Finite Group Databasebeta

← #196894

Subgroups — #196894

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C22×#383C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1515C212yesmaximal, complemented
5#9153C214yesmaximal, complemented
6#9143C214yesmaximal
7#8018C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#432C2212yes—
11#2857C412yescomplemented
12#2857C412yescomplemented
13#2855—21no—
14#2855C2212yes—
15#2851—22no—
16#2848—21no—
17#2848C2212yes—
18C4×C22⋊Q8—22no—
19C4×C22⋊Q8C414yescomplemented
20#2296—21no—
21#2285C2211yescharacteristic
22C2×#383—21no—
23C2×#383—21no—
24C2×#383—28no—
25C2×#383C2212yes—
26C2×#383C2212yes—
27#2280—21no—
28C22×#95—21no—
29C22×#95—21no—
30C22×#95C2212yes—
31#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34#1749—22no—
35#1515—28no—
36C2×#328C2218yes—
37#1294C2218yes—
38C24×C4—22no—
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42#533—24no—
43#533—24no—
44C2×C4⋊Q8C2×C411yescharacteristic, complemented
45C2×C4⋊Q8C2×C411yescharacteristic, complemented
46C2×#125—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C414yes—
50C2×C22⋊Q8C2312yes—
51C2×C4×Q8—21no—
52C2×C4×Q8C2×C412yescomplemented
53C2×#117—21no—
54C2×#117—21no—
55C22×C42—24no—
56C22×C42—24no—
57C22×C42C2×C412yes—
58C22×C42C2×C412yes—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65#432—24no—
66#432—24no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8C2314yes—
70C2×#97—22no—
71#391—22no—
72#391C2×C414yescomplemented
73#389—22no—
74#389—22no—
75#389—22no—
76#389C2×C414yescomplemented
77#387—24no—
78#387—24no—
79#385—24no—
80#385—22no—
81#385—24no—
82#385—22no—
83#384C2×C412yescomplemented
84#383—28no—
85#383—28no—
86#383—24no—
87#383—28no—
88#383D418yes—
89#382—21no—
90#381—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—21no—
95#380C2×C414yes—
96#380C2312yes—
97#379—22no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4C2×C412yescomplemented
101C4×C22⋊C4—24no—
102C2×#95—28no—
103C2×#95—28no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—28no—
110C2×#95C2314yes—
111#336—24no—
112#328—216no—
113C25C2×D411yescharacteristic
114C22×Q8—21no—
115C22×Q8C22×C412yes—
116C23×C4—24no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—28no—
124C23×C4—24no—
125C23×C4—42no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4—48no—
129C23×C4—48no—
130C23×C4—44no—
131C23×C4—44no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22×C412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147#125—44no—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—48no—
151C22⋊Q8—42no—
152C22⋊Q8—42no—
153C22⋊Q8—48no—
154C4×Q8—28no—
155C4×Q8—44no—
156#117—44no—
157#117—42no—
158#117—42no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C414yes—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—28no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22⋊C412yes—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22⋊C412yes—
190C2×C42C22×C412yes—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D414yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42#4114yes—
198C4⋊C8—44no—
199C22⋊C8—24no—
200C22⋊C8—24no—
201C22⋊C8—48no—
202#97—44no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—24no—
210#95—22no—
211#95—28no—
212#95—28no—
213#95—24no—
214#95—22no—
215#95C2×D418yes—
216C24—21no—
217C24—21no—
218C24—22no—
219C24—24no—
220C24—24no—
221C24C2×C22⋊C412yes—
222C24#13714yes—
223C24C22×D411yescharacteristic
224C2×Q8—24no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—416no—
271C22×C4—416no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—416no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—816no—
278C22×C4—816no—
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#12012yes—
287C22×C4#12012yes—
288C22×C4#12312yes—
289C22×C4#12312yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D412yes—
293C22×C4C2×#4112yes—
294C2×C8—48no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—44no—
306C4⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—44no—
319C42—44no—
320C42C4×D414yes—
321C42C4×D414yes—
322C42#12114yes—
323C42#12114yes—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—216no—
329C23—24no—
330C23—22no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—24no—
336C23—216no—
337C23—22no—
338C23#41014yes—
339C23#41914yes—
340C23#42114yes—
341C23C22×C22⋊C411yescharacteristic
342C23C2×#12011yescharacteristic
343C23C2×#12311yescharacteristic
344C23C2×#13712yes—
345C23#57512yes—
346Q8—84no—
347Q8—84no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—832no—
394C2×C4—88no—
395C2×C4—88no—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#38612yes—
400C2×C4C2×C4×D412yes—
401C2×C4C2×#12112yes—
402C8—84no—
403C22—216no—
404C22—216no—
405C22—24no—
406C22—24no—
407C22—24no—
408C22—24no—
409C22—24no—
410C22—24no—
411C22—21no—
412C22—21no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—24no—
417C22#151518yescomplemented
418C22C2×#38611yescharacteristic
419C22C2×#41012yes—
420C22C2×#41911yescharacteristic
421C22#289611yescharacteristic
422C22C2×#42112yes—
423C22#300712yes—
424C22#309512yes—
425C4—22no—
426C4—22no—
427C4—22no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—42no—
434C4—88no—
435C4—84no—
436C4—82no—
437C4—82no—
438C2—22no—
439C2—22no—
440C2—28no—
441C2C2×#151512yes—
442C2#1759312yes—
443C2#2091111yescharacteristic
444C2#2097911yescharacteristic
445C2#2098311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1515C2semidirect
5. #9153C2semidirect
7. #8018C2semidirect
11. #2857C4semidirect
12. #2857C4semidirect
19. C4×C22⋊Q8C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
45. C2×C4⋊Q8C2×C4semidirect
52. C2×C4×Q8C2×C4semidirect
72. #391C2×C4semidirect
76. #389C2×C4semidirect
83. #384C2×C4semidirect
100. C4×C4⋊C4C2×C4semidirect
417. C22#1515semidirect

Maximal subgroups

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

subgroupindex
#366252
C22×#3832
#207152
C2×#15152
#91532
#91432
#80182