Finite Group Databasebeta

← #238131

Subgroups — #238131

Order 512.

1036 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2#21860C211yesmaximal, characteristic
3#21385C211yesmaximal, characteristic
4#21312C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10355C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×#125—21no—
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2552—22no—
16#2552—22no—
17#2552—22no—
18#2552C2214yes—
19C2×#398—22no—
20C2×#389—21no—
21C2×#389—21no—
22C2×#384—21no—
23C2×#384C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1990C2212yes—
27#1989—22no—
28#1988C2212yes—
29#1987C2212yes—
30#1987C2212yes—
31#1981C2212yes—
32#1956C2212yes—
33#1872—22no—
34#1872—21no—
35#1872—21no—
36#1803—22no—
37C4×#103—22no—
38#1736—21no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44#533—22no—
45#533—22no—
46#533—22no—
47#533Q814yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×C4⋊Q8C2311yescharacteristic
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—21no—
53C2×#125—44no—
54C2×#125—44no—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8—21no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×#117—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4D411yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C42—21no—
71C22×C42—21no—
72C22×C42C2311yescharacteristic
73#431—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×#103—22no—
84C2×#103—22no—
85C2×#103—24no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#398—28no—
91#398—28no—
92#389—24no—
93#389—24no—
94#389—24no—
95#389—24no—
96#384—24no—
97#384—24no—
98#384—21no—
99#384—21no—
100#384D412yes—
101#384C2312yes—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4D412yes—
108C4×C4⋊C4C2312yes—
109C4×C4⋊C4C2312yes—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—22no—
117#340—22no—
118#340—22no—
119#340—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—21no—
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139C4⋊Q8—21no—
140C4⋊Q8—21no—
141C4⋊Q8—21no—
142C4⋊Q8C2×Q812yes—
143C4⋊Q8C2×Q812yes—
144#125—24no—
145#125—48no—
146#125—416no—
147#125—48no—
148#125—42no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—42no—
156C4×Q8—42no—
157#117—24no—
158#117—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—41no—
183C2×C4⋊C4—41no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—84no—
188C2×C4⋊C4SD1614yes—
189C2×C4⋊C4Q1614yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D414yes—
195C2×C22⋊C4—41no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—44no—
201C2×C42—41no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×Q812yes—
208C2×C42#4112yes—
209C2×C42#4112yes—
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—42no—
217C4⋊C8—42no—
218#103—44no—
219#103—44no—
220#103—84no—
221C22⋊C8—42no—
222C22⋊C8—42no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—42no—
226#95—28no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231C24#12011yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—42no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4C22⋊Q812yes—
270C22×C4C22⋊Q812yes—
271C22×C4C22⋊Q812yes—
272C22×C4#12412yes—
273C22×C4#12412yes—
274C22×C4#12412yes—
275C22×C4C2×SD1612yes—
276C22×C4C2×Q1612yes—
277C22×C4AGL(1,ℤ/8)12yes—
278C22×C4#13712yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—84no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—416no—
309C4⋊C4—416no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—44no—
320C4⋊C4—82no—
321C4⋊C4—88no—
322C4⋊C4C2×SD1614yes—
323C4⋊C4C2×Q1614yes—
324C22⋊C4—44no—
325C42—44no—
326C42—42no—
327C42—42no—
328C42—44no—
329C42—44no—
330C42—44no—
331C42C22⋊Q812yes—
332C42C22⋊Q812yes—
333C42C22⋊Q812yes—
334C42#12412yes—
335C42#12412yes—
336C42#12412yes—
337C42C22×Q811yescharacteristic
338C42C2×#4111yescharacteristic
339C42C2×#4111yescharacteristic
340C42C2×#4111yescharacteristic
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23#39312yes—
346C23#45012yes—
347C23#45112yes—
348C23C2×#12011yescharacteristic
349Q8—44no—
350Q8—44no—
351Q8—82no—
352Q8—82no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—84no—
383C2×C4—84no—
384C2×C4—88no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—81no—
388C2×C4—81no—
389C2×C4#39312yes—
390C2×C4#39312yes—
391C2×C4#39312yes—
392C2×C4#45012yes—
393C2×C4#45112yes—
394C2×C4#47514yes—
395C2×C4#47714yes—
396C2×C4#48714yes—
397C2×C4#48914yes—
398C2×C4C2×C22⋊Q811yescharacteristic
399C2×C4C2×C22⋊Q811yescharacteristic
400C2×C4C2×C22⋊Q811yescharacteristic
401C2×C4C2×#12411yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12411yescharacteristic
404C2×C4C22×SD1611yescharacteristic
405C2×C4C22×Q1611yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57512yes—
409C2×C4#57512yes—
410C8—82no—
411C8—82no—
412C8—82no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22#202414yes—
420C22C2×#39311yescharacteristic
421C22C2×#45011yescharacteristic
422C22C2×#45111yescharacteristic
423C22#300511yescharacteristic
424C22#300511yescharacteristic
425C22#301911yescharacteristic
426C22#302011yescharacteristic
427C4—22no—
428C4—21no—
429C4—21no—
430C4—22no—
431C4—44no—
432C4—44no—
433C4—82no—
434C4—82no—
435C4—84no—
436C4—82no—
437C4—82no—
438C4#202414yes—
439C4C2×#47511yescharacteristic
440C4C2×#47711yescharacteristic
441C4#308012yes—
442C4C2×#48711yescharacteristic
443C4C2×#48911yescharacteristic
444C4#314212yes—
445C2—24no—
446C2#1874911yescharacteristic
447C2#1875711yescharacteristic
448C2#1876711yescharacteristic
449C2#1877911yescharacteristic
450C2C2×#202411yescharacteristic
451C2#2345311yescharacteristic
452C2#2346211yescharacteristic
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
5. #10357C2semidirect
6. #10355C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#218602
#213852
#213122
#103572
#103552
#80432
#79012
#78922