Finite Group Databasebeta

← #314979

Subgroups — #314979

Order 512.

497 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21492C211yesmaximal, characteristic
2#20828C211yesmaximal, characteristic, complemented
3#10689C212yesmaximal, complemented
4#10659C212yesmaximal, complemented
5#9631C211yesmaximal, characteristic, complemented
6#9630C211yesmaximal, characteristic, complemented
7#9036C212yesmaximal
8#8863C212yesmaximal, complemented
9#8658C211yesmaximal, characteristic, complemented
10#8658C211yesmaximal, characteristic, complemented
11#8602C211yesmaximal, characteristic, complemented
12C2×#435—21no—
13C2×C8×D4C2211yescharacteristic
14#2921C2212yes—
15C2×#405C2211yescharacteristic
16#2392C2211yescharacteristic
17#1955C2211yescharacteristic
18#1940C2211yescharacteristic
19#1923—21no—
20#1869—21no—
21#1848C2211yescharacteristic
22#1848C2211yescharacteristic
23#1838C2212yes—
24#1811—21no—
25#1810—21no—
26#1798C2211yescharacteristic
27#1778—21no—
28C4×#108—21no—
29#1770C2211yescharacteristic
30#1752—21no—
31#1571C2212yes—
32#1570C2212yes—
33#1564C2211yescharacteristic
34#1564C2211yescharacteristic
35#1486C2212yes—
36#1486C2212yes—
37#1458C2212yes—
38#1303C2212yes—
39#1303C2212yes—
40#1301C2212yes—
41#1301C2212yes—
42#1275C2212yes—
43C22×M16D411yescharacteristic
44C23×C8—21no—
45C23×C8D411yescharacteristic
46C2×C4×D4—21no—
47C2×C4×D4C2311yescharacteristic
48C2×#117C2311yescharacteristic
49#435—44no—
50C8×D4—21no—
51C8×D4—21no—
52C8×D4—21no—
53C8×D4—21no—
54#433—42no—
55#433—42no—
56#431C2312yes—
57#431C2312yes—
58C2×#108—21no—
59C2×#108—21no—
60C2×#108—21no—
61C2×#108—21no—
62#423C2312yes—
63#423C2312yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67#407—22no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—21no—
72#405—21no—
73#405—21no—
74#405—21no—
75#405—21no—
76C4×M16—22no—
77C2×#97—21no—
78C2×#97D411yescharacteristic
79C2×#97C2311yescharacteristic
80C2×C4×C8D411yescharacteristic
81C2×C4×C8C2311yescharacteristic
82#398Q811yescharacteristic
83#397Q811yescharacteristic
84#396—21no—
85#394—21no—
86#394Q811yescharacteristic
87#392Q811yescharacteristic
88C4×C4⋊C4C2311yescharacteristic
89C4×C22⋊C4—21no—
90#341—22no—
91#341—22no—
92#336—21no—
93#336—21no—
94#336—21no—
95#335—21no—
96#335—21no—
97#335—42no—
98#334—21no—
99#334—21no—
100#334—42no—
101D4⋊C8—42no—
102D4⋊C8—42no—
103#324—22no—
104#323—22no—
105C22×D4—21no—
106C22×D4C2×Q811yescharacteristic
107C23×C4—21no—
108C23×C4C2×D411yescharacteristic
109C2×M16—21no—
110C2×M16—21no—
111C2×M16—22no—
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—22no—
115C22×C8—21no—
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8—41no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C4×D4—21no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—21no—
133C4×D4—42no—
134C4×D4—42no—
135#117—21no—
136#117—21no—
137#117—21no—
138#117—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4C2×Q811yescharacteristic
142C2×C4⋊C4C2×Q811yescharacteristic
143C2×C4⋊C4#4111yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146C2×C22⋊C4—21no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—21no—
149C2×C22⋊C4C2×Q811yescharacteristic
150C2×C22⋊C4#4111yescharacteristic
151C2×C42—21no—
152C2×C42C2×D411yescharacteristic
153C2×C42C2411yescharacteristic
154#108—42no—
155#108—42no—
156#108—42no—
157#108—42no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—21no—
161C4⋊C8—21no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—21no—
165C4⋊C8—22no—
166C4⋊C8—21no—
167C4⋊C8—22no—
168C4⋊C8—42no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—42no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—44no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177#97—21no—
178#97—21no—
179#97—21no—
180#97—21no—
181#97—42no—
182#97—42no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—42no—
188C4×C8#4111yescharacteristic
189C4×C8#4111yescharacteristic
190C4×C8#4111yescharacteristic
191C4×C8#4111yescharacteristic
192#95—21no—
193#95—21no—
194#95C2×Q811yescharacteristic
195#95C2×Q811yescharacteristic
196C24—21no—
197C24C22⋊Q811yescharacteristic
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—81no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×Q811yescharacteristic
224C22×C4C2×#4111yescharacteristic
225M16—42no—
226M16—42no—
227M16—42no—
228M16—42no—
229M16—42no—
230M16—42no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—21no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—82no—
252C2×C8—82no—
253C2×C8#12311yescharacteristic
254C2×C8#12311yescharacteristic
255C2×C8#12311yescharacteristic
256C2×C8#12311yescharacteristic
257C2×C8#13512yes—
258C2×C8AGL(1,ℤ/8)12yes—
259C4⋊C4—21no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—81no—
277C42—21no—
278C42—21no—
279C42—41no—
280C42—41no—
281C42C2×#4111yescharacteristic
282C42C2×#4111yescharacteristic
283C42H(5,𝔽2)11yescharacteristic
284C42H(5,𝔽2)11yescharacteristic
285C23—22no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—41no—
290C23—41no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23C2×C22⋊Q811yescharacteristic
298D4—22no—
299D4—22no—
300D4—41no—
301D4—41no—
302D4—84no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4C2×#12311yescharacteristic
330C2×C4#53511yescharacteristic
331C2×C4#55311yescharacteristic
332C2×C4#55311yescharacteristic
333C2×C4#55411yescharacteristic
334C2×C4#55411yescharacteristic
335C2×C4C2×#13511yescharacteristic
336C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
337C2×C4#57511yescharacteristic
338C2×C4#57511yescharacteristic
339C8—21no—
340C8—21no—
341C8—41no—
342C8—41no—
343C8—82no—
344C8—82no—
345C8—81no—
346C8—82no—
347C8—81no—
348C8—82no—
349C8#48212yes—
350C8#48212yes—
351C22—21no—
352C22—21no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—41no—
357C22—42no—
358C22—41no—
359C22—41no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22#248711yescharacteristic
366C22#307611yescharacteristic
367C22#307711yescharacteristic
368C4—21no—
369C4—21no—
370C4—21no—
371C4—21no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4C2×#48211yescharacteristic
379C4#309311yescharacteristic
380C4#321211yescharacteristic
381C4#321411yescharacteristic
382C2—21no—
383C2—21no—
384C2—81no—
385C2—82no—
386C2#2767911yescharacteristic
387C2#2769111yescharacteristic
388C2#2769411yescharacteristic
subgroupG/H
centerC2×C4#535
commutator, Frattini, cosocleC2×C42C24
socleC22#2487
prime radical#392Q8
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20828C2semidirect
3. #10689C2semidirect
4. #10659C2semidirect
5. #9631C2semidirect
6. #9630C2semidirect
8. #8863C2semidirect
9. #8658C2semidirect
10. #8658C2semidirect
11. #8602C2semidirect

Maximal subgroups

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

subgroupindex
#214922
#208282
#106892
#106592
#96312
#96302
#90362
#88632
#86582
#86022