Finite Group Databasebeta

← #241301

Subgroups — #241301

Order 512.

994 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#21170C211yesmaximal, characteristic
5#10581C214yesmaximal, complemented
6#10580C214yesmaximal, complemented
7#8043C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7892C211yesmaximal, characteristic
10C22×#125—21no—
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—21no—
14C2×#423C2211yescharacteristic
15#2552—22no—
16#2552—22no—
17#2552—22no—
18#2552C2214yes—
19C2×#398—22no—
20C2×#389—21no—
21C2×#389—21no—
22#2285C2211yescharacteristic
23C2×#384—21no—
24C2×#384C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#1949—22no—
28#1948—22no—
29#1948C2212yes—
30#1947C2212yes—
31#1947C2212yes—
32#1947C2212yes—
33#1853—22no—
34#1852—21no—
35#1852—21no—
36#1844C2212yes—
37#1834C2212yes—
38C4×#107—22no—
39#1736—21no—
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C22×M16—21no—
44C2×#125—24no—
45C2×#125—24no—
46C2×#125—48no—
47C2×#117—21no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4D411yescharacteristic
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D412yes—
58C22×C42—21no—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#431Q814yes—
64#431Q814yes—
65#428—44no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—22no—
69C2×#106—22no—
70C2×#106—24no—
71#423—22no—
72#423Q814yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—21no—
78C2×C22⋊C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#398—28no—
82#398—28no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#384—24no—
88#384—24no—
89#384—21no—
90#384—21no—
91#384D412yes—
92#384C2311yescharacteristic
93#384C2311yescharacteristic
94#384C2312yes—
95#382—21no—
96#380—21no—
97#380—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4C2312yes—
107C4×C4⋊C4C2312yes—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111#340—24no—
112#340—22no—
113#340—22no—
114#340—22no—
115#340—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—24no—
125C2×M16—42no—
126C2×M16—42no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130#125—416no—
131#125—48no—
132#125—48no—
133#117—42no—
134#117—42no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D414yes—
171C2×C22⋊C4—41no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—44no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—41no—
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×Q812yes—
184C2×C42C2×Q812yes—
185C2×C42C2×Q812yes—
186C2×C42#4112yes—
187C2×C42C2411yescharacteristic
188#107—44no—
189#107—44no—
190#107—44no—
191#106—44no—
192#106—84no—
193C4⋊C8—24no—
194C4⋊C8—24no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8C2×Q814yes—
198C4⋊C8C2×Q814yes—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C4×C8—24no—
202C4×C8—22no—
203C4×C8C2×Q814yes—
204#95—28no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—41no—
210#95—41no—
211C24#12011yescharacteristic
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4#12011yescharacteristic
239C22×C4#12011yescharacteristic
240C22×C4#12011yescharacteristic
241C22×C4C22⋊Q812yes—
242C22×C4C22⋊Q812yes—
243C22×C4C22⋊Q812yes—
244C22×C4C4⋊Q812yes—
245C22×C4C4⋊Q812yes—
246C22×C4C4⋊Q812yes—
247C22×C4#13512yes—
248C22×C4#13512yes—
249C22×C4AGL(1,ℤ/8)12yes—
250C22×C4#13712yes—
251C22×C4C22×D411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254M16—48no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—24no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—416no—
275C4⋊C4—416no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—88no—
292C4⋊C4AGL(1,ℤ/8)14yes—
293C4⋊C4#13714yes—
294C22⋊C4—82no—
295C42—44no—
296C42—42no—
297C42—42no—
298C42—44no—
299C42—44no—
300C42—82no—
301C42C22⋊Q812yes—
302C42C22⋊Q812yes—
303C42C22⋊Q812yes—
304C42C4⋊Q812yes—
305C42C4⋊Q812yes—
306C42C4⋊Q812yes—
307C42C22×Q811yescharacteristic
308C42C22×Q811yescharacteristic
309C42C22×Q811yescharacteristic
310C42C2×#4111yescharacteristic
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23#39512yes—
316C23#44912yes—
317C23#45212yes—
318C23C2×#12011yescharacteristic
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—48no—
332C2×C4—48no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4—88no—
348C2×C4—84no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4#39512yes—
356C2×C4#39512yes—
357C2×C4#39512yes—
358C2×C4#44912yes—
359C2×C4#45212yes—
360C2×C4#47814yes—
361C2×C4#47914yes—
362C2×C4#49914yes—
363C2×C4#50114yes—
364C2×C4C2×C22⋊Q811yescharacteristic
365C2×C4C2×C22⋊Q811yescharacteristic
366C2×C4C2×C22⋊Q811yescharacteristic
367C2×C4C2×C4⋊Q811yescharacteristic
368C2×C4C2×C4⋊Q811yescharacteristic
369C2×C4C2×C4⋊Q811yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4C2×#13511yescharacteristic
372C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)12yes—
374C2×C4C2×#13711yescharacteristic
375C2×C4C2×#13712yes—
376C8—44no—
377C8—44no—
378C8—44no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22#205914yes—
386C22C2×#39511yescharacteristic
387C22C2×#44911yescharacteristic
388C22C2×#45211yescharacteristic
389C22#300311yescharacteristic
390C22#300411yescharacteristic
391C22#301911yescharacteristic
392C22#302011yescharacteristic
393C4—22no—
394C4—21no—
395C4—21no—
396C4—22no—
397C4—44no—
398C4—44no—
399C4—82no—
400C4—84no—
401C4—82no—
402C4—161no—
403C4—161no—
404C4#205914yes—
405C4C2×#47811yescharacteristic
406C4C2×#47911yescharacteristic
407C4#307912yes—
408C4C2×#49911yescharacteristic
409C4C2×#50111yescharacteristic
410C4#316512yes—
411C2—24no—
412C2#1875211yescharacteristic
413C2#1876111yescharacteristic
414C2#1877111yescharacteristic
415C2#1878211yescharacteristic
416C2C2×#205911yescharacteristic
417C2#2355211yescharacteristic
418C2#2356111yescharacteristic
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. #10581C2semidirect
6. #10580C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25522
#219812
#213582
#211702
#105812
#105802
#80432
#79012
#78922