Finite Group Databasebeta

← #280995

Subgroups — #280995

Order 512.

654 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28227C211yesmaximal, characteristic
2#24966C212yesmaximal, complemented
3#22807C211yesmaximal, characteristic, complemented
4#10622C212yesmaximal, complemented
5#10431C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9615C211yesmaximal, characteristic, complemented
8#8893C212yesmaximal, complemented
9#8816C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#466—21no—
13C2×C4×D8C2211yescharacteristic
14C2×C8×D4C2211yescharacteristic
15#2537C2212yes—
16#2354C2212yes—
17#2077—22no—
18#2034—22no—
19#2033—22no—
20#1970—21no—
21#1954C2212yes—
22#1948C2212yes—
23#1925—22no—
24#1875—22no—
25#1845C2211yescharacteristic
26#1845C2211yescharacteristic
27#1834C2212yes—
28#1834C2212yes—
29#1825—22no—
30#1808—21no—
31#1802C2212yes—
32#1802C2212yes—
33#1802C2212yes—
34C4×#107—21no—
35C8×C4⋊C4C2211yescharacteristic
36C4×#102C2211yescharacteristic
37C2×#335C2211yescharacteristic
38C2×D4⋊C8C2211yescharacteristic
39C2×D4⋊C8C2212yes—
40C2×C82C2211yescharacteristic
41#1296C2212yes—
42#1296C2212yes—
43#1279C2211yescharacteristic
44#1277C2211yescharacteristic
45C22×D8D411yescharacteristic
46C23×C8—21no—
47C2×#121—22no—
48C2×#121—22no—
49C2×C4×D4C2312yes—
50C2×C4×D4C2311yescharacteristic
51C2×#117—22no—
52#466—44no—
53C4×D8—22no—
54C4×D8—22no—
55C8×D4—22no—
56C8×D4—22no—
57C2×#107—21no—
58C2×#107—21no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107D411yescharacteristic
62C2×#106—22no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×#102—22no—
67C2×#102—22no—
68C2×#102—22no—
69C2×#102D411yescharacteristic
70C2×#102D412yes—
71C2×#102D411yescharacteristic
72C2×C22⋊C8—21no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#394—42no—
80#390—22no—
81#388—42no—
82#384C2312yes—
83#384C2312yes—
84#380—22no—
85#380—22no—
86#380—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2311yescharacteristic
89#340—22no—
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—42no—
96#340D411yescharacteristic
97#340D411yescharacteristic
98#336—21no—
99#335—22no—
100D4⋊C8—22no—
101D4⋊C8—24no—
102C82—22no—
103C22×D4—42no—
104C22×D4C2×D412yes—
105C22×D4C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—21no—
108C2×D8—22no—
109C2×D8—22no—
110C2×D8—22no—
111C22×C8—21no—
112C22×C8—21no—
113C22×C8—21no—
114C22×C8—22no—
115C22×C8—42no—
116C22×C8—42no—
117C22×C8—42no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122#121—44no—
123#121—44no—
124#121—82no—
125#121—82no—
126C4×D4—21no—
127C4×D4—22no—
128C4×D4—21no—
129C4×D4—21no—
130C4×D4—22no—
131C4×D4—22no—
132C4×D4—21no—
133C4×D4—22no—
134#117—44no—
135#117—82no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D412yes—
144C2×C4⋊C4C2×D411yescharacteristic
145C2×C4⋊C4C2×D411yescharacteristic
146C2×C22⋊C4—22no—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—42no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—42no—
152C2×C42—21no—
153C2×C42—42no—
154C2×C42C2411yescharacteristic
155#107—22no—
156#107—42no—
157#107—42no—
158#107—81no—
159#107—81no—
160#106—82no—
161C4⋊C8—22no—
162C4⋊C8—21no—
163C4⋊C8—22no—
164C4⋊C8—21no—
165C4⋊C8—21no—
166C4⋊C8—21no—
167#102—22no—
168#102—24no—
169#102—22no—
170#102—44no—
171#102—82no—
172#102—82no—
173C22⋊C8—42no—
174C22⋊C8—82no—
175C22⋊C8—82no—
176C4×C8—21no—
177C4×C8—21no—
178C4×C8—21no—
179C4×C8—42no—
180C4×C8#4112yes—
