Finite Group Databasebeta

← #30611794

Subgroups — #30611794

Order 15616.

964 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
1#21462735C211yesmaximal, characteristic
2#21462466C211yesmaximal, characteristic
3#21461337C212yesmaximal, complemented
4#21461336C211yesmaximal, characteristic, complemented
5#21461333C211yesmaximal, characteristic, complemented
6#21461332C212yesmaximal, complemented
7#21461321C212yesmaximal, complemented
8#21461320C211yesmaximal, characteristic, complemented
9D61⋊#333C211yesmaximal, characteristic, complemented
10C2×#16840622C212yesmaximal, complemented
11#21459630C211yesmaximal, characteristic
12C2×#14519864—21no—
13#16840811—21no—
14#16840789—22no—
15D4×#92420—21no—
16#16840780C2211yescharacteristic
17#16840779C2212yes—
18#16840769—21no—
19#16840768—22no—
20#16840767C2211yescharacteristic
21#16840765C2212yes—
22C2×#14519844C2212yes—
23C2×#14519844C2212yes—
24D61⋊(C4⋊C8)C2211yescharacteristic
25D61⋊(C4⋊C8)C2211yescharacteristic
26#16840622C2211yescharacteristic
27#16840622C2212yes—
28#16840622C2212yes—
29#16840622C2212yes—
30#16840622C2212yes—
31#16840622C2211yescharacteristic
32#16840622C2212yes—
33#16840621C2211yescharacteristic
34#16840621C2211yescharacteristic
35#16840621C2212yes—
36#16840492—21no—
37#16840490C411yescharacteristic
38#16840487C411yescharacteristic
39#16840471—21no—
40#16840466—22no—
41D4×Dic122C411yescharacteristic
42Q8×C22⋊D61C412yes—
43#16840425C412yes—
44#16840391—22no—
45#16840389—21no—
46#16840316C2211yescharacteristic
47#16840314C2212yes—
48C2×#14519775C2212yes—
49C4⋊Q8×D61C2211yescharacteristic
50#16840241—21no—
51C4×#11758224C2211yescharacteristic
52D4×Q8×C61C411yescharacteristic
53#14519864—44no—
54C2×#11758213—21no—
55C2×#11758217—22no—
56C2×D61⋊C8—21no—
57C2×D61⋊C8—21no—
58C22×#92420—22no—
59C22×#92420—22no—
60#14519844—22no—
61#14519844—22no—
62#14519844—21no—
63#14519844C2311yescharacteristic
64#14519844C2312yes—
65#14519844C2311yescharacteristic
66#14519844C2311yescharacteristic
67#14519844C2312yes—
68#14519844C2311yescharacteristic
69C4⋊#92420—21no—
70C4⋊#92420—21no—
71C4⋊#92420—22no—
72#14519834—21no—
73C22⋊#92420—22no—
74C4×#92420—22no—
75C4×#92420—21no—
76C2×#11758224D411yescharacteristic
77C22×Dic122C2×C412yes—
78C2×Q8×D61C2×C412yes—
79#14519777C2×C411yescharacteristic
80#14519776—21no—
81#14519776—22no—
82#14519776—21no—
83#14519776—21no—
84#14519776C2311yescharacteristic
85#14519775—22no—
86#14519775—22no—
87#14519775—22no—
88#14519775C2×C412yes—
89#14519775C2×C412yes—
90#14519775C2312yes—
91#14519775C2311yescharacteristic
92#14519769—21no—
93#14519768—21no—
94#14519767—22no—
95#14519767—22no—
96#14519767—22no—
97#14519767—21no—
