Finite Group Databasebeta

← #190068

Subgroups — #190068

Order 512.

980 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34342C211yesmaximal, characteristic
2#21754C211yesmaximal, characteristic
3#21748C211yesmaximal, characteristic
4#8741C212yesmaximal, complemented
5#8740C212yesmaximal, complemented
6#8739C212yesmaximal, complemented
7#8738C212yesmaximal, complemented
8#7946C212yesmaximal, complemented
9#7943C212yesmaximal, complemented
10C2×#525C2211yescharacteristic
11C2×#407C2212yes—
12#2378—22no—
13#2377—22no—
14#2376—22no—
15#2375—22no—
16#2298C2212yes—
17#2297C2212yes—
18#2293C414yescomplemented
19#2290C414yescomplemented
20#2280C2211yescharacteristic
21#2280C2212yes—
22C2×#381C2211yescharacteristic
23#1901—22no—
24#1900—22no—
25#1899—22no—
26#1898—22no—
27#1822C2212yes—
28#1820C2211yescharacteristic
29#1820C2211yescharacteristic
30#1785C2212yes—
31#1782C2211yescharacteristic
32#1782C2211yescharacteristic
33#1284C2214yes—
34#1284C2214yes—
35#1284C2214yes—
36#1284C2214yes—
37C22×#41C2311yescharacteristic
38C24×C4C2311yescharacteristic
39C22×M16—22no—
40#525—22no—
41#525—22no—
42#525—22no—
43#525—22no—
44C2×#123D412yes—
45C2×C22⋊Q8D412yes—
46C2×#121D412yes—
47C2×#120D412yes—
48C2×C4×D4C2×C414yes—
49C2×#117—21no—
50C2×#117C2×C414yes—
51C2×#117C2×C412yescomplemented
52C2×#117C2311yescharacteristic
53C22×C22⋊C4—21no—
54C22×C42C2×C412yescomplemented
55C2×#101—22no—
56C2×#100—22no—
57C2×#99—22no—
58C2×#99—22no—
59#407—22no—
60#407—22no—
61#407—22no—
62#407C2314yes—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65#391—22no—
66#390—24no—
67#390—22no—
68#389—24no—
69#388—24no—
70#388—24no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—24no—
76#386—22no—
77#386—22no—
78#385—24no—
79#381—22no—
80#381—22no—
81#381—21no—
82#381C2311yescharacteristic
83#381C2311yescharacteristic
84#381C2311yescharacteristic
85#381C2311yescharacteristic
86#380—22no—
87#380—22no—
88#380—22no—
89#379—22no—
90#379—21no—
91C4×C4⋊C4C2×C414yescomplemented
92C4×C22⋊C4—22no—
93C4×C22⋊C4C2×C412yes—
94C4×C22⋊C4C2×C412yes—
95C4×C22⋊C4C2×C414yescomplemented
96C4×C22⋊C4C2314yes—
97#376C2×C412yescomplemented
98C2×#95—22no—
99C43C2×C412yescomplemented
100#343—24no—
101#342—22no—
102#342—22no—
103C25C2×D411yescharacteristic
104C2×#41—22no—
105C2×#41—22no—
106C2×#41—82no—
107C22×Q8C2×D411yescharacteristic
108C22×D4C22⋊C414yes—
109C22×D4C2×D412yes—
110C22×D4C2×D411yescharacteristic
111C23×C4—22no—
112C23×C4—41no—
113C23×C4—82no—
114C23×C4C22×C412yes—
115C23×C4C22×C412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4#4112yes—
119C23×C4#4112yes—
120C23×C4#4112yes—
121C23×C4#4112yes—
122C23×C4C2411yescharacteristic
123C2×M16—24no—
124C2×M16—42no—
125C2×M16—84no—
126C22×C8—42no—
127#123—24no—
128#123—22no—
129#123—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134#121—22no—
135#121—22no—
136#121—22no—
137#121—22no—
138#120—22no—
139#120—24no—
140#120—22no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145#117—22no—
146#117—22no—
147#117—84no—
148#117—82no—
149#117—82no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C2×D412yes—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4—21no—
159C2×C22⋊C4—21no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—82no—
169C2×C22⋊C4C22⋊C414yes—
170C2×C22⋊C4C22⋊C414yes—
171C2×C22⋊C4C2×D412yes—
172C2×C42—24no—
173C2×C42—22no—
174C2×C42—41no—
175C2×C42—82no—
176C2×C42C22×C412yes—
177C2×C42C22×C414yes—
178C2×C42C22×C414yes—
179#101—82no—
180#100—82no—
