes—
84D4×Dic61C2×C412yes—
85D4×Dic61C2312yes—
86#14519736—21no—
87#14519736—21no—
88#14519736—22no—
89#14519736—22no—
90#14519736—22no—
91#14519736—22no—
92#14519736C2×C412yes—
93#14519736C2×C412yes—
94C2×C4⋊Dic61—21no—
95C2×C4⋊Dic61—21no—
96C2×C4⋊Dic61C2×C411yescharacteristic
97C2×C4⋊Dic61C2×C411yescharacteristic
98C2×C22⋊Dic61—21no—
99C2×C22⋊Dic61—21no—
100C2×C22⋊Dic61—21no—
101C2×C4×Dic61—21no—
102C2×C4×Dic61—21no—
103C2×C4×Dic61—21no—
104C2×C4×Dic61—22no—
105C2×C4×Dic61—22no—
106C2×C4×Dic61—22no—
107C2×C4×Dic61—22no—
108C2×C4×Dic61C2×C411yescharacteristic
109C2×C4×Dic61C2×C411yescharacteristic
110C2×C4×Dic61C2×C411yescharacteristic
111C2×C4×Dic61D411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C2×C4×Dic61D411yescharacteristic
115C2×C4×Dic61D411yescharacteristic
116C2×C4×Dic61D411yescharacteristic
117C2×C4×Dic61C2311yescharacteristic
118C2×C4×Dic61C2311yescharacteristic
119C2×C4×Dic61C2311yescharacteristic
120C2×#41×C61—28no—
121C2×#41×C61—21no—
122C2×#41×C61—21no—
123C2×#41×C61C2×C412yes—
124C2×#41×C61C2×C412yes—
125C22×Q8×C61C2×C411yescharacteristic
126C22×D4×C61—21no—
127C22×D4×C61C2×C411yescharacteristic
128C23×C4×C61—21no—
129C23×C4×C61C2×C411yescharacteristic
130C24×C61—21no—
131C24×C61C22⋊C411yescharacteristic
132#41×C61—28no—
133#41×C61—28no—
134#41×C61—24no—
135#41×C61—24no—
136#41×C61—24no—
137#41×C61M1618yescomplemented
138C2×Q8×C61—24no—
139C2×Q8×C61—21no—
140C2×Q8×C61C22×C412yes—
141C2×D4×C61—28no—
142C2×D4×C61—22no—
143C2×D4×C61—22no—
144C2×D4×C61—24no—
145C2×D4×C61—21no—
146C2×D4×C61C22×C412yes—
147C22×C4×C61—28no—
148C22×C4×C61—21no—
149C22×C4×C61—22no—
150C22×C4×C61—22no—
151C22×C4×C61—24no—
152C22×C4×C61C22⋊C411yescharacteristic
153C22×C4×C61C22⋊C411yescharacteristic
154C22×C4×C61C22⋊C411yescharacteristic
155C22×C4×C61C22×C411yescharacteristic
156C22×C4×C61C22×C411yescharacteristic
157C22×C4×C61C22×C411yescharacteristic
158C22×Dic61—22no—
159C22×Dic61—22no—
160C22×Dic61—21no—
161C22×Dic61—21no—
162C22×Dic61—22no—
163C22×Dic61—22no—
164C22×Dic61—22no—
165C22×Dic61—22no—
166C22×Dic61—42no—
167C22×Dic61—44no—
168C22×Dic61—42no—
169C22×Dic61—42no—
170C22×Dic61C22⋊C411yescharacteristic
171C22×Dic61C22⋊C411yescharacteristic
172C22×Dic61C22⋊C411yescharacteristic
173C22×Dic61C22⋊C411yescharacteristic
174C22×Dic61C22⋊C411yescharacteristic
175C22×Dic61C22⋊C411yescharacteristic
176C22×Dic61C2×D411yescharacteristic
177C22×Dic61C2×D411yescharacteristic
178C22×Dic61C2×D411yescharacteristic
179C22×Dic61C2×D411yescharacteristic
180C22×Dic61C2×D411yescharacteristic
181C22×Dic61C2×D411yescharacteristic
182#11758217—44no—
183#11758217—44no—
184#11758217—44no—
185C2×#92420—24no—
186C2×#92420—24no—
187C2×#92420—44no—
188C22⋊Dic61—22no—
189C22⋊Dic61—22no—
190C22⋊Dic61—22no—
191C22⋊Dic61—22no—
192C22⋊Dic61—22no—
193C22⋊Dic61—22no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61—22no—
200C4⋊Dic61C22×C412yes—
201C4⋊Dic61C22×C412yes—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61—22no—
207C4×Dic61—22no—
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22×C411yescharacteristic
215C4×Dic61C22×C411yescharacteristic