98#14519757C2×C411yescharacteristic
99#14519756—22no—
100#14519756C2×C412yes—
101#14519755—21no—
102#14519755—22no—
103#14519754—22no—
104#14519754—22no—
105#14519754—22no—
106#14519753—22no—
107#14519753C2×C411yescharacteristic
108Q8×Dic61C2×C412yes—
109C4×Dic122—21no—
110C4×Dic122—21no—
111C4×Dic122C2×C411yescharacteristic
112C4×Dic122C2×C411yescharacteristic
113#14519742—21no—
114#14519742—22no—
115#14519742C2×C412yes—
116#14519742D411yescharacteristic
117D4×Dic61—21no—
118D4×Dic61D411yescharacteristic
119C4×C22⋊D61—22no—
120C4×C22⋊D61C2×C412yes—
121#14519737—22no—
122#14519737D412yes—
123#14519735D411yescharacteristic
124#14519734—21no—
125#14519733—22no—
126#14519733D412yes—
127C2×#11758196—22no—
128C2×#11758196—22no—
129C4⋊C4×D61—21no—
130C4⋊C4×D61—21no—
131C2×C4×Dic61C2312yes—
132C42×D61C2311yescharacteristic
133C22×Q8×C61C2×C412yes—
134C4⋊Q8×C61—21no—
135C4⋊Q8×C61C2×C411yescharacteristic
136C22⋊Q8×C61—22no—
137C22⋊Q8×C61C2×C412yes—
138C4×Q8×C61C2×C411yescharacteristic
139C4×D4×C61—21no—
140C4×D4×C61C2×C411yescharacteristic
141C2×Q8×C61—22no—
142C2×Q8×C61—82no—
143C2×Q8×C61C22×C412yes—
144C2×Q8×C61C22×C411yescharacteristic
145C2×D4×C61C22⋊C411yescharacteristic
146C22×C4×C61—22no—
147C22×C4×C61C22×C412yes—
148C2×C22⋊D61C22⋊C412yes—
149C22×Dic61—22no—
150C22×Dic61C2×D412yes—
151C22×Dic61C2×D412yes—
152Q8×D61—82no—
153#11758224SD1614yes—
154#11758224Q1614yes—
155C2×C4×D61—22no—
156C2×C4×D61C22×C411yescharacteristic
157C2×C4×D61C2×D411yescharacteristic
158C2×C4×D61C2×D411yescharacteristic
159C2×Dic122—21no—
160C2×Dic122—22no—
161C2×Dic122—22no—
162C2×Dic122—82no—
163C2×Dic122C22⋊C411yescharacteristic
164C2×Dic122C22×C412yes—
165C2×Dic122C22×C411yescharacteristic
166#11758217—44no—
167C2×#92420—21no—
168C2×#92420—22no—
169C2×#92420—21no—
170C2×#92420—21no—
171C2×#92420—21no—
172C2×#92420—22no—
173C2×#92420—44no—
174C2×#92420—82no—
175#11758213—42no—
176D61⋊C8—42no—
177D61⋊C8—81no—
178C4⋊C4×C61—21no—
179C4⋊C4×C61—21no—
180C4⋊C4×C61—21no—
181C4⋊C4×C61—22no—
182C4⋊C4×C61C22⋊C411yescharacteristic
183C4⋊C4×C61C22×C411yescharacteristic
184C22⋊C4×C61—22no—
185C22⋊C4×C61C22⋊C412yes—
186C42×C61—21no—
187C42×C61C22×C411yescharacteristic
188C22⋊Dic61—22no—
189C22⋊Dic61C22⋊C412yes—
190#11758198—21no—
191#11758198—21no—
192#11758198C22⋊C412yes—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—22no—
197C4⋊Dic61C22⋊C411yescharacteristic
198C4⋊Dic61C22×C411yescharacteristic
199#11758196—22no—
200#11758196—21no—
201#11758196—22no—
202#11758196—22no—
203#11758196—22no—
204#11758196—22no—
205#11758196—22no—
206#11758196—21no—
207#11758196—22no—
208#11758196—22no—
209#11758196—21no—
210#11758196—21no—
211#11758196—21no—
