Finite Group Databasebeta

← #29646949

Subgroups — #29646949

Order 15616.

926 conjugacy classes of subgroups.

The subgroup lattice has 602 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758222C211yesmaximal, characteristic
2#21454780C211yesmaximal, characteristic
3#21454487C211yesmaximal, characteristic
4#21444290C212yesmaximal, complemented
5#21444273C211yesmaximal, characteristic, complemented
6#21444264C212yesmaximal, complemented
7#21444238C212yesmaximal, complemented
8#21444223C211yesmaximal, characteristic, complemented
9#21444212C212yesmaximal, complemented
10C2×Q8⋊#92418C211yesmaximal, characteristic, complemented
11C61×#1487C211yesmaximal, characteristic, complemented
12Q8×Dic122C411yescharacteristic
13#16840490—21no—
14#16840490C411yescharacteristic
15#16840472—21no—
16#16840472C411yescharacteristic
17#16840467—22no—
18#16840467C412yes—
19Q8×D244C411yescharacteristic
20Q8×C22⋊D61C412yes—
21#16840319—21no—
22#16840319C2211yescharacteristic
23#16840316—22no—
24#16840316C2212yes—
25C4×#11758222—21no—
26C4×#11758222C2211yescharacteristic
27C4×Q8×D61C2212yes—
28#16839889—21no—
29#16839839—21no—
30#16839830—22no—
31#16839822—22no—
32#16839818C2212yes—
33#16839814C2212yes—
34#16839813C2212yes—
35C2×C4⋊#92418C2211yescharacteristic
36#16839730—21no—
37#16839720—21no—
38C4×#11758189C2212yes—
39C2×C4×#92418C2211yescharacteristic
40#16839494C2212yes—
41#16839494C2212yes—
42Q8⋊#92418C2211yescharacteristic
43Q8⋊#92418C2211yescharacteristic
44Q8⋊#92418C2211yescharacteristic
45Q8⋊#92418C2211yescharacteristic
46#16839492C2212yes—
47#16839492C2212yes—
48C2×C4×Q8×C61C2211yescharacteristic
49C61×#431C2211yescharacteristic
50C61×#423C2211yescharacteristic
51C61×Q8⋊C8C2212yes—
52C61×Q8⋊C8C2212yes—
53C2×#11758222D411yescharacteristic
54C2×Q8×D61C2×C412yes—
55C2×#11758189—22no—
56C22×#92418—21no—
57C22×#92418—21no—
58#14519777—21no—
59#14519777—21no—
60#14519777C2×C411yescharacteristic
61#14519777C2×C411yescharacteristic
62#14519776—22no—
63#14519776C2×C412yes—
64#14519775—22no—
65#14519775C2×C412yes—
66#14519769—21no—
67#14519769—21no—
68#14519769—21no—
69#14519767—22no—
70#14519767—22no—
71#14519767—22no—
72#14519757—22no—
73#14519757C2×C412yes—
74#14519756—21no—
75#14519756—22no—
76#14519756—22no—
77#14519756C2×C412yes—
78#14519755—22no—
79#14519755—22no—
80#14519755—22no—
81#14519753—22no—
82#14519753C2×C412yes—
83Q8×Dic61C2×C412yes—
84C4×Dic122—21no—
85C4×Dic122—21no—
86C4×Dic122—21no—
87C4×Dic122—22no—
88C4×Dic122—22no—
89C4×Dic122C2×C411yescharacteristic
90C4×Dic122D411yescharacteristic
91C4×Dic122C2312yes—
92#14519742—22no—
93#14519742C2×C412yes—
94C4×D244—21no—
95C4×D244—21no—
96C4×D244C2×C411yescharacteristic
97C4×D244D411yescharacteristic
98C4×C22⋊D61—22no—
99C4×C22⋊D61—22no—
100C4×C22⋊D61C2×C412yes—
101C4×C22⋊D61D412yes—
102#14519736—21no—
103#14519736D411yescharacteristic
104#14519734—22no—
105#14519734D412yes—
106C4⋊C4×D61—22no—
107C4⋊C4×D61C2×C412yes—
108C42×D61—22no—
109C42×D61C2312yes—
110C4⋊#92418—21no—
111C4⋊#92418C2311yescharacteristic
112C4⋊#92418C2311yescharacteristic
113#14519706—22no—
114#14519689—22no—
115#14519689—22no—
116#14519689—22no—
117#14519687—22no—
118#14519687—22no—
119C4×#92418—21no—
120C4×#92418C2311yescharacteristic
121C4×#92418C2311yescharacteristic
122C8×Dic61—22no—
123C22×Q8×C61C2×C411yescharacteristic
