Finite Group Databasebeta

← #214640

Subgroups — #214640

Order 512.

1122 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37302C211yesmaximal, characteristic
2#21755C211yesmaximal, characteristic, complemented
3#21750C211yesmaximal, characteristic, complemented
4#9463C214yesmaximal, complemented
5#9445C214yesmaximal, complemented
6#8741C211yesmaximal, characteristic, complemented
7#8740C211yesmaximal, characteristic, complemented
8#8518C211yesmaximal, characteristic, complemented
9#8518C211yesmaximal, characteristic, complemented
10C2×#540C2211yescharacteristic
11C2×#535C2211yescharacteristic
12#2879C2212yes—
13#2867C411yescharacteristic, complemented
14#2861C411yescharacteristic, complemented
15#2673—24no—
16#2669—24no—
17#2657C411yescharacteristic, complemented
18#2641C411yescharacteristic, complemented
19#2544C412yescomplemented
20#2541C412yescomplemented
21#2433C2212yes—
22#2431C2212yes—
23#2308C2212yes—
24#2298C2211yescharacteristic
25#1900—24no—
26#1898—24no—
27#1822C2211yescharacteristic
28#1822C2211yescharacteristic
29#1820C2211yescharacteristic
30#1820C2211yescharacteristic
31#1783C2212yes—
32#1782C2212yes—
33#1510C2214yes—
34#1510C2214yes—
35#1284C2214yes—
36#1284C2214yes—
37C22×#41C2311yescharacteristic
38C23×D4C2311yescharacteristic
39C22×M16—22no—
40#540—44no—
41#535—44no—
42C2×#127—24no—
43C2×#127C2×C411yescharacteristic
44C2×#127C2×C411yescharacteristic
45C2×#127C2×C412yes—
46C2×#127C2×C412yes—
47C2×#127C2×C411yescharacteristic, complemented
48C2×#127C2×C412yescomplemented
49C2×#125C2×C411yescharacteristic, complemented
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52C2×#124C2×C412yescomplemented
53C2×#123—22no—
54C2×#121—24no—
55C2×#121—22no—
56C2×#121D414yes—
57C2×#121C2312yes—
58C2×#120—24no—
59C2×#120D414yes—
60C2×C4×D4C2×C412yescomplemented
61C2×#117C2311yescharacteristic
62C22×C22⋊C4—22no—
63C2×#100—24no—
64C2×#99—24no—
65#408—44no—
66#407—22no—
67#407C2312yes—
68#407C2312yes—
69#394—24no—
70#394—24no—
71#392—24no—
72#392—24no—
73#392—24no—
74#392—24no—
75#391C2×C412yescomplemented
76#390C2×C412yes—
77#390C2×C412yes—
78#390C2×C412yescomplemented
79#388—24no—
80#388—22no—
81#387—22no—
82#386—24no—
83#386—24no—
84#386—44no—
85#381C2312yes—
86#381C2312yes—
87#379—22no—
88C4×C4⋊C4C2×C412yescomplemented
89C4×C22⋊C4C2312yes—
90#376C2×C411yescharacteristic, complemented
91C43C2×C411yescharacteristic, complemented
92#343—22no—
93#343—22no—
94#342—22no—
95#342—22no—
96#327—44no—
97#327—44no—
98C25C2×D412yes—
99C2×#41—24no—
100C2×#41—82no—
101C22×Q8C22×C411yescharacteristic
102C22×D4—24no—
103C22×D4—24no—
104C22×D4—22no—
105C22×D4—24no—
106C22×D4—22no—
107C22×D4—24no—
108C22×D4—82no—
109C22×D4C22⋊C414yes—
110C22×D4C22⋊C414yes—
111C22×D4C22×C412yes—
112C22×D4C22×C411yescharacteristic
113C22×D4C22×C411yescharacteristic
114C22×D4C22×C411yescharacteristic
115C22×D4C2×D412yes—
116C22×D4C2×D412yes—
117C23×C4—22no—
118C23×C4—42no—
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—42no—
124C2×M16—84no—
125C22×C8—42no—
126#127—42no—
127#127—42no—
128#127—42no—
129#127—82no—
130#127—82no—
131#127—82no—
132#127—88no—
133#125—42no—
134#124—42no—
135#124—42no—
136#124—82no—
137#123—48no—
138#121—24no—
139#121—24no—
140#121—48no—
141#121—44no—
142#121—44no—
143#121—88no—
144#120—44no—
145#120—44no—
146#120—88no—
147C4×D4—42no—
148C4×D4—42no—
149#117—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C2×D412yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—22no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—42no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—84no—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C22⋊C414yes—
