Finite Group Databasebeta

← #196865

Subgroups — #196865

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36625C211yesmaximal, characteristic
2C2×C4×#117C211yesmaximal, characteristic
3#20715C211yesmaximal, characteristic
4C2×#1506C212yesmaximal, complemented
5#9146C214yesmaximal
6#9144C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C2×#533C2211yescharacteristic
9C22×#117—21no—
10C23×C42C2211yescharacteristic
11C2×#432C2212yes—
12#2857C412yescomplemented
13#2857C412yescomplemented
14#2855—21no—
15#2855C2212yes—
16#2851—22no—
17#2848—21no—
18#2848C2212yes—
19C4×C22⋊Q8—22no—
20C4×C22⋊Q8C414yescomplemented
21#2296—21no—
22#2285C2211yescharacteristic
23#2280—21no—
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#1817C2212yes—
32#1797—22no—
33#1779C2212yes—
34#1749—22no—
35#1506—28no—
36C2×#328C2218yes—
37#1294C2218yes—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41#533—24no—
42#533—24no—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8C2×C414yes—
49C2×C22⋊Q8C2312yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C412yes—
52C2×#117—216no—
53C2×#117—28no—
54C2×#117—21no—
55C2×#117—21no—
56C22×C4⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C42—22no—
59C22×C42—24no—
60C22×C42—24no—
61C22×C42—22no—
62C22×C42—22no—
63C22×C42C2×C412yes—
64C22×C42C2×C412yes—
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#432—24no—
72#432—24no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8C2314yes—
76C2×#97—22no—
77#391—22no—
78#391C2×C414yes—
79#389—22no—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#387—24no—
84#387—24no—
85#385—24no—
86#385—22no—
87#385—24no—
88#385—22no—
89#384C2×C412yes—
90#382—21no—
91#381—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—21no—
96#380C2×C414yes—
97#380C2312yes—
98#379—22no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—28no—
103C4×C4⋊C4C2×C412yes—
104C4×C22⋊C4—28no—
105C4×C22⋊C4—24no—
106C4×C22⋊C4—28no—
107#376—24no—
108#376D418yes—
109C2×#95—22no—
110C2×#95C2314yes—
111C43—28no—
112#336—24no—
113#328—216no—
114C25C2×D411yescharacteristic
115C22×Q8—21no—
116C22×Q8C22×C412yes—
117C23×C4—28no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—44no—
127C23×C4—42no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—24no—
141C4⋊Q8—24no—
142C4⋊Q8—24no—
143#125—44no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—48no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—48no—
150C4×Q8—28no—
151C4×Q8—44no—
152#117—264no—
153#117—44no—
154#117—42no—
155#117—42no—
156C2×C4⋊C4—216no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C414yes—
169C2×C22⋊C4—216no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C42—216no—
176C2×C42—28no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—28no—
184C2×C42—28no—
185C2×C42—28no—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—28no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—44no—
192C2×C42—48no—
193C2×C42—44no—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D412yes—
202C2×C42C2×D414yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4114yes—
206C4⋊C8—44no—
207C22⋊C8—24no—
208C22⋊C8—24no—
209C22⋊C8—48no—
210#97—44no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—22no—
216#95—28no—
217#95—24no—
218#95—22no—
219#95C2×D418yes—
220C24—21no—
221C24—21no—
222C24—24no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24C22×D411yescharacteristic
228C2×Q8—24no—
229C2×Q8—44no—
230C2×Q8—42no—
231C2×Q8—44no—
232C22×C4—216no—
233C22×C4—216no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
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—216no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—48no—
272C22×C4—48no—
273C22×C4—48no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4#10418yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4#12012yes—
285C22×C4#12012yes—
286C22×C4#12312yes—
287C22×C4#12312yes—
288C22×C4#13714yes—
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4C22×D412yes—
292C22×C4C2×#4112yes—
293C2×C8—48no—
294C2×C8—44no—
295C4⋊C4—232no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4—44no—
299C4⋊C4—44no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C4⋊C4—48no—
307C22⋊C4—232no—
308C22⋊C4—48no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—48no—
313C42—232no—
314C42—24no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—416no—
321C42—416no—
322C42—44no—
323C42—44no—
324C42C4×D414yes—
325C42C4×D414yes—
326C42#12114yes—
327C42#12114yes—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—24no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—48no—
339C23—48no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23C2×#10414yes—
345C23C22×C22⋊C411yescharacteristic
346C23C2×#12011yescharacteristic
347C23C2×#12311yescharacteristic
348C23C2×#13712yes—
349C23#57512yes—
350Q8—84no—
351Q8—84no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—216no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—216no—
376C2×C4—24no—
377C2×C4—216no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—48no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—416no—
386C2×C4—416no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4#38612yes—
396C2×C4#38612yes—
397C2×C4#38612yes—
398C2×C4#38612yes—
399C2×C4#41014yes—
400C2×C4#41914yes—
401C2×C4C2×#10414yes—
402C2×C4C2×#10414yes—
403C2×C4C2×C4×D412yes—
404C2×C4C2×#12112yes—
405C8—84no—
406C22—24no—
407C22—24no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—24no—
413C22—24no—
414C22—24no—
415C22—48no—
416C22—48no—
417C22—42no—
418C22—42no—
419C22—42no—
420C22—42no—
421C22C2×#38611yescharacteristic
422C22C2×#41012yes—
423C22C2×#41911yescharacteristic
424C22#289611yescharacteristic
425C22C22×#10412yes—
426C22#300712yes—
427C22#309512yes—
428C4—22no—
429C4—22no—
430C4—24no—
431C4—22no—
432C4—24no—
433C4—22no—
434C4—24no—
435C4—22no—
436C4—416no—
437C4—84no—
438C4—82no—
439C4—82no—
440C4#189614yes—
441C2—22no—
442C2—22no—
443C2—44no—
444C2#1754312yes—
445C2#1754512yes—
446C2#2091111yescharacteristic
447C2C2×#189611yescharacteristic
448C2#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×#1506C2semidirect
6. #9144C2semidirect
7. #8022C2semidirect
12. #2857C4semidirect
13. #2857C4semidirect
20. C4×C22⋊Q8C4semidirect

Maximal subgroups

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

subgroupindex
#366252
C2×C4×#1172
#207152
C2×#15062
#91462
#91442
#80222