Finite Group Databasebeta

← #6519039

Subgroups — #6519039

Order 512.

2052 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#27604C214yesmaximal, complemented
4#25023C218yesmaximal, complemented
5#24978C218yesmaximal, complemented
6#24009C214yesmaximal, complemented
7#21986C212yesmaximal, complemented
8#21790C211yesmaximal, characteristic, complemented
9#20738C212yesmaximal, complemented
10C22×#121C2212yes—
11C22×C4×D4C2211yescharacteristic
12C22×C4×D4C2211yescharacteristic
13C2×#482C2218yescomplemented
14C2×#432C2212yes—
15#2535—24no—
16#2535C2218yes—
17C2×#397—21no—
18C2×#392—21no—
19#2357—24no—
20#2354C2214yescomplemented
21#2354C2214yescomplemented
22#2312—22no—
23#2309—21no—
24C2×#386—22no—
25#2285C2212yes—
26C2×#382C2211yescharacteristic
27#2280C2214yes—
28C2×#380C2212yes—
29#2274—21no—
30#2077C2218yescomplemented
31#2075C2218yescomplemented
32#2034C2218yescomplemented
33#2033C2218yescomplemented
34#1955C2218yescomplemented
35#1949C2218yescomplemented
36#1875C2218yescomplemented
37#1854C2218yescomplemented
38#1853C2214yes—
39#1852C2214yes—
40#1847C2214yescomplemented
41#1836C2214yescomplemented
42#1825C2218yescomplemented
43#1824C2214yes—
44#1802C2214yescomplemented
45#1802C2214yescomplemented
46#1786C2214yes—
47#1779C2214yes—
48#1762C2218yescomplemented
49C23×D4—22no—
50C23×D4D411yescharacteristic
51C23×D4C2311yescharacteristic
52C24×C4—21no—
53C24×C4C2312yes—
54C2×#121—24no—
55C2×#121—48no—
56C2×#121C2318yes—
57C2×C4×D4—22no—
58C2×C4×D4—28no—
59C2×C4×D4—28no—
60C2×C4×D4D414yes—
61C2×C4×D4C2314yes—
62C2×C4×D4C2314yes—
63C2×#117—22no—
64C2×#117C2314yes—
65C2×#117C2314yes—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#482—416no—
75#432—28no—
76C2×#107C2318yes—
77C2×#106C2318yes—
78C2×C4⋊C8C2314yes—
79C2×#102C2318yes—
80C2×#102C2318yes—
81C2×C22⋊C8C2314yes—
82C2×C22⋊C8C2318yes—
83C2×C22⋊C8C2318yes—
84C2×#97C2314yes—
85#397—28no—
86#392—28no—
87#390—24no—
88#390—24no—
89#388—24no—
90#388—24no—
91#388—24no—
92#388—24no—
93#386—28no—
94#386—28no—
95#384C2312yes—
96#384C2312yes—
97#382—22no—
98#382—22no—
99#382C2314yes—
100#381—24no—
101#380—22no—
102#380—24no—
103#380C2314yes—
104#380C2318yes—
105#380C2314yes—
106#380C2314yes—
107#379—24no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4C2314yes—
112C4×C22⋊C4—24no—
113C4×C22⋊C4—24no—
114C2×#95—21no—
115C2×#95—22no—
116C2×#95—21no—
117#340C2318yes—
118#340C2318yes—
119#340C2318yes—
120#340C2318yes—
121C25—21no—
122C25C2×D412yes—
123C22×D4—24no—
124C22×D4—28no—
125C22×D4—24no—
126C22×D4—21no—
127C22×D4—24no—
128C22×D4—42no—
129C22×D4—44no—
130C22×D4—48no—
131C22×D4—48no—
132C22×D4—84no—
133C22×D4C2×D414yes—
134C22×D4C2412yes—
135C23×C4—28no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—24no—
139C23×C4—44no—
140C23×C4—44no—
141C23×C4—44no—
142C23×C4—84no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D412yes—
145C23×C4C2×D414yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2414yes—
148C23×C4C2411yescharacteristic
149C22×C8C2418yes—
150#121—216no—
151#121—48no—
152#121—416no—
153C4×D4—216no—
154C4×D4—28no—
155C4×D4—28no—
156C4×D4—28no—
157C4×D4—28no—
158C4×D4—48no—
159#117—48no—
160#117—48no—
161#117—48no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2412yes—
173C2×C4⋊C4C2414yes—
174C2×C4⋊C4C2414yes—
175C2×C4⋊C4C2414yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—84no—
188C2×C22⋊C4C2×D414yes—
189C2×C22⋊C4C2×D414yes—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42C2×D412yes—
194C2×C42#4118yes—
195C2×C42C2412yes—
196#107—48no—
197#106—48no—
198C4⋊C8—28no—
199#102—216no—
200#102—48no—
201C22⋊C8—28no—
202C22⋊C8—48no—
203C22⋊C8—48no—
204#97—28no—