212#11758196C22⋊C412yes—
213#11758196C22⋊C412yes—
214C4×Dic61—21no—
215C4×Dic61—22no—
216C4×Dic61—22no—
217C4×Dic61C22×C412yes—
218C4×Dic61C2×D411yescharacteristic
219C4×Dic61C2×D412yes—
220C4×Dic61C2×D411yescharacteristic
221C4×Dic61C2×D412yes—
222C4×Dic61C2411yescharacteristic
223C23×C61C2×C22⋊C412yes—
224C22×D61C2×C22⋊C411yescharacteristic
225Q8×C61—42no—
226Q8×C61—41no—
227Q8×C61—82no—
228D4×C61#10314yes—
229C2×C4×C61—21no—
230C2×C4×C61—22no—
231C2×C4×C61—21no—
232C2×C4×C61—42no—
233C2×C4×C61—82no—
234C2×C4×C61C2×C22⋊C412yes—
235C2×C4×C61C2×C22⋊C411yescharacteristic
236C2×C4×C61C2×C22⋊C411yescharacteristic
237C2×C4×C61C23×C411yescharacteristic
238C22⋊D61#10318yes—
239C2×Dic61—22no—
240C2×Dic61—22no—
241C2×Dic61—21no—
242C2×Dic61—21no—
243C2×Dic61—22no—
244C2×Dic61—82no—
245C2×Dic61C2×C22⋊C412yes—
246C2×Dic61C2×C22⋊C411yescharacteristic
247C2×Dic61C2×C22⋊C412yes—
248C2×Dic61C2×SD1614yes—
249C2×Dic61C2×Q1614yes—
250C2×Dic61C22×D411yescharacteristic
251C2×Dic61C22×D411yescharacteristic
252C4×D61—21no—
253C4×D61—21no—
254C4×D61—41no—
255C4×D61—82no—
256C4×D61C2×SD1612yes—
257C4×D61C2×Q1612yes—
258Dic122—41no—
259Dic122—41no—
260Dic122—41no—
261Dic122—42no—
262Dic122—82no—
263Dic122—82no—
264Dic122#10314yes—
265#92420—41no—
266#92420—41no—
267#92420—42no—
268#92420—42no—
269#92420—41no—
270#92420—41no—
271#17722—611nomaximal, Sylow
272C22×C61—22no—
273C22×C61—22no—
274C22×C61C2×#10314yes—
275C22×C61C22×C22⋊C411yescharacteristic
276C2×D61—21no—
277C2×D61—21no—
278C2×D61C2×#10312yes—
279C4×C61—21no—
280C4×C61—22no—
281C4×C61—21no—
282C4×C61—42no—
283C4×C61—41no—
284C4×C61—41no—
285C4×C61C2×#10312yes—
286Dic61—21no—
287Dic61—22no—
288Dic61—42no—
289Dic61—41no—
290Dic61—41no—
291Dic61C2×#10314yes—
292Dic61C22×SD1611yescharacteristic
293Dic61C22×Q1611yescharacteristic
294Dic61#57711yescharacteristic
295Dic61#57811yescharacteristic
296#3503—611no—
297#2988—611no—
298#2906—611no—
299#1550—611no—
300#1550—612no—
301#1548—612no—
302#1548—611no—
303#1542—611no—
304#1542—612no—
305C2×#333—612no—
306C2×#333—611no—
307C2×C61—24no—
308C2×C61#288811yescharacteristic
309C2×C61C22×#10311yescharacteristic
310C2×C61#289811yescharacteristic
311D61—22no—
312C2×#131—1221no—
313Q82—1221no—
314#557—611no—
315#557—611no—
316#554—1221no—
317#554—1222no—
318D4×Q8—612no—
319D4×Q8—612no—
320D4×Q8—611no—
321D4×Q8#692311yescharacteristic
322#544—1222no—
323#544—1221no—
324#533—612no—
325#533—611no—
326C2×C4⋊Q8—612no—
327C2×C4⋊Q8—611no—
328#520—1221no—
329C4×#41—611no—
330C8×Q8—1221no—
331C8×D4—1221no—
332C8×D4—1222no—
333#431—612no—
334#431—611no—
335#424—1222no—
336#424—1221no—
337#423—612no—
338#423—611no—
339C2×C4⋊C8—611no—
340C2×C4⋊C8—612no—