181C4×C8#4112yes—
182C4×C8#4112yes—
183#95—21no—
184#95—42no—
185#95—42no—
186C24—21no—
187C24—22no—
188C2×D4—22no—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—22no—
198C2×D4—22no—
199C2×D4—84no—
200C2×D4—84no—
201C2×D4—162no—
202C2×D4#13512yes—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—41no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4H(5,𝔽2)11yescharacteristic
224D8—44no—
225C2×C8—21no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—21no—
229C2×C8—22no—
230C2×C8—21no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8—42no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—84no—
246C2×C8#12312yes—
247C2×C8#12312yes—
248C2×C8#13512yes—
249C2×C8AGL(1,ℤ/8)12yes—
250C4⋊C4—21no—
251C4⋊C4—22no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—21no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4#13512yes—
268C22⋊C4—41no—
269C22⋊C4—42no—
270C22⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C22⋊C4—162no—
276C42—41no—
277C42—41no—
278C42—84no—
279C42C2×#4111yescharacteristic
280C42C2×#4111yescharacteristic
281C42C2×#4111yescharacteristic
282C422−1+411yescharacteristic
283C23—22no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23D4211yescharacteristic
298D4—22no—
299D4—22no—
300D4—41no—
301D4—42no—
302D4—42no—
303D4—42no—
304D4—42no—
305D4—41no—
306D4—168no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4C2×#12311yescharacteristic
336C2×C4#53211yescharacteristic
337C2×C4#54211yescharacteristic
338C2×C4#54711yescharacteristic
339C2×C4#54711yescharacteristic
340C2×C4#55311yescharacteristic
341C2×C4C2×#13511yescharacteristic
342C2×C4C2×#13511yescharacteristic
343C2×C4C2×#13511yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4#57611yescharacteristic
346C2×C4#57611yescharacteristic
347C2×C4#57611yescharacteristic
348C2×C4#57811yescharacteristic
349C8—22no—
350C8—42no—
351C8—41no—
352C8—41no—
353C8—81no—
354C8—81no—
355C8—82no—
356C8#48214yes—
357C22—42no—
358C22—42no—
359C22—42no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22—82no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—82no—
370C22—82no—
371C22#242211yescharacteristic
372C22#327411yescharacteristic
373C22#328011yescharacteristic
374C22#328011yescharacteristic
375C22#328311yescharacteristic
376C22#329011yescharacteristic
377C22#329011yescharacteristic
378C4—21no—
379C4—21no—
380C4—42no—
381C4—81no—
382C4—81no—
383C4—82no—
384C4—162no—
385C4—162no—
386C4C2×#48211yescharacteristic
387C4#309811yescharacteristic
388C4#316811yescharacteristic
389C4#317611yescharacteristic
390C4#333911yescharacteristic
391C4#334111yescharacteristic
392C4#334111yescharacteristic
393C4#335311yescharacteristic
394C2—82no—
395C2—162no—
396C2#1990011yescharacteristic
397C2#1990811yescharacteristic
398C2#1998911yescharacteristic
399C2#1998911yescharacteristic
400C2#2501111yescharacteristic
401C2#2503211yescharacteristic
402C2#2505211yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radical#28227C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #24966C2semidirect
3. #22807C2semidirect
4. #10622C2semidirect
5. #10431C2semidirect
6. #9657C2semidirect
7. #9615C2semidirect
8. #8893C2semidirect
9. #8816C2semidirect
10. #8644C2semidirect
11. C8×#102C2semidirect

Maximal subgroups

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

subgroupindex
#282272
#249662
#228072
#106222
#104312
#96572
#96152
#88932
#88162
#86442
C8×#1022