Finite Group Databasebeta

← #280989

Subgroups — #280989

Order 512.

704 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28227C211yesmaximal, characteristic
2#24965C212yesmaximal, complemented
3C4⋊C4×D8C211yesmaximal, characteristic, complemented
4#10621C212yesmaximal, complemented
5#10431C212yesmaximal, complemented
6#9657C211yesmaximal, characteristic, complemented
7#9614C211yesmaximal, characteristic, complemented
8#8890C212yesmaximal, complemented
9#8816C211yesmaximal, characteristic, complemented
10#8644C211yesmaximal, characteristic, complemented
11C8×#102C211yesmaximal, characteristic, complemented
12C2×#466—21no—
13C2×C4×D8—21no—
14C2×C4×D8C2211yescharacteristic
15C2×C8×D4C2211yescharacteristic
16#2537C2212yes—
17D4×C4⋊C4C2212yes—
18#2075—22no—
19#2033—22no—
20#2033—22no—
21#1954C2212yes—
22#1947C2212yes—
23#1942—21no—
24#1874—22no—
25#1862—22no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1834C2212yes—
29#1834C2212yes—
30#1823—22no—
31#1808—21no—
32#1802C2212yes—
33#1802C2212yes—
34#1802C2212yes—
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×D4—22no—
50C2×C4×D4C2312yes—
51C2×C4×D4C2311yescharacteristic
52C22×C4⋊C4—22no—
53#466—44no—
54C4×D8—22no—
55C4×D8—22no—
56C4×D8—44no—
57C8×D4—22no—
58C8×D4—22no—
59C2×#107—21no—
60C2×#107—21no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107—22no—
64C2×#107D411yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×C4⋊C8C2312yes—
68C2×#102—22no—
69C2×#102—22no—
70C2×#102—22no—
71C2×#102D412yes—
72C2×#102D411yescharacteristic
73C2×#102D411yescharacteristic
74C2×C22⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×C22⋊C8—22no—
77C2×C4×C8—21no—
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80C2×C4×C8C2311yescharacteristic
81#394—42no—
82#389—22no—
83#385—42no—
84#384C2312yes—
85#384C2312yes—
86#380—22no—
87#380—22no—
88#380—22no—
89C4×C4⋊C4C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—42no—
97#340D411yescharacteristic
98#340D411yescharacteristic
99#336—21no—
100#335—22no—
101D4⋊C8—22no—
102D4⋊C8—24no—
103C82—22no—
104C22×D4—42no—
105C22×D4C2×D412yes—
106C22×D4C2×D411yescharacteristic
107C23×C4—22no—
108C23×C4—21no—
109C23×C4—42no—
110C2×D8—22no—
111C2×D8—22no—
112C2×D8D814yes—
113C22×C8—21no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—42no—
118C22×C8—42no—
119C22×C8—42no—
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C22×C8C2×D411yescharacteristic
123C22×C8C2×D411yescharacteristic
124#121—44no—
125#121—44no—
126#121—82no—
127#121—82no—
128C4×D4—21no—
129C4×D4—22no—
130C4×D4—21no—
131C4×D4—21no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—21no—
136C4×D4—44no—
137C4×D4—82no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—44no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—82no—
147C2×C4⋊C4C2×D411yescharacteristic
148C2×C4⋊C4C2×D412yes—
149C2×C4⋊C4C2×D411yescharacteristic
150C2×C4⋊C4C2×D411yescharacteristic
151C2×C22⋊C4—22no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C42—21no—
157C2×C42C2411yescharacteristic
158#107—22no—
159#107—42no—
160#107—42no—
161#107—82no—
162#107—81no—
163#107—81no—
164C4⋊C8—22no—
165C4⋊C8—21no—
166C4⋊C8—22no—
167C4⋊C8—21no—
168C4⋊C8—21no—
169C4⋊C8—21no—
170#102—22no—
171#102—24no—
172#102—22no—
173#102—44no—
174#102—44no—
175#102—82no—
176C22⋊C8—42no—
177C22⋊C8—82no—
178C22⋊C8—82no—
179C4×C8—21no—
180C4×C8—21no—
181C4×C8—21no—
182C4×C8—42no—
183C4×C8#4112yes—
184C4×C8#4112yes—
185C4×C8#4112yes—
186#95—21no—