124C2×M16×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2×C411yescharacteristic
128C4×Q8×C61C2×C411yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131C2×C4⋊C4×C61—21no—
132C2×C4⋊C4×C61C2×C411yescharacteristic
133C2×C42×C61—21no—
134C2×C42×C61C2311yescharacteristic
135C4⋊C8×C61—21no—
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2312yes—
138C22⋊C8×C61—21no—
139C4×C8×C61C2312yes—
140C2×Q8×C61—21no—
141C2×Q8×C61—22no—
142C2×Q8×C61—41no—
143C2×Q8×C61C22×C411yescharacteristic
144C2×Q8×C61C22×C411yescharacteristic
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61C22×C411yescharacteristic
148C22×C4×C61C2×D411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C2×C22⋊D61C22⋊C412yes—
151Q8×D61—24no—
152Q8×D61—42no—
153#11758222—22no—
154#11758222—22no—
155C2×D244C22⋊C411yescharacteristic
156C2×C4×D61—22no—
157C2×C4×D61—22no—
158C2×C4×D61C22×C412yes—
159C2×C4×D61C2×D412yes—
160C2×C4×D61C2×D412yes—
161C2×Dic122—22no—
162C2×Dic122—21no—
163C2×Dic122—22no—
164C2×Dic122C22⋊C411yescharacteristic
165C2×Dic122C22×C412yes—
166M16×C61—42no—
167C2×C8×C61—22no—
168C2×C8×C61—22no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61C22×C411yescharacteristic
175C4⋊C4×C61C22×C411yescharacteristic
176C42×C61—21no—
177C42×C61—21no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61C22×C411yescharacteristic
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61C2411yescharacteristic
184C22⋊Dic61—22no—
185C22⋊Dic61C22⋊C412yes—
186#11758198—22no—
187#11758198—22no—
188#11758198C22⋊C412yes—
189#11758198C22⋊C412yes—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—22no—
194C4⋊Dic61—22no—
195C4⋊Dic61C22⋊C411yescharacteristic
196C4⋊Dic61C22⋊C411yescharacteristic
197C4⋊Dic61C22×C412yes—
198#11758196—22no—
199#11758196—22no—
200#11758196—22no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196C22⋊C412yes—
207#11758196C22⋊C412yes—
208C4×Dic61—22no—
209C4×Dic61—22no—
210C4×Dic61—22no—
211C4×Dic61C22×C412yes—
212C4×Dic61C2×D412yes—
213C4×Dic61C2×D412yes—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418—21no—
218C2×#92418—42no—
219C2×#92418—42no—
220#11758189—44no—
221C23×C61C2×C22⋊C411yescharacteristic
222C22×D61C2×C22⋊C412yes—
223Q8×C61—21no—
224Q8×C61—21no—
225Q8×C61—21no—
226Q8×C61—42no—
227Q8×C61#13112yes—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—42no—
236C2×C4×C61—81no—
237C2×C4×C61C2×C22⋊C411yescharacteristic
238C2×C4×C61C2×C22⋊C411yescharacteristic
239C2×C4×C61C2×C22⋊C411yescharacteristic
240C2×C4×C61#13712yes—
241C2×C4×C61#13712yes—
242C2×C4×C61C23×C411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C22×D411yescharacteristic
245C22⋊D61—24no—
246C2×Dic61—22no—
247C2×Dic61—22no—
248C2×Dic61C2×C22⋊C412yes—
249C2×Dic61C2×C22⋊C412yes—
250C2×Dic61C2×C22⋊C412yes—
251D244—22no—
252C4×D61—22no—
253C4×D61—22no—
254C4×D61—22no—
255C4×D61—22no—
256C4×D61—42no—
257C4×D61—44no—
258C4×D61—82no—
259Dic122—22no—
260Dic122—22no—
261Dic122—22no—
262Dic122—44no—
263Dic122—82no—
264Dic122—81no—
265C8×C61—42no—
266C8×C61—42no—
267#92418—22no—
268#92418—41no—
269#92418—41no—
270#92418—41no—
271#17514—611nomaximal, Sylow
272C22×C61—21no—
273C22×C61—21no—
274C22×C61#41912yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C2×D61—22no—
277C2×D61—22no—
278C2×D61—22no—
279C4×C61—22no—
280C4×C61—21no—
281C4×C61—21no—
