Finite Group Databasebeta

← #29646887

Subgroups — #29646887

Order 15616.

994 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11758222C211yesmaximal, characteristic
2C8×#11758222C211yesmaximal, characteristic
3#21454384C211yesmaximal, characteristic
4#21444275C211yesmaximal, characteristic, complemented
5#21444272C212yesmaximal, complemented
6#21444248C211yesmaximal, characteristic, complemented
7#21444246C212yesmaximal, complemented
8#21444221C211yesmaximal, characteristic, complemented
9#21444220C212yesmaximal, complemented
10Q8⋊C8×D61C212yesmaximal, complemented
11C2×C61×Q8⋊C8C211yesmaximal, characteristic, complemented
12C2×#14519783—21no—
13Q8×Dic122C411yescharacteristic
14#16840490—21no—
15#16840490C411yescharacteristic
16#16840472—21no—
17#16840472C411yescharacteristic
18#16840467—22no—
19#16840467C412yes—
20Q8×D244C411yescharacteristic
21Q8×C22⋊D61C412yes—
22#16840319—21no—
23#16840319C2211yescharacteristic
24#16840316—22no—
25#16840316C2212yes—
26C4×#11758222—21no—
27C4×#11758222C2211yescharacteristic
28C4×Q8×D61C2212yes—
29C8×Dic122—21no—
30C8×D244—21no—
31C8×C22⋊D61—22no—
32#16839815C2211yescharacteristic
33#16839813C2212yes—
34#16839788—21no—
35#16839787—22no—
36#16839786C2211yescharacteristic
37#16839784C2212yes—
38C4⋊C8×D61C2212yes—
39C4×C8×D61C2212yes—
40#16839494C2212yes—
41#16839494C2212yes—
42Q8⋊#92418C2212yes—
43Q8⋊#92418C2212yes—
44#16839492C2212yes—
45#16839492C2212yes—
46C2×C4×Q8×C61C2211yescharacteristic
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×C4×C8×C61C2211yescharacteristic
49C61×Q8⋊C8C2211yescharacteristic
50C61×Q8⋊C8C2211yescharacteristic
51C61×Q8⋊C8C2211yescharacteristic
52C61×Q8⋊C8C2211yescharacteristic
53C2×#11758222D411yescharacteristic
54C2×Q8×D61C2×C412yes—
55#14519783—44no—
56C2×#11758194—21no—
57C2×#11758189—22no—
58C2×C8×D61—22no—
59C2×C8×D61—22no—
60#14519777—21no—
61#14519777—21no—
62#14519777C2×C411yescharacteristic
63#14519777C2×C411yescharacteristic
64#14519776—22no—
65#14519776C2×C412yes—
66#14519775—22no—
67#14519775C2×C412yes—
68#14519769—21no—
69#14519769—21no—
70#14519769—21no—
71#14519767—22no—
72#14519767—22no—
73#14519767—22no—
74#14519757—22no—
75#14519757C2×C412yes—
76#14519756—21no—
77#14519756—22no—
78#14519756—22no—
79#14519756C2×C412yes—
80#14519755—22no—
81#14519755—22no—
82#14519755—22no—
83#14519753—22no—
84#14519753C2×C412yes—
85Q8×Dic61C2×C412yes—
86C4×Dic122—21no—
87C4×Dic122—21no—
88C4×Dic122—21no—
89C4×Dic122—22no—
90C4×Dic122—22no—
91C4×Dic122C2×C411yescharacteristic
92C4×Dic122D411yescharacteristic
93C4×Dic122C2312yes—
94#14519742—22no—
95#14519742C2×C412yes—
96C4×D244—21no—
97C4×D244—21no—
98C4×D244C2×C411yescharacteristic
99C4×D244D411yescharacteristic
100C4×C22⋊D61—22no—
101C4×C22⋊D61—22no—
102C4×C22⋊D61C2×C412yes—
103C4×C22⋊D61D412yes—
104#14519736—21no—
105#14519736D411yescharacteristic
106#14519734—22no—
107#14519734D412yes—
108C4⋊C4×D61—22no—
109C4⋊C4×D61C2×C412yes—
110C42×D61—22no—
111C42×D61C2312yes—
112C4⋊#92418—21no—
113C4⋊#92418—21no—
114C4⋊#92418C2312yes—
115#14519706—22no—
116#14519706—22no—
117#14519706—22no—
118C22⋊#92418—21no—
119#14519689—22no—
120C4×#92418C2312yes—
121C8×Dic61—22no—
122C22×Q8×C61C2×C411yescharacteristic
123C22×C8×C61—21no—
124C22×C8×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×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138C4×C8×C61—21no—
