Finite Group Databasebeta

← #29670230

Subgroups — #29670230

Order 15616.

580 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458727C211yesmaximal, characteristic
2Q8×#11758202C211yesmaximal, characteristic, complemented
3#21457695C211yesmaximal, characteristic
4#21445493C211yesmaximal, characteristic, complemented
5#21445322C211yesmaximal, characteristic, complemented
6#21445275C211yesmaximal, characteristic, complemented
7#21445223C211yesmaximal, characteristic, complemented
8#21445216C211yesmaximal, characteristic
9#21445039C211yesmaximal, characteristic, complemented
10#21444984C211yesmaximal, characteristic
11#21444959C211yesmaximal, characteristic, complemented
12#21444951C211yesmaximal, characteristic
13#21444669C211yesmaximal, characteristic, complemented
14SD16⋊#92418C211yesmaximal, characteristic
15Q8×SD16×C61C211yesmaximal, characteristic
16Q8×Dic122C2211yescharacteristic
17#16840469C2211yescharacteristic
18#16840169—21no—
19#16840167C2211yescharacteristic
20#16840165—21no—
21#16840165C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840114C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840095—21no—
26#16840071—21no—
27#16840057—21no—
28#16840057C2211yescharacteristic
29#16840057C2211yescharacteristic
30#16840057C2211yescharacteristic
31#16840054—21no—
32#16840054C2211yescharacteristic
33#16840052C2211yescharacteristic
34#16840040—21no—
35#16840001—21no—
36#16839969C2211yescharacteristic
37#16839918—21no—
38#16839892C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839863—21no—
41C4×#11758202—21no—
42C4×#11758202C2211yescharacteristic
43D4×#92418C2211yescharacteristic
44C2×#14519701—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53#16839492C2211yescharacteristic
54D4⋊#92418C2211yescharacteristic
55#16839483C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#498—21no—
59C61×#498C2211yescharacteristic
60C61×#476—21no—
61C61×#476C2211yescharacteristic
62C61×#475—21no—
63C61×#475C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×SD16×C61—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758202—41no—
69C2×#11758202D411yescharacteristic
70C22×#92418—21no—
71#14519777—21no—
72#14519777C2311yescharacteristic
73#14519777C2311yescharacteristic
74#14519776—21no—
75#14519776—41no—
76#14519768—21no—
77#14519756—21no—
78#14519754—41no—
79Q8×Dic61—21no—
80C4×Dic122—21no—
81C4×Dic122C2311yescharacteristic
82D4×Dic61—21no—
83C2×C4⋊Dic61—21no—
84#14519711—21no—
85#14519711—21no—
86#14519710—21no—
87#14519710—21no—
88#14519710—21no—
89#14519710—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519708—21no—
94C4⋊#92418—21no—
95C4⋊#92418C2311yescharacteristic
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98#14519706—21no—
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519700—41no—
108#14519700D411yescharacteristic
109#14519698—21no—
110C22⋊#92418—21no—
111#14519687—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C22×Q8×C61—21no—
116C2×SD16×C61D411yescharacteristic
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61C2311yescharacteristic
122C4⋊Q8×C61C2311yescharacteristic
123C22⋊Q8×C61—21no—
124C22⋊Q8×C61—41no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61C2311yescharacteristic
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61—21no—
130C4×D4×C61C2311yescharacteristic
131#106×C61—21no—
132#106×C61—21no—
133#106×C61—21no—
134#106×C61—21no—
135#106×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61D411yescharacteristic
140#102×C61—21no—
141#102×C61D411yescharacteristic
142C4×C8×C61—21no—
143C4×C8×C61C2311yescharacteristic
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—41no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C22×Dic61—41no—
156C2×Dic122—21no—
157C2×Dic122—21no—
158C2×Dic122—41no—
159C2×Dic122C2×D411yescharacteristic
160SD16×C61Q1614yescomplemented
161C2×C8×C61—21no—
