Finite Group Databasebeta

← #29670231

Subgroups — #29670231

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#21458725C211yesmaximal, characteristic
2Q8×#11758201C211yesmaximal, characteristic
3#21457697C211yesmaximal, characteristic
4#21445495C211yesmaximal, characteristic, complemented
5#21445327C211yesmaximal, characteristic, complemented
6#21445277C211yesmaximal, characteristic, complemented
7#21445221C211yesmaximal, characteristic, complemented
8#21445215C211yesmaximal, characteristic
9#21445040C211yesmaximal, characteristic, complemented
10#21444982C211yesmaximal, characteristic
11#21444959C211yesmaximal, characteristic, complemented
12#21444952C211yesmaximal, characteristic
13#21444670C211yesmaximal, characteristic, complemented
14#21444629C211yesmaximal, characteristic
15Q8×Q16×C61C211yesmaximal, characteristic, complemented
16#16840472C2211yescharacteristic
17Q8×D244C2211yescharacteristic
18#16840168—21no—
19#16840165C2211yescharacteristic
20#16840158—21no—
21#16840158C2211yescharacteristic
22#16840141C2211yescharacteristic
23#16840113C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840099—21no—
26#16840069—21no—
27#16840057C2211yescharacteristic
28#16840057C2211yescharacteristic
29#16840049—21no—
30#16840049C2211yescharacteristic
31#16840049C2211yescharacteristic
32#16840047—21no—
33#16840047C2211yescharacteristic
34#16840039—21no—
35#16840002—21no—
36#16839972C2211yescharacteristic
37#16839916—21no—
38#16839893C2211yescharacteristic
39Q8×#92418C2211yescharacteristic
40#16839860—21no—
41C4×#11758201—21no—
42C4×#11758201C2211yescharacteristic
43C8×D244C2211yescharacteristic
44#103×D61—21no—
45#16839515C2211yescharacteristic
46#16839514C2211yescharacteristic
47#16839511C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53Q8⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55#16839484C2211yescharacteristic
56Q82×C61C2211yescharacteristic
57Q82×C61C2211yescharacteristic
58C61×#500—21no—
59C61×#500C2211yescharacteristic
60C61×#477—21no—
61C61×#477—21no—
62C61×#477C2211yescharacteristic
63C61×#477C2211yescharacteristic
64C8×Q8×C61C2211yescharacteristic
65C4×Q16×C61—21no—
66C4×Q16×C61C2211yescharacteristic
67C2×Q8×D61—21no—
68C2×#11758202—21no—
69C2×#11758202—41no—
70C2×#11758201D411yescharacteristic
71C2×C8×D61—21no—
72#14519777—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519757—21no—
77#14519757—41no—
78#14519755—41no—
79#14519753—21no—
80C4×D244—21no—
81C4×D244—21no—
82C4×D244C2311yescharacteristic
83C4⋊C4×D61—21no—
84#14519710—21no—
85#14519710—21no—
86#14519710—21no—
87#14519710—21no—
88#14519708—21no—
89#14519708—21no—
90#14519708—21no—
91#14519708—21no—
92#14519708—21no—
93#14519708D411yescharacteristic
94C4⋊#92418—21no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519702—21no—
100#14519702—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—41no—
105#14519701D411yescharacteristic
106#14519700—21no—
107#14519697—21no—
108#14519697—21no—
109#14519697D411yescharacteristic
110#14519689—21no—
111#14519688—21no—
112C4×#92418—21no—
113C4×#92418C2311yescharacteristic
114C4×#92418C2311yescharacteristic
115C2×Q16×C61D411yescharacteristic
116C4⋊Q8×C61—21no—
117C4⋊Q8×C61—21no—
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61—41no—
121C4⋊Q8×C61—41no—
122C4⋊Q8×C61C2311yescharacteristic
123C4⋊Q8×C61C2311yescharacteristic
124C4×Q8×C61—21no—
125C4×Q8×C61—21no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×Q8×C61C2311yescharacteristic
131#107×C61—21no—
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61D411yescharacteristic
136C4⋊C8×C61—21no—