341C2×C4⋊C8—611no—
342C2×C4⋊C8—612no—
343#333—611no—
344#333—612no—
345#333—612no—
346#333—612no—
347#333—612no—
348#333—611no—
349#333—612no—
350#332—611no—
351#332—611no—
352#332—612no—
353C61#1772211yescharacteristic, complemented, Sylow
354C2×#41—611no—
355C22×Q8—612no—
356C22×Q8—612no—
357C22×Q8C2×#692312yes—
358#131—2444no—
359C2×M16—1222no—
360C2×M16—1221no—
361C22×C8—1222no—
362C22×C8—1221no—
363C22×C8—1222no—
364C22×C8—1221no—
365C4⋊Q8—21no—
366C4⋊Q8—611no—
367C4⋊Q8—612no—
368C4⋊Q8—611no—
369C4⋊Q8—612no—
370C4⋊Q8—612no—
371C4⋊Q8—611no—
372C4⋊Q8—1221no—
373C4⋊Q8—1221no—
374C4⋊Q8—1222no—
375C4⋊Q8—1222no—
376C4⋊Q8—1222no—
377C4⋊Q8—1222no—
378C4⋊Q8—1221no—
379C4⋊Q8C2×#692311yescharacteristic
380#125—1221no—
381#125—1222no—
382#125—1221no—
383#125—1222no—
384#125—1222no—
385#125—1221no—
386C22⋊Q8—22no—
387C22⋊Q8—611no—
388C22⋊Q8—612no—
389C22⋊Q8—611no—
390C22⋊Q8—1222no—
391C22⋊Q8—1222no—
392C22⋊Q8—1222no—
393C22⋊Q8—1222no—
394C22⋊Q8—1222no—
395C22⋊Q8—1221no—
396C22⋊Q8—1222no—
397C22⋊Q8C2×#692312yes—
398C4×Q8—611no—
399C4×Q8—611no—
400C4×Q8—612no—
401C4×Q8—612no—
402C4×Q8—611no—
403C4×Q8—1221no—
404C4×Q8—1221no—
405C4×Q8—1222no—
406C4×Q8—1221no—
407C4×Q8C2×#692311yescharacteristic
408C4×D4—21no—
409C4×D4—611no—
410C4×D4—612no—
411C4×D4—612no—
412C4×D4—1222no—
413C4×D4—1222no—
414C4×D4—1221no—
415C4×D4C2×#692311yescharacteristic
416#117—611no—
417#117—612no—
418#117—1221no—
419#117—1222no—
420C2×C4⋊C4—1221no—
421C2×C4⋊C4—1221no—
422C2×C4⋊C4—1222no—
423C2×C4⋊C4—1222no—
424C2×C42—612no—
425C2×C42—611no—
426C4⋊C8—611no—
427C4⋊C8—612no—
428C4⋊C8—611no—
429C4⋊C8—611no—
430C4⋊C8—612no—
431C4⋊C8—611no—
432C4⋊C8—1221no—
433C4⋊C8—1221no—
434C4⋊C8—1222no—
435C4⋊C8—1222no—
436C4⋊C8—1222no—
437C4⋊C8—1221no—
438C22⋊C8—1222no—
439C22⋊C8—1221no—
440C4×C8—1222no—
441C4×C8—1221no—
442#41—614no—
443#41—614no—
444C2×Q8—22no—
445C2×Q8—82no—
446C2×Q8—611no—
447C2×Q8—612no—
448C2×Q8—611no—
449C2×Q8—1222no—
450C2×Q8—1221no—
451C2×Q8—1222no—
452C2×Q8—4882no—
453C2×Q8—4882no—
454C2×Q8C22×#692312yes—
455C2×Q8C22×#692311yescharacteristic
456C2×D4—612no—
457C2×D4C22⋊#692311yescharacteristic
458C22×C4—22no—
459C22×C4—612no—
460C22×C4—611no—
461C22×C4—612no—
462C22×C4—611no—
463C22×C4—611no—
464C22×C4—1222no—
465C22×C4—1222no—
466C22×C4C22×#692312yes—
467M16—2442no—
468M16—2444no—
469C2×C8—1221no—
470C2×C8—1222no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8—1221no—
474C2×C8—1222no—
475C2×C8—4881no—
476C2×C8—4882no—
477C2×C8—2442no—
478C2×C8—2444no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—22no—
482C4⋊C4—21no—
483C4⋊C4—612no—
484C4⋊C4—612no—
485C4⋊C4—611no—
486C4⋊C4—611no—
487C4⋊C4—1221no—
488C4⋊C4—1222no—
489C4⋊C4—1221no—
