Finite Group Databasebeta

← #225558

Subgroups — #225558

Order 512.

737 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#106C211yesmaximal, characteristic, complemented
2#22703C211yesmaximal, characteristic
3C4⋊C4×SD16C211yesmaximal, characteristic
4#22671C211yesmaximal, characteristic
5C2×SD16⋊C8C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic, complemented
7#9615C211yesmaximal, characteristic, complemented
8#8896C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic
10#8888C211yesmaximal, characteristic, complemented
11#8886C211yesmaximal, characteristic
12#8819C211yesmaximal, characteristic, complemented
13#8647C211yesmaximal, characteristic, complemented
14#8642C211yesmaximal, characteristic
15#8290C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×#440—21no—
18C2×C4×SD16—21no—
19C2×C4×SD16C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21Q8×C4⋊C4C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24D4×C4⋊C4C2211yescharacteristic
25#1948—21no—
26#1948—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1944—21no—
32#1941—21no—
33#1865—21no—
34#1864—21no—
35#1863—21no—
36#1862—21no—
37#1853—21no—
38#1853—21no—
39#1845C2211yescharacteristic
40#1844—21no—
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1807—21no—
47#1803C2211yescharacteristic
48#1803C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51C4×#106—21no—
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C2×#340—21no—
56SD16⋊C8C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×SD16D411yescharacteristic
69C23×C8—21no—
70C2×C4×Q8—21no—
71C2×C4×Q8—21no—
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2311yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C8×Q8—22no—
79C8×Q8C2×C414yescomplemented
80#440—44no—
81C4×SD16—44no—
82C4×SD16C2×C414yes—
83C4×SD16Q814yes—
84C8×D4—22no—
85C8×D4C2×C414yes—
86C2×#107—21no—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—41no—
97C2×#106D411yescharacteristic
98C2×#106D411yescharacteristic
99C2×#106D411yescharacteristic
100C2×C4⋊C8—21no—
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×#103—21no—
106C2×#103—21no—
107C2×#103D411yescharacteristic
108C2×#102—21no—
109C2×#102—21no—
110C2×#102D411yescharacteristic
111C2×C22⋊C8—21no—
112C2×#97—21no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#389—21no—
119#389—21no—
120#389—21no—
121#389—21no—
122#389—41no—
123#389—41no—
124#385—41no—
125#385—41no—
126#384—21no—
127#384—21no—
128#384C2311yescharacteristic
129#384C2311yescharacteristic
130#384C2311yescharacteristic
131#384C2311yescharacteristic
132#380—21no—
133#380—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—41no—
149#340D411yescharacteristic
150#340D411yescharacteristic
151#336—21no—
152#336—21no—
153#335C2×C414yescomplemented
154#334—22no—
155Q8⋊C8C2×C414yescomplemented
156D4⋊C8C2×C414yes—
157#322C2×C414yescomplemented
158C22×Q8C2×D411yescharacteristic
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—41no—
162C23×C4—41no—
163C2×SD16C4⋊C414yes—
164C2×SD16SD1614yes—
165C2×SD16SD1614yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—24no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—81no—
173C22×C8—81no—
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×Q8—81no—
183C4×Q8—81no—
184C4×Q8C22×C412yes—
185C4×Q8C2×Q812yes—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—81no—
191C4×D4—81no—
192C4×D4C22×C412yes—
193C4×D4C2×Q812yes—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—41no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4—81no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C4⋊C4C2×D411yescharacteristic
222C2×C4⋊C4C2×D411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—41no—
225C2×C22⋊C4—41no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—41no—
230C2×C42—41no—
231C2×C42C2411yescharacteristic
232#107—42no—
233#107—81no—
234#106—22no—
235#106—41no—
236#106—42no—
237#106—42no—
238#106—81no—
239#106—81no—
240#106—81no—
241#106—81no—