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#103×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×C61C2×D411yescharacteristic
149C2×Q8×C61C2×D411yescharacteristic
150C2×Q8×C61C2×D411yescharacteristic
151Q8×D61—42no—
152Q8×D61—81no—
153C2×D244C2×D411yescharacteristic
154C2×C4×D61—21no—
155C2×C4×D61—41no—
156C2×C4×D61—41no—
157C2×Dic122—21no—
158C2×Dic122—21no—
159C2×Dic122—41no—
160Q16×C61—22no—
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—21no—
171C4⋊C4×C61—41no—
172C4⋊C4×C61—41no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C42×C61—21no—
179C42×C61—21no—
180C42×C61—21no—
181C42×C61—41no—
182C42×C61—41no—
183C42×C61C2411yescharacteristic
184#11758202—81no—
185#11758202—82no—
186#11758201Q1614yescomplemented
187#11758198—21no—
188#11758198—41no—
189#11758198—41no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194C4⋊Dic61—41no—
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—41no—
197#11758196—41no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418C2×D411yescharacteristic
203C2×#92418C2×D411yescharacteristic
204C8×D61—81no—
205C22×D61—21no—
206Q8×C61—21no—
207Q8×C61—21no—
208Q8×C61—21no—
209Q8×C61—21no—
210Q8×C61—81no—
211Q8×C61—81no—
212Q8×C61—161no—
213Q8×C61—161no—
214Q8×C61C2×Q1612yes—
215Q8×C61#13712yes—
216C2×C4×C61—21no—
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×C61C22×D411yescharacteristic
225C2×C4×C61C22×D411yescharacteristic
226C2×C4×C61H(5,𝔽2)11yescharacteristic
227C2×Dic61—21no—
228C2×Dic61—41no—
229C2×Dic61—41no—
230D244—21no—
231D244C2×Q1612yes—
232C4×D61—41no—
233C4×D61—41no—
234C4×D61—82no—
235C4×D61—161no—
236Dic122—41no—
237Dic122—41no—
238Dic122—82no—
239Dic122—81no—
240C8×C61—21no—
241C8×C61—21no—
242C8×C61—41no—
243#92418—21no—
244#92418—41no—
245#92418—41no—
246#92418—41no—
247#92418—41no—
248#92418C2×Q1612yes—
249#19813—611nomaximal, Sylow
250C22×C61D4211yescharacteristic
251C2×D61—22no—
252C2×D61—41no—
253C2×D61—41no—
254C4×C61—22no—
255C4×C61—22no—
256C4×C61—21no—
257C4×C61—41no—
258C4×C61—41no—
259C4×C61—41no—
260C4×C61—41no—
261C4×C61—81no—
262C4×C61—81no—
263C4×C61C22×Q1611yescharacteristic
264C4×C61C2×#13711yescharacteristic
265C4×C61#57711yescharacteristic
266C4×C61#57811yescharacteristic
267Dic61—41no—
268Dic61—81no—
269Dic61—81no—
270#3385—611no—
271Q8×Q16D6111yescharacteristic, complemented
272Q8×SD16—611no—
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#326611yescharacteristic
286C2×C61D4×Q1611yescharacteristic
287C2×C61#328811yescharacteristic
288D61—42no—
289Q82C2×D6111yescharacteristic
290Q82C2×D6111yescharacteristic
291#554—611no—
292D4×Q8—611no—
293#501—1221no—
294#500—21no—
295#500—611no—
296#500C2×D6111yescharacteristic
297#498—611no—
298#498—1221no—
299#494—611no—
300#487—611no—
301#487—611no—
302#484—1221no—
303#479—1221no—
304#477—21no—
305#477—21no—
306#477—611no—
307#477—611no—
308#477C2×D6111yescharacteristic
309#477C2×D6111yescharacteristic
310#476—611no—
311#476—611no—
312#476—1221no—
313#475—611no—
314#475—1221no—
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—21no—
324C4×Q16C2×D6111yescharacteristic
325C4×SD16—611no—
326C4×SD16—1221no—
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—1221no—
342C2×Q16—1221no—
343C2×Q16—2441no—
344C2×Q16C22⋊D6111yescharacteristic
345C2×SD16—611no—
346C22×C8—1221no—
347C4⋊Q8—21no—
348C4⋊Q8—21no—
349C4⋊Q8—21no—
350C4⋊Q8—21no—
351C4⋊Q8—41no—
352C4⋊Q8—41no—
353C4⋊Q8—611no—
354C4⋊Q8—611no—
355C4⋊Q8—1221no—
356C4⋊Q8C22×D6111yescharacteristic
357C4⋊Q8C22×D6111yescharacteristic
358#125—1221no—
359C22⋊Q8—1221no—
360C22⋊Q8—1221no—
361C22⋊Q8—2441no—
362C22⋊Q8—2441no—
363C4×Q8—21no—
364C4×Q8—21no—
365C4×Q8—21no—