168C2×C22⋊C4C2×D412yes—
169C2×C42—22no—
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C412yes—
173C2×C42C22×C412yes—
174#100—84no—
175#99—84no—
176C22⋊C8—24no—
177C22⋊C8—24no—
178C22⋊C8—44no—
179#95—22no—
180#95—22no—
181#95—24no—
182#95—22no—
183#95—24no—
184#95—24no—
185C24—22no—
186C24—22no—
187C24—22no—
188C24—44no—
189C24—84no—
190C24C2×C22⋊C412yes—
191C24C2×C22⋊C412yes—
192C24C2×C22⋊C412yes—
193C24C22×D411yescharacteristic
194#41—84no—
195#41—84no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—24no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—48no—
215C2×D4—44no—
216C2×D4—48no—
217C2×D4—44no—
218C2×D4—84no—
219C2×D4—84no—
220C2×D4—84no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4—84no—
226C2×D4—84no—
227C2×D4—84no—
228C2×D4—84no—
229C2×D4—84no—
230C2×D4—84no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—44no—
237C22×C4—42no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—84no—
242C22×C4—84no—
243C22×C4—164no—
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4H(5,𝔽2)14yes—
250M16—44no—
251M16—44no—
252M16—84no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—84no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C22⋊C4—24no—
262C22⋊C4—44no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C22⋊C4—44no—
266C22⋊C4—48no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—84no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—164no—
275C42—22no—
276C42—22no—
277C42—42no—
278C42—42no—
279C42—44no—
280C23—22no—
281C23—22no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—24no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23—42no—
297C23—42no—
298C23—42no—
299C23—84no—
300C23—84no—
301C23—168no—
302C23C22×C22⋊C411yescharacteristic
303C23#53412yes—
304Q8—82no—
305D4—84no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—82no—
311D4—84no—
312D4—164no—
313D4—164no—
314D4—168no—
315D4—168no—
316D4—164no—
317D4—164no—
318D4—168no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—24no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—164no—
344C2×C4—168no—
345C2×C4#51812yes—
346C2×C4#53412yes—
347C8—82no—
348C8—82no—
349C22—22no—
350C22—24no—
351C22—22no—
352C22—44no—
353C22—44no—
354C22—44no—
355C22—42no—
356C22—42no—
357C22—42no—
358C22—82no—
359C22—84no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22—84no—
365C22—84no—
366C22—82no—
367C22—82no—
368C22—82no—
369C22—164no—
370C22#230811yescharacteristic
371C22#288312yes—
372C4—22no—
373C4—22no—
374C4—44no—
375C4—42no—
376C4—42no—
377C4—42no—
378C4—82no—
379C4—82no—
380C4—162no—
381C4—162no—
382C2—22no—
383C2—42no—
384C2—82no—
385C2—82no—
386C2—82no—
387C2—162no—
388C2#2175011yescharacteristic
389C2#2178212yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radical#37302C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21755C2semidirect
3. #21750C2semidirect
4. #9463C2semidirect
5. #9445C2semidirect
6. #8741C2semidirect
7. #8740C2semidirect
8. #8518C2semidirect
9. #8518C2semidirect
13. #2867C4semidirect
14. #2861C4semidirect
17. #2657C4semidirect
18. #2641C4semidirect
19. #2544C4semidirect
20. #2541C4semidirect
47. C2×#127C2×C4semidirect
48. C2×#127C2×C4semidirect
49. C2×#125C2×C4semidirect
52. C2×#124C2×C4semidirect
60. C2×C4×D4C2×C4semidirect
75. #391C2×C4semidirect
78. #390C2×C4semidirect
88. C4×C4⋊C4C2×C4semidirect
90. #376C2×C4semidirect
91. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#373022
#217552
#217502
#94632
#94452
#87412
#87402
#85182