Finite Group Databasebeta

← #190809

Subgroups — #190809

Order 512.

1108 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34207C211yesmaximal, characteristic
2#21924C212yesmaximal, complemented
3#8725C211yesmaximal, characteristic, complemented
4#8723C211yesmaximal, characteristic, complemented
5#8594C211yesmaximal, characteristic, complemented
6#8588C212yesmaximal, complemented
7#8587C211yesmaximal, characteristic, complemented
8#8344C212yesmaximal
9#8235C212yesmaximal, complemented
10#8231C211yesmaximal, characteristic, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#532C2211yescharacteristic
13C2×#520C412yes—
14#2361—22no—
15#2361—22no—
16#2354—22no—
17D4×C4⋊C4—22no—
18#2326C412yes—
19#2326C412yes—
20#2311C412yes—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×#127C2211yescharacteristic
24C4×#127C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1923—22no—
29#1917—22no—
30#1916—22no—
31#1842C2212yes—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36C4×C4⋊C8C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#331C2212yes—
40C2×#331C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2211yescharacteristic
46#1296C2212yes—
47C22×#41C2×C412yes—
48#532—22no—
49#532—22no—
50#532—24no—
51C2×C4⋊Q8C2311yescharacteristic
52C2×#127C2311yescharacteristic
53C2×#124C2×C412yes—
54C2×#121—22no—
55#520—48no—
56C2×C4×Q8C2×C412yes—
57C2×C4×Q8C2×C412yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4C2×C412yes—
64C2×C4×D4C2×C412yes—
65C2×#117—22no—
66C2×#117C2×C411yescharacteristic
67C2×#117C2×C411yescharacteristic
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4C2×C412yes—
71C22×C42C2311yescharacteristic
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#391C2×C412yes—
83#391C2×C412yes—
84#390—22no—
85#390—22no—
86#390C2×C412yes—
87#390C2×C412yes—
88#389—22no—
89#389—22no—
90#388—24no—
91#388—24no—
92#386—24no—
93#385—24no—
94#385—24no—
95#384—22no—
96#384C2×C412yes—
97#384C2311yescharacteristic
98#384C2311yescharacteristic
99#380—24no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C22⋊C4D412yes—
105C4×C22⋊C4D412yes—
106#376C2×C411yescharacteristic
107#376C2×C411yescharacteristic
108C43C2311yescharacteristic
109C43C2311yescharacteristic
110#336—22no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#333—24no—
117#333—24no—
118#331—24no—
119#331—24no—
120C2×#41—24no—
121C2×#41—24no—
122C2×#41—44no—
123C2×#41—44no—
124C22×Q8C22×C412yes—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4C22×C412yes—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D412yes—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C4⋊Q8—22no—
139C4⋊Q8#4112yes—
140C4⋊Q8#4112yes—
141#127—21no—
142#127—21no—
143#127#4114yes—
144#124—24no—
145#124—24no—
146#121—28no—
147#121—28no—
148C4×Q8—44no—
149C4×Q8—44no—
150C4×D4—28no—
151C4×D4—28no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—48no—
155C4×D4—48no—
156C4×D4—82no—
157C4×D4—82no—
158C4×D4—82no—
159C4×D4—82no—
160#117—24no—
161#117—42no—
162#117—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4C22⋊C411yescharacteristic
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C411yescharacteristic
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—82no—
212C4⋊C8—82no—
213C22⋊C8—82no—
214C22⋊C8—82no—
215#97—82no—
216C4×C8—82no—
217#95—24no—
218#95—24no—
219#95C22⋊C412yes—
220#95C22⋊C412yes—
221#95C22⋊C412yes—
222#95C22⋊C412yes—
223C24—22no—
224C24C2×C22⋊C411yescharacteristic
225#41—416no—
226#41—416no—
227C2×Q8—22no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8#11714yes—
231C2×D4—28no—
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—24no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—44no—
238C2×D4—44no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4#11714yes—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—28no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4—84no—
265C22×C4—84no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C23×C411yescharacteristic
275C22×C4C22×D412yes—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—84no—
303C22⋊C4—216no—
304C22⋊C4—22no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—84no—
309C22⋊C4—84no—
310C22⋊C4#10014yes—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—41no—
327C42—44no—
328C42—44no—
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C42C2×#4112yes—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—21no—
336C23—24no—
337C23—42no—
338C23—42no—
339C23—44no—
340C23—44no—
341C23C2×#10012yes—
342C23C22×C22⋊C411yescharacteristic
343Q8—44no—
344Q8—44no—
345D4—44no—
346D4—44no—
347D4—416no—
348D4—44no—
349D4—44no—
350D4—44no—
351D4—44no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—416no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—164no—
386C2×C4—164no—
387C2×C4—164no—
388C2×C4C2×#10012yes—
389C2×C4C2×#10012yes—
390C2×C4#41312yes—
391C2×C4C2×#11711yescharacteristic
392C2×C4C2×#11711yescharacteristic
393C2×C4#52512yes—
394C2×C4#52512yes—
395C2×C4#57612yes—
396C2×C4#57712yes—
397C2×C4#57712yes—
398C2×C4#57812yes—
399C8—82no—
400C8—82no—
401C8—82no—
402C8—82no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—42no—
407C22—42no—
408C22—48no—
409C22—44no—
410C22—82no—
411C22—84no—
412C22—82no—
413C22#228011yescharacteristic
414C22C22×#10011yescharacteristic
415C22C2×#41311yescharacteristic
416C22#289712yes—
417C22#289812yes—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—42no—
424C4—41no—
425C4—42no—
426C4—42no—
427C4—41no—
428C4—44no—
429C4—82no—
430C4—82no—
431C4#311912yes—
432C4#312012yes—
433C4#312412yes—
434C4#312512yes—
435C2—42no—
436C2—84no—
437C2#1777011yescharacteristic
438C2#1777212yes—
439C2#1777911yescharacteristic
440C2#2072611yescharacteristic
441C2#2074612yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21924C2semidirect
3. #8725C2semidirect
4. #8723C2semidirect
5. #8594C2semidirect
6. #8588C2semidirect
7. #8587C2semidirect
9. #8235C2semidirect
10. #8231C2semidirect
11. C4×#331C2semidirect

Maximal subgroups

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

subgroupindex
#342072
#219242
#87252
#87232
#85942
#85882
#85872
#83442
#82352
#82312
C4×#3312