Finite Group Databasebeta

← #7542904

Subgroups — #7542904

Order 512.

1737 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59827C211yesmaximal, characteristic
2#36559C211yesmaximal, characteristic
3#32259C214yesmaximal, complemented
4#29206C212yesmaximal, complemented
5#26298C214yesmaximal, complemented
6#25341C214yesmaximal, complemented
7#25014C214yesmaximal, complemented
8#24097C214yesmaximal, complemented
9#23173C211yesmaximal, characteristic
10D4×#106C214yesmaximal, complemented
11C4×#475C212yesmaximal, complemented
12C2×#559—21no—
13C2×#554C2212yes—
14C2×#483C2214yes—
15C2×#475C2212yes—
16C2×C8×D4C2214yes—
17C22×#106C2214yes—
18#2868C2211yescharacteristic
19#2859—21no—
20#2724—22no—
21#2582—21no—
22#2566—22no—
23#2563C2212yes—
24#2547C2214yes—
25#2538C2214yes—
26#2486C2214yes—
27#2452—24no—
28#2447—24no—
29D4×C4⋊C4C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8C2212yes—
32C4×#127—21no—
33C4×#127C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35#2077C2218yes—
36#2077C2218yes—
37#2058C2218yes—
38#2035C2218yes—
39#1990C2214yes—
40#1988C2212yes—
41#1954C2214yes—
42#1948C2214yes—
43#1948C2214yes—
44#1948C2212yes—
45#1919C2214yes—
46#1892C2214yes—
47#1869C2214yes—
48#1865C2214yes—
49#1853C2218yes—
50#1853C2218yes—
51#1844C2214yes—
52#1824C2214yes—
53#1807C2218yes—
54#1803C2212yes—
55C4×#106C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C23×C8C2314yes—
59#559—48no—
60#554—44no—
61#554—44no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#127D411yescharacteristic
66C2×#125—22no—
67C2×#124—24no—
68C2×#123—24no—
69C2×C22⋊Q8—24no—
70C2×C22⋊Q8C2314yes—
71C2×C4×Q8—22no—
72C2×C4×Q8C2312yes—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—22no—
75C2×C4×D4—24no—
76C2×C4×D4—24no—
77C2×C4×D4C2314yes—
78C2×C4×D4C2312yes—
79C22×C4⋊C4C2314yes—
80C22×C4⋊C4C2314yes—
81#483—48no—
82#475—28no—
83C8×D4—216no—
84C2×#106—48no—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×#106C2318yes—
88C2×#106C2318yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103C2314yes—
92C2×#103C2314yes—
93C2×C22⋊C8C2318yes—
94C2×C22⋊C8C2314yes—
95C2×C4×C8C2314yes—
96#398—24no—
97#397—24no—
98#397—28no—
99#396—28no—
100#393—24no—
101#391—24no—
102#391—22no—
103#390—22no—
104#390—24no—
105#390D412yes—
106#390D414yes—
107#389—24no—
108#389—24no—
109#389—24no—
110#389—22no—
111#389C2314yes—
112#385—28no—
113#385—28no—
114#384—21no—
115#384—21no—
116#384D411yescharacteristic
117#384C2312yes—
118#384C2314yes—
119#382—22no—
120#381—24no—
121#380C2318yes—
122#380C2318yes—
123#380C2314yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2312yes—
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×C4—24no—
146C23×C4C2414yes—
147C22×C8—216no—
148C22×C8—44no—
149C22×C8C2414yes—
150C22×C8C2414yes—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—84no—
156#127SD1618yes—
157#125—48no—
158#124—416no—
159#123—416no—
160C22⋊Q8—28no—
161C22⋊Q8—44no—
162C22⋊Q8—416no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—48no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—48no—
173C4×D4—416no—
174C4×D4—88no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—28no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
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⋊C4—44no—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2411yescharacteristic
198C2×C22⋊C4—24no—
199C2×C22⋊C4—28no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4C2×D418yes—
203C2×C22⋊C4C2×D414yes—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—24no—
207C2×C42—22no—
208C2×C42—21no—
209C2×C42C2×D412yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C2×C42C2412yes—
214#106—28no—
215#106—28no—
216#106—48no—
217#106—48no—
218C4⋊C8—24no—
219C4⋊C8#4118yes—
220#103—28no—
221#103—28no—
222C22⋊C8—216no—
223C22⋊C8—44no—
224C4×C8—28no—
225#95—22no—
226#95—21no—
227#95—24no—
228#95—24no—
229#95—28no—
230#95—28no—
231#95—24no—
232C24C22×D414yes—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—84no—
241C2×Q8H(5,𝔽2)12yes—
242C2×D4—28no—
243C2×D4—28no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4C2×SD16116yes—
247C2×D4C2×SD1618yes—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—28no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—84no—
261C22×C4—88no—
262C22×C4—84no—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D414yes—
265C22×C4C22×D412yes—
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C2511yescharacteristic
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—416no—
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—48no—
290C4⋊C4—44no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—82no—
297C4⋊C4—84no—
298C4⋊C4—84no—
299C4⋊C4H(5,𝔽2)12yes—
300C22⋊C4—216no—
301C22⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—416no—
304C22⋊C4—48no—
305C22⋊C4—88no—
306C42—24no—
307C42—24no—
308C42—21no—
309C42—21no—
310C42—41no—
311C42—42no—
312C42—44no—
313C42—48no—
314C42—44no—
315C42C2×SD1614yes—
316C42C2×#4114yes—
317C422−1+414yes—
318C23—24no—
319C23—28no—
320C23—24no—
321C23—28no—
322C23C22×SD1618yes—
323C23C23×D411yescharacteristic
324Q8—24no—
325Q8—41no—
326Q8—41no—
327Q8—88no—
328Q8—164no—
329D4—232no—
330D4—48no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—28no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—168no—
358C2×C4—168no—
359C2×C4#54714yes—
360C2×C4#54714yes—
361C2×C4C22×SD1612yes—
362C2×C4C22×SD1614yes—
363C2×C4#57714yes—
364C2×C4C22×#4112yes—
365C2×C4C2×H(5,𝔽2)11yescharacteristic
366C2×C4C2×H(5,𝔽2)11yescharacteristic
367C2×C4C2×2−1+412yes—
368C8—28no—
369C8—44no—
370C22—216no—
371C22—28no—
372C22—48no—
373C22—44no—
374C22—44no—
375C22C2×#54712yes—
376C22#348411yescharacteristic
377C22C23×SD1611yescharacteristic
378C22C2×#57712yes—
379C22#358611yescharacteristic
380C4—24no—
381C4—22no—
382C4—44no—
383C4—42no—
384C4—41no—
385C4—42no—
386C4—84no—
387C4—82no—
388C4—84no—
389C4—84no—
390C4#333518yes—
391C4#348111yescharacteristic
392C4#352212yes—
393C4#355011yescharacteristic
394C2—48no—
395C2#5907011yescharacteristic
396C2C2×#333512yes—
397C2#6128112yes—
398C2#6156211yescharacteristic
399C2#6166211yescharacteristic
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. #32259C2semidirect
4. #29206C2semidirect
5. #26298C2semidirect
6. #25341C2semidirect
7. #25014C2semidirect
8. #24097C2semidirect
10. D4×#106C2semidirect
11. C4×#475C2semidirect

Maximal subgroups

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

subgroupindex
#598272
#365592
#322592
#292062
#262982
#253412
#250142
#240972
#231732
D4×#1062
C4×#4752