Finite Group Databasebeta

← #30616461

Subgroups — #30616461

Order 15616.

1250 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
1D61⋊#423C211yesmaximal, characteristic
2C2×C4×C4⋊#6923C211yesmaximal, characteristic
3#21461999C214yesmaximal, complemented
4#21461998C212yesmaximal, complemented
5#21461997C212yesmaximal, complemented
6#21461508C214yesmaximal
7#533×D61C211yesmaximal, characteristic
8C2×D61⋊M16C2212yes—
9C22×C4⋊#6923C2212yes—
10C22×C4×#6923—21no—
11#16840767—21no—
12#16840766—21no—
13#16840765—21no—
14#16840765—21no—
15D61⋊(C4⋊C8)C2212yes—
16D61⋊(C4⋊C8)C2212yes—
17C4×C4⋊#6923—21no—
18C4×C4⋊#6923—21no—
19C4×C4⋊#6923C2212yes—
20C4×C4⋊#6923C2212yes—
21C2×(C2×#6923)⋊C4—21no—
22#16840635C2218yes—
23#16840635C2218yes—
24#16840319C411yescharacteristic, complemented
25#16840318C412yescomplemented
26#16840317C411yescharacteristic, complemented
27#16840317C411yescharacteristic, complemented
28#16840316—21no—
29#16840316—21no—
30#16840315C412yescomplemented
31#16840314—21no—
32#16840314—21no—
33C4⋊Q8×D61C2211yescharacteristic
34C4⋊Q8×D61C2211yescharacteristic
35#125×D61C2212yes—
36C22⋊Q8×D61—21no—
37C22⋊Q8×D61—21no—
38C4×Q8×D61—21no—
39C4×Q8×D61—21no—
40#117×D61C2212yes—
41C2×C42×D61C2211yescharacteristic
42C61×#533C411yescharacteristic, complemented
43C23×#6923—22no—
44C2×#11758213—22no—
45D61⋊M16—28no—
46C2×#11758217—22no—
47C2×D61⋊C8C2314yes—
48C2×C4⋊#6923—22no—
49C2×C4⋊#6923—24no—
50C2×C4⋊#6923Q818yescomplemented
51C2×C4⋊#6923C2314yes—
52C2×C4×#6923—22no—
53C2×C4×#6923—22no—
54C2×C4×#6923—44no—
55#14519844—22no—
56#14519844—22no—
57C4⋊#92420—22no—
58C4⋊#92420—22no—
59(C2×#6923)⋊C4—22no—
60(C2×#6923)⋊C4—22no—
61C2×Q8×D61—21no—
62C2×Q8×D61—21no—
63C22×C4×D61D411yescharacteristic
64C22×C4×D61D411yescharacteristic
65C22×C4×D61C2311yescharacteristic
66#14519777C2×C411yescharacteristic, complemented
67#14519777C2×C411yescharacteristic, complemented
68#14519776—21no—
69#14519776—21no—
70#14519776C2×C412yescomplemented
71#14519776C2×C412yescomplemented
72#14519775—21no—
73#14519775—21no—
74#14519769C2×C412yescomplemented
75#14519768C2×C412yescomplemented
76#14519768C2×C412yescomplemented
77#14519767—22no—
78#14519767—22no—
79#14519757—21no—
80#14519757—21no—
81#14519756—21no—
82#14519756—21no—
83#14519755—22no—
84#14519755—22no—
85#14519754—22no—
86#14519754—22no—
87#14519753—21no—
88#14519753—21no—
89Q8×Dic61—21no—
90Q8×Dic61—21no—
91C4×Dic122—21no—
92C4×Dic122—21no—
93C4×Dic122—22no—
94C4×Dic122—22no—
95#14519742—21no—
96#14519742—21no—
97#14519742—22no—
98#14519742—22no—
99#14519736C2×C412yes—
100#14519735C2×C412yes—
101#14519735C2×C412yes—
102#14519734—22no—
103#14519733—22no—
104C4⋊C4×D61—22no—
105C4⋊C4×D61—22no—
106C4⋊C4×D61—21no—
107C4⋊C4×D61—21no—
108C4⋊C4×D61C2312yes—
109C4⋊C4×D61C2312yes—
110C22⋊C4×D61—22no—
111C2×C4×Dic61C2×C411yescharacteristic
112C2×C4×Dic61D411yescharacteristic
113C2×C4×Dic61D411yescharacteristic
114C42×D61—22no—