162C2×C8×C61C2×D411yescharacteristic
163C2×C8×C61C2×D411yescharacteristic
164C4⋊C4×C61—21no—
165C4⋊C4×C61—21no—
166C4⋊C4×C61—21no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—41no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C4⋊C4×C61C2×D411yescharacteristic
176C22⋊C4×C61—21no—
177C22⋊C4×C61—41no—
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—41no—
181C42×C61C2411yescharacteristic
182C22⋊Dic61—41no—
183#11758202—22no—
184#11758202—81no—
185#11758202—82no—
186C4⋊Dic61—21no—
187C4⋊Dic61—21no—
188C4⋊Dic61—21no—
189C4⋊Dic61—21no—
190C4⋊Dic61—41no—
191C4⋊Dic61—41no—
192C4⋊Dic61C2×D411yescharacteristic
193#11758196—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—21no—
197C4×Dic61—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—81no—
203C2×#92418C2×D411yescharacteristic
204C2×#92418C2×D411yescharacteristic
205C23×C61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—41no—
209Q8×C61—41no—
210Q8×C61—82no—
211Q8×C61—81no—
212Q8×C61—161no—
213Q8×C61C2×Q1612yes—
214Q8×C61#13712yes—
215D4×C61—21no—
216D4×C61C2×Q1612yes—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—41no—
223C2×C4×C61—41no—
224C2×C4×C61—41no—
225C2×C4×C61—41no—
226C2×C4×C61—82no—
227C2×C4×C61C22×D411yescharacteristic
228C2×C4×C61C22×D411yescharacteristic
229C2×C4×C61H(5,𝔽2)11yescharacteristic
230C2×Dic61—21no—
231C2×Dic61—21no—
232C2×Dic61—41no—
233C2×Dic61—41no—
234C2×Dic61—161no—
235Dic122—21no—
236Dic122—21no—
237Dic122—81no—
238Dic122—81no—
239Dic122—161no—
240C8×C61—21no—
241C8×C61—41no—
242C8×C61C2×Q1612yes—
243#92418—21no—
244#92418—21no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#92418—41no—
249#19813—611nomaximal, Sylow
250C22×C61—22no—
251C22×C61—41no—
252C22×C61—41no—
253C22×C61D4211yescharacteristic
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—81no—
261C4×C61—81no—
262C4×C61C22×Q1611yescharacteristic
263C4×C61C2×#13711yescharacteristic
264C4×C61#57711yescharacteristic
265C4×C61#57811yescharacteristic
266Dic61—41no—
267Dic61—41no—
268Dic61—81no—
269Dic61—81no—
270#3385—611no—
271Q8×Q16—611no—
272Q8×SD16D6111yescharacteristic
273#3299—611no—
274#1723—611no—
275#1692—611no—
276#1684—611no—
277#1676—611no—
278#1675—611no—
279#1645—611no—
280#1637—611no—
281#1633—611no—
282#1632—611no—
283#1584—611no—
284SD16⋊C8—611no—
285C2×C61—42no—
286C2×C61#326611yescharacteristic
287C2×C61D4×Q1611yescharacteristic
288C2×C61#328811yescharacteristic
289Q82—611no—
290Q82C2×D6111yescharacteristic
291#554—611no—
292D4×Q8C2×D6111yescharacteristic
293#501—1221no—
294#500—611no—
295#500—611no—
296#500—1221no—
297#498—21no—
298#498C2×D6111yescharacteristic
299#494—611no—
300#487—611no—
301#487—611no—
302#484—1221no—
303#479—1221no—
304#477—611no—
305#477—611no—
306#477—611no—
307#477—611no—
308#477—1221no—
309#477—1221no—
310#476—21no—
311#476—611no—
312#476C2×D6111yescharacteristic
313#475—21no—
314#475C2×D6111yescharacteristic
315#474—1221no—
316#467—1221no—
317#462—611no—
318#451—1221no—
319#446—611no—
320C8×Q8—611no—
321C8×Q8C2×D6111yescharacteristic
322#440—1221no—
323C4×Q16—611no—
324C4×Q16—1221no—
325C4×SD16—21no—
326C4×SD16C2×D6111yescharacteristic
327C8×D4—611no—
328C2×#103—1221no—
329#335—611no—
330#335—611no—
331#333—611no—
332#333—611no—
333#333—611no—
334#332—611no—
335Q8⋊C8—611no—
336Q8⋊C8—611no—
337Q8⋊C8—611no—
338D4⋊C8—611no—
339#322—611no—
340C61#1981311yescharacteristic, complemented, Sylow
341C22×Q8—21no—
342C2×Q16—611no—
343C2×Q16—1221no—
344C2×Q16—2441no—
345C2×SD16C22⋊D6111yescharacteristic
346C22×C8—1221no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—41no—
351C4⋊Q8—611no—
352C4⋊Q8—611no—
353C4⋊Q8—1221no—
354C4⋊Q8—1221no—
355C4⋊Q8—2441no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#125—1221no—
359C22⋊Q8—21no—
360C22⋊Q8—41no—
361C22⋊Q8—1221no—
362C22⋊Q8—2441no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—611no—
366C4×Q8—1221no—
367C4×Q8—1221no—
