Finite Group Databasebeta

← #225654

Subgroups — #225654

Order 512.

675 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22805C211yesmaximal, characteristic
2#22729C211yesmaximal, characteristic, complemented
3#22713C211yesmaximal, characteristic
4#22671C211yesmaximal, characteristic
5C2×#1592C211yesmaximal, characteristic
6#9618C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8895C211yesmaximal, characteristic, complemented
9#8894C211yesmaximal, characteristic
10#8889C211yesmaximal, characteristic
11#8888C211yesmaximal, characteristic, complemented
12#8819C211yesmaximal, characteristic, complemented
13#8655C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic
15#8291C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×#440—21no—
18C2×C4×SD16C2211yescharacteristic
19C2×#436C2211yescharacteristic
20#2355C2211yescharacteristic
21#2354C2211yescharacteristic
22Q8×C4⋊C4C2211yescharacteristic
23D4×C4⋊C4C2211yescharacteristic
24#1968—21no—
25#1949—21no—
26#1948—21no—
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1947C2211yescharacteristic
30#1947C2211yescharacteristic
31#1944—21no—
32#1927—21no—
33#1926—21no—
34#1863—21no—
35#1862—21no—
36#1854—21no—
37#1852—21no—
38#1847—21no—
39#1845C2211yescharacteristic
40#1844C2211yescharacteristic
41#1844C2211yescharacteristic
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1834C2211yescharacteristic
45#1809—21no—
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1776—21no—
51C4×#107C2211yescharacteristic
52C4×#106—21no—
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1736—21no—
56#1592C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
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×Q8—21no—
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C4⋊C4—21no—
76#446—22no—
77#446C2×C414yescomplemented
78#440—44no—
79C4×SD16C2×C414yes—
80C4×SD16Q814yes—
81#436—22no—
82#436C2×C414yes—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107D411yescharacteristic
88C2×#107D411yescharacteristic
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—41no—
96C2×#106D411yescharacteristic
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#103—21no—
103C2×#103—21no—
104C2×#103D411yescharacteristic
105C2×#102—21no—
106C2×#102—21no—
107C2×#102D411yescharacteristic
108C2×C22⋊C8—21no—
109C2×#97—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391—21no—
116#390—21no—
117#389—21no—
118#389—21no—
119#389—41no—
120#388—41no—
121#387—41no—
122#385—41no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#384C2311yescharacteristic
128#382—21no—
129#380—21no—
130#380—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#335C2×C414yescomplemented
154Q8⋊C8C2×C414yescomplemented
155D4⋊C8C2×C414yes—
156#322C2×C414yescomplemented
157C22×Q8C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—41no—
161C2×SD16—22no—
162C2×SD16—22no—
163C2×SD16C4⋊C414yes—
164C2×M16—24no—
165C2×M16—81no—
166C2×M16—81no—
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×Q8—42no—
179C4×Q8—81no—
180C4×Q8C22×C412yes—
181C4×Q8C2×Q812yes—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—42no—
185C4×D4—81no—
186C4×D4C22×C412yes—
187C4×D4C2×Q812yes—
188#117—42no—
189#117—81no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—41no—
222C2×C42—41no—
223C2×C42—41no—
224C2×C42C2411yescharacteristic
225#107—22no—
226#107—41no—
227#107—42no—
228#107—81no—
229#107—81no—
230#107—81no—
231#106—42no—
232#106—42no—
233#106—81no—
234#106—81no—
235#106—81no—
236#106—81no—
237#106—82no—
238#106C4⋊C414yes—
239C4⋊C8—24no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248#103—42no—
249#103—81no—
250#103C4⋊C414yes—
251#102—42no—
252#102—81no—
253#102C4⋊C414yes—
254C22⋊C8—24no—
255#97—24no—
256#97—42no—
257C4×C8—21no—
258C4×C8—42no—
259C4×C8C22×C412yes—
260C4×C8C22×C412yes—
261C4×C8C22×C412yes—
262C4×C8C2×Q812yes—
263C4×C8#4112yes—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272C24—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8C2×C4⋊C412yes—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4C2×C4⋊C412yes—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—161no—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4H(5,𝔽2)11yescharacteristic
308SD16—28no—
309M16—84no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
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×C8—41no—
330C2×C8—41no—
331C2×C8C2×C4⋊C412yes—
332C2×C8C2×C4⋊C412yes—
333C2×C8C4×D412yes—
334C2×C8C4×D412yes—
335C2×C8AGL(1,ℤ/8)12yes—
336C2×C8#13712yes—
337C4⋊C4—22no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
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—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—81no—
365C4⋊C4—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—82no—
369C4⋊C4—82no—
370C4⋊C4—161no—
371C4⋊C4C2×C4⋊C412yes—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4#13712yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C22⋊C4—161no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—82no—
386C42—82no—
387C42—161no—
388C42C23×C411yescharacteristic
389C42C22×Q811yescharacteristic
390C42C2×#4111yescharacteristic
391C422−1+411yescharacteristic
392C23—22no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—41no—
399C23D4211yescharacteristic
400Q8—22no—
401Q8—22no—
402Q8—41no—
403Q8—41no—
404D4—22no—
405D4—22no—
406D4—41no—
407D4—41no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
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—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—164no—
448C2×C4C22×C4⋊C411yescharacteristic
449C2×C4C2×C4×D411yescharacteristic
450C2×C4#52011yescharacteristic
451C2×C4#54711yescharacteristic
452C2×C4D4×Q811yescharacteristic
453C2×C4#55411yescharacteristic
454C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C2×C4#57511yescharacteristic
458C2×C4#57611yescharacteristic
459C2×C4#57711yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C8—22no—
463C8—22no—
464C8—22no—
465C8—42no—
466C8—42no—
467C8—42no—
468C8—81no—
469C8—81no—
470C8#42814yes—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—81no—
475C22—81no—
476C22—81no—
477C22—81no—
478C22D4×C4⋊C411yescharacteristic
479C22#326611yescharacteristic
480C22#326811yescharacteristic
481C22#327211yescharacteristic
482C22#327811yescharacteristic
483C22#328811yescharacteristic
484C22#328911yescharacteristic
485C4—21no—
486C4—21no—
487C4—42no—
488C4—42no—
489C4—42no—
490C4—42no—
491C4—81no—
492C4—81no—
493C4—81no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4—161no—
498C4C2×#42811yescharacteristic
499C4#291211yescharacteristic
500C4C4×#13711yescharacteristic
501C4#294611yescharacteristic
502C4#335411yescharacteristic
503C4#335611yescharacteristic
504C4#338511yescharacteristic
505C4#338811yescharacteristic
506C2—82no—
507C2#2017911yescharacteristic
508C2#2018811yescharacteristic
509C2#2025911yescharacteristic
510C2#2026811yescharacteristic
511C2#2266311yescharacteristic
512C2#2267811yescharacteristic
513C2#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
2. #22729C2semidirect
6. #9618C2semidirect
7. #9614C2semidirect
8. #8895C2semidirect
11. #8888C2semidirect
12. #8819C2semidirect
13. #8655C2semidirect
15. #8291C2semidirect
56. #1592C4semidirect
77. #446C2×C4semidirect
153. #335C2×C4semidirect
154. Q8⋊C8C2×C4semidirect
156. #322C2×C4semidirect

Maximal subgroups

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

subgroupindex
#228052
#227292
#227132
#226712
C2×#15922
#96182
#96142
#88952
#88942
#88892
#88882
#88192
#86552
#86502
#82912