Finite Group Databasebeta

← #7542937

Subgroups — #7542937

Order 512.

1651 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic, complemented
2#36187C211yesmaximal, characteristic
3#32268C214yesmaximal, complemented
4#29208C212yesmaximal, complemented
5#26299C214yesmaximal, complemented
6#25346C214yesmaximal, complemented
7#25010C214yesmaximal, complemented
8#24105C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic, complemented
10#22665C214yesmaximal, complemented
11C4×#476C212yesmaximal, complemented
12C2×#555C2212yes—
13C2×#480C2214yes—
14C2×#476C2212yes—
15C2×#436C2214yes—
16C2×#428C2214yes—
17#2868C2211yescharacteristic
18#2856—21no—
19#2804—22no—
20#2580—24no—
21#2571—21no—
22#2563C2214yes—
23#2563C2212yes—
24#2537C2214yes—
25#2500—22no—
26#2491—24no—
27#2487C2214yes—
28#2366—21no—
29#2354C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8C2212yes—
32C4×#127C2211yescharacteristic
33#2285—21no—
34C42×D4C2211yescharacteristic
35#2077C2218yes—
36#2075C2218yes—
37#2057C2218yes—
38#2033C2218yes—
39#1990C2214yes—
40#1987C2212yes—
41#1954C2214yes—
42#1948C2212yes—
43#1947C2214yes—
44#1947C2214yes—
45#1920C2214yes—
46#1895C2214yes—
47#1870C2214yes—
48#1864C2214yes—
49#1854C2218yes—
50#1852C2218yes—
51#1845C2214yes—
52#1825C2214yes—
53#1809C2218yes—
54#1802C2212yes—
55C4×#106C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#102C2214yes—
58C22×M16C2314yes—
59#555—44no—
60#555—44no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2312yes—
63C2×C4⋊Q8C2312yes—
64C2×#127D411yescharacteristic
65C2×#124—24no—
66C2×#121—22no—
67C2×#121—24no—
68C2×#121C2314yes—
69C2×C4×Q8—22no—
70C2×C4×D4—24no—
71C2×C4×D4C2312yes—
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2314yes—
74C2×C4×D4C2312yes—
75C2×#117—22no—
76C2×#117C2314yes—
77C22×C4⋊C4C2314yes—
78C22×C42—21no—
79#480—48no—
80#476—28no—
81#436—216no—
82#428—48no—
83C2×#107C2318yes—
84C2×#107C2318yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#102C2314yes—
90C2×#102C2314yes—
91C2×C22⋊C8C2318yes—
92C2×C22⋊C8C2314yes—
93C2×C4×C8C2314yes—
94#400—24no—
95#399—24no—
96#394—28no—
97#393—28no—
98#393—24no—
99#391—22no—
100#390—24no—
101#390—22no—
102#390—24no—
103#390D414yes—
104#390D412yes—
105#389—24no—
106#389—22no—
107#389C2314yes—
108#388—28no—
109#386—24no—
110#385—24no—
111#385—28no—
112#384—21no—
113#384—21no—
114#384D411yescharacteristic
115#384C2312yes—
116#384C2314yes—
117#382—24no—
118#382—22no—
119#380—24no—
120#380C2314yes—
121#380C2318yes—
122#380C2318yes—
123C4×C4⋊C4—24no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2314yes—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2312yes—
128C4×C22⋊C4—22no—
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
131#340C2318yes—
132#340C2318yes—
133#340C2314yes—
134#340C2314yes—
135#340C2314yes—
136#340C2314yes—
137#336C2314yes—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×D4—24no—
141C22×D4C2×D414yes—
142C22×D4C2×D412yes—
143C22×D4C2411yescharacteristic
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—22no—
147C23×C4C2414yes—
148C2×M16—216no—
149C2×M16—44no—
150C22×C8C2414yes—
151C22×C8C2414yes—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—84no—
157#127—24no—
158#124—416no—
159#121—28no—
160#121—44no—
161#121—48no—
162#121—88no—
163C4×Q8—84no—
164C4×D4—22no—
165C4×D4—22no—
166C4×D4—28no—
167C4×D4—24no—
168C4×D4—28no—
169C4×D4—24no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—88no—
173#117—28no—
174#117—44no—
175#117—48no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C2×D414yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2412yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—28no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C2×D418yes—