139C4×C8×C61C2311yescharacteristic
140C4×C8×C61C2311yescharacteristic
141C2×Q8×C61—21no—
142C2×Q8×C61—22no—
143C2×Q8×C61—41no—
144C2×Q8×C61C22×C411yescharacteristic
145C2×Q8×C61C22×C411yescharacteristic
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61C22×C411yescharacteristic
149C22×C4×C61C2×D411yescharacteristic
150C22×C4×C61C2×D411yescharacteristic
151C2×C22⋊D61C22⋊C412yes—
152Q8×D61—24no—
153Q8×D61—42no—
154#11758222SD1614yes—
155#11758222Q1614yes—
156C2×D244C22⋊C411yescharacteristic
157C2×C4×D61—22no—
158C2×C4×D61—22no—
159C2×C4×D61C22×C412yes—
160C2×C4×D61C2×D412yes—
161C2×C4×D61C2×D412yes—
162C2×Dic122—22no—
163C2×Dic122—21no—
164C2×Dic122—22no—
165C2×Dic122C22⋊C411yescharacteristic
166C2×Dic122C22×C412yes—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61—21no—
171C2×C8×C61—42no—
172C2×C8×C61—42no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61C22×C411yescharacteristic
179C4⋊C4×C61C22×C411yescharacteristic
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C22×C411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61C2411yescharacteristic
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C412yes—
190#11758198—22no—
191#11758198—22no—
192#11758198C22⋊C412yes—
193#11758198C22⋊C412yes—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—22no—
198C4⋊Dic61—22no—
199C4⋊Dic61C22⋊C411yescharacteristic
200C4⋊Dic61C22⋊C411yescharacteristic
201C4⋊Dic61C22×C412yes—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—22no—
207#11758196—22no—
208#11758196—22no—
209#11758196—22no—
210#11758196C22⋊C412yes—
211#11758196C22⋊C412yes—
212C4×Dic61—22no—
213C4×Dic61—22no—
214C4×Dic61—22no—
215C4×Dic61C22×C412yes—
216C4×Dic61C2×D412yes—
217C4×Dic61C2×D412yes—
218#11758194—42no—
219C2×#92418—22no—
220C2×#92418—22no—
221#11758189—44no—
222C8×D61—44no—
223C8×D61—44no—
224C23×C61C2×C22⋊C411yescharacteristic
225C22×D61C2×C22⋊C412yes—
226Q8×C61—21no—
227Q8×C61—21no—
228Q8×C61—21no—
229Q8×C61—42no—
230Q8×C61#13112yescomplemented
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—21no—
235C2×C4×C61—21no—
236C2×C4×C61—21no—
237C2×C4×C61—41no—
238C2×C4×C61—42no—
239C2×C4×C61—81no—
240C2×C4×C61C2×C22⋊C411yescharacteristic
241C2×C4×C61C2×C22⋊C411yescharacteristic
242C2×C4×C61C2×C22⋊C411yescharacteristic
243C2×C4×C61C2×SD1612yes—
244C2×C4×C61C2×Q1612yes—
245C2×C4×C61C23×C411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C22×D411yescharacteristic
248C22⋊D61#10318yes—
249C2×Dic61—22no—
250C2×Dic61—22no—
251C2×Dic61C2×C22⋊C412yes—
252C2×Dic61C2×C22⋊C412yes—
253C2×Dic61C2×C22⋊C412yes—
254D244#10314yes—
255C4×D61—22no—
256C4×D61—22no—
257C4×D61—42no—
258C4×D61—44no—
259C4×D61—82no—
260C4×D61C2×SD1614yes—
261C4×D61C2×Q1614yes—
262Dic122—22no—
263Dic122—22no—
264Dic122—44no—
265Dic122—82no—
266Dic122—81no—
267Dic122#10314yes—
268C8×C61—22no—
269C8×C61—41no—
270C8×C61—41no—
271C8×C61—41no—
272#92418—42no—
273#92418—42no—
274#17509—611nomaximal, Sylow
275C22×C61—21no—
276C22×C61—21no—
277C22×C61C2×#10312yes—
278C22×C61C22×C22⋊C411yescharacteristic
279C2×D61—22no—
280C2×D61—22no—
281C2×D61C2×#10314yes—
282C4×C61—22no—
283C4×C61—21no—
284C4×C61—21no—
285C4×C61—21no—
286C4×C61—41no—
287C4×C61—41no—
288C4×C61—41no—
289C4×C61C2×#10312yes—
290C4×C61C2×#13111yescharacteristic
291C4×C61#56811yescharacteristic
292C4×C61C22×SD1611yescharacteristic