282C4×C61—21no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61—41no—
286C4×C61#41912yes—
287C4×C61C2×#13111yescharacteristic
288C4×C61#56811yescharacteristic
289C4×C61C2×#13711yescharacteristic
290C4×C61C2×#13711yescharacteristic
291Dic61—22no—
292Dic61—22no—
293Dic61—22no—
294Dic61—42no—
295Dic61—42no—
296Q8×#41—611no—
297#2965—611no—
298#2925—611no—
299#1497—612no—
300#1493—611no—
301#1491—612no—
302#1487D6111yescharacteristic, complemented
303#1485—612no—
304#1481—611no—
305#1479—612no—
306C2×Q8⋊C8—611no—
307C2×C61—22no—
308C2×C61#288011yescharacteristic
309C2×C61C2×#41911yescharacteristic
310C2×C61#290111yescharacteristic
311D61—42no—
312Q82—611no—
313#557—611no—
314#557—1221no—
315#554—611no—
316#554—612no—
317#554—1222no—
318#554—1221no—
319D4×Q8—612no—
320D4×Q8—611no—
321#533—612no—
322#533—611no—
323#533—1221no—
324#533—1222no—
325C4×#41—611no—
326C4×#41—1221no—
327C2×C4×Q8—612no—
328C2×C4×Q8C2×D6111yescharacteristic
329#446—1221no—
330#436—1221no—
331#435—1222no—
332#433—1222no—
333#432—612no—
334#431—612no—
335#431—612no—
336#431C2×D6111yescharacteristic
337#423C2×D6111yescharacteristic
338C2×C4⋊C8—611no—
339#408—1221no—
340#405—1221no—
341C4×M16—612no—
342C2×C4×C8—611no—
343Q8⋊C8—611no—
344Q8⋊C8—611no—
345Q8⋊C8—611no—
346Q8⋊C8—611no—
347Q8⋊C8—612no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—612no—
351Q8⋊C8C2×D6112yes—
352Q8⋊C8C2×D6112yes—
353C61#1751411yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C22×Q8—612no—
356C22×Q8C4×D6111yescharacteristic
357C2×M16—21no—
358C2×M16—1222no—
359C22×C8—1221no—
360C22×C8—1221no—
361C4⋊Q8—612no—
362C4⋊Q8—612no—
363C4⋊Q8—611no—
364C4⋊Q8—611no—
365C4⋊Q8—1221no—
366C4⋊Q8—1221no—
367C4⋊Q8—1222no—
368C4⋊Q8—1222no—
369#125—1221no—
370#125—1221no—
371#125—1222no—
372#125—1222no—
373#125—1222no—
374#125—1221no—
375C22⋊Q8—612no—
376C22⋊Q8—612no—
377C22⋊Q8—612no—
378C22⋊Q8—1222no—
379C22⋊Q8—1222no—
380C22⋊Q8—1222no—
381C22⋊Q8—1222no—
382C22⋊Q8—1222no—
383C22⋊Q8—1222no—
384C22⋊Q8—1222no—
385C22⋊Q8—1221no—
386C4×Q8—21no—
387C4×Q8—21no—
388C4×Q8—611no—
389C4×Q8—611no—
390C4×Q8—612no—
391C4×Q8—612no—
392C4×Q8—612no—
393C4×Q8—1222no—
394C4×Q8—1222no—
395C4×Q8—1221no—
396C4×Q8—1221no—
397C4×Q8—1222no—
398C4×Q8—1221no—
399C4×Q8C4×D6111yescharacteristic
400C4×Q8C4×D6111yescharacteristic
401C4×Q8C22×D6111yescharacteristic
402C4×Q8C22×D6111yescharacteristic
403C4×D4—612no—
404C4×D4—612no—
405C4×D4—611no—
406C4×D4—611no—
407C4×D4—1221no—
408C4×D4—1221no—
409C4×D4—1222no—
410C4×D4—1222no—
411#117—611no—
412#117—612no—
413#117—1222no—
414#117—1221no—
415C2×C4⋊C4—21no—
416C2×C4⋊C4—612no—
417C2×C4⋊C4—1222no—
418C2×C4⋊C4C4×D6111yescharacteristic
419C2×C42—21no—
420C2×C42—612no—
421C2×C42—1222no—
422C2×C42C22×D6111yescharacteristic
423C4⋊C8—21no—
424C4⋊C8—21no—
425C4⋊C8—611no—
426C4⋊C8—611no—
427C4⋊C8—1222no—
428C4⋊C8—1221no—
429C4⋊C8C22×D6112yes—
430C22⋊C8—21no—
431C22⋊C8—1222no—
432C22⋊C8—1222no—
433C22⋊C8—1222no—
434#97—1222no—
435#97—1222no—
436C4×C8—611no—
437C4×C8—611no—
438C4×C8—1222no—
439C4×C8—1221no—
440C4×C8C22×D6112yes—
441#41—1222no—
442#41—1222no—
443C2×Q8—21no—
444C2×Q8—22no—
445C2×Q8—41no—
446C2×Q8—611no—
447C2×Q8—612no—
448C2×Q8—1222no—
449C2×Q8—1222no—
450C2×Q8—1221no—
451C2×Q8—1224no—
452C2×Q8—2442no—
453C2×Q8C2×C4×D6111yescharacteristic
454C2×Q8C2×C4×D6111yescharacteristic
