Finite Group Databasebeta

← #481499

Subgroups — #481499

Order 512.

1236 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35702C211yesmaximal, characteristic
2#20354C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#11717C212yesmaximal, complemented
5#11716C214yesmaximal, complemented
6#11715C212yesmaximal, complemented
7C2×#1351C214yesmaximal, complemented
8C2×D4×Q8C411yescharacteristic, complemented
9C2×#518—21no—
10C2×#518C2211yescharacteristic
11#2972—21no—
12#2628C2211yescharacteristic
13#2626C2212yes—
14#2580C411yescharacteristic, complemented
15#2564C412yescomplemented
16#2563C411yescharacteristic, complemented
17#2452—21no—
18#2450—22no—
19#2439—21no—
20#2437—22no—
21#2409C412yescomplemented
22#2396C2211yescharacteristic
23C2×#393C2212yes—
24#2349—21no—
25#2349C411yescharacteristic, complemented
26D4×C22⋊C4—21no—
27#1740—21no—
28#1740C2212yes—
29C2×#342—22no—
30C2×#342C2214yes—
31#1460—21no—
32#1459—22no—
33#1458C2212yes—
34C2×#323C2214yes—
35#1352C2214yes—
36#1351C2214yes—
37#1351C2218yes—
38C23×Q8C2×C412yescomplemented
39C23×D4D411yescharacteristic
40D4×Q8—44no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic, complemented
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8C2×C412yescomplemented
46#518—28no—
47#518—44no—
48#518—44no—
49C2×C4×Q8C2×C411yescharacteristic, complemented
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—44no—
53C2×C4×D4C2×C412yes—
54C2×C4×D4C2×C411yescharacteristic
55C2×C4×D4C2×C411yescharacteristic
56C2×C4×D4D412yes—
57C2×C4×D4D411yescharacteristic
58C2×C4×D4D411yescharacteristic
59C2×#117—22no—
60C2×#117D411yescharacteristic
61C2×#117C2311yescharacteristic
62C22×C22⋊C4—24no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D412yes—
65C22×C22⋊C4C2312yes—
66C8×D4—44no—
67#431—22no—
68C2×C22⋊C8—24no—
69#400—21no—
70#400C2×C411yescharacteristic, complemented
71#399—21no—
72#399C2×C412yescomplemented
73#399C2311yescharacteristic
74#398—21no—
75#396—22no—
76#396—22no—
77#396—21no—
78#395C2×C412yescomplemented
79#395C2×C411yescharacteristic, complemented
80#393—22no—
81#393—22no—
82#393C2×C412yescomplemented
83#393C2×C412yescomplemented
84#393C2×C412yescomplemented
85#393C2312yes—
86#393C2311yescharacteristic
87#391—22no—
88#390—21no—
89#390—21no—
90#389—21no—
91#389—21no—
92#389—21no—
93#387—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385—22no—
100#385—22no—
101#382—21no—
102#382C2×C411yescharacteristic, complemented
103#380—22no—
104#380—21no—
105#380—22no—
106#380C2×C412yescomplemented
107#379—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4C2×C411yescharacteristic, complemented
110C4×C22⋊C4—21no—
111C2×#95—22no—
112C2×#95C2×C412yescomplemented
113#342—22no—
114#342—24no—
115#342C2314yes—
116#324—22no—
117#323—24no—
118#323C2314yes—
119C25C2×D412yes—
120C22×Q8—22no—
121C22×Q8—44no—
122C22×Q8—84no—
123C22×Q8C22×C411yescharacteristic
124C22×Q8C22×C412yes—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4C22⋊C412yes—
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22⋊C411yescharacteristic
130C22×D4SD1614yes—
131C22×D4Q1614yes—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—44no—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C22×C8—44no—
141C4⋊Q8—42no—
142C4⋊Q8—44no—
143C22⋊Q8—48no—
144C22⋊Q8—44no—
145C22⋊Q8—84no—
146C4×Q8—42no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—42no—
150C4×D4—42no—
151C4×D4—44no—
152C4×D4—44no—
153C4×D4—432no—
154C4×D4—44no—
155C4×D4—44no—
156#117—28no—
157#117—42no—
158#117—42no—
159#117—44no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22⋊C411yescharacteristic
173C2×C4⋊C4C22⋊C411yescharacteristic
174C2×C4⋊C4C22×C411yescharacteristic
175C2×C4⋊C4C22×C411yescharacteristic
176C2×C4⋊C4C2×D411yescharacteristic
177C2×C22⋊C4—216no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—21no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4—84no—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22×C412yes—
196C2×C22⋊C4C2×D411yescharacteristic
197C2×C22⋊C4C2×D412yes—
198C2×C22⋊C4C2×D412yes—
199C2×C22⋊C4C2411yescharacteristic