490C4⋊C4—1222no—
491C4⋊C4—1222no—
492C4⋊C4—1222no—
493C4⋊C4—1222no—
494C4⋊C4—1222no—
495C4⋊C4—1221no—
496C4⋊C4—1222no—
497C4⋊C4—1221no—
498C4⋊C4—1222no—
499C4⋊C4—1221no—
500C4⋊C4—1221no—
501C4⋊C4—1221no—
502C4⋊C4—1221no—
503C4⋊C4—1222no—
504C4⋊C4C22⋊#692311yescharacteristic
505C4⋊C4C22×#692311yescharacteristic
506C22⋊C4—22no—
507C22⋊C4—612no—
508C22⋊C4—612no—
509C22⋊C4—1221no—
510C22⋊C4—1221no—
511C22⋊C4—1222no—
512C22⋊C4C22⋊#692312yes—
513C42—21no—
514C42—612no—
515C42—611no—
516C42—612no—
517C42—611no—
518C42—611no—
519C42—612no—
520C42—1222no—
521C42—1222no—
522C42—1221no—
523C42C22×#692311yescharacteristic
524C23—611no—
525C23C2×C22⋊#692312yes—
526Q8—41no—
527Q8—42no—
528Q8—82no—
529Q8—614no—
530Q8—4882no—
531Q8—4882no—
532Q8—2441no—
533Q8—2442no—
534Q8—2441no—
535Q8—2441no—
536D4—618no—
537D4#1451984014yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—42no—
542C2×C4—82no—
543C2×C4—614no—
544C2×C4—614no—
545C2×C4—612no—
546C2×C4—611no—
547C2×C4—611no—
548C2×C4—612no—
549C2×C4—612no—
550C2×C4—612no—
551C2×C4—611no—
552C2×C4—1221no—
553C2×C4—1221no—
554C2×C4—1222no—
555C2×C4—1222no—
556C2×C4—1221no—
557C2×C4—1221no—
558C2×C4—1222no—
559C2×C4—4882no—
560C2×C4—4882no—
561C2×C4—2441no—
562C2×C4C2×C22⋊#692312yes—
563C2×C4C2×C22⋊#692311yescharacteristic
564C2×C4C2×C22⋊#692311yescharacteristic
565C2×C4C23×#692311yescharacteristic
566C8—2441no—
567C8—2442no—
568C8—2441no—
569C8—2441no—
570C8—2441no—
571C8—2442no—
572C22—22no—
573C22—22no—
574C22—612no—
575C22—1221no—
576C22—1221no—
577C22C2×#1451984014yes—
578C22C22×C22⋊#692311yescharacteristic
579C4—21no—
580C4—22no—
581C4—21no—
582C4—41no—
583C4—42no—
584C4—41no—
585C4—611no—
586C4—611no—
587C4—614no—
588C4—611no—
589C4—611no—
590C4—1221no—
591C4—1222no—
592C4—2442no—
593C4—2441no—
594C4—2441no—
595C4C2×#1451984012yes—
596C2—24no—
597C2—1222no—
598C2#2146242911yescharacteristic
599C2C22×#1451984011yescharacteristic
600C2#2146244111yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6923
commutatorC2×C4×C61C23×C4
FrattiniC2×C4C23×#6923
FittingD4×Q8×C61C4
socleC22×C61C22×C22⋊C4
prime radicalC2×#11758224D4
cosocleC4×Dic61C24
hypercenterD4×Q8#6923
hypocenterC61#17722
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
3. #21461337C2semidirect
4. #21461336C2semidirect
5. #21461333C2semidirect
6. #21461332C2semidirect
7. #21461321C2semidirect
8. #21461320C2semidirect
9. D61⋊#333C2semidirect
10. C2×#16840622C2semidirect
353. C61#17722semidirect

Maximal subgroups

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

subgroupindex
#214627352
#214624662
#214613372
#214613362
#214613332
#214613322
#214613212
#214613202
D61⋊#3332
C2×#168406222
#214596302
#1772261