Finite Group Databasebeta

← #238725

Subgroups — #238725

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35179C211yesmaximal, characteristic
2#21885C211yesmaximal, characteristic
3#20816C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1493C211yesmaximal, characteristic
6C22×Q8⋊C8C211yesmaximal, characteristic
7#10438C214yesmaximal, complemented
8#10437C214yesmaximal, complemented
9#8045C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#419—21no—
15C22×C4×C8C2211yescharacteristic
16#2855—22no—
17#2627—22no—
18#2582C2214yes—
19#2581—22no—
20#2568—21no—
21#2567—21no—
22#2565—22no—
23#2317—21no—
24#2310—21no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#1987—21no—
28#1981—21no—
29#1892—22no—
30#1844C2212yes—
31#1834C2212yes—
32#1807—22no—
33#1805—22no—
34#1803C2214yes—
35#1803C2214yes—
36C4×#106—22no—
37#1736—21no—
38#1493—24no—
39C2×Q8⋊C8—24no—
40C2×Q8⋊C8C2212yes—
41C2×Q8⋊C8C2212yes—
42C2×Q8⋊C8C2212yes—
43C2×Q8⋊C8C2212yes—
44#1296C2214yes—
45C23×Q8D411yescharacteristic
46C22×M16—21no—
47C23×C8—21no—
48C23×C8—21no—
49#533—22no—
50#533—22no—
51#533—22no—
52#533—22no—
53C2×C4⋊Q8D411yescharacteristic
54C2×C4⋊Q8D411yescharacteristic
55C2×#125—21no—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8D412yes—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8—21no—
61C2×C4×Q8—22no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×C4×Q8D412yes—
66C2×C4×Q8C2312yes—
67C2×C4×Q8C2312yes—
68C2×#117—21no—
69C2×#117—21no—
70C2×#117—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75C2×#106—24no—
76#423—22no—
77#423—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84#419—44no—
85C2×#103—24no—
86C2×#103—24no—
87C2×C4×C8—21no—
88C2×C4×C8—22no—
89C2×C4×C8D414yes—
90C2×C4×C8C2311yescharacteristic
91C2×C4×C8C2311yescharacteristic
92#400—24no—
93#398—28no—
94#395—24no—
95#391—22no—
96#391—22no—
97#391—24no—
98#391—24no—
99#389—24no—
100#389—24no—
101#387—28no—
102#385—24no—
103#384—22no—
104#384C2312yes—
105#384C2312yes—
106#382—22no—
107#382—21no—
108#382—21no—
109#381—24no—
110#380—21no—
111#380—21no—
112#380—21no—
113#380D412yes—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4D412yes—
118#340—24no—
119#340—24no—
120#340—24no—
121Q8⋊C8—24no—
122Q8⋊C8—24no—
123Q8⋊C8—24no—
124Q8⋊C8—24no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8SD1614yes—
133C22×Q8C2×D412yes—
134C22×Q8C2×D412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C2×M16—42no—
141C2×M16—42no—
142C2×M16—42no—
143C22×C8—21no—
144C22×C8—24no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—44no—
149C22×C8—42no—
150C22×C8—82no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155#125—42no—
156#125—42no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—22no—
168C4×Q8—22no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—42no—
172C4×Q8—42no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×Q8—42no—
177#117—42no—
178#117—42no—
179#117—42no—
180#117—42no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4C2×D412yes—
202C2×C4⋊C4C2×D412yes—
203C2×C4⋊C4C2×D414yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—41no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—41no—
214C2×C42C2×D412yes—
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42C2411yescharacteristic
222#106—48no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8—42no—
229#103—48no—
230#103—84no—
231C4×C8—24no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8C2×D412yes—
235C4×C8C2×D412yes—
236#95—28no—
237#95—28no—
238#95—28no—
239#95—22no—
240#95—22no—
241#95—41no—
242#95—41no—
243C24#12011yescharacteristic
244C2×Q8—22no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—28no—
255C2×Q8—28no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—82no—
262C2×Q8C2×SD1614yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—82no—
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12312yes—
284C22×C4#12312yes—
285C22×C4#12412yes—
286C22×C4C2×SD1612yes—
287C22×C4#13512yes—
288C22×C4#13712yes—
289C22×C4#13712yes—
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293M16—84no—
294C2×C8—24no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—48no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—44no—
304C2×C8—84no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—88no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—82no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—44no—
338C42—44no—
339C42—82no—
340C42#12112yes—
341C42#12112yes—
342C42#12112yes—
343C42#12312yes—
344C42#12312yes—
345C42#12412yes—
346C42C22×D411yescharacteristic
347C42C2×#4111yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—22no—
351C23—22no—
352C23#39412yes—
353C23#44812yes—
354C23#45212yes—
355C23C2×#12011yescharacteristic
356Q8—24no—
357Q8—24no—
358Q8—42no—
359Q8—42no—
360Q8—48no—
361Q8—44no—
362Q8—42no—
363Q8—42no—
364Q8—84no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#39412yes—
391C2×C4#39412yes—
392C2×C4#39412yes—
393C2×C4#44812yes—
394C2×C4#45212yes—
395C2×C4#46514yes—
396C2×C4#47014yes—
397C2×C4#48314yes—
398C2×C4#48514yes—
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12311yescharacteristic
403C2×C4C2×#12311yescharacteristic
404C2×C4C2×#12411yescharacteristic
405C2×C4C22×SD1611yescharacteristic
406C2×C4C2×#13511yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4C2×#13711yescharacteristic
409C2×C4#57712yes—
410C2×C4#57812yes—
411C8—24no—
412C8—42no—
413C8—82no—
414C8—82no—
415C22—22no—
416C22—22no—
417C22—24no—
418C22—24no—
419C22#203514yes—
420C22C2×#39411yescharacteristic
421C22C2×#44811yescharacteristic
422C22C2×#45211yescharacteristic
423C22#301312yes—
424C22#301812yes—
425C4—22no—
426C4—21no—
427C4—21no—
428C4—22no—
429C4—44no—
430C4—82no—
431C4—82no—
432C4—82no—
433C4—84no—
434C4—162no—
435C4#203514yes—
436C4C2×#46511yescharacteristic
437C4C2×#47011yescharacteristic
438C4#305711yescharacteristic
439C4#305911yescharacteristic
440C4C2×#48311yescharacteristic
441C4C2×#48511yescharacteristic
442C4#309712yes—
443C2—24no—
444C2#1866712yes—
445C2#1867612yes—
446C2C2×#203511yescharacteristic
447C2#2348711yescharacteristic
448C2#2349811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10438C2semidirect
8. #10437C2semidirect

Maximal subgroups

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

subgroupindex
#351792
#218852
#208162
#206472
C2×#14932
C22×Q8⋊C82
#104382
#104372
#80452