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—42no—
203C2×C42C22⋊C412yes—
204C2×C42C22×C411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C4⋊C8—42no—
207C22⋊C8—24no—
208C22⋊C8—44no—
209C4×C8—42no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221#95C22×C411yescharacteristic
222#95C22×C412yes—
223#95C22×C411yescharacteristic
224C24—24no—
225C24—24no—
226C24—82no—
227C24—82no—
228C24C2×C22⋊C412yes—
229C24C2×C22⋊C412yes—
230C24C2×SD1614yes—
231C24C2×Q1614yes—
232C24C22×D411yescharacteristic
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—41no—
237C2×Q8—42no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—216no—
245C2×D4—216no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4#9914yes—
249C2×D4#10318yes—
250C2×D4#10314yes—
251C2×D4#10314yes—
252C2×D4#10414yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—416no—
269C22×C4—44no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4—164no—
278C22×C4—164no—
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×SD1612yes—
286C22×C4C2×Q1612yes—
287C22×C4C23×C411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C2×C8—44no—
290C2×C8—164no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—48no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—82no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C22⋊C4—24no—
309C22⋊C4—24no—
310C22⋊C4—28no—
311C22⋊C4—41no—
312C22⋊C4—42no—
313C22⋊C4—41no—
314C22⋊C4—42no—
315C22⋊C4—42no—
316C22⋊C4—42no—
317C22⋊C4—44no—
318C22⋊C4—416no—
319C22⋊C4—44no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C22⋊C4—81no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—22no—
328C42—22no—
329C42—42no—
330C42—48no—
331C42—42no—
332C42—42no—
333C23—21no—
334C23—22no—
335C23—21no—
336C23—22no—
337C23—28no—
338C23—28no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—42no—
344C23—41no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—84no—
349C23—164no—
350C23C2×#9914yes—
351C23C2×#10314yes—
352C23C2×#10314yes—
353C23C2×#10414yes—
354C23C22×C22⋊C411yescharacteristic
355C23C22×SD1611yescharacteristic
356C23C22×Q1611yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—88no—
360Q8—84no—
361D4—216no—
362D4—28no—
363D4#328116yes—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—48no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—88no—
402C2×C4—84no—
403C2×C4—84no—
404C2×C4—162no—
405C2×C4—162no—
406C2×C4—164no—
407C2×C4—1616no—
408C2×C4C2×#9912yes—
409C2×C4C2×#10312yes—
410C2×C4C2×#10312yes—
411C2×C4C2×#10412yes—
412C8—84no—
413C22—21no—
414C22—24no—
415C22—21no—
416C22—24no—
417C22—21no—
418C22—21no—
419C22—24no—
420C22—44no—
421C22—44no—
422C22—41no—
423C22—41no—
424C22—44no—
425C22—41no—
426C22—42no—
427C22—41no—
428C22—42no—
429C22—88no—
430C22—82no—
431C22—81no—
432C22—88no—
433C22C2×#32818yes—
434C22C22×#9911yescharacteristic
435C22C22×#10311yescharacteristic
436C22C22×#10411yescharacteristic
437C4—22no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—42no—
444C4—41no—
445C4—81no—
446C4—82no—
447C4—81no—
448C4—88no—
449C4—82no—
450C4—81no—
451C4C2×#32814yes—
452C2—21no—
453C2—21no—
454C2—28no—
455C2—41no—
456C2—44no—
457C2—41no—
458C2C22×#32811yescharacteristic
459C2#3334811yescharacteristic
460C2#3336111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×#518 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11717C2semidirect
5. #11716C2semidirect
6. #11715C2semidirect
7. C2×#1351C2semidirect
8. C2×D4×Q8C4semidirect
14. #2580C4semidirect
15. #2564C4semidirect
16. #2563C4semidirect
21. #2409C4semidirect
25. #2349C4semidirect
38. C23×Q8C2×C4semidirect
42. C2×C4⋊Q8C2×C4semidirect
45. C2×C22⋊Q8C2×C4semidirect
49. C2×C4×Q8C2×C4semidirect
70. #400C2×C4semidirect
72. #399C2×C4semidirect
78. #395C2×C4semidirect
79. #395C2×C4semidirect
82. #393C2×C4semidirect
83. #393C2×C4semidirect
84. #393C2×C4semidirect
102. #382C2×C4semidirect
106. #380C2×C4semidirect
109. C4×C4⋊C4C2×C4semidirect
112. C2×#95C2×C4semidirect

Maximal subgroups

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

subgroupindex
#357022
#203542
#173502
#117172
#117162
#117152
C2×#13512