Finite Group Databasebeta

← #7542940

Subgroups — #7542940

Order 512.

1671 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#36594C211yesmaximal, characteristic
3#32271C214yesmaximal, complemented
4#29208C212yesmaximal, complemented
5#26301C214yesmaximal, complemented
6#25348C214yesmaximal, complemented
7#25016C214yesmaximal, complemented
8#24107C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic
10#22665C214yesmaximal, complemented
11C4×#475C212yesmaximal, complemented
12C2×#559—21no—
13C2×#556C2212yes—
14C2×#484C2214yes—
15C2×#475C2212yes—
16C2×#436C2214yes—
17C2×#428C2214yes—
18#2868C2211yescharacteristic
19#2858—21no—
20#2724—22no—
21#2582—21no—
22#2547C2214yes—
23#2547C2212yes—
24#2545—22no—
25#2538C2214yes—
26#2512—24no—
27#2486C2214yes—
28#2447—24no—
29#2354C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35#2077C2218yes—
36#2076C2218yes—
37#2057C2218yes—
38#2036C2218yes—
39#1990C2214yes—
40#1988C2212yes—
41#1954C2214yes—
42#1948C2212yes—
43#1947C2214yes—
44#1947C2214yes—
45#1919C2214yes—
46#1894C2214yes—
47#1870C2214yes—
48#1865C2214yes—
49#1854C2218yes—
50#1852C2218yes—
51#1845C2214yes—
52#1825C2214yes—
53#1809C2218yes—
54#1803C2212yes—
55C4×#106C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C22×M16C2314yes—
59#559—48no—
60#556—44no—
61#556—44no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—21no—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2312yes—
66C2×#127D411yescharacteristic
67C2×#126—24no—
68C2×#124—22no—
69C2×#124—24no—
70C2×#124—22no—
71C2×#123—24no—
72C2×C22⋊Q8C2314yes—
73C2×C4×Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8C2312yes—
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4—24no—
78C2×C4×D4C2314yes—
79C2×C4×D4C2312yes—
80C2×#117C2314yes—
81C22×C4⋊C4C2314yes—
82#484—48no—
83#475—28no—
84#436—216no—
85#428—48no—
86C2×#107C2318yes—
87C2×#107C2318yes—
88C2×#106C2314yes—
89C2×#106C2314yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103C2314yes—
93C2×#103C2314yes—
94C2×C22⋊C8C2318yes—
95C2×C22⋊C8C2314yes—
96C2×C4×C8C2314yes—
97#398—24no—
98#397—24no—
99#397—28no—
100#397—24no—
101#396—28no—
102#391—22no—
103#391—24no—
104#391—22no—
105#390—24no—
106#390—24no—
107#390D414yes—
108#390D412yes—
109#389—24no—
110#389—24no—
111#389—22no—
112#389C2314yes—
113#388—28no—
114#385—28no—
115#384—22no—
116#384D411yescharacteristic
117#384C2312yes—
118#384C2314yes—
119#381—24no—
120#380C2318yes—
121#380C2318yes—
122#380C2314yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2312yes—
129C4×C4⋊C4C2311yescharacteristic
130C43—21no—
131C43C2311yescharacteristic
132#340C2318yes—
133#340C2318yes—
134#340C2314yes—
135#340C2314yes—
136#340C2314yes—
137#340C2314yes—
138#336C2314yes—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8C2411yescharacteristic
142C22×D4C2×D414yes—
143C22×D4C2×D412yes—
144C23×C4—24no—
145C23×C4C2414yes—
146C2×M16—216no—
147C2×M16—44no—
148C22×C8C2414yes—
149C22×C8C2414yes—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—44no—
155C4⋊Q8—82no—
156#127—24no—
157#126—416no—
158#124—48no—
159#124—416no—
160#124—84no—
161#123—416no—
162C22⋊Q8—28no—
163C22⋊Q8—44no—
164C4×Q8—22no—
165C4×Q8—22no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—48no—
169C4×Q8—48no—
170C4×D4—28no—
171C4×D4—28no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—416no—
175#117—28no—
176#117—44no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2412yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2411yescharacteristic
197C2×C22⋊C4—24no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4C2×D418yes—
203C2×C22⋊C4C2×D414yes—
204C2×C42—21no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2411yescharacteristic
214C2×C42C2412yes—
215#107—48no—
216#107—48no—
217#106—28no—
218#106—28no—
219C4⋊C8—24no—
220C4⋊C8#4118yes—
221#103—28no—
222#103—28no—
223C22⋊C8—216no—
224C22⋊C8—44no—
225C4×C8—28no—
226#95—22no—
227#95—21no—
228#95—28no—
229#95—24no—
230#95—24no—
231#95—28no—
232#95—24no—
233C24C22×D414yes—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8H(5,𝔽2)12yes—
244C2×D4—28no—
245C2×D4—28no—
246C2×D4—24no—
247C2×D4—28no—
248C2×D4—42no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—48no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4C22×D414yes—
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C2511yescharacteristic
269M16—416no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8C2×#4118yes—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—21no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—24no—
287C4⋊C4—42no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4H(5,𝔽2)12yes—
299C22⋊C4—216no—
300C22⋊C4—44no—
301C22⋊C4—48no—
302C22⋊C4—48no—
303C22⋊C4—416no—
304C22⋊C4—48no—
305C22⋊C4—88no—
306C42—24no—
307C42—24no—
308C42—21no—
309C42—21no—
310C42—41no—
311C42—48no—
312C42—42no—
313C42—44no—
314C42—48no—
315C42—44no—
316C42AGL(1,ℤ/8)14yes—
317C42C2×#4114yes—
318C422−1+414yes—
319C23—24no—
320C23—28no—
321C23—24no—
322C23—24no—
323C23—28no—
324C23C23×D411yescharacteristic
325Q8—24no—
326Q8—41no—
327Q8—41no—
328Q8—88no—
329Q8—88no—
330D4—416no—
331D4—84no—
332C2×C4—21no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—28no—
337C2×C4—22no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—42no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—168no—
360C2×C4#54714yes—
361C2×C4#54714yes—
362C2×C4C2×AGL(1,ℤ/8)12yes—
363C2×C4C2×AGL(1,ℤ/8)14yes—
364C2×C4#57814yes—
365C2×C4C22×#4112yes—
366C2×C4C2×H(5,𝔽2)11yescharacteristic
367C2×C4C2×H(5,𝔽2)11yescharacteristic
368C2×C4C2×2−1+412yes—
369C8—28no—
370C8—44no—
371C22—48no—
372C22—48no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22C2×#54712yes—
377C22#348411yescharacteristic
378C22C22×AGL(1,ℤ/8)11yescharacteristic
379C22C2×#57812yes—
380C22#358511yescharacteristic
381C4—24no—
382C4—22no—
383C4—44no—
384C4—42no—
385C4—41no—
386C4—42no—
387C4—84no—
388C4—82no—
389C4—84no—
390C4—84no—
391C4#335318yes—
392C4#348111yescharacteristic
393C4#352212yes—
394C4#355011yescharacteristic
395C2—84no—
396C2#5907011yescharacteristic
397C2C2×#335312yes—
398C2#6128212yes—
399C2#6158711yescharacteristic
400C2#6166511yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32271C2semidirect
4. #29208C2semidirect
5. #26301C2semidirect
6. #25348C2semidirect
7. #25016C2semidirect
8. #24107C2semidirect
10. #22665C2semidirect
11. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#365942
#322712
#292082
#263012
#253482
#250162
#241072
#231762
#226652
C4×#4752