115C42×D61C2×C411yescharacteristic
116C42×D61C2×C411yescharacteristic
117C42×D61D411yescharacteristic
118C42×D61D411yescharacteristic
119C42×D61D411yescharacteristic
120C42×D61D411yescharacteristic
121C42×D61C2311yescharacteristic
122C42×D61C2311yescharacteristic
123C4⋊Q8×C61C2×C411yescharacteristic, complemented
124C4⋊Q8×C61C2×C411yescharacteristic, complemented
125#125×C61C2×C412yescomplemented
126C22⋊Q8×C61—21no—
127C22⋊Q8×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130#117×C61C2×C412yes—
131C2×C42×C61C2×C411yescharacteristic
132C23×D61C2×D411yescharacteristic
133C22×#6923—28no—
134C22×#6923—24no—
135C22×#6923—44no—
136C22×#6923—44no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C22×C4×C61C22⋊C411yescharacteristic
140C22×C4×C61C22⋊C411yescharacteristic
141C22×C4×C61C22×C411yescharacteristic
142C22×Dic61C22⋊C411yescharacteristic
143C22×Dic61C22⋊C411yescharacteristic
144C22×Dic61C2×D411yescharacteristic
145Q8×D61—44no—
146Q8×D61—42no—
147Q8×D61—42no—
148C2×C4×D61—22no—
149C2×C4×D61—22no—
150C2×C4×D61—22no—
151C2×C4×D61—22no—
152C2×C4×D61—42no—
153C2×C4×D61C22⋊C411yescharacteristic
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22×C411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×C4×D61C2×D411yescharacteristic
160C2×C4×D61C2×D411yescharacteristic
161C2×C4×D61C2×D411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×Q814yes—
165C2×C4×D61#4114yes—
166C2×C4×D61C2411yescharacteristic
167C2×Dic122—21no—
168C2×Dic122—21no—
169C2×Dic122—22no—
170C2×Dic122—22no—
171#11758217—44no—
172C2×#92420—24no—
173C4⋊#6923—28no—
174C4⋊#6923—24no—
175C4⋊#6923C2×Q818yes—
176C4×#6923—44no—
177C4×#6923—44no—
178#11758213—44no—
179D61⋊C8—28no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61—21no—
183C4⋊C4×C61—21no—
184C4⋊C4×C61C22×C412yes—
185C4⋊C4×C61C22×C412yes—
186C22⋊C4×C61—22no—
187C42×C61—22no—
188C42×C61C22⋊C411yescharacteristic
189C42×C61C22⋊C411yescharacteristic
190C42×C61C22×C411yescharacteristic
191C42×C61C22×C411yescharacteristic
192C22⋊Dic61—22no—
193#11758198—22no—
194#11758198—22no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22×C412yes—
200C4⋊Dic61C22×C412yes—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61—22no—
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C22⋊C411yescharacteristic
215C4×Dic61C22⋊C411yescharacteristic
216C4×Dic61C22⋊C411yescharacteristic
217C4×Dic61C22⋊C411yescharacteristic
218C4×Dic61C22×C411yescharacteristic
219C4×Dic61C22×C411yescharacteristic
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D411yescharacteristic
222C4×Dic61C2×D411yescharacteristic
223C4×Dic61C2×D411yescharacteristic
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61—21no—
226C22×D61—21no—
227C22×D61—22no—
228C22×D61C2×C22⋊C411yescharacteristic
229C22×D61C22⋊Q814yes—
230C22×D61C22×D411yescharacteristic
231C2×#6923—28no—
232C2×#6923—416no—
233C2×#6923—44no—
234C2×#6923—44no—
235Q8×C61—42no—
236Q8×C61—42no—
237C2×C4×C61—22no—
