Finite Group Databasebeta

← #29646145

Subgroups — #29646145

Order 15616.

960 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840318C211yesmaximal, characteristic
2#21454447C211yesmaximal, characteristic
3#432×D61C211yesmaximal, characteristic
4#21444181C214yesmaximal
5#21444176C211yesmaximal, characteristic, complemented
6#21444175C211yesmaximal, characteristic, complemented
7#21444174C211yesmaximal, characteristic, complemented
8#21444161C214yesmaximal, complemented
9C61×#1470C211yesmaximal, characteristic, complemented
10#16840318—21no—
11#16840318—21no—
12#16840318—21no—
13#16840318—21no—
14#16840318—21no—
15#16840318—21no—
16#16840318—21no—
17#16840318—21no—
18C2×#14519776C411yescharacteristic
19C2×#14519776C411yescharacteristic
20C2×#14519769C2211yescharacteristic
21C2×#14519755C412yes—
22C2×#14519755C2212yes—
23C2×#14519742C412yes—
24C2×#14519742C2212yes—
25C2×#14519735C411yescharacteristic
26C2×#14519735C411yescharacteristic
27C2×C42×D61C2211yescharacteristic
28#16839819C2211yescharacteristic
29#16839818—21no—
30#16839818—21no—
31#16839817—21no—
32#16839816C2212yes—
33#16839816C2212yes—
34#16839815C2211yescharacteristic
35#16839813—21no—
36#16839784—21no—
37C4⋊C8×D61—21no—
38#16839724C2212yes—
39C22⋊C8×D61C2212yes—
40C4×#11758189—21no—
41#97×D61—21no—
42#16839488C2214yes—
43#16839487C2214yes—
44#16839487C2214yes—
45C2×#125×C61C2211yescharacteristic
46C61×#432C2211yescharacteristic
47C61×#431C2211yescharacteristic
48C61×#324C2214yes—
49C22×Dic122C2×C412yes—
50C22×Dic122C2×C412yes—
51C22×C4×D61C2×C411yescharacteristic
52C22×C4×D61C2×C411yescharacteristic
53C22×C4×D61C2311yescharacteristic
54C2×#11758189—22no—
55C2×C8×D61—22no—
56#14519776—21no—
57#14519776—21no—
58#14519776—21no—
59#14519776—21no—
60#14519776—21no—
61#14519776—21no—
62#14519776—21no—
63#14519776—21no—
64#14519769—21no—
65#14519769—21no—
66#14519769—21no—
67#14519769—21no—
68#14519755—22no—
69#14519755—22no—
70#14519755—22no—
71#14519755—22no—
72#14519755—22no—
73#14519755—22no—
74#14519755—22no—
75#14519755—22no—
76#14519742—22no—
77#14519742—22no—
78#14519742—22no—
79#14519742—22no—
80#14519742—22no—
81#14519742—22no—
82#14519742—22no—
83#14519742—22no—
84#14519735—21no—
85#14519735—21no—
86#14519735—21no—
87#14519735—21no—
88#14519735—21no—
89#14519735—21no—
90#14519735—21no—
91#14519735—21no—
92C2×C4⋊Dic61C2×C411yescharacteristic
93C2×C4⋊Dic61C2×C411yescharacteristic
94C2×#11758196C2×C412yes—
95C2×#11758196C2×C412yes—
96C2×#11758196C2×C412yes—
97C2×#11758196C2312yes—
98C2×#11758198C2×C412yes—
99C2×#11758198C2×C412yes—
100C2×C4×Dic61C2×C412yes—
101C2×C4×Dic61C2×C412yes—
102C2×C4×Dic61C2×C411yescharacteristic
103C2×C4×Dic61C2×C411yescharacteristic
104C2×C4×Dic61C2311yescharacteristic
105C42×D61D411yescharacteristic
106C42×D61D411yescharacteristic
107C42×D61D411yescharacteristic
108C42×D61D411yescharacteristic
109C42×D61D411yescharacteristic
110C42×D61D411yescharacteristic
111C42×D61D411yescharacteristic
112C42×D61D411yescharacteristic
113C4⋊#92418—21no—
114C4⋊#92418—21no—
115#14519706—22no—
116#14519706—22no—
117C22⋊#92418C2312yes—
118C22⋊#92418C2312yes—
119#14519689—22no—
120#14519689—22no—
121#14519688—21no—
122#14519687—22no—
123#14519687—22no—
124C4×#92418—21no—
125#125×C61—21no—
126#125×C61—21no—
127#125×C61—21no—
128#125×C61—21no—
129C2×C4⋊C4×C61C2×C411yescharacteristic