201C2×C22⋊C4C2×D414yes—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—44no—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211C2×C42C2412yes—
212#107—48no—
213#107—48no—
214#106—28no—
215#106—28no—
216C4⋊C8—24no—
217C4⋊C8#4118yes—
218#102—28no—
219#102—28no—
220C22⋊C8—216no—
221C22⋊C8—44no—
222C4×C8—28no—
223#95—22no—
224#95—21no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—28no—
230C24—21no—
231C24C22×D414yes—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—82no—
235C2×Q8—82no—
236C2×Q8—84no—
237C2×D4—28no—
238C2×D4—28no—
239C2×D4—24no—
240C2×D4—28no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—48no—
247C2×D4—88no—
248C2×D4H(5,𝔽2)12yes—
249C22×C4—24no—
250C22×C4—21no—
251C22×C4—28no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—28no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—84no—
265C22×C4—88no—
266C22×C4C22×D414yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D412yes—
271C22×C4C2511yescharacteristic
272M16—416no—
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8C2×#4118yes—
279C4⋊C4—28no—
280C4⋊C4—21no—
281C4⋊C4—24no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—42no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—84no—
295C4⋊C4—82no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4H(5,𝔽2)12yes—
299C22⋊C4—24no—
300C22⋊C4—216no—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—48no—
304C22⋊C4—44no—
305C22⋊C4—84no—
306C22⋊C4—88no—
307C22⋊C4—88no—
308C42—24no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—48no—
313C42—44no—
314C42—42no—
315C42—48no—
316C42#13714yes—
317C42C2×#4114yes—
318C422−1+414yes—
319C23—24no—
320C23—28no—
321C23—22no—
322C23—24no—
323C23—24no—
324C23—28no—
325C23—41no—
326C23—42no—
327C23—41no—
328C23—42no—
329C23C23×D411yescharacteristic
330Q8—88no—
331Q8—164no—
332D4—24no—
333D4—416no—
334D4—41no—
335D4—41no—
336D4—84no—
337D4—168no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—28no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—48no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—48no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—88no—
365C2×C4—168no—
366C2×C4#54714yes—
367C2×C4#54714yes—
368C2×C4C2×#13712yes—
369C2×C4C2×#13714yes—
370C2×C4#57614yes—
371C2×C4C22×#4112yes—
372C2×C4C2×H(5,𝔽2)11yescharacteristic
373C2×C4C2×H(5,𝔽2)11yescharacteristic
374C2×C4C2×2−1+412yes—
375C8—28no—
376C8—44no—
377C22—48no—
378C22—48no—
379C22—42no—
380C22—44no—
381C22—44no—
382C22—42no—
383C22—44no—
384C22—42no—
385C22—82no—
386C22—82no—
387C22C2×#54712yes—
388C22#348411yescharacteristic
389C22C22×#13711yescharacteristic
390C22C2×#57612yes—
391C22#358611yescharacteristic
392C4—24no—
393C4—22no—
394C4—42no—
395C4—44no—
396C4—41no—
397C4—82no—
398C4—84no—
399C4—84no—
400C4—84no—
401C4#335718yes—
402C4#348111yescharacteristic
403C4#352212yes—
404C4#355011yescharacteristic
405C2—84no—
406C2—82no—
407C2#5907011yescharacteristic
408C2C2×#335712yes—
409C2#6127812yes—
410C2#6158611yescharacteristic
411C2#6166111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#36187C2
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
1. #59864C2semidirect
3. #32268C2semidirect
4. #29208C2semidirect
5. #26299C2semidirect
6. #25346C2semidirect
7. #25010C2semidirect
8. #24105C2semidirect
9. #23176C2semidirect
10. #22665C2semidirect
11. C4×#476C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#361872
#322682
#292082
#262992
#253462
#250102
#241052
#231762
#226652
C4×#4762