366C4×Q8—21no—
367C4×Q8C22×D6111yescharacteristic
368C4×Q8C22×D6111yescharacteristic
369C4×Q8C22×D6111yescharacteristic
370C4×D4—611no—
371C4×D4—1221no—
372C4×D4—1221no—
373C2×C4⋊C4—1221no—
374#107—21no—
375#107—21no—
376#107—21no—
377#107—21no—
378#107—1221no—
379#107—1221no—
380#107—1221no—
381#107—1221no—
382#107C22⋊D6111yescharacteristic
383#106—611no—
384#106—1221no—
385#106—1221no—
386#106—1221no—
387#106—1221no—
388#106—1221no—
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—21no—
398#103—611no—
399#103—1221no—
400#103—1221no—
401#103—1221no—
402#103—1221no—
403#103—1221no—
404#103—1221no—
405#103—2441no—
406#103C22⋊D6111yescharacteristic
407#103C22⋊D6111yescharacteristic
408#102—611no—
409#102—1221no—
410#102—1221no—
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—1221no—
423C2×Q8—1221no—
424C2×Q8—4881no—
425C2×Q8—2441no—
426C2×Q8—2442no—
427C2×Q8D4×D6111yescharacteristic
428C2×Q8C2×C22⋊D6111yescharacteristic
429C2×Q8C2×C22⋊D6111yescharacteristic
430C2×D4—611no—
431C22×C4—1221no—
432C22×C4—2441no—
433C22×C4—2441no—
434Q16—22no—
435Q16—4881no—
436Q16—4882no—
437SD16—614no—
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—21no—
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—41no—
458C4⋊C4—41no—
459C4⋊C4—611no—
460C4⋊C4—1221no—
461C4⋊C4—1221no—
462C4⋊C4—1221no—
463C4⋊C4—1221no—
464C4⋊C4—2441no—
465C4⋊C4—2441no—
466C4⋊C4—2441no—
467C4⋊C4D4×D6111yescharacteristic
468C4⋊C4C2×C22⋊D6111yescharacteristic
469C4⋊C4C2×C22⋊D6111yescharacteristic
470C22⋊C4—1221no—
471C22⋊C4—2441no—
472C22⋊C4—2441no—
473C42—21no—
474C42—21no—
475C42—21no—
476C42—41no—
477C42—41no—
478C42C23×D6111yescharacteristic
479C23—1221no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—21no—
484Q8—81no—
485Q8—81no—
486Q8—161no—
487Q8—161no—
488Q8—4881no—
489Q8—4882no—
490Q8—2441no—
491Q8—2441no—
492Q8Q16×D6112yes—
493Q8#1451980812yes—
494D4—612no—
495D4—1221no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—1221no—
505C2×C4—9761no—
506C2×C4—4882no—
507C2×C4—2441no—
508C2×C4—2441no—
509C2×C4—2441no—
510C2×C4—2441no—
511C2×C4C2×D4×D6111yescharacteristic
512C2×C4C22×C22⋊D6111yescharacteristic
513C2×C4#1451982211yescharacteristic
514C8—21no—
515C8—21no—
516C8—41no—
517C8—612no—
518C8—1221no—
519C8—2441no—
520C8—2441no—
521C8—2441no—
522C8—2441no—
523C22—1222no—
524C22—2441no—
525C22—2441no—
526C22D4×C22⋊D6111yescharacteristic
527C4—22no—
528C4—21no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—81no—
535C4—81no—
536C4—4881no—
537C4—4881no—
538C4—2441no—
539C4C2×Q16×D6111yescharacteristic
540C4C2×#1451980811yescharacteristic
541C4#1684057011yescharacteristic
542C4#1684057511yescharacteristic
543C2—2442no—
544C2#2145733011yescharacteristic
545C2Q16×C22⋊D6111yescharacteristic
546C2#2145758311yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingQ8×Q16×C61C2
socleC22×C61D42
prime radicalC2×D244C2×D4
hypercenterQ8×Q16D61
hypocenterC61#19813
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ Q8×Q16

Complements

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

NH ≅ G/Ntype
4. #21445495C2semidirect
5. #21445327C2semidirect
6. #21445277C2semidirect
7. #21445221C2semidirect
9. #21445040C2semidirect
11. #21444959C2semidirect
13. #21444670C2semidirect
15. Q8×Q16×C61C2semidirect
186. #11758201Q16semidirect
271. Q8×Q16D61semidirect
340. C61#19813semidirect

Maximal subgroups

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

subgroupindex
#214587252
Q8×#117582012
#214576972
#214454952
#214453272
#214452772
#214452212
#214452152
#214450402
#214449822
#214449592
#214449522
#214446702
#214446292
Q8×Q16×C612
#1981361