368C4×Q8C22×D6111yescharacteristic
369C4×Q8C22×D6111yescharacteristic
370C4×D4—21no—
371C4×D4—1221no—
372C4×D4C22×D6111yescharacteristic
373C2×C4⋊C4—1221no—
374#107—611no—
375#107—1221no—
376#107—1221no—
377#107—1221no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107—1221no—
383#106—21no—
384#106—21no—
385#106—21no—
386#106—21no—
387#106—1221no—
388#106C22⋊D6111yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—611no—
391C4⋊C8—611no—
392C4⋊C8—611no—
393C4⋊C8—1221no—
394C4⋊C8—1221no—
395C4⋊C8C22×D6111yescharacteristic
396#103—21no—
397#103—611no—
398#103—611no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—1221no—
406#103—2441no—
407#103C22⋊D6111yescharacteristic
408#102—21no—
409#102—1221no—
410#102C22⋊D6111yescharacteristic
411C22⋊C8—1221no—
412#97—1221no—
413C4×C8—21no—
414C4×C8—611no—
415C4×C8—611no—
416C4×C8—1221no—
417C4×C8C22×D6111yescharacteristic
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—41no—
421C2×Q8—41no—
422C2×Q8—42no—
423C2×Q8—81no—
424C2×Q8—611no—
425C2×Q8—1221no—
426C2×Q8—1221no—
427C2×Q8—2441no—
428C2×Q8D4×D6111yescharacteristic
429C2×Q8C2×C22⋊D6111yescharacteristic
430C2×D4C2×C22⋊D6111yescharacteristic
431C22×C4—21no—
432C22×C4—41no—
433C22×C4—2441no—
434Q16—1222no—
435Q16—4881no—
436Q16—4882no—
437SD16#1175820214yescomplemented
438C2×C8—21no—
439C2×C8—611no—
440C2×C8—611no—
441C2×C8—1221no—
442C2×C8—1221no—
443C2×C8—1221no—
444C2×C8—1221no—
445C2×C8—4881no—
446C2×C8C2×C22⋊D6111yescharacteristic
447C2×C8C2×C22⋊D6111yescharacteristic
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—611no—
458C4⋊C4—1221no—
459C4⋊C4—1221no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—2441no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4D4×D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C4⋊C4C2×C22⋊D6111yescharacteristic
470C22⋊C4—21no—
471C22⋊C4—41no—
472C22⋊C4—2441no—
473C42—21no—
474C42—21no—
475C42—41no—
476C42—1221no—
477C42—2441no—
478C42C23×D6111yescharacteristic
479C23—21no—
480Q8—21no—
481Q8—21no—
482Q8—41no—
483Q8—41no—
484Q8—82no—
485Q8—81no—
486Q8—161no—
487Q8—1221no—
488Q8—1221no—
489Q8—9761no—
490Q8—4881no—
491Q8—4881no—
492Q8C2×#1175820212yes—
493Q8#1451980612yes—
494D4—21no—
495D4C2×#1175820212yes—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—82no—
506C2×C4—1221no—
507C2×C4—1221no—
508C2×C4—9761no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4#1451982211yescharacteristic
514C8—21no—
515C8—41no—
516C8—1221no—
517C8—1221no—
518C8—2441no—
519C8—2441no—
520C8—2441no—
521C8—2441no—
522C8C2×#1175820212yes—
523C22—22no—
524C22—41no—
525C22—41no—
526C22D4×C22⋊D6111yescharacteristic
527C4—22no—
528C4—21no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—81no—
534C4—81no—
535C4—4881no—
536C4—4881no—
537C4—2441no—
538C4—2441no—
539C4C22×#1175820211yescharacteristic
540C4C2×#1451980611yescharacteristic
541C4#1684056811yescharacteristic
542C4#1684057711yescharacteristic
543C2—42no—
544C2#2145732911yescharacteristic
545C2D4×#1175820211yescharacteristic
546C2#2145758111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×SD16×C61C2
socleC22×C61D42
prime radicalC2×D4×C61C2×D4
hypercenterQ8×SD16D61
hypocenterC61#19813
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×SD16

Complements

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

NH ≅ G/Ntype
2. Q8×#11758202C2semidirect
4. #21445493C2semidirect
5. #21445322C2semidirect
6. #21445275C2semidirect
7. #21445223C2semidirect
9. #21445039C2semidirect
11. #21444959C2semidirect
13. #21444669C2semidirect
160. SD16×C61Q16semidirect
340. C61#19813semidirect
437. SD16#11758202semidirect

Maximal subgroups

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

subgroupindex
#214587272
Q8×#117582022
#214576952
#214454932
#214453222
#214452752
#214452232
#214452162
#214450392
#214449842
#214449592
#214449512
#214446692
SD16⋊#924182
Q8×SD16×C612
#1981361