Finite Group Databasebeta

← #194764

Subgroups — #194764

Order 512.

1338 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2599C211yesmaximal, characteristic
2#34177C211yesmaximal, characteristic
3#20735C212yesmaximal, complemented
4C2×#1458C211yesmaximal, characteristic
5#9026C212yesmaximal, complemented
6#9025C212yesmaximal
7#9014C212yesmaximal, complemented
8#9013C212yesmaximal
9#8025C212yesmaximal, complemented
10C22×C4×Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#431C2212yes—
13#2599—24no—
14#2599C418yescomplemented
15C2×#398C2211yescharacteristic
16C2×#397C2211yescharacteristic
17C2×#395C2211yescharacteristic
18C2×#393C2211yescharacteristic
19#2343—24no—
20#2343—24no—
21#2315C2212yes—
22#2313—21no—
23#2313C2212yes—
24C2×#388—21no—
25C2×#387—21no—
26C2×#387—21no—
27C2×#385—21no—
28#2283—21no—
29#2283C2212yes—
30C2×#383C2211yescharacteristic
31C2×#381—21no—
32#2273—21no—
33#1823C2214yes—
34#1805—24no—
35#1787C2214yes—
36C4×#103—24no—
37#1458—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C24×C4D411yescharacteristic
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yescomplemented
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—24no—
52C2×#117C2×C414yes—
53C2×#117D414yes—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—24no—
69#431—24no—
70C2×C4⋊C8—22no—
71C2×#103—28no—
72C2×C22⋊C8C2314yes—
73C2×C4×C8—22no—
74#398—22no—
75#398C2×C414yescomplemented
76#397—22no—
77#397C2×C414yes—
78#395—22no—
79#395C2×C414yescomplemented
80#393—22no—
81#393C2×C414yes—
82#388—28no—
83#387—28no—
84#387—28no—
85#387—28no—
86#387—28no—
87#385—28no—
88#385—28no—
89#383—22no—
90#383—22no—
91#383—22no—
92#383C2×C414yescomplemented
93#382—22no—
94#382D414yes—
95#381—22no—
96#381—22no—
97#381—22no—
98#381—22no—
99#380—22no—
100#380C2314yes—
101C4×C4⋊C4—22no—
102C4×C4⋊C4D414yes—
103C4×C22⋊C4—28no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106#376—22no—
107#376—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2311yescharacteristic
117C2×#95C2311yescharacteristic
118#340—28no—
119#324—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—44no—
126C22×Q8C22×C412yes—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—42no—
137C23×C4C22⋊C414yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C4×Q8—28no—
146C4×Q8—28no—
147C4×Q8—44no—
148C4×Q8—44no—
149#117—28no—
150#117—28no—
151#117—28no—
152#117—28no—
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—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22⋊C414yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D418yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4C22⋊C414yes—
187C2×C22⋊C4C22×C412yes—
188C2×C22⋊C4C2×D412yes—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C22⋊C414yes—
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42#4114yes—
203C4⋊C8—44no—
204#103—416no—
205C22⋊C8—24no—
206C22⋊C8—24no—
207C4×C8—44no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—21no—
222#95—21no—
223#95—24no—
224#95C22×C412yes—
225#95C22×C412yes—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—42no—
230C24—42no—
231C24C2×C22⋊C412yes—
232C24#13514yes—
233C24C22×D411yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—44no—
239C2×Q8—48no—
240C2×Q8—42no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—28no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—28no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—48no—
282C22×C4—84no—
283C22×C4—84no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12012yes—
292C22×C4#12312yes—
293C22×C4#12312yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4112yes—
298C2×C8—48no—
299C2×C8—44no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—24no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—84no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—28no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—48no—
319C42—24no—
320C42—24no—
321C42—48no—
322C42—84no—
323C42—84no—
324C42C4×D414yes—
325C42C4×D414yes—
326C42#12114yes—
327C42#12114yes—
328C23—22no—
329C23—22no—
330C23—24no—
331C23—22no—
332C23—24no—
333C23—22no—
334C23—42no—
335C23—42no—
336C23—48no—
337C23—42no—
338C23—42no—
339C23—84no—
340C23—81no—
341C23—81no—
342C23#41614yes—
343C23C22×C22⋊C411yescharacteristic
344C23C2×#12011yescharacteristic
345C23C2×#12311yescharacteristic
346C23C2×#13512yes—
347C23#57512yes—
348Q8—28no—
349Q8—48no—
350Q8—44no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
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—48no—
390C2×C4—48no—
391C2×C4—48no—
392C2×C4—42no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—84no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—168no—
400C2×C4#38612yes—
401C2×C4#38612yes—
402C2×C4#38614yes—
403C2×C4C2×C4×D412yes—
404C2×C4C2×#12112yes—
405C8—84no—
406C22—24no—
407C22—24no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—24no—
413C22—44no—
414C22—48no—
415C22—42no—
416C22—42no—
417C22—82no—
418C22—84no—
419C22C2×#38611yescharacteristic
420C22C2×#41611yescharacteristic
421C22#289611yescharacteristic
422C22#300212yes—
423C22#302212yes—
424C22#302512yes—
425C22#309112yes—
426C4—22no—
427C4—22no—
428C4—24no—
429C4—22no—
430C4—22no—
431C4—42no—
432C4—42no—
433C4—44no—
434C4—42no—
435C4—82no—
436C4—82no—
437C4—88no—
438C4—82no—
439C2—21no—
440C2—21no—
441C2—22no—
442C2—44no—
443C2#1820914yes—
444C2#2090811yescharacteristic
445C2#2095511yescharacteristic
446C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20735C2semidirect
5. #9026C2semidirect
7. #9014C2semidirect
9. #8025C2semidirect
14. #2599C4semidirect
48. C2×C4×Q8C2×C4semidirect
75. #398C2×C4semidirect
79. #395C2×C4semidirect
92. #383C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25992
#341772
#207352
C2×#14582
#90262
#90252
#90142
#90132
#80252