216C4×Dic61C22×C411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2411yescharacteristic
224C23×C61—24no—
225C23×C61—22no—
226C23×C61—22no—
227C23×C61—22no—
228C23×C61—41no—
229C23×C61—42no—
230C23×C61—42no—
231C23×C61C2×C22⋊C411yescharacteristic
232C23×C61C2×C22⋊C411yescharacteristic
233C23×C61C2×C22⋊C411yescharacteristic
234Q8×C61—24no—
235Q8×C61—22no—
236Q8×C61C2×M1614yescomplemented
237D4×C61—28no—
238D4×C61—24no—
239D4×C61—28no—
240D4×C61—22no—
241D4×C61C2×M1614yescomplemented
242C2×C4×C61—24no—
243C2×C4×C61—28no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—22no—
248C2×C4×C61—22no—
249C2×C4×C61—28no—
250C2×C4×C61—22no—
251C2×C4×C61C2×C22⋊C411yescharacteristic
252C2×C4×C61C2×C22⋊C411yescharacteristic
253C2×C4×C61C2×C22⋊C411yescharacteristic
254C2×C4×C61C2×M1614yes—
255C2×C4×C61C23×C411yescharacteristic
256C2×Dic61—22no—
257C2×Dic61—21no—
258C2×Dic61—21no—
259C2×Dic61—21no—
260C2×Dic61—21no—
261C2×Dic61—22no—
262C2×Dic61—22no—
263C2×Dic61—22no—
264C2×Dic61—22no—
265C2×Dic61—22no—
266C2×Dic61—22no—
267C2×Dic61—22no—
268C2×Dic61—41no—
269C2×Dic61—41no—
270C2×Dic61—41no—
271C2×Dic61—44no—
272C2×Dic61—42no—
273C2×Dic61—42no—
274C2×Dic61—42no—
275C2×Dic61—42no—
276C2×Dic61—44no—
277C2×Dic61—44no—
278C2×Dic61C2×C22⋊C411yescharacteristic
279C2×Dic61C2×C22⋊C411yescharacteristic
280C2×Dic61C2×C22⋊C411yescharacteristic
281C2×Dic61C2×C22⋊C411yescharacteristic
282C2×Dic61C2×C22⋊C411yescharacteristic
283C2×Dic61C2×C22⋊C411yescharacteristic
284C2×Dic61AGL(1,ℤ/8)12yes—
285C2×Dic61#13712yes—
286C2×Dic61C22×D411yescharacteristic
287C2×Dic61C22×D411yescharacteristic
288#92420—44no—
289#92420—44no—
290#17451—611nomaximal, Sylow
291C22×C61—21no—
292C22×C61—21no—
293C22×C61—21no—
294C22×C61—21no—
295C22×C61—21no—
296C22×C61—24no—
297C22×C61—21no—
298C22×C61—21no—
299C22×C61—44no—
300C22×C61—48no—
301C22×C61—42no—
302C22×C61—42no—
303C22×C61#40712yes—
304C22×C61#41712yes—
305C22×C61C22×C22⋊C411yescharacteristic
306C4×C61—21no—
307C4×C61—24no—
308C4×C61—21no—
309C4×C61#40712yes—
310C4×C61C22×M1612yes—
311Dic61—21no—
312Dic61—21no—
313Dic61—21no—
314Dic61—21no—
315Dic61—21no—
316Dic61—42no—
317Dic61—42no—
318Dic61#41712yes—
319Dic61#42112yes—
320Dic61C2×AGL(1,ℤ/8)11yescharacteristic
321Dic61C2×#13711yescharacteristic
322C2×C4×#41—611no—
323#2920—611no—
324C2×#423—611no—
325#1489—614no—
326#1487—612no—
327#1486—612no—
328#1477—614no—
329C2×C61—22no—
330C2×C61—22no—
331C2×C61—41no—
332C2×C61—44no—
333C2×C61C2×#40711yescharacteristic
334C2×C61C2×#41711yescharacteristic
335C2×C61C2×#42111yescharacteristic
336C22×#41#692311yescharacteristic
337C22×M16—1221no—
338C4×#41—612no—
339C4×#41—612no—
340C4×#41—612no—
341C4×#41—612no—
342C4×#41—1221no—
343C4×#41—1221no—
344C4×#41—1221no—
345C4×#41—1221no—
346C2×C4×Q8—611no—
347C2×C4×Q8—611no—
348C2×C4×D4—611no—
349C2×C4×D4—611no—
350C2×C4×D4—1221no—
351C2×C4×D4—1221no—
352C2×#117—611no—
353C2×#117—1221no—
354C22×C42—611no—
355C22×C42—1221no—
356#432—1221no—
357#432—1221no—
358#431—612no—
359#431—612no—
360#423—612no—
361#423—612no—
362#423—1221no—
363#423—1221no—
364#423—1221no—
365C2×C4⋊C8—612no—
366C2×C4⋊C8—1221no—