238C2×C4×C61—22no—
239C2×C4×C61—22no—
240C2×C4×C61—41no—
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×C22⋊C411yescharacteristic
244C2×C4×C61C2×C22⋊C411yescharacteristic
245C2×C4×C61C2×C22⋊C411yescharacteristic
246C2×C4×C61C4×Q814yes—
247C2×C4×C61C23×C411yescharacteristic
248C2×Dic61—22no—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61—41no—
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C2×Dic61C2×C22⋊C411yescharacteristic
255C2×Dic61C2×C22⋊C411yescharacteristic
256C2×Dic61C2×C22⋊C411yescharacteristic
257C2×Dic61#12412yes—
258C2×Dic61C4⋊Q812yes—
259C2×Dic61C22×D411yescharacteristic
260C4×D61—21no—
261C4×D61—21no—
262C4×D61—22no—
263C4×D61—22no—
264C4×D61—22no—
265C4×D61—22no—
266C4×D61—41no—
267C4×D61—41no—
268C4×D61—42no—
269C4×D61—42no—
270C4×D61—44no—
271C4×D61C4×Q814yes—
272C4×D61C22⋊Q814yes—
273C4×D61#12412yes—
274C4×D61C4⋊Q812yes—
275C4×D61AGL(1,ℤ/8)12yes—
276C4×D61#13712yes—
277C4×D61C22×Q812yes—
278C4×D61C2×#4112yes—
279Dic122—44no—
280Dic122—42no—
281Dic122—42no—
282Dic122—44no—
283#92420—44no—
284#21312—611nomaximal, Sylow
285C22×C61—21no—
286C22×C61—21no—
287C22×C61#39112yes—
288C22×C61C22×C22⋊C411yescharacteristic
289C2×D61—21no—
290C2×D61—21no—
291C2×D61—24no—
292C2×D61—24no—
293C2×D61#39112yes—
294C2×D61C2×C22⋊Q812yes—
295C2×D61C2×AGL(1,ℤ/8)11yescharacteristic
296C2×D61C2×#13711yescharacteristic
297C2×D61#57512yes—
298#6923—48no—
299C4×C61—21no—
300C4×C61—21no—
301C4×C61—21no—
302C4×C61—42no—
303C4×C61—42no—
304C4×C61#39112yes—
305C4×C61#41712yes—
306C4×C61C2×C4×Q812yes—
307Dic61—21no—
308Dic61—21no—
309Dic61—21no—
310Dic61—42no—
311Dic61—42no—
312Dic61#39112yes—
313Dic61#41712yes—
314Dic61C2×#12411yescharacteristic
315Dic61C2×C4⋊Q811yescharacteristic
316C2×#533—611no—
317C2×#423—611no—
318C2×C4×C4⋊C4—611no—
319#1989—614no—
320#1988—612no—
321#1987—612no—
322#1736—614no—
323C2×C61—22no—
324C2×C61C2×#39111yescharacteristic
325C2×C61C2×#41711yescharacteristic
326C2×C61#289611yescharacteristic
327D61—22no—
328D61#307712yes—
329D61#307812yes—
330C22×M16—612no—
331#533—611no—
332#533—611no—
333#533—612no—
334#533—612no—
335#533—611no—
336#533—1221no—
337#533—1221no—
338#533—1221no—
339#533—1221no—
340#533#692311yescharacteristic, complemented
341C2×C4⋊Q8—611no—
342C2×C4⋊Q8—611no—
343C2×#125—612no—
344C2×C22⋊Q8—1221no—
345C2×C22⋊Q8—1221no—
346C2×C4×Q8—1221no—
347C2×C4×Q8—1221no—
348C2×#117—612no—
349C22×C4⋊C4—612no—
350C22×C42—611no—
351C22×C42—1221no—
352#423—1221no—
353#423—1221no—
354#423—1221no—
355#423—1221no—
356C2×C4⋊C8—612no—
357C2×C4⋊C8—612no—
358C4×C4⋊C4—612no—
359C4×C4⋊C4—612no—
360C4×C4⋊C4—1221no—
361C4×C4⋊C4—1221no—
362C2×#95—1221no—
363#340—618no—
364#340—618no—
365C61#2131211yescharacteristic, complemented, Sylow
366C22×Q8—1221no—
367C22×Q8—1221no—
368C23×C4—611no—
369C23×C4—611no—
370C23×C4—611no—
371C23×C4—1222no—
372C2×M16—1222no—
373C2×M16—1228no—
374C2×M16—1222no—
375C22×C8—614no—
376C4⋊Q8—612no—
