Finite Group Databasebeta

← #193262

Subgroups — #193262

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34206C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic, complemented
3#21410C211yesmaximal, characteristic
4#21358C211yesmaximal, characteristic, complemented
5C2×#1553C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8922C212yesmaximal, complemented
8#8921C212yesmaximal, complemented
9#8920C212yesmaximal, complemented
10#7904C211yesmaximal, characteristic, complemented
11#7896C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×C4×M16C2211yescharacteristic
17#2367—22no—
18#2366C2212yes—
19#2365C2212yes—
20#2354—21no—
21#2352—22no—
22D4×C4⋊C4—21no—
23#2325—22no—
24#2311—21no—
25C4×#121—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1948C2212yes—
34#1947C2212yes—
35#1947C2212yes—
36#1947C2212yes—
37#1895—22no—
38#1854—22no—
39#1852—21no—
40#1852—21no—
41#1847—21no—
42#1836—21no—
43C4×#107—22no—
44#1762—22no—
45#1553C418yescomplemented
46C2×#332C2212yes—
47C2×#331C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×#41—21no—
53C22×M16—21no—
54#532—22no—
55#532—22no—
56#532C2×C414yes—
57#532Q814yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×#127C2311yescharacteristic
60C2×#124—21no—
61C2×#121—21no—
62C2×#121D412yes—
63#520—48no—
64C2×C4×Q8—22no—
65C2×C4×D4—24no—
66C2×C4×D4—22no—
67C2×C4×D4—22no—
68C2×C4×D4—21no—
69C2×C4×D4D412yes—
70C2×#117—22no—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432C2×C414yescomplemented
80#431—22no—
81#431C2×C414yescomplemented
82C2×#107—22no—
83C2×#107—22no—
84C2×#107—22no—
85C2×#106—22no—
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#102—22no—
89C2×#102—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C4×M16—22no—
93C4×M16—22no—
94C2×#97—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#391—22no—
100#390—22no—
101#390—22no—
102#389—22no—
103#388—24no—
104#388—24no—
105#387—24no—
106#386—24no—
107#386—24no—
108#385—24no—
109#384D411yescharacteristic
110#384D411yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#382—21no—
115#380—22no—
116#380—22no—
117#380—21no—
118#380—21no—
119#380—21no—
120#380D412yes—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2312yes—
130C4×C22⋊C4—24no—
131C2×#95—21no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137#340—22no—
138#332C2×C418yescomplemented
139#331C2×C418yescomplemented
140C2×#41—24no—
141C2×#41—24no—
142C2×#41—44no—
143C2×#41—44no—
144C22×Q8—21no—
145C22×D4—22no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C2×D412yes—
149C23×C4—21no—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C2×M16—42no—
157C2×M16—42no—
158C2×M16—42no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—22no—
162C22×C8—22no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8C22×C412yes—
166C4⋊Q8C2×Q812yes—
167#127—21no—
168#127—21no—
169#127C22×C412yes—
170#127C2×Q812yes—
171#124—24no—
172#124—24no—
173#121—24no—
174#121—24no—
175#121—24no—
176#121C4⋊C418yes—
177C4×Q8—44no—
178C4×Q8—44no—
179C4×D4—24no—
180C4×D4—24no—
181C4×D4—24no—
182C4×D4—44no—
183C4×D4—416no—
184C4×D4—44no—
185C4×D4—82no—
186C4×D4—82no—
187C4×D4C4⋊C418yes—
188#117—48no—
189#117—44no—
190#117—82no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
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—42no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—41no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—41no—
225C2×C42—82no—
226C2×C42C22×C412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q812yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233C2×C42C2411yescharacteristic
234#107—44no—
235#107—44no—
236#107—82no—
237#106—82no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8C22×C414yes—
241C4⋊C8C22×C414yes—
242#102—44no—
243#102—44no—
244C22⋊C8—24no—
245C22⋊C8—24no—
246#97—24no—
247#97—42no—
248C4×C8—24no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—41no—
260#95—41no—
261C24—22no—
262C24#12011yescharacteristic
263#41—416no—
264#41—416no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—42no—
268C2×Q8—42no—
269C2×D4—28no—
270C2×D4—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—42no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—42no—
283C2×D4C2×C4⋊C414yes—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—28no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—48no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—81no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4C2×C4⋊C412yes—
310C22×C4C4×D412yes—
311C22×C4C4×D412yes—
312C22×C4#12011yescharacteristic
313C22×C4#12011yescharacteristic
314C22×C4#12011yescharacteristic
315C22×C4C22⋊Q812yes—
316C22×C4C22⋊Q812yes—
317C22×C4#12312yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—48no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—24no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—48no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—81no—
355C4⋊C4—81no—
356C4⋊C4C2×C4⋊C414yes—
357C4⋊C4AGL(1,ℤ/8)14yes—
358C22⋊C4—28no—
359C22⋊C4—28no—
360C22⋊C4—48no—
361C22⋊C4—44no—
362C22⋊C4—84no—
363C22⋊C4—84no—
364C22⋊C4—82no—
365C42—48no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C42—82no—
371C42—82no—
372C42C2×C4⋊C412yes—
373C42C4×D412yes—
374C42C4×D412yes—
375C42C22⋊Q812yes—
376C42C22⋊Q812yes—
377C42#12312yes—
378C42C23×C411yescharacteristic
379C42C22×Q811yescharacteristic
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—24no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23#38512yes—
396C23C2×#12011yescharacteristic
397Q8—44no—
398Q8—44no—
399D4—44no—
400D4—416no—
401D4—44no—
402D4—44no—
403D4—44no—
404D4—44no—
405D4—82no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—24no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—416no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—84no—
437C2×C4—84no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—168no—
445C2×C4#38512yes—
446C2×C4#38512yes—
447C2×C4#38512yes—
448C2×C4C22×C4⋊C411yescharacteristic
449C2×C4C2×C4×D411yescharacteristic
450C2×C4C2×C4×D411yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×C22⋊Q811yescharacteristic
453C2×C4C2×#12311yescharacteristic
454C2×C4C2×AGL(1,ℤ/8)12yes—
455C2×C4#57512yes—
456C2×C4#57612yes—
457C2×C4#57812yes—
458C8—42no—
459C8—42no—
460C8—44no—
461C8—44no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22C2×#38511yescharacteristic
477C22#300311yescharacteristic
478C22#300511yescharacteristic
479C22#300811yescharacteristic
480C22#301111yescharacteristic
481C22#301811yescharacteristic
482C22#301811yescharacteristic
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—44no—
489C4—44no—
490C4—82no—
491C4—82no—
492C4—82no—
493C4—161no—
494C4—161no—
495C4C4×AGL(1,ℤ/8)12yes—
496C4#295112yes—
497C4#307712yes—
498C4#308312yes—
499C2—42no—
500C2—84no—
501C2#1859811yescharacteristic
502C2#1860311yescharacteristic
503C2#1861111yescharacteristic
504C2#1861411yescharacteristic
505C2#2085511yescharacteristic
506C2#2087311yescharacteristic
507C2#2088111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
2. #21988C2semidirect
4. #21358C2semidirect
6. #8926C2semidirect
7. #8922C2semidirect
8. #8921C2semidirect
9. #8920C2semidirect
10. #7904C2semidirect
11. #7896C2semidirect
45. #1553C4semidirect
79. #432C2×C4semidirect
81. #431C2×C4semidirect
138. #332C2×C4semidirect
139. #331C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342062
#219882
#214102
#213582
C2×#15532
#89262
#89222
#89212
#89202
#79042
#78962