Finite Group Databasebeta

← #193277

Subgroups — #193277

Order 512.

1184 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
1#34206C211yesmaximal, characteristic
2#21964C211yesmaximal, characteristic, complemented
3#21863C211yesmaximal, characteristic
4#21365C211yesmaximal, characteristic, complemented
5C2×#1557C211yesmaximal, characteristic
6#8926C212yesmaximal, complemented
7#8923C212yesmaximal, complemented
8#8921C212yesmaximal, complemented
9#8921C212yesmaximal, complemented
10#7901C211yesmaximal, characteristic, complemented
11#7900C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520—21no—
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#423C2211yescharacteristic
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#1948—21no—
34#1948C2212yes—
35#1948C2212yes—
36#1947—21no—
37#1947C2212yes—
38#1947C2212yes—
39#1875—22no—
40#1853—21no—
41#1852—21no—
42#1809—22no—
43#1804—22no—
44#1776—22no—
45#1557C418yescomplemented
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#431—22no—
82C2×#107—22no—
83C2×#107—22no—
84C2×#106—22no—
85C2×#106—22no—
86#423—22no—
87#423C2×C414yescomplemented
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#102—22no—
93C2×#102—22no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—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—24no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—21no—
162C22×C8—21no—
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—81no—
226C2×C42—81no—
227C2×C42C22×C412yes—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×Q812yes—
232C2×C42#4112yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#107—44no—
236#107—82no—
237#106—44no—
238#106—82no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8—42no—
243C4⋊C8C22×C414yes—
244C4⋊C8C22×C414yes—
245#102—82no—
246#102—82no—
247C22⋊C8—24no—
248C22⋊C8—42no—
249#97—24no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254#95—24no—
255#95—24no—
256#95—22no—
257#95—22no—
258#95—22no—
259#95—22no—
260#95—41no—
261#95—41no—
262C24—22no—
263C24#12011yescharacteristic
264#41—416no—
265#41—416no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—42no—
269C2×Q8—42no—
270C2×D4—28no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—22no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4C2×C4⋊C414yes—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—24no—
296C22×C4—28no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—48no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C4⋊C412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12011yescharacteristic
318C22×C4#12011yescharacteristic
319C22×C4#12011yescharacteristic
320C22×C4C22⋊Q812yes—
321C22×C4C22⋊Q812yes—
322C22×C4#12312yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326M16—84no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C4⋊C4—24no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—81no—
363C4⋊C4—81no—
364C4⋊C4C2×C4⋊C414yes—
365C4⋊C4AGL(1,ℤ/8)14yes—
366C22⋊C4—28no—
367C22⋊C4—28no—
368C22⋊C4—48no—
369C22⋊C4—44no—
370C22⋊C4—84no—
371C22⋊C4—84no—
372C22⋊C4—82no—
373C42—48no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—82no—
379C42—82no—
380C42C2×C4⋊C412yes—
381C42C4×D412yes—
382C42C4×D412yes—
383C42C22⋊Q812yes—
384C42C22⋊Q812yes—
385C42#12312yes—
386C42C23×C411yescharacteristic
387C42C22×Q811yescharacteristic
388C42C2×#4111yescharacteristic
389C42C2×#4111yescharacteristic
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—24no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23#38512yes—
404C23C2×#12011yescharacteristic
405Q8—44no—
406Q8—44no—
407D4—44no—
408D4—416no—
409D4—44no—
410D4—42no—
411D4—42no—
412D4—44no—
413D4—42no—
414D4—42no—
415D4—82no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—24no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—416no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—168no—
457C2×C4#38512yes—
458C2×C4#38512yes—
459C2×C4#38512yes—
460C2×C4C22×C4⋊C411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C22⋊Q811yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×AGL(1,ℤ/8)12yes—
467C2×C4#57512yes—
468C2×C4#57611yescharacteristic
469C2×C4#57711yescharacteristic
470C2×C4#57711yescharacteristic
471C2×C4#57811yescharacteristic
472C8—44no—
473C8—44no—
474C8—81no—
475C8—81no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—44no—
484C22—44no—
485C22—44no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22C2×#38511yescharacteristic
491C22#300311yescharacteristic
492C22#300511yescharacteristic
493C22#301211yescharacteristic
494C22#301311yescharacteristic
495C22#302111yescharacteristic
496C22#302211yescharacteristic
497C4—21no—
498C4—21no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—44no—
503C4—44no—
504C4—82no—
505C4—82no—
506C4—82no—
507C4—161no—
508C4—161no—
509C4C4×AGL(1,ℤ/8)12yes—
510C4#294611yescharacteristic
511C4#294711yescharacteristic
512C4#307712yes—
513C4#308311yescharacteristic
514C4#308411yescharacteristic
515C2—42no—
516C2—84no—
517C2#1884611yescharacteristic
518C2#1884811yescharacteristic
519C2#1885411yescharacteristic
520C2#1885611yescharacteristic
521C2#2085811yescharacteristic
522C2#2087311yescharacteristic
523C2#2087811yescharacteristic
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. #21964C2semidirect
4. #21365C2semidirect
6. #8926C2semidirect
7. #8923C2semidirect
8. #8921C2semidirect
9. #8921C2semidirect
10. #7901C2semidirect
11. #7900C2semidirect
45. #1557C4semidirect
79. #432C2×C4semidirect
87. #423C2×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
#219642
#218632
#213652
C2×#15572
#89262
#89232
#89212
#79012
#79002