Finite Group Databasebeta

← #416684

Subgroups — #416684

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1947C211yesmaximal, characteristic
3C2×#1845C211yesmaximal, characteristic
4#13911C211yesmaximal, characteristic
5#11324C214yesmaximal
6#11219C214yesmaximal
7#10849C211yesmaximal, characteristic
8#10834C211yesmaximal, characteristic
9C4×#340C211yesmaximal, characteristic
10C22×#125—21no—
11C22×#107C2214yes—
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4×C8C2211yescharacteristic
15#2552—24no—
16#2552—24no—
17C2×#398—22no—
18C2×#389—21no—
19C2×#389C2212yes—
20C2×#384C2211yescharacteristic
21C2×#384C2212yes—
22C2×#384C2211yescharacteristic
23C2×#382—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1947—24no—
28#1845—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2214yes—
33C2×#336C2214yes—
34#1442—22no—
35C4⋊#95—21no—
36C4⋊#95C2214yes—
37C23×C8C2312yes—
38C23×C8C2312yes—
39C2×#125—28no—
40C2×#125—44no—
41C2×#125—44no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42—21no—
58C22×C42C2312yes—
59C22×C42C2311yescharacteristic
60C2×#107—28no—
61C2×#107—28no—
62C2×#107—24no—
63C2×#107—44no—
64C2×#107—44no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8Q814yes—
68C2×C4×C8—22no—
69C2×C4×C8—22no—
70C2×C4×C8Q814yes—
71#398—216no—
72#389—28no—
73#389—24no—
74#389—24no—
75#384—24no—
76#384—24no—
77#384—22no—
78#384—22no—
79#384—22no—
80#384D414yes—
81#382—44no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D414yes—
87C4×C4⋊C4Q814yes—
88C2×#95—21no—
89C2×#95—22no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95D412yes—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—44no—
98#336—28no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—22no—
103C23×C4—22no—
104C23×C4C2×D412yes—
105C23×C4C2×D412yes—
106C23×C4C2×D411yescharacteristic
107C23×C4C2×D411yescharacteristic
108C23×C4C2411yescharacteristic
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—24no—
115C22×C8—22no—
116C22×C8—24no—
117C22×C8C2×Q814yes—
118C22×C8C2×Q814yes—
119C22×C8#4114yes—
120C22×C8#4114yes—
121#125—432no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—24no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—24no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—44no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—44no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—84no—
162C2×C4⋊C4D814yes—
163C2×C4⋊C4Q1614yes—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D414yes—
167C2×C4⋊C4C2×Q814yes—
168C2×C42—24no—
169C2×C42—24no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×Q812yes—
181C2×C42C2×Q812yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184#107—416no—
185#107—48no—
186C4⋊C8—28no—
187C4×C8—28no—
188#95—28no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95—48no—
193#95—84no—
194C24C22×D411yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—24no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—44no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—44no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4#12012yes—
233C22×C4#12012yes—
234C22×C4C22⋊Q812yes—
235C22×C4C22⋊Q812yes—
236C22×C4C4⋊Q812yes—
237C22×C4C4⋊Q812yes—
238C22×C4C2×D814yes—
239C22×C4C2×D812yes—
240C22×C4C2×Q1614yes—
241C22×C4C2×Q1612yes—
242C22×C4AGL(1,ℤ/8)12yes—
243C22×C4#13712yes—
244C22×C4C22×D411yescharacteristic
245C22×C4C22×D411yescharacteristic
246C22×C4C22×Q811yescharacteristic
247C22×C4C22×Q811yescharacteristic
248C22×C4C22×Q811yescharacteristic
249C22×C4C2×#4111yescharacteristic
250C22×C4C2×#4112yes—
251C22×C4C2×#4112yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+412yes—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—28no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—44no—
265C2×C8#12518yescomplemented
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—24no—
274C4⋊C4—28no—
275C4⋊C4—416no—
276C4⋊C4—416no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—84no—
288C4⋊C4—84no—
289C4⋊C4—88no—
290C4⋊C4—84no—
291C42—42no—
292C42—42no—
293C42—44no—
294C42—48no—
295C42—48no—
296C42C22⋊Q814yes—
297C42C22⋊Q814yes—
298C42C4⋊Q814yes—
299C42C4⋊Q814yes—
300C23C2×#12011yescharacteristic
301C23C2×C22⋊Q811yescharacteristic
302C23C2×C4⋊Q811yescharacteristic
303C23#54712yes—
304C23#55012yes—
305C23C22×D811yescharacteristic
306C23C22×Q1611yescharacteristic
307C23C2×AGL(1,ℤ/8)11yescharacteristic
308C23C2×#13711yescharacteristic
309C23#57612yes—
310C23#57812yes—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4#39514yes—
353C2×C4#39514yes—
354C2×C4#44714yes—
355C2×C4#45114yes—
356C2×C4#47414yes—
357C2×C4#47714yes—
358C2×C4#50014yes—
359C2×C4#50114yes—
360C2×C4C2×C22⋊Q811yescharacteristic
361C2×C4C2×C22⋊Q811yescharacteristic
362C2×C4C2×#12512yes—
363C2×C4C2×C4⋊Q811yescharacteristic
364C2×C4C2×C4⋊Q811yescharacteristic
365C2×C4#53211yescharacteristic
366C2×C4#53211yescharacteristic
367C2×C4#53312yes—
368C2×C4#53511yescharacteristic
369C2×C4#53611yescharacteristic
370C2×C4#54412yes—
371C2×C4#54412yes—
372C2×C4#54612yes—
373C2×C4#54812yes—
374C2×C4#55212yes—
375C2×C4#55312yes—
376C2×C4#55412yes—
377C2×C4#55712yes—
378C8—28no—
379C8—44no—
380C22C2×#39511yescharacteristic
381C22#243111yescharacteristic
382C22#243211yescharacteristic
383C22#244112yes—
384C22#255212yes—
385C22C2×#44711yescharacteristic
386C22C2×#45111yescharacteristic
387C22#300811yescharacteristic
388C22#301111yescharacteristic
389C22C2×#47411yescharacteristic
390C22C2×#47711yescharacteristic
391C22#308312yes—
392C22C2×#50011yescharacteristic
393C22C2×#50111yescharacteristic
394C22#316412yes—
395C22#333412yes—
396C22#333812yes—
397C22#335312yes—
398C22#335712yes—
399C22#339112yes—
400C22#339512yes—
401C22#339812yes—
402C22#339912yes—
403C4—24no—
404C4—44no—
405C4—44no—
406C4—82no—
407C4—84no—
408C4—82no—
409C4—84no—
410C4—162no—
411C4#205718yes—
412C4#244212yes—
413C4#255012yes—
414C4#256511yescharacteristic
415C4#256611yescharacteristic
416C4#266011yescharacteristic
417C4#285911yescharacteristic
418C2#1522811yescharacteristic
419C2C2×#205711yescharacteristic
420C2#2355511yescharacteristic
421C2#2563111yescharacteristic
422C2#2563411yescharacteristic
423C2#2570311yescharacteristic
424C2#2570611yescharacteristic
425C2#2608612yes—
426C2#2609212yes—
427C2#2882712yes—
428C2#2883512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
265. C2×C8#125semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
C2×#19472
C2×#18452
#139112
#113242
#112192
#108492
#108342
C4×#3402