130C2×C4⋊C4×C61C2×C411yescharacteristic
131C2×C4⋊C4×C61C2×C412yes—
132C2×C4⋊C4×C61C2312yes—
133C2×C42×C61C2311yescharacteristic
134C4⋊C8×C61—21no—
135C4⋊C8×C61—21no—
136C22⋊C8×C61C2312yes—
137C22⋊C8×C61C2312yes—
138#97×C61—21no—
139C4×C8×C61—21no—
140C23×D61C22×C411yescharacteristic
141C22×C4×C61C22×C412yes—
142C22×C4×C61C22×C411yescharacteristic
143C22×C4×C61C22×C411yescharacteristic
144C22×C4×C61C22×C412yes—
145C22×C4×C61C2411yescharacteristic
146C22×Dic61C22×C412yes—
147C22×Dic61C22×C411yescharacteristic
148C22×Dic61C22×C411yescharacteristic
149C22×Dic61C22×C411yescharacteristic
150C22×Dic61C22×C412yes—
151C2×C4×D61—24no—
152C2×C4×D61—42no—
153C2×C4×D61—42no—
154C2×C4×D61C22⋊C411yescharacteristic
155C2×C4×D61C22⋊C411yescharacteristic
156C2×C4×D61C22⋊C411yescharacteristic
157C2×C4×D61C22⋊C411yescharacteristic
158C2×C4×D61C22⋊C411yescharacteristic
159C2×C4×D61C22⋊C411yescharacteristic
160C2×C4×D61C22⋊C411yescharacteristic
161C2×C4×D61C22⋊C411yescharacteristic
162C2×C4×D61C2×D411yescharacteristic
163C2×C4×D61C2×D411yescharacteristic
164C2×C4×D61C2×D411yescharacteristic
165C2×C4×D61C2×D411yescharacteristic
166C2×Dic122—22no—
167C2×Dic122—22no—
168C2×Dic122—22no—
169C2×Dic122—22no—
170C2×Dic122—44no—
171C2×Dic122—44no—
172C2×C8×C61—22no—
173C2×C8×C61—22no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—22no—
179C4⋊C4×C61—22no—
180C4⋊C4×C61—22no—
181C4⋊C4×C61—22no—
182C42×C61C2×D411yescharacteristic
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186#11758198—22no—
187#11758198—22no—
188#11758198—22no—
189#11758198—22no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758196—22no—
195#11758196—22no—
196#11758196—22no—
197#11758196—22no—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202C4×Dic61—22no—
203C4×Dic61—22no—
204C4×Dic61—22no—
205C4×Dic61—22no—
206C4×Dic61C22⋊C411yescharacteristic
207C4×Dic61C22⋊C411yescharacteristic
208C4×Dic61C22⋊C411yescharacteristic
209C4×Dic61C22⋊C411yescharacteristic
210C4×Dic61C22⋊C411yescharacteristic
211C4×Dic61C22⋊C411yescharacteristic
212C4×Dic61C22⋊C411yescharacteristic
213C4×Dic61C22⋊C411yescharacteristic
214C4×Dic61C2×D411yescharacteristic
215C4×Dic61C2×D411yescharacteristic
216C4×Dic61C2×D411yescharacteristic
217C4×Dic61C2×D411yescharacteristic
218C2×#92418—22no—
219C2×#92418—22no—
220#11758189—44no—
221C8×D61—44no—
222C23×C61C23×C411yescharacteristic
223C22×D61—22no—
224C22×D61—22no—
225C22×D61—41no—
226C22×D61C2×C22⋊C411yescharacteristic
227C22×D61C2×C22⋊C411yescharacteristic
228C2×C4×C61—22no—
229C2×C4×C61—22no—
230C2×C4×C61—22no—
231C2×C4×C61—22no—
232C2×C4×C61—24no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61C22×D411yescharacteristic
241C2×Dic61—22no—
242C2×Dic61—22no—
243C2×Dic61—22no—
244C2×Dic61—22no—
245C2×Dic61—22no—
246C2×Dic61—24no—
247C2×Dic61—42no—
248C2×Dic61C2×C22⋊C411yescharacteristic
249C2×Dic61C2×C22⋊C411yescharacteristic
250C2×Dic61C2×C22⋊C411yescharacteristic
251C2×Dic61C2×C22⋊C411yescharacteristic
252C2×Dic61C2×C22⋊C411yescharacteristic
253C2×Dic61C2×C22⋊C411yescharacteristic
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—22no—
258C4×D61—22no—
259C4×D61—41no—
260C4×D61—41no—
261C4×D61—41no—
262C4×D61—41no—
263C4×D61—41no—
264C4×D61—41no—
265C4×D61#10114yes—
266Dic122—44no—
267Dic122—44no—
268Dic122—44no—
269Dic122—44no—
270C8×C61—42no—
271C8×C61—42no—
272#92418—42no—
273#92418—42no—