377C4⋊Q8—612no—
378C4⋊Q8—611no—
379C4⋊Q8—611no—
380C4⋊Q8—1221no—
381C4⋊Q8—1221no—
382C4⋊Q8—1221no—
383C4⋊Q8—1221no—
384C4⋊Q8C2×#692311yescharacteristic, complemented
385C4⋊Q8C2×#692311yescharacteristic, complemented
386#125—612no—
387#125—612no—
388#125—612no—
389#125—1222no—
390#125—1222no—
391#125C2×#692312yescomplemented
392C22⋊Q8—21no—
393C22⋊Q8—21no—
394C22⋊Q8—1222no—
395C22⋊Q8—1222no—
396C22⋊Q8—1222no—
397C22⋊Q8—1222no—
398C22⋊Q8—1221no—
399C22⋊Q8—1221no—
400C22⋊Q8—1221no—
401C22⋊Q8—1221no—
402C22⋊Q8—1221no—
403C22⋊Q8—1221no—
404C4×Q8—21no—
405C4×Q8—21no—
406C4×Q8—1222no—
407C4×Q8—1222no—
408C4×Q8—1222no—
409C4×Q8—1222no—
410C4×Q8—1221no—
411C4×Q8—1221no—
412C4×Q8—1221no—
413C4×Q8—1221no—
414C4×Q8—1221no—
415C4×Q8—1221no—
416#117—612no—
417#117—612no—
418#117—612no—
419#117—1222no—
420#117—1222no—
421#117C2×#692312yes—
422C2×C4⋊C4—618no—
423C2×C4⋊C4—614no—
424C2×C4⋊C4—612no—
425C2×C4⋊C4—612no—
426C2×C4⋊C4—1222no—
427C2×C4⋊C4—1222no—
428C2×C4⋊C4—1221no—
429C2×C4⋊C4—1221no—
430C2×C4⋊C4—1224no—
431C2×C4⋊C4—1222no—
432C2×C22⋊C4—1222no—
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—611no—
443C2×C42—611no—
444C2×C42—1222no—
445C2×C42—1222no—
446C2×C42—1222no—
447C2×C42—2444no—
448C2×C42C2×#692311yescharacteristic
449C4⋊C8—1222no—
450C4⋊C8—1222no—
451C4⋊C8—1222no—
452C4⋊C8—1222no—
453#95—1222no—
454#95—1222no—
455C24—611no—
456C2×Q8—21no—
457C2×Q8—21no—
458C2×Q8—1222no—
459C2×Q8—1222no—
460C2×Q8—1221no—
461C2×Q8—1221no—
462C2×Q8—2442no—
463C2×Q8—2442no—
464C2×Q8—2444no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—614no—
470C22×C4—614no—
471C22×C4—611no—
472C22×C4—611no—
473C22×C4—611no—
474C22×C4—611no—
475C22×C4—611no—
476C22×C4—611no—
477C22×C4—611no—
478C22×C4—611no—
479C22×C4—611no—
480C22×C4—611no—
481C22×C4—611no—
482C22×C4—1222no—
483C22×C4—1222no—
484C22×C4—1222no—
485C22×C4—1222no—
486C22×C4—1224no—
487C22×C4—1228no—
488C22×C4—2442no—
489C22×C4—2444no—
490C22×C4—2444no—
491C22×C4C22⋊#692311yescharacteristic
492C22×C4C22⋊#692311yescharacteristic
493C22×C4C22×#692311yescharacteristic
494M16—2444no—
495M16—2444no—
496C2×C8—1224no—
497C2×C8—1228no—
498C4⋊C4—22no—
499C4⋊C4—22no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—618no—
503C4⋊C4—612no—
504C4⋊C4—612no—
505C4⋊C4—1222no—
506C4⋊C4—1222no—
507C4⋊C4—1222no—
508C4⋊C4—1222no—
509C4⋊C4—1222no—
510C4⋊C4—1222no—
511C4⋊C4—1222no—
512C4⋊C4—1222no—
513C4⋊C4—1228no—
514C4⋊C4—1224no—
515C4⋊C4—1222no—
516C4⋊C4—1222no—
517C4⋊C4—1221no—
518C4⋊C4—1221no—
519C4⋊C4C22×#692312yes—
520C4⋊C4C22×#692312yes—
521C22⋊C4—22no—
522C22⋊C4—1222no—
523C22⋊C4—1222no—
524C22⋊C4—1222no—
525C42—22no—
526C42—611no—
527C42—611no—
528C42—611no—
529C42—611no—
530C42—611no—
531C42—611no—
532C42—611no—
533C42—611no—
534C42—611no—
535C42—611no—
536C42—611no—
537C42—611no—
538C42—1222no—
539C42—1222no—
540C42—1222no—
541C42—2444no—
542C42—2444no—
543C42C22⋊#692311yescharacteristic
544C42C22⋊#692311yescharacteristic
