Finite Group Databasebeta

← #220086

Subgroups — #220086

Order 512.

1209 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36127C211yesmaximal, characteristic
2#22750C212yesmaximal, complemented
3C2×#1551C212yesmaximal, complemented
4#9702C212yesmaximal, complemented
5#9136C212yesmaximal, complemented
6#9127C212yesmaximal, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8344C212yesmaximal, complemented
9#8231C211yesmaximal, characteristic, complemented
10C2×#547C412yes—
11C2×#532C2212yes—
12C2×#432C2212yes—
13C22×C4⋊C8C2212yes—
14#2868C2211yescharacteristic
15#2857C2212yes—
16#2547C412yes—
17#2537—22no—
18#2432C414yes—
19#2422—22no—
20#2422—22no—
21D4×C4⋊C4—22no—
22C4×#127C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1917—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1752—22no—
32#1551—28no—
33C2×#333C2212yes—
34C2×#333C2212yes—
35C2×#333C2212yes—
36C2×#333C2212yes—
37C2×#331C2211yescharacteristic
38C2×#331C2212yes—
39C2×#331C2211yescharacteristic
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C22×#41C2×C414yes—
44C23×C8—22no—
45#547—48no—
46#532—24no—
47#532—24no—
48#532—42no—
49#532—42no—
50C2×C4⋊Q8C2×C412yes—
51C2×C4⋊Q8C2×C412yes—
52C2×C4⋊Q8C2312yes—
53C2×#127C2311yescharacteristic
54C2×#124C2×C414yes—
55C2×#123—24no—
56C2×C22⋊Q8C2×C414yes—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C411yescharacteristic
65C2×C4×D4D412yes—
66C22×C4⋊C4—22no—
67C22×C42C2312yes—
68#432—24no—
69#432—42no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×#97—22no—
82#397—22no—
83#397—22no—
84#394—24no—
85#394—24no—
86#391C2×C414yes—
87#390—22no—
88#390C2×C412yes—
89#389—22no—
90#389—22no—
91#385—24no—
92#385—24no—
93#384—22no—
94#384—22no—
95#384C2311yescharacteristic
96#380—22no—
97#380—22no—
98#380—24no—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4D412yes—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#333—24no—
111#333—24no—
112#333—42no—
113#333—42no—
114#331—24no—
115#331—22no—
116#331D414yescomplemented
117C2×#41—28no—
118C2×#41—44no—
119C2×#41—48no—
120C2×#41—84no—
121C22×Q8C22×C414yes—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22×C412yes—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4C22×C412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—44no—
136C22×C8—44no—
137C22×C8—82no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—44no—
143C4⋊Q8—82no—
144#127—22no—
145#127C2×D412yes—
146#127#4112yes—
147#124—28no—
148#124—44no—
149#123—416no—
150C22⋊Q8—48no—
151#121—44no—
152#121—44no—
153#121—44no—
154#121—44no—
155#121—48no—
156C4×D4—24no—
157C4×D4—42no—
158C4×D4—42no—
159C4×D4—44no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—44no—
163C4×D4—44no—
164C4×D4—44no—
165C4×D4—44no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—48no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2411yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—44no—
205C4⋊C8C2×D414yes—
206C22⋊C8—44no—
207C22⋊C8—44no—
208C22⋊C8—82no—
209#97—44no—
210#97—82no—
211#95—24no—
212#95—24no—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215#95C22⋊C411yescharacteristic
216#95C22⋊C411yescharacteristic
217C24—22no—
218C24C2×C22⋊C412yes—
219#41—816no—
220#41—816no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—24no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—48no—
232C2×D4—48no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—84no—
238C2×D4#10014yes—
239C2×D4C4×D414yes—
240C22×C4—24no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—84no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D412yes—
271C2×C8—24no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—88no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—48no—
288C4⋊C4—44no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C22⋊C4—24no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—48no—
301C22⋊C4—48no—
302C22⋊C4—48no—
303C22⋊C4—44no—
304C22⋊C4—48no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—41no—
317C42C22×D411yescharacteristic
318C42C2×#4111yescharacteristic
319C42H(5,𝔽2)12yes—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—24no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23—44no—
331C23C2×#10014yes—
332C23C22×C22⋊C411yescharacteristic
333Q8—84no—
334Q8—84no—
335D4—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—44no—
340D4—44no—
341D4—44no—
342D4—816no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—88no—
376C2×C4—816no—
377C2×C4C2×#10012yes—
378C2×C4C2×C4×D411yescharacteristic
379C2×C4#51811yescharacteristic
380C2×C4D4212yes—
381C2×C4#54612yes—
382C2×C4#57512yes—
383C2×C4#57512yes—
384C2×C4#57712yes—
385C2×C4#57812yes—
386C8—44no—
387C8—82no—
388C8—82no—
389C8—82no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22—48no—
399C22—82no—
400C22—84no—
401C22—82no—
402C22D4×C22⋊C411yescharacteristic
403C22C22×#10011yescharacteristic
404C22#288811yescharacteristic
405C22#289612yes—
406C22#289812yes—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—42no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—84no—
417C4#326712yes—
418C4#326912yes—
419C4#330712yes—
420C4#331512yes—
421C2—44no—
422C2—84no—
423C2#1775411yescharacteristic
424C2#1776812yes—
425C2#1777911yescharacteristic
426C2D4×#10011yescharacteristic
427C2#2242012yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36127C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22750C2semidirect
3. C2×#1551C2semidirect
4. #9702C2semidirect
5. #9136C2semidirect
6. #9127C2semidirect
7. #8592C2semidirect
8. #8344C2semidirect
9. #8231C2semidirect
116. #331D4semidirect

Maximal subgroups

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

subgroupindex
#361272
#227502
C2×#15512
#97022
#91362
#91272
#85922
#83442
#82312