181#99—82no—
182#99—82no—
183C22⋊C8—24no—
184C22⋊C8—24no—
185C22⋊C8—42no—
186C22⋊C8—42no—
187#95—24no—
188#95—22no—
189#95—22no—
190#95—24no—
191#95—24no—
192#95—24no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—42no—
198C24—42no—
199C24—82no—
200C24C2×C22⋊C412yes—
201C24C2×C22⋊C412yes—
202C24#12311yescharacteristic
203C24#12312yes—
204C24#12311yescharacteristic
205C24C22×D411yescharacteristic
206#41—84no—
207#41—84no—
208C2×Q8—22no—
209C2×Q8—42no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—24no—
213C2×D4—24no—
214C2×D4—22no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—84no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—82no—
233C22×C4—82no—
234C22×C4—82no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4—84no—
238C22×C4—82no—
239C22×C4—84no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4#11712yes—
244C22×C4#11712yes—
245C22×C4#11712yes—
246C22×C4#11712yes—
247C22×C4#12312yes—
248C22×C4#12312yes—
249C22×C4#12312yes—
250C22×C4#12312yes—
251C22×C4#12312yes—
252C22×C4#12311yescharacteristic
253C22×C4#12311yescharacteristic
254C22×C4C23×C411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4112yes—
257C22×C4C2×#4112yes—
258M16—44no—
259M16—44no—
260M16—84no—
261C2×C8—44no—
262C2×C8—84no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—84no—
268C4⋊C4—82no—
269C4⋊C4—84no—
270C4⋊C4—82no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—24no—
274C22⋊C4—24no—
275C22⋊C4—24no—
276C22⋊C4—24no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—42no—
280C22⋊C4—42no—
281C22⋊C4—84no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C22⋊C4—82no—
287C22⋊C4—84no—
288C22⋊C4—82no—
289C42—24no—
290C42—44no—
291C42—44no—
292C42—82no—
293C42—84no—
294C42—82no—
295C23—22no—
296C23—24no—
297C23—22no—
298C23—22no—
299C23—22no—
300C23—41no—
301C23—42no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—44no—
311C23—88no—
312C23#38112yes—
313C23#38112yes—
314C23C22×C22⋊C411yescharacteristic
315C23C2×#12311yescharacteristic
316C23C2×#12311yescharacteristic
317Q8—82no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4—84no—
322D4—84no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—22no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—88no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—82no—
360C2×C4—164no—
361C2×C4#38112yes—
362C2×C4#38112yes—
363C2×C4C2×#11712yes—
364C2×C4C2×#12312yes—
365C8—84no—
366C22—22no—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—22no—
376C22—22no—
377C22—44no—
378C22—42no—
379C22—44no—
380C22—82no—
381C22—84no—
382C22—82no—
383C22—82no—
384C22—82no—
385C22—84no—
386C22C2×#38111yescharacteristic
387C22#288312yes—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—22no—
392C4—42no—
393C4—44no—
394C4—84no—
395C4—84no—
396C4—161no—
397C4—161no—
398C4—161no—
399C4—161no—
400C2—22no—
401C2—41no—
402C2—41no—
403C2—42no—
404C2—42no—
405C2—82no—
406C2#2068911yescharacteristic
407C2#2070812yes—
subgroupG/H
center, socleC22C2×#381
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8741C2semidirect
5. #8740C2semidirect
6. #8739C2semidirect
7. #8738C2semidirect
8. #7946C2semidirect
9. #7943C2semidirect
18. #2293C4semidirect
19. #2290C4semidirect
51. C2×#117C2×C4semidirect
54. C22×C42C2×C4semidirect
91. C4×C4⋊C4C2×C4semidirect
95. C4×C22⋊C4C2×C4semidirect
97. #376C2×C4semidirect
99. C43C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343422
#217542
#217482
#87412
#87402
#87392
#87382
#79462
#79432