Finite Group Databasebeta

← #192727

Subgroups — #192727

Order 512.

1018 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#21170C211yesmaximal, characteristic
4#20816C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8892C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8888C212yesmaximal, complemented
9#8887C212yesmaximal, complemented
10#8045C211yesmaximal, characteristic
11C2×#1279C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2370—22no—
18#2369—21no—
19#2365—21no—
20#2355—22no—
21#2355C2212yes—
22Q8×C4⋊C4C2212yes—
23#2329—22no—
24#2317—21no—
25C4×C22⋊Q8—22no—
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274—21no—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1987—21no—
31#1981—21no—
32#1949—22no—
33#1948C2212yes—
34#1947C2212yes—
35#1892—22no—
36#1844C2211yescharacteristic
37#1844C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1807—22no—
41#1803C2212yes—
42#1803C2212yes—
43C4×#106—22no—
44#1736—21no—
45#1493C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×M16—21no—
53C23×C8—21no—
54C23×C8—21no—
55#533—22no—
56#533—22no—
57#533C2×C414yes—
58#533Q814yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8D412yes—
64C2×C4×Q8—22no—
65C2×C4×Q8—21no—
66C2×C4×Q8C2312yes—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117—21no—
70C22×C4⋊C4D411yescharacteristic
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#428—44no—
75C2×#107—22no—
76C2×#106—22no—
77C2×#106—22no—
78C2×#106—22no—
79#423—22no—
80#423C2×C414yes—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87C2×#103—22no—
88C2×#103—22no—
89C2×C4×C8—21no—
90C2×C4×C8—22no—
91C2×C4×C8C2×C414yes—
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#391—24no—
95#391—22no—
96#389—22no—
97#389—22no—
98#389—24no—
99#389—22no—
100#387—24no—
101#387—24no—
102#387—24no—
103#385—24no—
104#384—22no—
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#382—22no—
110#382—21no—
111#382—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116C4×C4⋊C4—24no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
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⋊C4D412yes—
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—
138Q8⋊C8C2×C418yescomplemented
139Q8⋊C8C2×C418yescomplemented
140C22×Q8—21no—
141C22×Q8C2×D412yes—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C22×C8—21no—
151C22×C8—24no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—42no—
156C22×C8—42no—
157C22×C8—42no—
158C22×C8—81no—
159C22×C8—81no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8C4⋊C414yes—
163C4⋊Q8C4⋊C414yes—
164#125—24no—
165#125—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8C4⋊C418yes—
170C4×Q8—24no—
171C4×Q8—24no—
172C4×Q8—22no—
173C4×Q8—22no—
174C4×Q8—82no—
175C4×Q8—82no—
176C4×Q8C22×C414yes—
177C4×Q8C2×Q814yes—
178#117—24no—
179#117—24no—
180#117—42no—
181#117—42no—
182#117—42no—
183#117—42no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
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—21no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4SD1614yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—41no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42—22no—
221C2×C42—41no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42C22×C412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×Q812yes—
229C2×C42#4112yes—
230C2×C42#4112yes—
231C2×C42C2411yescharacteristic
232#107—44no—
233#106—44no—
234#106—44no—
235#106—44no—
236C4⋊C8—24no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8—42no—
241C4⋊C8C22×C414yes—
242#103—82no—
243#103—82no—
244C4×C8—24no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8C22×C412yes—
248C4×C8C22×C412yes—
249#95—24no—
250#95—24no—
251#95—24no—
252#95—24no—
253#95—22no—
254#95—22no—
255#95—22no—
256#95—22no—
257#95—41no—
258#95—41no—
259C24#12011yescharacteristic
260C2×Q8—22no—
261C2×Q8—24no—
262C2×Q8—22no—
263C2×Q8—22no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—81no—
267C2×Q8—81no—
268C2×Q8C2×C4⋊C414yes—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4—81no—
288C22×C4—81no—
289C22×C4C2×C4⋊C412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12011yescharacteristic
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4C22⋊Q812yes—
296C22×C4C22⋊Q812yes—
297C22×C4#12312yes—
298C22×C4C2×SD1612yes—
299C22×C4#13512yes—
300C22×C4#13712yes—
301C22×C4#13712yes—
302C22×C4C22×D411yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305M16—48no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—48no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—84no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—28no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
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—84no—
353C4⋊C4—84no—
354C4⋊C4C2×C4⋊C414yes—
355C4⋊C4C2×SD1614yes—
356C22⋊C4—28no—
357C22⋊C4—44no—
358C22⋊C4—82no—
359C42—24no—
360C42—24no—
361C42—42no—
362C42—42no—
363C42—44no—
364C42—82no—
365C42—84no—
366C42—84no—
367C42—82no—
368C42C2×C4⋊C412yes—
369C42C4×D412yes—
370C42C4×D412yes—
371C42C22⋊Q812yes—
372C42C22⋊Q812yes—
373C42#12312yes—
374C42C23×C411yescharacteristic
375C42C22×Q811yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23#38512yes—
383C23#44812yes—
384C23#45212yes—
385C23C2×#12011yescharacteristic
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—42no—
390Q8—84no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—84no—
422C2×C4—84no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#38512yes—
430C2×C4#38512yes—
431C2×C4#38512yes—
432C2×C4C4×SD1614yes—
433C2×C4#44114yes—
434C2×C4#44812yes—
435C2×C4#45212yes—
436C2×C4#47514yes—
437C2×C4#47914yes—
438C2×C4C22×C4⋊C411yescharacteristic
439C2×C4C2×C4×D411yescharacteristic
440C2×C4C2×C4×D411yescharacteristic
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4C22×SD1611yescharacteristic
445C2×C4C2×#13511yescharacteristic
446C2×C4C2×#13711yescharacteristic
447C2×C4C2×#13711yescharacteristic
448C2×C4#57611yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57711yescharacteristic
451C2×C4#57811yescharacteristic
452C8—24no—
453C8—42no—
454C8—44no—
455C8—82no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22#186514yes—
463C22C2×#38511yescharacteristic
464C22C2×#44811yescharacteristic
465C22C2×#45211yescharacteristic
466C22#301211yescharacteristic
467C22#301311yescharacteristic
468C22#301811yescharacteristic
469C22#301811yescharacteristic
470C4—22no—
471C4—21no—
472C4—21no—
473C4—22no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4—161no—
481C4—161no—
482C4#186514yes—
483C4#291211yescharacteristic
484C4#291311yescharacteristic
485C4C2×C4×SD1611yescharacteristic
486C4C2×#44111yescharacteristic
487C4C2×#47511yescharacteristic
488C4C2×#47911yescharacteristic
489C4#309611yescharacteristic
490C4#309711yescharacteristic
491C2—24no—
492C2#1866411yescharacteristic
493C2#1866711yescharacteristic
494C2#1867611yescharacteristic
495C2#1867911yescharacteristic
496C2#2084911yescharacteristic
497C2#2086311yescharacteristic
498C2C2×#186511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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. #8892C2semidirect
7. #8891C2semidirect
8. #8888C2semidirect
9. #8887C2semidirect
45. #1493C4semidirect
138. Q8⋊C8C2×C4semidirect
139. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#218852
#211702
#208162
C2×#14932
#88922
#88912
#88882
#88872
#80452
C2×#12792