Finite Group Databasebeta

← #192717

Subgroups — #192717

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34214C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21171C211yesmaximal, characteristic
4C2×#1845C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6#8884C214yesmaximal, complemented
7#8883C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7890C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#428—21no—
13C22×#107—22no—
14C2×#423C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2360—22no—
17#2360—22no—
18#2355—21no—
19#2355C2212yes—
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4C2212yes—
22#2330—22no—
23#2317—21no—
24C4×C4⋊Q8—21no—
25C4×C4⋊Q8—21no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274—21no—
29C4×#117C2211yescharacteristic
30#1990—21no—
31#1956—21no—
32#1948—21no—
33#1948C2212yes—
34#1947—21no—
35#1947C2212yes—
36#1872—22no—
37#1845—22no—
38#1845C2214yes—
39#1803C2214yes—
40#1736—21no—
41#1493C418yescomplemented
42C2×Q8⋊C8C2214yes—
43#1296C2214yes—
44#1279C2212yes—
45#1279C2212yes—
46C22×M16—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533Q814yes—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×#125—21no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8D412yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2312yes—
61C2×#117—21no—
62C2×#117D411yescharacteristic
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D412yes—
70C22×C42C2311yescharacteristic
71#431—22no—
72#431—22no—
73#428—44no—
74C2×#107—28no—
75C2×#107—22no—
76C2×#107—48no—
77C2×#106—22no—
78#423—22no—
79#423C2×C414yes—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×#103—24no—
84C2×C22⋊C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—22no—
87C2×C4×C8—21no—
88C2×C4×C8C2×C414yes—
89C2×C4×C8C2312yes—
90#391—22no—
91#391—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—24no—
96#389—22no—
97#389—24no—
98#387—24no—
99#387—24no—
100#385—24no—
101#384—22no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384D412yes—
108#384C2312yes—
109#384C2312yes—
110#382—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4D411yescharacteristic
118C4×C4⋊C4D411yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—24no—
122C4×C22⋊C4—21no—
123C4×C22⋊C4—21no—
124#376—21no—
125C43—21no—
126#340—24no—
127#340—22no—
128#340—22no—
129Q8⋊C8C2×C4116yescomplemented
130C22×Q8—21no—
131C22×Q8C2×D412yes—
132C23×C4—22no—
133C23×C4—41no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—24no—
138C2×M16—42no—
139C2×M16—42no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8C4⋊C414yes—
149C4⋊Q8C4⋊C414yes—
150#125—24no—
151#125—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8C4⋊C418yes—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—84no—
161C4×Q8—84no—
162C4×Q8C22×C414yes—
163C4×Q8C2×Q814yes—
164#117—24no—
165#117—24no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117Q1614yes—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D414yes—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42C22×C412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×Q812yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#107—44no—
223#107—416no—
224#106—44no—
225C4⋊C8—24no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8C22×C414yes—
230#103—84no—
231C22⋊C8—42no—
232C4×C8—24no—
233C4×C8—22no—
234C4×C8—42no—
235C4×C8C22×C414yes—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24#12011yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8C2×C4⋊C414yes—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—21no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C4⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4C22⋊Q812yes—
277C22×C4C22⋊Q812yes—
278C22×C4#12312yes—
279C22×C4C2×Q1612yes—
280C22×C4#13512yes—
281C22×C4AGL(1,ℤ/8)12yes—
282C22×C4#13712yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—48no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—216no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—28no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4C2×C4⋊C414yes—
327C4⋊C4C2×Q1612yes—
328C4⋊C4C2×Q1612yes—
329C22⋊C4—28no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—42no—
336C42—42no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—84no—
341C42—84no—
342C42C2×C4⋊C412yes—
343C42C4×D412yes—
344C42C4×D412yes—
345C42C22⋊Q812yes—
346C42C22⋊Q812yes—
347C42#12312yes—
348C42C23×C411yescharacteristic
349C42C22×Q811yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23#38512yes—
357C23#44912yes—
358C23#45112yes—
359C23C2×#12011yescharacteristic
360Q8—28no—
361Q8—44no—
362Q8—48no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—44no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
395C2×C4—164no—
396C2×C4—164no—
397C2×C4#38512yes—
398C2×C4#38512yes—
399C2×C4#38512yes—
400C2×C4C4×Q1614yes—
401C2×C4#44014yes—
402C2×C4#44912yes—
403C2×C4#45112yes—
404C2×C4#45312yes—
405C2×C4#45612yes—
406C2×C4#47714yes—
407C2×C4#47914yes—
408C2×C4C22×C4⋊C411yescharacteristic
409C2×C4C2×C4×D411yescharacteristic
410C2×C4C2×C4×D411yescharacteristic
411C2×C4C2×C22⋊Q811yescharacteristic
412C2×C4C2×C22⋊Q811yescharacteristic
413C2×C4C2×#12311yescharacteristic
414C2×C4C22×Q1611yescharacteristic
415C2×C4C2×#13511yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
417C2×C4C2×#13711yescharacteristic
418C2×C4#57612yes—
419C2×C4#57812yes—
420C8—28no—
421C8—44no—
422C8—82no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22#186314yes—
430C22C2×#38511yescharacteristic
431C22C2×#45111yescharacteristic
432C22C2×#44911yescharacteristic
433C22#300811yescharacteristic
434C22#301111yescharacteristic
435C22C2×#45311yescharacteristic
436C22C2×#45611yescharacteristic
437C4—22no—
438C4—21no—
439C4—21no—
440C4—44no—
441C4—44no—
442C4—42no—
443C4—42no—
444C4—81no—
445C4—84no—
446C4—81no—
447C4—162no—
448C4#185914yes—
449C4#186314yes—
450C4#291312yes—
451C4C2×C4×Q1611yescharacteristic
452C4C2×#44011yescharacteristic
453C4C2×#47711yescharacteristic
454C4C2×#47911yescharacteristic
455C4#309812yes—
456C2—24no—
457C2#1862311yescharacteristic
458C2#1862411yescharacteristic
459C2#1865311yescharacteristic
460C2#1865411yescharacteristic
461C2#2084711yescharacteristic
462C2C2×#185911yescharacteristic
463C2C2×#186311yescharacteristic
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. #8884C2semidirect
7. #8883C2semidirect
41. #1493C4semidirect
129. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342142
#218602
#211712
C2×#18452
C2×#14932
#88842
#88832
#80432
#78902