274#17404—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C22×C22⋊C411yescharacteristic
278C22×C61#56812yes—
279C2×D61—21no—
280C2×D61—21no—
281C2×D61—21no—
282C2×D61—21no—
283C2×D61—21no—
284C2×D61—44no—
285C2×D61C2×#10112yes—
286C4×C61—21no—
287C4×C61—21no—
288C4×C61—21no—
289C4×C61—21no—
290C4×C61—21no—
291C4×C61—42no—
292C4×C61—42no—
293C4×C61C2×#10112yes—
294Dic61—21no—
295Dic61—21no—
296Dic61—21no—
297Dic61—21no—
298Dic61—42no—
299Dic61—42no—
300Dic61#41012yes—
301Dic61C2×#10112yes—
302C2×#533—611no—
303#2920—611no—
304C2×#432—611no—
305#1472—614no—
306#1470—611no—
307#1470—611no—
308#1470—611no—
309#1470D6111yescharacteristic, complemented
310#1467—614no—
311C2×C61—22no—
312C2×C61#287911yescharacteristic
313C2×C61C2×#41011yescharacteristic
314C2×C61C22×#10111yescharacteristic
315D61—22no—
316D61—41no—
317#533—1221no—
318#533—1221no—
319#533—1221no—
320#533—1221no—
321#533—1221no—
322#533—1221no—
323#533—1221no—
324#533—1221no—
325C2×C4⋊Q8—611no—
326C2×C4⋊Q8—611no—
327C2×#125—611no—
328C2×#125C2×D6111yescharacteristic
329C2×C22⋊Q8—612no—
330C2×C22⋊Q8—612no—
331C2×C4×Q8—612no—
332C2×C4×Q8—612no—
333C2×#117—611no—
334C2×#117—611no—
335C22×C42—611no—
336#432—611no—
337#432—612no—
338#432—612no—
339#432—1221no—
340#432—1221no—
341#432—1221no—
342#432C2×D6111yescharacteristic
343#431—611no—
344#431—1221no—
345#431C2×D6111yescharacteristic
346#423—1221no—
347C2×C4⋊C8—1221no—
348#407—612no—
349C2×C22⋊C8—612no—
350C4×M16—1221no—
351C2×#97—1221no—
352#324—614no—
353#324—614no—
354#324—614no—
355#324C2×D6114yes—
356C61#1740411yescharacteristic, complemented, Sylow
357C22×Q8—612no—
358C22×Q8—612no—
359C23×C4—611no—
360C23×C4—611no—
361C23×C4—611no—
362C2×M16—1222no—
363C22×C8—1222no—
364C4⋊Q8—1221no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—1221no—
368C4⋊Q8—1221no—
369C4⋊Q8—1221no—
370C4⋊Q8—1221no—
371C4⋊Q8—1221no—
372#125—21no—
373#125—21no—
374#125—21no—
375#125—21no—
376#125—1221no—
377#125—1221no—
378#125—1221no—
379#125—1221no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C4×Q8—1222no—
389C4×Q8—1222no—
390C4×Q8—1222no—
391C4×Q8—1222no—
392C4×Q8—1222no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1222no—
396#117—1221no—
397#117—1221no—
398#117—1221no—
399#117—1221no—
400#117—1221no—
401#117—1221no—
402#117—1221no—
403#117—1221no—
404C2×C4⋊C4—611no—
405C2×C4⋊C4—611no—
406C2×C4⋊C4—612no—
407C2×C4⋊C4—612no—
408C2×C4⋊C4—612no—
409C2×C4⋊C4—612no—
410C2×C4⋊C4C4×D6111yescharacteristic
411C2×C4⋊C4C4×D6111yescharacteristic
412C2×C4⋊C4C4×D6112yes—
413C2×C4⋊C4C22×D6112yes—
414C2×C22⋊C4—612no—
415C2×C22⋊C4—612no—
416C2×C42—611no—
417C2×C42—611no—
418C2×C42—611no—
419C2×C42—611no—
420C2×C42—611no—
421C2×C42—611no—
422C2×C42—611no—
423C2×C42—611no—
424C2×C42—611no—
425C2×C42—611no—
426C2×C42—611no—
427C2×C42—612no—
428C2×C42—612no—
429C2×C42C22×D6111yescharacteristic
430C4⋊C8—21no—
431C4⋊C8—21no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C22⋊C8—612no—
437C22⋊C8—612no—
438C22⋊C8—1222no—
439C22⋊C8—1222no—
440C22⋊C8C22×D6112yes—
441C22⋊C8C22×D6112yes—
442#97—21no—
443#97—1222no—
444#97—1222no—
445#97—1221no—
446C4×C8—21no—
447C4×C8—1221no—
448C24—611no—
449C2×Q8—1222no—
450C2×Q8—1222no—
451C2×Q8—1222no—
452C2×Q8—1222no—
453C2×Q8—2444no—
454C2×Q8—2444no—
455C22×C4—611no—
456C22×C4—611no—
457C22×C4—611no—
458C22×C4—611no—
459C22×C4—611no—
460C22×C4—611no—