293C4×C61C22×Q1611yescharacteristic
294Dic61—22no—
295Dic61—22no—
296Dic61—42no—
297Dic61—42no—
298Dic61C2×#10314yes—
299Q8×#41—611no—
300C8×#41—611no—
301#2906—611no—
302#1493—611no—
303#1493—612no—
304#1487—612no—
305#1487—611no—
306#1481—611no—
307#1481—612no—
308C2×Q8⋊C8—612no—
309C2×Q8⋊C8D6111yescharacteristic, complemented
310C2×C61—22no—
311C2×C61#288011yescharacteristic
312C2×C61C22×#10311yescharacteristic
313C2×C61#290111yescharacteristic
314D61—24no—
315C2×#131—1221no—
316Q82—611no—
317#557—611no—
318#557—1221no—
319#554—611no—
320#554—612no—
321#554—1222no—
322#554—1221no—
323D4×Q8—612no—
324D4×Q8—611no—
325#533—612no—
326#533—611no—
327#533—1221no—
328#533—1222no—
329C4×#41—611no—
330C4×#41—1221no—
331C2×C4×Q8—612no—
332C2×C4×Q8C2×D6111yescharacteristic
333C8×Q8—1221no—
334C8×D4—1221no—
335C8×D4—1222no—
336#431—611no—
337#431—612no—
338#424—1222no—
339#424—1221no—
340#423—611no—
341#423—612no—
342C2×C4⋊C8—612no—
343C2×C4⋊C8C2×D6111yescharacteristic
344C2×C4×C8—612no—
345C2×C4×C8C2×D6111yescharacteristic
346Q8⋊C8—612no—
347Q8⋊C8—612no—
348Q8⋊C8—612no—
349Q8⋊C8—612no—
350Q8⋊C8—612no—
351Q8⋊C8—612no—
352Q8⋊C8C2×D6111yescharacteristic
353Q8⋊C8C2×D6111yescharacteristic
354Q8⋊C8C2×D6111yescharacteristic
355Q8⋊C8C2×D6111yescharacteristic
356C61#1750911yescharacteristic, complemented, Sylow
357C2×#41—611no—
358C22×Q8—612no—
359C22×Q8C4×D6111yescharacteristic
360#131—2444no—
361C2×M16—1222no—
362C2×M16—1221no—
363C22×C8—21no—
364C22×C8—21no—
365C22×C8—1222no—
366C22×C8—1222no—
367C4⋊Q8—612no—
368C4⋊Q8—612no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371C4⋊Q8—1221no—
372C4⋊Q8—1221no—
373C4⋊Q8—1222no—
374C4⋊Q8—1222no—
375#125—1221no—
376#125—1221no—
377#125—1222no—
378#125—1222no—
379#125—1222no—
380#125—1221no—
381C22⋊Q8—612no—
382C22⋊Q8—612no—
383C22⋊Q8—612no—
384C22⋊Q8—1222no—
385C22⋊Q8—1222no—
386C22⋊Q8—1222no—
387C22⋊Q8—1222no—
388C22⋊Q8—1222no—
389C22⋊Q8—1222no—
390C22⋊Q8—1222no—
391C22⋊Q8—1221no—
392C4×Q8—21no—
393C4×Q8—21no—
394C4×Q8—611no—
395C4×Q8—611no—
396C4×Q8—612no—
397C4×Q8—612no—
398C4×Q8—612no—
399C4×Q8—1222no—
400C4×Q8—1222no—
401C4×Q8—1221no—
402C4×Q8—1221no—
403C4×Q8—1222no—
404C4×Q8—1221no—
405C4×Q8C4×D6111yescharacteristic
406C4×Q8C4×D6111yescharacteristic
407C4×Q8C22×D6111yescharacteristic
408C4×Q8C22×D6111yescharacteristic
409C4×D4—612no—
410C4×D4—612no—
411C4×D4—611no—
412C4×D4—611no—
413C4×D4—1221no—
414C4×D4—1221no—
415C4×D4—1222no—
416C4×D4—1222no—
417#117—611no—
418#117—612no—
419#117—1222no—
420#117—1221no—
421C2×C4⋊C4—21no—
422C2×C4⋊C4—612no—
423C2×C4⋊C4—1222no—
424C2×C4⋊C4C4×D6111yescharacteristic
425C2×C42—21no—
426C2×C42—612no—
427C2×C42—1222no—
428C2×C42C22×D6111yescharacteristic
429C4⋊C8—21no—
430C4⋊C8—612no—
431C4⋊C8—1222no—
432C4⋊C8—1222no—
433C4⋊C8—1222no—
434C4⋊C8—1221no—
435C4⋊C8—1221no—
436C4⋊C8C22×D6111yescharacteristic
437C4⋊C8C22×D6111yescharacteristic
438C22⋊C8—1222no—
439C22⋊C8—1221no—
440C4×C8—21no—
441C4×C8—612no—
442C4×C8—1222no—
443C4×C8C22×D6111yescharacteristic
444C4×C8C22×D6111yescharacteristic
445#41—614no—
446#41—614no—
447C2×Q8—21no—
448C2×Q8—22no—
449C2×Q8—41no—
450C2×Q8—611no—
451C2×Q8—612no—
452C2×Q8—1222no—
453C2×Q8—1222no—
454C2×Q8—1221no—
455C2×Q8—1224no—
456C2×Q8—2442no—
457C2×Q8C2×C4×D6111yescharacteristic
458C2×Q8C2×C4×D6111yescharacteristic
459C2×D4—612no—
