Finite Group Databasebeta

← #215849

Subgroups — #215849

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36640C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3#13757C211yesmaximal, characteristic
4#9910C214yesmaximal
5#9886C214yesmaximal
6#8717C212yesmaximal
7#8716C212yesmaximal
8#2864—21no—
9#2864—21no—
10#2859C411yescharacteristic, complemented
11#2859C411yescharacteristic, complemented
12#2859C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2851—21no—
15#2851—21no—
16#2851C2212yes—
17#2851C2212yes—
18C4×#125—21no—
19C4×#125—21no—
20C4×#125C412yescomplemented
21C4×#125C412yescomplemented
22#2296—21no—
23#2296—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26C2×#376—21no—
27C2×C43C2211yescharacteristic
28#1839—21no—
29#1831—21no—
30#1817C2212yes—
31#1797C2212yes—
32C4×C4⋊C8—21no—
33C4×C22⋊C8C2212yes—
34#1744—21no—
35#1434—21no—
36#1434—21no—
37#1434C2212yes—
38#1434C2212yes—
39C4×#95—21no—
40C4×#95C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C2×#125—21no—
44C2×#125—21no—
45C2×#125C2×C412yes—
46C2×#125C2×C412yes—
47C2×#117—21no—
48C2×#117—21no—
49C22×C42—22no—
50C22×C42D411yescharacteristic
51C22×C42D411yescharacteristic
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C2×C4⋊C8—22no—
58C2×C22⋊C8C2314yes—
59C2×C4×C8—22no—
60#389—21no—
61#389—21no—
62#389—21no—
63#389—21no—
64#389—21no—
65#389—21no—
66#389—21no—
67#389—21no—
68#389—21no—
69#389—21no—
70#389—21no—
71#389—21no—
72#389C2×C412yes—
73#389C2×C412yes—
74#389C2×C412yes—
75#389C2×C412yes—
76#387—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#387—22no—
84#385—22no—
85#385—22no—
86#385—22no—
87#385—22no—
88#384C2×C411yescharacteristic
89#384C2×C411yescharacteristic
90#384C2×C411yescharacteristic
91#384C2×C411yescharacteristic
92#382—21no—
93#382—21no—
94#382—21no—
95#382—21no—
96#382C2×C412yes—
97#382C2×C412yes—
98#380—21no—
99#380—21no—
100#380C2312yes—
101#380C2312yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110#376—28no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95C2314yes—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129C23×C4—24no—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—24no—
138#125—44no—
139#125—44no—
140#125—44no—
141#125—44no—
142#117—44no—
143#117—44no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C42—24no—
167C2×C42—24no—
168C2×C42—24no—
169C2×C42—28no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—42no—
173C2×C42—42no—
174C2×C42—42no—
175C2×C42—42no—
176C2×C42—44no—
177C2×C42—44no—
178C2×C42C22⋊C411yescharacteristic
179C2×C42C22⋊C411yescharacteristic
180C2×C42C22⋊C411yescharacteristic
181C2×C42C22⋊C411yescharacteristic
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C4⋊C8—44no—
207C22⋊C8—28no—
208C4×C8—44no—
209#95—28no—
210#95—28no—
211#95—22no—
212#95—22no—
213#95—22no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—28no—
224#95—44no—
225#95—44no—
226#95—44no—
227C24C22×D411yescharacteristic
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—28no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C411yescharacteristic
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4#13514yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C42−1+414yes—
274C2×C8—44no—
275C2×C8—44no—
276C2×C8—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C22⋊C4—44no—
298C22⋊C4—44no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42—41no—
321C42—41no—
322C42—41no—
323C42—41no—
324C42—41no—
325C42—41no—
326C42—416no—
327C42—44no—
328C42—44no—
329C42#10414yes—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23C22×C22⋊C411yescharacteristic
335C23#53612yes—
336C23C2×#13512yes—
337C23#57512yes—
338C2×C4—28no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—28no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—48no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—84no—
396C2×C4#41012yes—
397C2×C4#41012yes—
398C2×C4#41612yes—
399C2×C4#41612yes—
400C2×C4C2×#10412yes—
401C2×C4C2×#10412yes—
402C2×C4C2×#10412yes—
403C2×C4#42112yes—
404C2×C4#51912yes—
405C2×C4#53612yes—
406C8—84no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22#231011yescharacteristic
414C22C2×#41011yescharacteristic
415C22C2×#41011yescharacteristic
416C22C2×#41611yescharacteristic
417C22#289611yescharacteristic
418C22C22×#10411yescharacteristic
419C22C2×#42111yescharacteristic
420C22#323214yes—
421C4—22no—
422C4—22no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—41no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—88no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4—82no—
440C2—24no—
441C2#1753311yescharacteristic
442C2#1754111yescharacteristic
443C2#1757311yescharacteristic
444C2#1758611yescharacteristic
445C2#2182611yescharacteristic
446C2#2183611yescharacteristic
447C2#2184411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
10. #2859C4semidirect
11. #2859C4semidirect
12. #2859C4semidirect
13. #2859C4semidirect
20. C4×#125C4semidirect
21. C4×#125C4semidirect

Maximal subgroups

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

subgroupindex
#366402
#207772
#137572
#99102
#98862
#87172
#87162