Finite Group Databasebeta

← #6528592

Subgroups — #6528592

Order 512.

1838 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35166C211yesmaximal, characteristic
2C2×#2571C211yesmaximal, characteristic, complemented
3#29334C214yesmaximal, complemented
4#27982C214yesmaximal, complemented
5#27755C218yesmaximal, complemented
6#27736C218yesmaximal, complemented
7#21859C212yesmaximal, complemented
8#21386C212yesmaximal, complemented
9#20536C211yesmaximal, characteristic, complemented
10C2×#537—21no—
11C2×#537—21no—
12C2×#532C2212yes—
13C2×#431C2212yes—
14#2856C2214yes—
15#2656—22no—
16#2571—24no—
17#2571C2218yes—
18#2542C2214yes—
19#2541C2214yes—
20#2540—24no—
21C2×#400—22no—
22C2×#389—21no—
23C2×#389C2212yes—
24C2×#384C2211yescharacteristic
25C2×#382—21no—
26C2×#382C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#2100C2218yes—
30#2073C22116yes—
31#2041C2218yes—
32#1990C2214yes—
33#1988C2214yes—
34#1982C2214yes—
35#1961C2218yes—
36#1957C2214yes—
37#1936C22116yes—
38#1874C2214yes—
39#1874C2218yes—
40#1874C2214yes—
41#1836C2218yes—
42#1802C2218yes—
43#1770C2218yes—
44C4×#102C2218yes—
45#1749C2214yes—
46C22×#41—22no—
47#537—48no—
48#537—48no—
49#532—28no—
50#532—28no—
51C2×C4⋊Q8C2312yes—
52C2×#127C2312yes—
53C2×#127C2314yes—
54C2×#124—22no—
55C2×C22⋊Q8—24no—
56C2×C22⋊Q8—24no—
57C2×#121—24no—
58C2×#121—24no—
59C2×#121—22no—
60C2×#121—24no—
61C2×#121—24no—
62C2×#121C2314yes—
63C2×C4×D4—22no—
64C2×C4×D4C2314yes—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#431—28no—
77C2×C4⋊C8C2314yes—
78C2×#102C23116yes—
79C2×#102C2318yes—
80C2×#102C2318yes—
81C2×C22⋊C8C2314yes—
82C2×#97C2318yes—
83C2×C4×C8C2314yes—
84#400—216no—
85#399—28no—
86#392—28no—
87#390—24no—
88#390—24no—
89#390C2318yes—
90#389—28no—
91#389—24no—
92#389C2318yes—
93#384—22no—
94#384C2314yes—
95#384C2314yes—
96#382—28no—
97#382—24no—
98#382C2318yes—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C4×C4⋊C4C2314yes—
102C4×C4⋊C4C2314yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95D412yes—
108#340C23116yes—
109#340C2318yes—
110#340C2318yes—
111#336C2318yes—
112C2×#41—48no—
113C2×#41—48no—
114C2×#41—48no—
115C22×Q8—22no—
116C22×D4—24no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4C2414yes—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—21no—
125C23×C4—24no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8C2418yes—
132C4⋊Q8—24no—
133C4⋊Q8—24no—
134#127—24no—
135#127—24no—
136#127—48no—
137#124—48no—
138C22⋊Q8—416no—
139C22⋊Q8—416no—
140#121—28no—
141#121—28no—
142#121—416no—
143#121—416no—
144#121—416no—
145#121—416no—
146#121—48no—
147C4×D4—28no—
148C4×D4—28no—
149C4×D4—48no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—82no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×D414yes—
168C2×C4⋊C4C2×D414yes—
169C2×C4⋊C4C2412yes—
170C2×C4⋊C4C2412yes—
171C2×C4⋊C4C2414yes—
172C2×C4⋊C4C2414yes—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C42—24no—
178C2×C42—44no—
179C2×C42—44no—
180C2×C42C2×D414yes—
181C2×C42C2×D412yes—
182C2×C42#4118yes—
183C2×C42C2414yes—
184C2×C42C2412yes—
185C4⋊C8—28no—
186#102—48no—
187#102—416no—
188#102—48no—
189C22⋊C8—28no—
190#97—48no—
191C4×C8—28no—
192#95—24no—
193#95—28no—
194#95—28no—
195#95—28no—
