Finite Group Databasebeta

← #226125

Subgroups — #226125

Order 512.

674 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22805C211yesmaximal, characteristic
2#22770C211yesmaximal, characteristic
3#22725C211yesmaximal, characteristic, complemented
4#22665C211yesmaximal, characteristic
5C2×#1583C211yesmaximal, characteristic
6#9613C211yesmaximal, characteristic, complemented
7#9611C211yesmaximal, characteristic, complemented
8#8891C211yesmaximal, characteristic, complemented
9#8890C211yesmaximal, characteristic
10#8889C211yesmaximal, characteristic
11#8887C211yesmaximal, characteristic, complemented
12#8819C211yesmaximal, characteristic, complemented
13#8650C211yesmaximal, characteristic
14#8645C211yesmaximal, characteristic, complemented
15#8285C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C4×SD16C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×#428—21no—
20#2355C2211yescharacteristic
21#2355C2211yescharacteristic
22#2354C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1968—21no—
25#1949—21no—
26#1949—21no—
27#1948C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1933—21no—
32#1927—21no—
33#1926—21no—
34#1919—21no—
35#1877—21no—
36#1854—21no—
37#1852—21no—
38#1846—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50C4×#106—21no—
51C4×#106—21no—
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102—21no—
56#1583C418yescomplemented
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
69C22×M16—21no—
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C22×C4⋊C4—21no—
75C8×Q8—22no—
76C8×Q8C2×C414yescomplemented
77C4×SD16C2×C414yes—
78C4×SD16Q814yes—
79#436—22no—
80#436C2×C414yes—
81#428—44no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—41no—
93C2×#106D411yescharacteristic
94C2×#106D411yescharacteristic
95C2×#106D411yescharacteristic
96C2×C4⋊C8—21no—
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×#103—21no—
102C2×#103—21no—
103C2×#103D411yescharacteristic
104C2×#102—21no—
105C2×#102—21no—
106C2×#102D411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391—21no—
116#390—21no—
117#389—21no—
118#388—41no—
119#387—41no—
120#387—41no—
121#385—41no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#382—21no—
127#382—21no—
128#380—21no—
129#380—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137#340—21no—
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—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#335—22no—
153#334C2×C414yescomplemented
154Q8⋊C8C2×C414yescomplemented
155D4⋊C8C2×C414yes—
156#322C2×C414yescomplemented
157C22×Q8C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—41no—
161C2×SD16—41no—
162C2×SD16—41no—
163C2×SD16C4⋊C414yes—
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8C22×C412yes—
179C4×Q8C2×Q812yes—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—42no—
183C4×D4—81no—
184C4×D4C22×C412yes—
185C4×D4C2×Q812yes—
186#117—42no—
187#117—81no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
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—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—81no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C22⋊C4—21no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—41no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42—41no—
220C2×C42C2411yescharacteristic
221#107—42no—
222#107—42no—
223#107—81no—
224#107—81no—
225#107—81no—
226#106—22no—
227#106—42no—
228#106—42no—
229#106—42no—
230#106—81no—
231#106—81no—
232#106—82no—
233#106C4⋊C414yes—
234#106SD1614yescomplemented
235C4⋊C8—24no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8C22×C412yes—
243C4⋊C8C22×C412yes—
244#103—81no—
245#103—81no—
246#103C4⋊C414yes—
247#102—42no—
248#102—81no—
249#102C4⋊C414yes—
250C22⋊C8—24no—
251C4×C8—24no—
252C4×C8—21no—
253C4×C8—42no—
254C4×C8—81no—
255C4×C8C22×C412yes—
256C4×C8C22×C412yes—
257C4×C8C22×C412yes—
258C4×C8C2×Q812yes—
259C4×C8#4112yes—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8C2×C4⋊C412yes—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4C2×C4⋊C412yes—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—82no—
301C22×C4—82no—
302C22×C4—161no—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4H(5,𝔽2)11yescharacteristic
306SD16—44no—
307M16—48no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8C2×C4⋊C412yes—
330C2×C8C2×C4⋊C412yes—
331C2×C8C4×D412yes—
332C2×C8C4×D412yes—
333C2×C8C2×SD1612yes—
334C2×C8#13512yes—
335C4⋊C4—22no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—22no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—82no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4C2×C4⋊C412yes—
367C4⋊C4C2×C4⋊C412yes—
368C4⋊C4C2×SD1612yes—
369C4⋊C4C2×SD1612yes—
370C22⋊C4—22no—
371C22⋊C4—22no—
372C22⋊C4—82no—
373C22⋊C4—82no—
374C22⋊C4—161no—
375C42—22no—
376C42—22no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—82no—
382C42—161no—
383C42—161no—
384C42C23×C411yescharacteristic
385C42C22×Q811yescharacteristic
386C42C2×#4111yescharacteristic
387C422−1+411yescharacteristic
388C23—22no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23D4211yescharacteristic
396Q8—22no—
397Q8—22no—
398Q8—41no—
399Q8—41no—
400D4—22no—
401D4—22no—
402D4—41no—
403D4—41no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—164no—
444C2×C4C22×C4⋊C411yescharacteristic
445C2×C4C2×C4×D411yescharacteristic
446C2×C4#52011yescharacteristic
447C2×C4#54711yescharacteristic
448C2×C4D4×Q811yescharacteristic
449C2×C4#55411yescharacteristic
450C2×C4C22×SD1611yescharacteristic
451C2×C4C2×#13511yescharacteristic
452C2×C4#57511yescharacteristic
453C2×C4#57511yescharacteristic
454C2×C4#57611yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57711yescharacteristic
457C2×C4#57811yescharacteristic
458C8—22no—
459C8—22no—
460C8—22no—
461C8—42no—
462C8—42no—
463C8—42no—
464C8—81no—
465C8—81no—
466C8C4×SD1614yes—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—81no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22D4×C4⋊C411yescharacteristic
475C22#326811yescharacteristic
476C22#326911yescharacteristic
477C22D4×SD1611yescharacteristic
478C22#328411yescharacteristic
479C22#328811yescharacteristic
480C22#328911yescharacteristic
481C4—21no—
482C4—21no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—161no—
491C4—161no—
492C4—161no—
493C4—161no—
494C4#291111yescharacteristic
495C4#291311yescharacteristic
496C4C2×C4×SD1611yescharacteristic
497C4#295211yescharacteristic
498C4#334711yescharacteristic
499C4#334911yescharacteristic
500C4Q8×SD1611yescharacteristic
501C4#338111yescharacteristic
502C2—82no—
503C2#1980711yescharacteristic
504C2#1980811yescharacteristic
505C2#1987911yescharacteristic
506C2#1988011yescharacteristic
507C2#2266011yescharacteristic
508C2C4⋊C4×SD1611yescharacteristic
509C2#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
3. #22725C2semidirect
6. #9613C2semidirect
7. #9611C2semidirect
8. #8891C2semidirect
11. #8887C2semidirect
12. #8819C2semidirect
14. #8645C2semidirect
15. #8285C2semidirect
56. #1583C4semidirect
76. C8×Q8C2×C4semidirect
153. #334C2×C4semidirect
154. Q8⋊C8C2×C4semidirect
156. #322C2×C4semidirect
234. #106SD16semidirect

Maximal subgroups

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

subgroupindex
#228052
#227702
#227252
#226652
C2×#15832
#96132
#96112
#88912
#88902
#88892
#88872
#88192
#86502
#86452
#82852