Finite Group Databasebeta

← #196867

Subgroups — #196867

Order 512.

1808 conjugacy classes of subgroups.

The subgroup lattice has 463 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#20682C211yesmaximal, characteristic
4C2×#1508C212yesmaximal
5#9145C214yesmaximal, complemented
6#9144C214yesmaximal, complemented
7#8019C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C22×#117—21no—
10C23×C42C2211yescharacteristic
11C2×#407C2212yes—
12#2855—22no—
13#2855—21no—
14#2855—21no—
15#2855C414yescomplemented
16#2855C2212yes—
17#2855C2212yes—
18C2×#391C412yescomplemented
19C2×#389C412yescomplemented
20C2×#387—22no—
21C2×#385—21no—
22C2×#381—21no—
23C2×#380C2211yescharacteristic
24C4×#117—28no—
25C2×C4×C4⋊C4—21no—
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C22⋊C4—21no—
28C2×C4×C22⋊C4C2212yes—
29C2×#376C2212yes—
30C2×C43—21no—
31#1817—22no—
32#1817C2212yes—
33#1817C2212yes—
34#1728—22no—
35#1508—28no—
36C2×#328C2218yes—
37#1294C2218yes—
38C23×Q8C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4D411yescharacteristic
41C24×C4C2311yescharacteristic
42C22×M16—22no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8C2×C412yes—
48C2×C22⋊Q8C2312yes—
49C2×#117—216no—
50C2×#117—28no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C42—24no—
60C22×C42—22no—
61C22×C42—24no—
62C22×C42—22no—
63C22×C42—22no—
64C22×C42C2×C412yes—
65C22×C42C2×C412yes—
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D412yes—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72#407—24no—
73#407—24no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2314yes—
76#391—22no—
77#391—22no—
78#391C2×C414yes—
79#391C2×C414yes—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#389C2×C414yes—
84#387—24no—
85#387—24no—
86#387—24no—
87#387—24no—
88#385—24no—
89#385—22no—
90#385—22no—
91#381—24no—
92#381—22no—
93#381—22no—
94#380—21no—
95#380—21no—
96#380C2×C412yes—
97#380C2312yes—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—28no—
100C4×C22⋊C4—28no—
101C4×C22⋊C4—24no—
102C4×C22⋊C4D418yes—
103#376—28no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95C2314yes—
110C43—28no—
111#336—24no—
112#336—24no—
113#328—216no—
114C25C2×D411yescharacteristic
115C22×Q8—21no—
116C22×Q8—24no—
117C22×Q8—44no—
118C22×Q8C22×C412yes—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—28no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—44no—
130C23×C4—42no—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D414yes—
145C23×C4#4114yes—
146C23×C4C2411yescharacteristic
147C2×M16—48no—
148C2×M16—44no—
149C22×C8—24no—
150C22⋊Q8—28no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—42no—
154C22⋊Q8—44no—
155C22⋊Q8—42no—
156#117—264no—
157C2×C4⋊C4—216no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—48no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C414yes—
171C2×C22⋊C4—216no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—42no—
180C2×C42—28no—
181C2×C42—216no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—44no—
197C2×C42—48no—
198C2×C42—44no—
199C2×C42C22⋊C412yes—
200C2×C42C22⋊C412yes—
201C2×C42C22⋊C412yes—
202C2×C42C22⋊C412yes—
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—44no—
210#95—28no—
211#95—22no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—22no—
219#95—24no—
220#95C2×D418yes—
221C24—22no—
222C24—22no—
223C24—24no—
224C24—42no—
225C24C2×C22⋊C411yescharacteristic
226C24C2×C22⋊C411yescharacteristic
227C24#12012yes—
228C24#12312yes—
229C24C22×D411yescharacteristic
230C2×Q8—24no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—88no—
235C22×C4—216no—
236C22×C4—216no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—216no—
255C22×C4—21no—
256C22×C4—28no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4—48no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4#10418yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C4×D414yes—
294C22×C4C4×D414yes—
295C22×C4#12012yes—
296C22×C4#12114yes—
297C22×C4#12114yes—
298C22×C4#12312yes—
299C22×C4#13714yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C2×#4112yes—
304M16—88no—
305C2×C8—48no—
306C2×C8—44no—
307C4⋊C4—232no—
308C4⋊C4—24no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—48no—
315C4⋊C4—48no—
316C22⋊C4—232no—
317C22⋊C4—44no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C42—232no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—416no—
332C42—416no—
333C23—22no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—24no—
339C23—22no—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—24no—
344C23—24no—
345C23—24no—
346C23—42no—
347C23—42no—
348C23—48no—
349C23—48no—
350C23#38612yes—
351C23#38612yes—
352C23C2×#10414yes—
353C23C22×C22⋊C411yescharacteristic
354C23C2×#12011yescharacteristic
355C23C2×#12311yescharacteristic
356C23C2×#13712yes—
357C23#57512yes—
358Q8—84no—
359Q8—84no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—216no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—216no—
373C2×C4—24no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—28no—
379C2×C4—28no—
380C2×C4—24no—
381C2×C4—216no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—416no—
393C2×C4—416no—
394C2×C4—42no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—88no—
405C2×C4—84no—
406C2×C4—84no—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4#41014yes—
410C2×C4#41914yes—
411C2×C4C2×#10414yes—
412C2×C4C2×#10414yes—
413C2×C4C2×C4×D412yes—
414C2×C4C2×#12112yes—
415C8—84no—
416C22—24no—
417C22—24no—
418C22—24no—
419C22—24no—
420C22—22no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—21no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—48no—
430C22—48no—
431C22—42no—
432C22—42no—
433C22#189614yes—
434C22C2×#38611yescharacteristic
435C22C2×#41012yes—
436C22C2×#41911yescharacteristic
437C22#289611yescharacteristic
438C22C22×#10412yes—
439C22#300712yes—
440C22#309512yes—
441C4—24no—
442C4—24no—
443C4—24no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—416no—
448C4—42no—
449C4—84no—
450C4—82no—
451C4—82no—
452C4#189914yes—
453C2—22no—
454C2—22no—
455C2—24no—
456C2—42no—
457C2#1753712yes—
458C2#1754512yes—
459C2#2091111yescharacteristic
460C2C2×#189611yescharacteristic
461C2C2×#189911yescharacteristic
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
5. #9145C2semidirect
6. #9144C2semidirect
15. #2855C4semidirect
18. C2×#391C4semidirect
19. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
C2×C4×#1172
#206822
C2×#15082
#91452
#91442
#80192