367#407—1221no—
368C4×M16—612no—
369Q8⋊C8—618no—
370D4⋊C8—618no—
371C61#1745111yescharacteristic, complemented, Sylow
372C2×#41—28no—
373C2×#41—21no—
374C2×#41—21no—
375C2×#41C2×#692312yes—
376C2×#41C2×#692312yes—
377C22×Q8C2×#692311yescharacteristic
378C22×D4—21no—
379C22×D4C2×#692311yescharacteristic
380C23×C4—21no—
381C23×C4—611no—
382C23×C4—611no—
383C23×C4—1221no—
384C23×C4—1221no—
385C23×C4C2×#692311yescharacteristic
386C2×M16—1222no—
387C2×M16—1222no—
388C2×M16—1222no—
389C2×M16—2444no—
390C22×C8—1222no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—612no—
394C4×Q8—612no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1221no—
398C4×Q8—1221no—
399C4×D4—612no—
400C4×D4—612no—
401C4×D4—612no—
402C4×D4—612no—
403C4×D4—1221no—
404C4×D4—1221no—
405C4×D4—1221no—
406C4×D4—1221no—
407C4×D4—1222no—
408C4×D4—1222no—
409C4×D4—1222no—
410C4×D4—1222no—
411C4×D4—1222no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1222no—
415#117—612no—
416#117—612no—
417#117—1222no—
418#117—1222no—
419#117—1222no—
420#117—1222no—
421#117—1221no—
422#117—1221no—
423C2×C4⋊C4—611no—
424C2×C4⋊C4—611no—
425C2×C4⋊C4—1221no—
426C2×C4⋊C4—1221no—
427C2×C22⋊C4—1221no—
428C2×C22⋊C4—1221no—
429C2×C22⋊C4—1221no—
430C2×C42—611no—
431C2×C42—611no—
432C2×C42—611no—
433C2×C42—611no—
434C2×C42—611no—
435C2×C42—611no—
436C2×C42—611no—
437C2×C42—611no—
438C2×C42—611no—
439C2×C42—611no—
440C2×C42—611no—
441C2×C42—611no—
442C2×C42—1222no—
443C2×C42—1222no—
444C2×C42—1222no—
445C2×C42—1222no—
446C2×C42—1221no—
447C2×C42—1221no—
448C2×C42—1221no—
449C4⋊C8—614no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453C4⋊C8—1222no—
454C4⋊C8—1222no—
455C22⋊C8—1222no—
456C22⋊C8—1222no—
457#97—1222no—
458C4×C8—614no—
459C24—21no—
460C24C22⋊#692311yescharacteristic
461#41—28no—
462#41—28no—
463#41—24no—
464#41—24no—
465#41—24no—
466#41#1175821718yescomplemented
467C2×Q8—24no—
468C2×Q8—21no—
469C2×Q8C22×#692312yes—
470C2×D4—28no—
471C2×D4—22no—
472C2×D4—22no—
473C2×D4—24no—
474C2×D4—21no—
475C2×D4C22×#692312yes—
476C22×C4—28no—
477C22×C4—21no—
478C22×C4—22no—
479C22×C4—22no—
480C22×C4—24no—
481C22×C4—611no—
482C22×C4—611no—
483C22×C4—611no—
484C22×C4—611no—
485C22×C4—611no—
486C22×C4—611no—
487C22×C4—611no—
488C22×C4—611no—
489C22×C4—611no—
490C22×C4—611no—
491C22×C4—611no—
492C22×C4—611no—
493C22×C4—1222no—
494C22×C4—1222no—
495C22×C4—1222no—
496C22×C4—1222no—
497C22×C4—1222no—
498C22×C4—1222no—
499C22×C4—1221no—
500C22×C4—1221no—
501C22×C4—2442no—
502C22×C4—2442no—
503C22×C4—2442no—
504C22×C4—2444no—
505C22×C4C22⋊#692311yescharacteristic
506C22×C4C22⋊#692311yescharacteristic
507C22×C4C22⋊#692311yescharacteristic
508C22×C4C22×#692311yescharacteristic
509C22×C4C22×#692311yescharacteristic
510C22×C4C22×#692311yescharacteristic
511M16—2444no—
512M16—2444no—
513M16—2444no—
514C2×C8—1224no—
515C2×C8—1224no—
516C2×C8—2444no—
517C4⋊C4—612no—
518C4⋊C4—612no—
519C4⋊C4—1222no—
520C4⋊C4—1222no—
521C4⋊C4—1222no—
522C4⋊C4—1222no—
523C4⋊C4—1221no—
524C4⋊C4—1221no—
525C22⋊C4—1222no—
526C22⋊C4—1222no—
527C22⋊C4—1222no—
528C22⋊C4—1222no—
529C22⋊C4—1222no—