461C22×C4—611no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—611no—
465C22×C4—611no—
466C22×C4—611no—
467C22×C4—611no—
468C22×C4—611no—
469C22×C4—611no—
470C22×C4—612no—
471C22×C4—612no—
472C22×C4—1224no—
473C22×C4—2442no—
474C22×C4—2442no—
475C22×C4C2×C4×D6112yes—
476C22×C4C2×C4×D6111yescharacteristic
477C22×C4C2×C4×D6111yescharacteristic
478C22×C4C2×C4×D6112yes—
479C22×C4C23×D6111yescharacteristic
480M16—2444no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—1222no—
484C2×C8—1222no—
485C2×C8—2444no—
486C4⋊C4—22no—
487C4⋊C4—22no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1222no—
498C4⋊C4—1222no—
499C4⋊C4—1222no—
500C4⋊C4—1222no—
501C4⋊C4—1222no—
502C4⋊C4—1221no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C22⋊C4—1222no—
509C22⋊C4—1222no—
510C42—611no—
511C42—611no—
512C42—611no—
513C42—611no—
514C42—611no—
515C42—611no—
516C42—611no—
517C42—611no—
518C42—611no—
519C42—611no—
520C42—611no—
521C42—611no—
522C42—1222no—
523C42—1222no—
524C42—1222no—
525C42—1222no—
526C42D4×D6111yescharacteristic
527C42D4×D6111yescharacteristic
528C42D4×D6111yescharacteristic
529C42D4×D6111yescharacteristic
530C23—611no—
531C23—611no—
532C23—1222no—
533C23—1222no—
534C23—2441no—
535C23C22×C4×D6111yescharacteristic
536Q8—2444no—
537Q8—2444no—
538Q8—2444no—
539Q8—2444no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—41no—
546C2×C4—42no—
547C2×C4—611no—
548C2×C4—611no—
549C2×C4—614no—
550C2×C4—611no—
551C2×C4—611no—
552C2×C4—611no—
553C2×C4—611no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1222no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—1222no—
562C2×C4—1222no—
563C2×C4—1222no—
564C2×C4—1224no—
565C2×C4—2441no—
566C2×C4—2441no—
567C2×C4—2441no—
568C2×C4—2441no—
569C2×C4—2441no—
570C2×C4—2441no—
571C2×C4—2442no—
572C2×C4C22⋊C4×D6111yescharacteristic
573C2×C4C22⋊C4×D6111yescharacteristic
574C2×C4C22⋊C4×D6111yescharacteristic
575C2×C4C22⋊C4×D6111yescharacteristic
576C2×C4C2×D4×D6111yescharacteristic
577C2×C4C2×D4×D6111yescharacteristic
578C8—42no—
579C8—42no—
580C8—2442no—
581C8—2442no—
582C22—21no—
583C22—21no—
584C22—612no—
585C22—1221no—
586C22—1221no—
587C22—1221no—
588C22—1221no—
589C22—1221no—
590C22—2444no—
591C22C2×C22⋊C4×D6111yescharacteristic
592C22#1684052712yes—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—21no—
597C4—21no—
598C4—42no—
599C4—42no—
600C4—612no—
601C4—612no—
602C4—1221no—
603C4—1221no—
604C4—1221no—
605C4—1221no—
606C4—2442no—
607C4—2442no—
608C4#101×D6112yes—
609C2—22no—
610C2—1222no—
611C2—2441no—
612C2#2145426111yescharacteristic
613C2#410×D6111yescharacteristic
614C2C2×#101×D6111yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D61
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1470C2
socleC22×C61C22×C22⋊C4
prime radicalC23×D61C22×C4
cosocleC22×C4×C61C24
hypercenter#1470D61
hypocenterC61#17404
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1470

Complements

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

NH ≅ G/Ntype
5. #21444176C2semidirect
6. #21444175C2semidirect
7. #21444174C2semidirect
8. #21444161C2semidirect
9. C61×#1470C2semidirect
309. #1470D61semidirect
356. C61#17404semidirect

Maximal subgroups

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

subgroupindex
C2×#168403182
#214544472
#432×D612
#214441812
#214441762
#214441752
#214441742
#214441612
C61×#14702
#1740461