460C2×D4—611no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—612no—
464C22×C4—612no—
465C22×C4—612no—
466C22×C4—1222no—
467C22×C4—1222no—
468C22×C4C2×C4×D6111yescharacteristic
469C22×C4D4×D6111yescharacteristic
470C22×C4D4×D6111yescharacteristic
471M16—2444no—
472M16—2442no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—42no—
478C2×C8—42no—
479C2×C8—1222no—
480C2×C8—1222no—
481C2×C8—2444no—
482C2×C8—2444no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—612no—
489C4⋊C4—612no—
490C4⋊C4—611no—
491C4⋊C4—611no—
492C4⋊C4—612no—
493C4⋊C4—1222no—
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—1222no—
503C4⋊C4—1221no—
504C4⋊C4—1221no—
505C4⋊C4—1221no—
506C4⋊C4C2×C4×D6111yescharacteristic
507C4⋊C4C2×C4×D6111yescharacteristic
508C22⋊C4—612no—
509C22⋊C4—612no—
510C22⋊C4—612no—
511C22⋊C4—1222no—
512C22⋊C4—1222no—
513C22⋊C4—1222no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—21no—
518C42—612no—
519C42—612no—
520C42—612no—
521C42—1222no—
522C42—1222no—
523C42—1222no—
524C42C2×C4×D6111yescharacteristic
525C42D4×D6111yescharacteristic
526C42D4×D6111yescharacteristic
527C42C23×D6111yescharacteristic
528C23—612no—
529C23C22⋊C4×D6111yescharacteristic
530Q8—21no—
531Q8—21no—
532Q8—21no—
533Q8—42no—
534Q8—614no—
535Q8—1222no—
536Q8—1222no—
537Q8—4881no—
538Q8—4882no—
539Q8—2444no—
540Q8#1451978312yescomplemented
541D4—618no—
542D4—614no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—41no—
550C2×C4—42no—
551C2×C4—81no—
552C2×C4—612no—
553C2×C4—614no—
554C2×C4—614no—
555C2×C4—612no—
556C2×C4—612no—
557C2×C4—1222no—
558C2×C4—1222no—
559C2×C4—1222no—
560C2×C4—1222no—
561C2×C4—4882no—
562C2×C4—2442no—
563C2×C4—2444no—
564C2×C4C22⋊C4×D6111yescharacteristic
565C2×C4C22⋊C4×D6111yescharacteristic
566C2×C4C22⋊C4×D6111yescharacteristic
567C2×C4SD16×D6112yes—
568C2×C4Q16×D6112yes—
569C2×C4C22×C4×D6111yescharacteristic
570C2×C4C2×D4×D6111yescharacteristic
571C2×C4C2×D4×D6111yescharacteristic
572C8—22no—
573C8—41no—
574C8—41no—
575C8—41no—
576C8—2442no—
577C8—2442no—
578C22—21no—
579C22—21no—
580C22—614no—
581C22—1222no—
582C22—1222no—
583C22#103×D6112yes—
584C22C2×C22⋊C4×D6111yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—614no—
593C4—1222no—
594C4—1222no—
595C4—2442no—
596C4—2442no—
597C4#103×D6112yes—
598C4C2×#1451978311yescharacteristic
599C4#1684052811yescharacteristic
600C4C2×SD16×D6111yescharacteristic
601C4C2×Q16×D6111yescharacteristic
602C2—22no—
603C2—1224no—
604C2#2145426511yescharacteristic
605C2C2×#103×D6111yescharacteristic
606C2#2145436311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D61
commutatorC2×C4×C61C23×C4
FrattiniC42C23×D61
FittingC2×C61×Q8⋊C8C2
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758222D4
cosocleC42×C61C24
hypercenterC2×Q8⋊C8D61
hypocenterC61#17509
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #21444275C2semidirect
5. #21444272C2semidirect
6. #21444248C2semidirect
7. #21444246C2semidirect
8. #21444221C2semidirect
9. #21444220C2semidirect
10. Q8⋊C8×D61C2semidirect
11. C2×C61×Q8⋊C8C2semidirect
230. Q8×C61#131semidirect
309. C2×Q8⋊C8D61semidirect
356. C61#17509semidirect
540. Q8#14519783semidirect

Maximal subgroups

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

subgroupindex
Q8×#117582222
C8×#117582222
#214543842
#214442752
#214442722
#214442482
#214442462
#214442212
#214442202
Q8⋊C8×D612
C2×C61×Q8⋊C82
#1750961