205#95—28no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212C24—24no—
213C24—24no—
214C24—22no—
215C24—24no—
216C24—24no—
217C24—42no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—44no—
222C24—42no—
223C24—42no—
224C24—44no—
225C24C22×D414yes—
226C24C22×D411yescharacteristic
227C2×D4—24no—
228C2×D4—28no—
229C2×D4—24no—
230C2×D4—22no—
231C2×D4—24no—
232C2×D4—416no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—416no—
236C2×D4—42no—
237C2×D4—416no—
238C2×D4—416no—
239C2×D4—42no—
240C2×D4—88no—
241C2×D4—832no—
242C2×D4H(5,𝔽2)14yes—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—28no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—48no—
266C22×C4—816no—
267C22×C4—84no—
268C22×C4—84no—
269C22×C4—816no—
270C22×C4#12314yes—
271C22×C4#12314yes—
272C22×C4C22×D414yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C22×D412yes—
275C22×C4C22×D412yes—
276C22×C4C2×#4114yes—
277C22×C4C2511yescharacteristic
278C2×C8—28no—
279C2×C8—28no—
280C2×C8—28no—
281C4⋊C4—28no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—28no—
285C4⋊C4—28no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—82no—
291C4⋊C4—82no—
292C22⋊C4—28no—
293C22⋊C4—28no—
294C22⋊C4—416no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C22⋊C4—88no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C42—48no—
303C42—84no—
304C42—84no—
305C42#12318yes—
306C42C2×#4114yes—
307C42C2×#4114yes—
308C23—22no—
309C23—21no—
310C23—21no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—24no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—48no—
320C23—48no—
321C23—416no—
322C23—48no—
323C23—44no—
324C23—48no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—84no—
329C23—84no—
330C23—84no—
331C23—84no—
332C23—84no—
333C23—88no—
334C23—84no—
335C23C2×#12312yes—
336C23#57514yes—
337C23C23×D411yescharacteristic
338D4—28no—
339D4—416no—
340D4—48no—
341D4—44no—
342D4—832no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—44no—
355C2×C4—416no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—88no—
368C2×C4—816no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—88no—
372C2×C4C2×#12312yes—
373C2×C4C2×#12314yes—
374C2×C4C2×#12314yes—
375C2×C4C22×#4112yes—
376C2×C4C2×H(5,𝔽2)12yes—
377C2×C4#58512yes—
378C8—48no—
379C22—24no—
380C22—22no—
381C22—24no—
382C22—22no—
383C22—44no—
384C22—44no—
385C22—44no—
386C22—44no—
387C22—44no—
388C22—48no—
389C22—44no—
390C22—44no—
391C22—44no—
392C22—816no—
393C22—816no—
394C22—84no—
395C22—82no—
396C22—82no—
397C22#309314yes—
398C22C22×#12311yescharacteristic
399C22#348411yescharacteristic
400C22#348511yescharacteristic
401C22C2×#57512yes—
402C22#358512yes—
403C4—24no—
404C4—24no—
405C4—22no—
406C4—44no—
407C4—84no—
408C4—82no—
409C4—84no—
410C4—84no—
411C4#349414yes—
412C2—24no—
413C2—84no—
414C2—84no—
415C2#3554011yescharacteristic
416C2C2×#309311yescharacteristic
417C2#6051811yescharacteristic
418C2#6157612yes—
419C2#6171312yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#2535C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34186C2semidirect
3. #27604C2semidirect
4. #25023C2semidirect
5. #24978C2semidirect
6. #24009C2semidirect
7. #21986C2semidirect
8. #21790C2semidirect
9. #20738C2semidirect
13. C2×#482C22semidirect
20. #2354C22semidirect
21. #2354C22semidirect
30. #2077C22semidirect
31. #2075C22semidirect
32. #2034C22semidirect
33. #2033C22semidirect
34. #1955C22semidirect
35. #1949C22semidirect
36. #1875C22semidirect
37. #1854C22semidirect
40. #1847C22semidirect
41. #1836C22semidirect
42. #1825C22semidirect
44. #1802C22semidirect
45. #1802C22semidirect
48. #1762C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
#341862
#276042
#250232
#249782
#240092
#219862
#217902
#207382