187#95—42no—
188#95—42no—
189C24—22no—
190C24—21no—
191C2×D4—22no—
192C2×D4—21no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—21no—
196C2×D4—22no—
197C2×D4—21no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—84no—
202C2×D4—84no—
203C2×D4—162no—
204C2×D4C2×D814yes—
205C2×D4#13512yes—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—84no—
225C22×C4—84no—
226C22×C4—162no—
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4H(5,𝔽2)11yescharacteristic
230D8—28no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—41no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—41no—
251C2×C8—84no—
252C2×C8#12312yes—
253C2×C8#12312yes—
254C2×C8C2×D812yes—
255C2×C8#13712yes—
256C4⋊C4—21no—
257C4⋊C4—22no—
258C4⋊C4—21no—
259C4⋊C4—22no—
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—84no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4#13512yes—
275C22⋊C4—42no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—84no—
280C22⋊C4—84no—
281C22⋊C4—84no—
282C42—41no—
283C42—41no—
284C42C2×#4111yescharacteristic
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C422−1+411yescharacteristic
288C23—24no—
289C23—22no—
290C23—42no—
291C23—41no—
292C23—41no—
293C23—42no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23—41no—
298C23—42no—
299C23—41no—
300C23—42no—
301C23—41no—
302C23D4211yescharacteristic
303D4—24no—
304D4—22no—
305D4—24no—
306D4—22no—
307D4—41no—
308D4—42no—
309D4—42no—
310D4—41no—
311D4—168no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—82no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—168no—
341C2×C4C2×#12311yescharacteristic
342C2×C4#53211yescharacteristic
343C2×C4#54211yescharacteristic
344C2×C4#54711yescharacteristic
345C2×C4#54711yescharacteristic
346C2×C4#55311yescharacteristic
347C2×C4C22×D811yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×#13511yescharacteristic
350C2×C4C2×#13711yescharacteristic
351C2×C4#57611yescharacteristic
352C2×C4#57611yescharacteristic
353C2×C4#57611yescharacteristic
354C2×C4#57811yescharacteristic
355C8—22no—
356C8—42no—
357C8—41no—
358C8—41no—
359C8—81no—
360C8—81no—
361C8—82no—
362C8#48014yes—
363C22—44no—
364C22—42no—
365C22—44no—
366C22—44no—
367C22—42no—
368C22—42no—
369C22—81no—
370C22—81no—
371C22—82no—
372C22—81no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—81no—
377C22#242211yescharacteristic
378C22#327611yescharacteristic
379C22D4×D811yescharacteristic
380C22#328011yescharacteristic
381C22#328011yescharacteristic
382C22#329011yescharacteristic
383C22#329011yescharacteristic
384C4—21no—
385C4—21no—
386C4—42no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4—162no—
391C4—162no—
392C4C2×#48011yescharacteristic
393C4#309811yescharacteristic
394C4#316811yescharacteristic
395C4#317611yescharacteristic
396C4#333411yescharacteristic
397C4#333911yescharacteristic
398C4#334111yescharacteristic
399C4#335711yescharacteristic
400C2—84no—
401C2—82no—
402C2#1990311yescharacteristic
403C2#1990511yescharacteristic
404C2#1996811yescharacteristic
405C2#1996811yescharacteristic
406C2#2501011yescharacteristic
407C2#2503211yescharacteristic
408C2#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. #24965C2semidirect
3. C4⋊C4×D8C2semidirect
4. #10621C2semidirect
5. #10431C2semidirect
6. #9657C2semidirect
7. #9614C2semidirect
8. #8890C2semidirect
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
#249652
C4⋊C4×D82
#106212
#104312
#96572
#96142
#88902
#88162
#86442
C8×#1022