242#106—81no—
243#106—81no—
244#106—82no—
245#106C4⋊C414yes—
246C4⋊C8—24no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8C22×C412yes—
254C4⋊C8C22×C412yes—
255#103—42no—
256#103—42no—
257#103C4⋊C414yes—
258#102—42no—
259#102—42no—
260#102C4⋊C414yes—
261C22⋊C8—24no—
262#97—42no—
263C4×C8—24no—
264C4×C8—21no—
265C4×C8—42no—
266C4×C8C22×C412yes—
267C4×C8C22×C412yes—
268C4×C8C22×C412yes—
269C4×C8C2×Q812yes—
270C4×C8#4112yes—
271#95—21no—
272#95—21no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277C24—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8C2×C4⋊C412yes—
282C2×Q8C2×SD1612yes—
283C2×Q8C2×SD1612yes—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4C2×C4⋊C412yes—
288C2×D4C2×SD1612yes—
289C2×D4C2×SD1612yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—161no—
312C22×C4—161no—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4H(5,𝔽2)11yescharacteristic
316SD16#106116yescomplemented
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—84no—
339C2×C8C2×C4⋊C412yes—
340C2×C8C2×C4⋊C412yes—
341C2×C8C4×D412yes—
342C2×C8C4×D412yes—
343C2×C8C2×SD1612yes—
344C2×C8C2×SD1612yes—
345C4⋊C4—22no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4—82no—
375C4⋊C4—81no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—161no—
381C4⋊C4—161no—
382C4⋊C4C2×C4⋊C412yes—
383C4⋊C4C2×C4⋊C412yes—
384C4⋊C4#13712yes—
385C22⋊C4—22no—
386C22⋊C4—22no—
387C22⋊C4—82no—
388C22⋊C4—82no—
389C42—22no—
390C42—22no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—82no—
396C42—82no—
397C42C23×C411yescharacteristic
398C42C22×Q811yescharacteristic
399C42C2×#4111yescharacteristic
400C422−1+411yescharacteristic
401C23—22no—
402C23—22no—
403C23—22no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23D4211yescharacteristic
409Q8—22no—
410Q8—22no—
411Q8—22no—
412Q8C2×#10614yes—
413D4—22no—
414D4—22no—
415D4—22no—
416D4C2×#10614yes—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—164no—
457C2×C4—164no—
458C2×C4C22×C4⋊C411yescharacteristic
459C2×C4C2×C4×D411yescharacteristic
460C2×C4#52011yescharacteristic
461C2×C4#54711yescharacteristic
462C2×C4D4×Q811yescharacteristic
463C2×C4#55411yescharacteristic
464C2×C4C22×SD1611yescharacteristic
465C2×C4C22×SD1611yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57611yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57711yescharacteristic
472C8—22no—
473C8—22no—
474C8—22no—
475C8—42no—
476C8—42no—
477C8—42no—
478C8—81no—
479C8—81no—
480C8C2×#10614yes—
481C22—24no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22D4×C4⋊C411yescharacteristic
489C22#326611yescharacteristic
490C22#326811yescharacteristic
491C22D4×SD1611yescharacteristic
492C22D4×SD1611yescharacteristic
493C22#328811yescharacteristic
494C22#328911yescharacteristic
495C4—24no—
496C4—21no—
497C4—21no—
498C4—42no—
499C4—42no—
500C4—42no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—161no—
505C4—161no—
506C4—161no—
507C4—161no—
508C4C22×#10611yescharacteristic
509C4#291211yescharacteristic
510C4C4×#13711yescharacteristic
511C4#294611yescharacteristic
512C4#333511yescharacteristic
513C4#333511yescharacteristic
514C4#338511yescharacteristic
515C4#338811yescharacteristic
516C2—44no—
517C2#1976511yescharacteristic
518C2#1976711yescharacteristic
519C2#1980511yescharacteristic
520C2#1980711yescharacteristic
521C2D4×#10611yescharacteristic
522C2#2267811yescharacteristic
523C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×#106C2semidirect
6. #9618C2semidirect
7. #9615C2semidirect
8. #8896C2semidirect
10. #8888C2semidirect
12. #8819C2semidirect
13. #8647C2semidirect
15. #8290C2semidirect
56. SD16⋊C8C4semidirect
79. C8×Q8C2×C4semidirect
153. #335C2×C4semidirect
155. Q8⋊C8C2×C4semidirect
157. #322C2×C4semidirect
316. SD16#106semidirect

Maximal subgroups

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

subgroupindex
Q8×#1062
#227032
C4⋊C4×SD162
#226712
C2×SD16⋊C82
#96182
#96152
#88962
#88942
#88882
#88862
#88192
#86472
#86422
#82902