455C2×D4—612no—
456C2×D4—611no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—612no—
460C22×C4—612no—
461C22×C4—612no—
462C22×C4—1222no—
463C22×C4—1222no—
464C22×C4C2×C4×D6111yescharacteristic
465C22×C4D4×D6111yescharacteristic
466C22×C4D4×D6111yescharacteristic
467M16—42no—
468M16—2444no—
469C2×C8—22no—
470C2×C8—22no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1221no—
475C2×C8—2442no—
476C2×C8—2442no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—612no—
483C4⋊C4—612no—
484C4⋊C4—611no—
485C4⋊C4—611no—
486C4⋊C4—612no—
487C4⋊C4—1222no—
488C4⋊C4—1222no—
489C4⋊C4—1222no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1222no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—1221no—
500C4⋊C4C2×C4×D6111yescharacteristic
501C4⋊C4C2×C4×D6111yescharacteristic
502C22⋊C4—612no—
503C22⋊C4—612no—
504C22⋊C4—612no—
505C22⋊C4—1222no—
506C22⋊C4—1222no—
507C22⋊C4—1222no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—21no—
512C42—612no—
513C42—612no—
514C42—612no—
515C42—1222no—
516C42—1222no—
517C42—1222no—
518C42C2×C4×D6111yescharacteristic
519C42D4×D6111yescharacteristic
520C42D4×D6111yescharacteristic
521C42C23×D6111yescharacteristic
522C23—612no—
523C23C22⋊C4×D6111yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—21no—
527Q8—42no—
528Q8—1222no—
529Q8—1222no—
530Q8—1222no—
531Q8—4881no—
532Q8—4882no—
533Q8—2444no—
534Q8#1451978512yes—
535D4—1224no—
536D4—1222no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—41no—
544C2×C4—42no—
545C2×C4—81no—
546C2×C4—612no—
547C2×C4—612no—
548C2×C4—612no—
549C2×C4—1222no—
550C2×C4—1222no—
551C2×C4—1222no—
552C2×C4—1222no—
553C2×C4—1222no—
554C2×C4—1222no—
555C2×C4—4882no—
556C2×C4—2442no—
557C2×C4—2444no—
558C2×C4C22⋊C4×D6111yescharacteristic
559C2×C4C22⋊C4×D6111yescharacteristic
560C2×C4C22⋊C4×D6111yescharacteristic
561C2×C4#1451980612yes—
562C2×C4#1451980912yes—
563C2×C4C22×C4×D6111yescharacteristic
564C2×C4C2×D4×D6111yescharacteristic
565C2×C4C2×D4×D6111yescharacteristic
566C8—42no—
567C8—42no—
568C8—1222no—
569C8—2441no—
570C8—2441no—
571C8—2441no—
572C22—21no—
573C22—21no—
574C22—1222no—
575C22—1222no—
576C22—1222no—
577C22#1683976912yes—
578C22C2×C22⋊C4×D6111yescharacteristic
579C4—21no—
580C4—21no—
581C4—21no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—1222no—
587C4—1222no—
588C4—1222no—
589C4—2442no—
590C4—2442no—
591C4#1683976912yes—
592C4C2×#1451978511yescharacteristic
593C4#1684052811yescharacteristic
594C4C2×#1451980611yescharacteristic
595C4C2×#1451980911yescharacteristic
596C2—22no—
597C2—2442no—
598C2#2145426911yescharacteristic
599C2C2×#1683976911yescharacteristic
600C2#2145436511yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC61×#1487C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222D4
cosocleC42×C61C24
hypercenter#1487D61
hypocenterC61#17514
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
4. #21444290C2semidirect
5. #21444273C2semidirect
6. #21444264C2semidirect
7. #21444238C2semidirect
8. #21444223C2semidirect
9. #21444212C2semidirect
10. C2×Q8⋊#92418C2semidirect
11. C61×#1487C2semidirect
302. #1487D61semidirect
353. C61#17514semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582222
#214547802
#214544872
#214442902
#214442732
#214442642
#214442382
#214442232
#214442122
C2×Q8⋊#924182
C61×#14872
#1751461