530C22⋊C4—1222no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—611no—
539C42—611no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—611no—
544C42—611no—
545C42—611no—
546C42—611no—
547C42—1222no—
548C42—1222no—
549C42—1222no—
550C42—1222no—
551C42—1222no—
552C42—1222no—
553C23—24no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—41no—
558C23—42no—
559C23—42no—
560C23C2×C22⋊#692311yescharacteristic
561C23C2×C22⋊#692311yescharacteristic
562C23C2×C22⋊#692311yescharacteristic
563Q8—24no—
564Q8—22no—
565Q8C2×#1175821714yescomplemented
566D4—28no—
567D4—24no—
568D4—28no—
569D4—22no—
570D4C2×#1175821714yescomplemented
571C2×C4—24no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—28no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—612no—
581C2×C4—612no—
582C2×C4—611no—
583C2×C4—611no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—611no—
587C2×C4—611no—
588C2×C4—611no—
589C2×C4—611no—
590C2×C4—1221no—
591C2×C4—1221no—
592C2×C4—1222no—
593C2×C4—1221no—
594C2×C4—1221no—
595C2×C4—1222no—
596C2×C4—1222no—
597C2×C4—1222no—
598C2×C4—1222no—
599C2×C4—1222no—
600C2×C4—1222no—
601C2×C4—1222no—
602C2×C4—2442no—
603C2×C4—2442no—
604C2×C4—2444no—
605C2×C4—2444no—
606C2×C4—2441no—
607C2×C4—2441no—
608C2×C4—2441no—
609C2×C4—2444no—
610C2×C4—2442no—
611C2×C4—2442no—
612C2×C4C2×C22⋊#692311yescharacteristic
613C2×C4C2×C22⋊#692311yescharacteristic
614C2×C4C2×C22⋊#692311yescharacteristic
615C2×C4C2×#1175821714yes—
616C2×C4C23×#692311yescharacteristic
617C8—2444no—
618C8—2444no—
619C22—21no—
620C22—21no—
621C22—21no—
622C22—21no—
623C22—21no—
624C22—24no—
625C22—21no—
626C22—21no—
627C22—44no—
628C22—48no—
629C22—42no—
630C22—42no—
631C22#1684073212yes—
632C22#1684075312yes—
633C22C22×C22⋊#692311yescharacteristic
634C4—21no—
635C4—24no—
636C4—21no—
637C4—611no—
638C4—611no—
639C4—612no—
640C4—612no—
641C4—1221no—
642C4—1221no—
643C4—1221no—
644C4—1221no—
645C4—1221no—
646C4—2442no—
647C4—2442no—
648C4#1684073212yes—
649C4C22×#1175821712yes—
650C2—22no—
651C2—22no—
652C2—41no—
653C2—44no—
654C2C2×#1684073211yescharacteristic
655C2C2×#1684075311yescharacteristic
656C2D61⋊#42111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
Fitting, prime radicalC22×#41×C61C4
socleC22×C61C22×C22⋊C4
cosocleC4×Dic61C24
hypercenterC22×#41#6923
hypocenterC61#17451
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#41

Complements

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

NH ≅ G/Ntype
1. #21462495C2semidirect
2. C2×#16840765C2semidirect
3. #21461154C2semidirect
4. #21461147C2semidirect
5. #21461144C2semidirect
6. #21461114C2semidirect
12. #16840778C22semidirect
17. #16840765C22semidirect
19. C2×C4⋊#92420C22semidirect
21. C4×#11758217C22semidirect
22. Q8⋊#92420C22semidirect
23. D4⋊#92420C22semidirect
137. #41×C61M16semidirect
236. Q8×C61C2×M16semidirect
241. D4×C61C2×M16semidirect
371. C61#17451semidirect
466. #41#11758217semidirect
565. Q8C2×#11758217semidirect
570. D4C2×#11758217semidirect

Maximal subgroups

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

subgroupindex
#214624952
C2×#168407652
#214611542
#214611472
#214611442
#214611142
C2×#41×Dic612
#1745161