Finite Group Databasebeta

← #203861

Subgroups — #203861

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21362C211yesmaximal, characteristic
2#21169C211yesmaximal, characteristic
3#20824C211yesmaximal, characteristic
4#20480C211yesmaximal, characteristic
5C2×#1565C211yesmaximal, characteristic
6#9622C211yesmaximal, characteristic, complemented
7#9621C211yesmaximal, characteristic, complemented
8#9619C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9614C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9612C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#8044C211yesmaximal, characteristic
16C2×#432C2211yescharacteristic
17C2×#431C2211yescharacteristic
18C2×#427—21no—
19C22×C4⋊C8C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23C4×#117C2211yescharacteristic
24#1949—21no—
25#1949—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1854—21no—
35#1854—21no—
36#1845C2211yescharacteristic
37#1844C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1809—21no—
41#1809—21no—
42#1776—21no—
43#1776—21no—
44C4×#107—21no—
45C4×#106—21no—
46#1770C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C8×C4⋊C4C2211yescharacteristic
50#1753—21no—
51#1753—21no—
52#1745—21no—
53#1735—21no—
54C4×#97—21no—
55#1565C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#334C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C23×C8—21no—
71C2×#117—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C42C2311yescharacteristic
76#432—22no—
77#432Q814yes—
78#431—22no—
79#431C2×C414yes—
80#427—44no—
81C2×#107—21no—
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—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8C2×C414yes—
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16C2×C414yes—
105C4×M16Q814yes—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8—21no—
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#382—21no—
121#382—21no—
122#380—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D411yescharacteristic
132C4×C4⋊C4C2311yescharacteristic
133C4×C4⋊C4C2311yescharacteristic
134C4×C22⋊C4—21no—
135C4×C22⋊C4—21no—
136#376—21no—
137C43—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—21no—
148#340—21no—
149#340—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#335C2×C414yes—
155#335C2×C414yes—
156#334C2×C414yes—
157#334C2×C414yes—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—41no—
164C2×M16—41no—
165C2×M16C4⋊C414yes—
166C2×M16C4⋊C414yes—
167C22×C8—24no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180#117—22no—
181#117—22no—
182#117—41no—
183#117—41no—
184#117—42no—
185#117—42no—
186#117—82no—
187C2×C4⋊C4—21no—
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⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42C22×C412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42C2×Q812yes—
220C2×C42#4112yes—
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—42no—
225#107—42no—
226#107—42no—
227#107—42no—
228#107—42no—
229#106—42no—
230#106—42no—
231#106—42no—
232#106—42no—
233#106—42no—
234#106—42no—
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—
244C4⋊C8C22×C412yes—
245C4⋊C8C22×C412yes—
246C4⋊C8C2×Q812yes—
247C4⋊C8C2×Q812yes—
248C22⋊C8—24no—
249C22⋊C8—24no—
250#97—24no—
251#97—42no—
252#97—42no—
253#97C4⋊C414yes—
254#97C4⋊C414yes—
255C4×C8—24no—
256C4×C8—42no—
257C4×C8—42no—
258C4×C8C22×C412yes—
259C4×C8C22×C412yes—
260C4×C8C2×Q812yes—
261C4×C8C2×Q812yes—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—41no—
267#95—41no—
268#95—41no—
269#95—41no—
270C24#12111yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C2×C4⋊C412yes—
290C22×C4C4×D412yes—
291C22×C4C4×Q812yes—
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22⋊Q812yes—
296C22×C4#12512yes—
297C22×C4C4⋊Q812yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C2×#4111yescharacteristic
301M16—28no—
302M16—44no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—48no—
316C2×C8—41no—
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×C8C2×C4⋊C412yes—
329C2×C8C2×C4⋊C412yes—
330C2×C8C2×C4⋊C412yes—
331C2×C8C2×C4⋊C412yes—
332C2×C8#13512yes—
333C2×C8#13512yes—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
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—81no—
357C4⋊C4—81no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4AGL(1,ℤ/8)12yes—
361C4⋊C4#13712yes—
362C22⋊C4—42no—
363C22⋊C4—42no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—42no—
367C42—42no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—81no—
373C42—81no—
374C42—82no—
375C42—82no—
376C42C2×C4⋊C412yes—
377C42C4×D412yes—
378C42C4×Q812yes—
379C42C22⋊Q812yes—
380C42#12512yes—
381C42C4⋊Q812yes—
382C42C23×C411yescharacteristic
383C42C22×Q811yescharacteristic
384C42C22×Q811yescharacteristic
385C42C2×#4111yescharacteristic
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23#38912yes—
393C23C2×#12111yescharacteristic
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
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—41no—
423C2×C4—41no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4#38912yes—
437C2×C4#38912yes—
438C2×C4#38912yes—
439C2×C4C22×C4⋊C411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C4×Q811yescharacteristic
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12511yescharacteristic
444C2×C4C2×C4⋊Q811yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×#13511yescharacteristic
447C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57511yescharacteristic
450C2×C4#57511yescharacteristic
451C2×C4#57511yescharacteristic
452C2×C4#57511yescharacteristic
453C8—22no—
454C8—22no—
455C8—22no—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C8—42no—
461C8—42no—
462C8#42714yescomplemented
463C22—22no—
464C22—22no—
465C22—22no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22C2×#38911yescharacteristic
471C22#303511yescharacteristic
472C22#303511yescharacteristic
473C22#305511yescharacteristic
474C22#305611yescharacteristic
475C22#306811yescharacteristic
476C22#306911yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—22no—
482C4—42no—
483C4—42no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4C2×#42711yescharacteristic
491C4#291111yescharacteristic
492C4C4×AGL(1,ℤ/8)11yescharacteristic
493C4C4×#13711yescharacteristic
494C4#307611yescharacteristic
495C4#307611yescharacteristic
496C4#316511yescharacteristic
497C4#316611yescharacteristic
498C2—42no—
499C2#1918011yescharacteristic
500C2#1918111yescharacteristic
501C2#1918211yescharacteristic
502C2#1918311yescharacteristic
503C2#2115911yescharacteristic
504C2#2116911yescharacteristic
505C2#2118211yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9622C2semidirect
7. #9621C2semidirect
8. #9619C2semidirect
9. #9617C2semidirect
10. #9616C2semidirect
11. #9614C2semidirect
12. #9613C2semidirect
13. #9612C2semidirect
55. #1565C4semidirect
462. C8#427semidirect

Maximal subgroups

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

subgroupindex
#213622
#211692
#208242
#204802
C2×#15652
#96222
#96212
#96192
#96172
#96162
#96142
#96132
#96122
#80462
#80442