545C42C22×#692311yescharacteristic
546C42C22×#692311yescharacteristic
547C23—611no—
548C23—614no—
549C23—611no—
550C23—1221no—
551C23—1221no—
552C23—1222no—
553C23C2×C22⋊#692311yescharacteristic
554Q8—42no—
555Q8—42no—
556Q8—2442no—
557Q8—2444no—
558Q8—2444no—
559Q8—2442no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—41no—
564C2×C4—612no—
565C2×C4—612no—
566C2×C4—611no—
567C2×C4—611no—
568C2×C4—611no—
569C2×C4—614no—
570C2×C4—612no—
571C2×C4—612no—
572C2×C4—612no—
573C2×C4—612no—
574C2×C4—614no—
575C2×C4—612no—
576C2×C4—612no—
577C2×C4—611no—
578C2×C4—611no—
579C2×C4—611no—
580C2×C4—1222no—
581C2×C4—1221no—
582C2×C4—1221no—
583C2×C4—1222no—
584C2×C4—1222no—
585C2×C4—1222no—
586C2×C4—1222no—
587C2×C4—1228no—
588C2×C4—1222no—
589C2×C4—1222no—
590C2×C4—2441no—
591C2×C4—2441no—
592C2×C4—2442no—
593C2×C4—2442no—
594C2×C4—2444no—
595C2×C4—2441no—
596C2×C4—2444no—
597C2×C4—2444no—
598C2×C4—24416no—
599C2×C4C2×C22⋊#692311yescharacteristic
600C2×C4C2×C22⋊#692311yescharacteristic
601C2×C4C2×C22⋊#692311yescharacteristic
602C2×C4C2×C22⋊#692311yescharacteristic
603C2×C4C2×C22⋊#692311yescharacteristic
604C2×C4Q8×#692314yes—
605C2×C4C23×#692311yescharacteristic
606C8—2444no—
607C22—21no—
608C22—21no—
609C22—612no—
610C22—611no—
611C22—611no—
612C22—612no—
613C22—612no—
614C22—1224no—
615C22—1224no—
616C22—1221no—
617C22—1221no—
618C22(C2×#6923)⋊Q812yes—
619C22C22×C22⋊#692311yescharacteristic
620C4—21no—
621C4—21no—
622C4—21no—
623C4—42no—
624C4—42no—
625C4—611no—
626C4—611no—
627C4—612no—
628C4—612no—
629C4—1221no—
630C4—1221no—
631C4—1221no—
632C4—2448no—
633C4—2442no—
634C4—2442no—
635C4(C2×#6923)⋊Q812yes—
636C4#1684075312yes—
637C4C2×Q8×#692312yes—
638C2—22no—
639C2—612no—
640C2—612no—
641C2—1222no—
642C2C2×(C2×#6923)⋊Q811yescharacteristic
643C2C2×#1684075311yescharacteristic
644C2#2146243911yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingC61×#533C4
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC2×C4×D61C24
hypercenter#533#6923
hypocenterC61#21312
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
3. #21461999C2semidirect
4. #21461998C2semidirect
5. #21461997C2semidirect
24. #16840319C4semidirect
25. #16840318C4semidirect
26. #16840317C4semidirect
27. #16840317C4semidirect
30. #16840315C4semidirect
42. C61×#533C4semidirect
50. C2×C4⋊#6923Q8semidirect
66. #14519777C2×C4semidirect
67. #14519777C2×C4semidirect
70. #14519776C2×C4semidirect
71. #14519776C2×C4semidirect
74. #14519769C2×C4semidirect
75. #14519768C2×C4semidirect
76. #14519768C2×C4semidirect
123. C4⋊Q8×C61C2×C4semidirect
124. C4⋊Q8×C61C2×C4semidirect
125. #125×C61C2×C4semidirect
340. #533#6923semidirect
365. C61#21312semidirect
384. C4⋊Q8C2×#6923semidirect
385. C4⋊Q8C2×#6923semidirect
391. #125C2×#6923semidirect

Maximal subgroups

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

subgroupindex
D61⋊#4232
C2×C4×C4⋊#69232
#214619992
#214619982
#214619972
#214615082
#533×D612
#2131261