196#95C2×D418yes—
197C24—24no—
198C24C22×D411yescharacteristic
199#41—832no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—28no—
206C2×D4—28no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—48no—
211C2×D4—48no—
212C2×D4—44no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—82no—
217C2×D4—82no—
218C2×D4—84no—
219C2×D4—84no—
220C2×D4—84no—
221C2×D4—84no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—44no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—48no—
248C22×C4#12414yes—
249C22×C4#12414yes—
250C22×C4AGL(1,ℤ/8)14yes—
251C22×C4AGL(1,ℤ/8)14yes—
252C22×C4C22×D414yes—
253C22×C4C22×D414yes—
254C22×C4C22×D412yes—
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C2×#4114yes—
258C22×C4C2511yescharacteristic
259C2×C8—28no—
260C2×C8—28no—
261C2×C8—28no—
262C4⋊C4—28no—
263C4⋊C4—28no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—24no—
268C4⋊C4—28no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—44no—
272C4⋊C4—48no—
273C4⋊C4—48no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—84no—
277C4⋊C4—88no—
278C22⋊C4—416no—
279C22⋊C4—416no—
280C22⋊C4—48no—
281C22⋊C4—48no—
282C42—28no—
283C42—44no—
284C42—48no—
285C42#12418yes—
286C42C2×#4114yes—
287C42C2×#4114yes—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23—44no—
296C23—44no—
297C23—44no—
298C23—44no—
299C23C2×#12412yes—
300C23C2×AGL(1,ℤ/8)12yes—
301C23C2×AGL(1,ℤ/8)12yes—
302C23C23×D411yescharacteristic
303Q8—88no—
304D4—48no—
305D4—48no—
306D4—88no—
307D4—816no—
308D4—88no—
309D4—168no—
310D4—168no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—28no—
316C2×C4—24no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—44no—
323C2×C4—42no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—84no—
338C2×C4—84no—
339C2×C4—88no—
340C2×C4—816no—
341C2×C4#490116yes—
342C2×C4C2×#12412yes—
343C2×C4C2×#12414yes—
344C2×C4C2×#12414yes—
345C2×C4C2×AGL(1,ℤ/8)14yes—
346C2×C4C2×AGL(1,ℤ/8)14yes—
347C2×C4C22×#4112yes—
348C2×C4#58512yes—
349C2×C4#58512yes—
350C8—48no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—84no—
358C22—84no—
359C22—84no—
360C22—84no—
361C22—84no—
362C22—84no—
363C22—84no—
364C22C2×#49014yes—
365C22C22×#12411yescharacteristic
366C22#348511yescharacteristic
367C22#348711yescharacteristic
368C22C22×AGL(1,ℤ/8)11yescharacteristic
369C22C22×AGL(1,ℤ/8)11yescharacteristic
370C22#358511yescharacteristic
371C22#358611yescharacteristic
372C4—22no—
373C4—24no—
374C4—44no—
375C4—44no—
376C4—82no—
377C4—84no—
378C4—82no—
379C4—84no—
380C4C2×#49018yes—
381C4#351214yes—
382C2—24no—
383C2—164no—
384C2#3564911yescharacteristic
385C2C22×#49011yescharacteristic
386C2#6062111yescharacteristic
387C2#6172011yescharacteristic
388C2#6172211yescharacteristic
389C2#6188811yescharacteristic
390C2#6189111yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#35166C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2571C2semidirect
3. #29334C2semidirect
4. #27982C2semidirect
5. #27755C2semidirect
6. #27736C2semidirect
7. #21859C2semidirect
8. #21386C2semidirect
9. #20536C2semidirect

Maximal subgroups

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

subgroupindex
#351662
C2×#25712
#293342
#279822
#277552
#277362
#218592
#213862
#205362