Finite Group Databasebeta

← #196875

Subgroups — #196875

Order 512.

1613 conjugacy classes of subgroups.

The subgroup lattice has 433 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36626C211yesmaximal, characteristic
2C22×#383C211yesmaximal, characteristic, complemented
3#20682C211yesmaximal, characteristic, complemented
4C2×#1504C212yesmaximal
5#9157C214yesmaximal, complemented
6#9153C214yesmaximal, complemented
7#8020C212yesmaximal, complemented
8C2×#525C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#407C2212yes—
11#2855C412yescomplemented
12#2855C2212yes—
13#2854C412yescomplemented
14#2854C2212yes—
15#2853—22no—
16#2849—21no—
17#2849—21no—
18C4×#123—22no—
19C4×#123—22no—
20#2297C412yescomplemented
21#2296C412yescomplemented
22C2×#383—22no—
23C2×#383—28no—
24C2×#383C2214yes—
25#2280C2211yescharacteristic
26C22×#95—21no—
27C22×#95—21no—
28C22×#95C2212yes—
29#1817—22no—
30#1817C2214yes—
31#1728—22no—
32#1504—28no—
33C2×#328C2214yes—
34C2×#327C2214yes—
35#1294C2218yes—
36C22×#41C2×C411yescharacteristic
37C24×C4—22no—
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42#525—24no—
43#525—24no—
44C2×#123—22no—
45C2×#123—22no—
46C2×#123—21no—
47C2×C22⋊Q8C2×C411yescharacteristic
48C2×C22⋊Q8C2311yescharacteristic
49C2×#121C2×C411yescharacteristic
50C2×#121C2311yescharacteristic
51C2×#120—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C412yes—
56C22×C42—24no—
57C22×C42—24no—
58C22×C42C2×C412yes—
59C22×C42C2×C412yes—
60C22×C42D412yes—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42C2312yes—
64#407—24no—
65#407—24no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#391C2×C414yescomplemented
69#390C2×C414yescomplemented
70#389C2×C414yescomplemented
71#389C2×C414yescomplemented
72#388—24no—
73#388—24no—
74#387—24no—
75#387—24no—
76#386—22no—
77#386—22no—
78#386—22no—
79#386—22no—
80#385—22no—
81#385—22no—
82#385—22no—
83#385—22no—
84#383—216no—
85#383—216no—
86#381—22no—
87#381—22no—
88#381—21no—
89#380C2×C412yes—
90#380C2312yes—
91#379—21no—
92C4×C4⋊C4—24no—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—21no—
95C4×C22⋊C4—24no—
96C2×#95—28no—
97C2×#95—28no—
98C2×#95—24no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—24no—
102C2×#95D418yes—
103C2×#95C2314yes—
104#336—28no—
105#328—28no—
106#327—28no—
107C25C2×D411yescharacteristic
108C2×#41—24no—
109C2×#41—44no—
110C22×Q8C22×C411yescharacteristic
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4C22×C411yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—28no—
120C23×C4—24no—
121C23×C4—21no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—48no—
126C23×C4—48no—
127C23×C4—48no—
128C23×C4C22⋊C412yes—
129C23×C4C22⋊C412yes—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D414yes—
140C23×C4#4114yes—
141C23×C4C2411yescharacteristic
142C2×M16—48no—
143C2×M16—44no—
144C22×C8—24no—
145#123—48no—
146#123—48no—
147#123—44no—
148C22⋊Q8—24no—
149C22⋊Q8—22no—
150C22⋊Q8—22no—
151#121—24no—
152#121—22no—
153#121—22no—
154#120—42no—
155#120—42no—
156C4×D4—42no—
157C4×D4—44no—
158C4×D4—42no—
159#117—24no—
160#117—44no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4C22×C411yescharacteristic
166C2×C4⋊C4C22×C411yescharacteristic
167C2×C4⋊C4C22×C414yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—21no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—21no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—21no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—24no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—24no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42C22⋊C414yes—
189C2×C42C22⋊C414yes—
190C2×C42C22×C414yes—
191C2×C42C2×D414yes—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8—44no—
195#95—216no—
196#95—216no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—28no—
203#95—24no—
204#95—22no—
205#95C2×D418yes—
206C24—24no—
207C24—24no—
208C24—24no—
209C24—21no—
210C24C2×C22⋊C411yescharacteristic
211C24C2×C22⋊C411yescharacteristic
212C24#12012yes—
213C24#12312yes—
214C24C22×D411yescharacteristic
215#41—88no—
216C2×Q8—22no—
217C2×Q8—42no—
218C2×D4—22no—
219C2×D4—42no—
220C2×D4—41no—
221C2×D4—41no—
222C2×D4—44no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—44no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—28no—
246C22×C4—44no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—48no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—416no—
264C22×C4—416no—
265C22×C4—416no—
266C22×C4—48no—
267C22×C4—42no—
268C22×C4—832no—
269C22×C4C2×C22⋊C414yes—
270C22×C4C2×C22⋊C414yes—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C4×D414yes—
274C22×C4C4×D414yes—
275C22×C4#12012yes—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4#12312yes—
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D412yes—
282C22×C4C2×#4112yes—
283M16—88no—
284C2×C8—48no—
285C2×C8—44no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C22⋊C4—48no—
294C22⋊C4—48no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—44no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C42—28no—
305C42—28no—
306C42—28no—
307C42—28no—
308C42—44no—
309C42—42no—
310C23—232no—
311C23—24no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—24no—
321C23—24no—
322C23—24no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—41no—
327C23#38612yes—
328C23#38612yes—
329C23#41014yes—
330C23#42114yes—
331C23C22×C22⋊C411yescharacteristic
332C23C2×#12011yescharacteristic
333C23C2×#12311yescharacteristic
334C23#57514yes—
335Q8—82no—
336D4—82no—
337D4—82no—
338D4—84no—
339D4—84no—
340D4—82no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—48no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—84no—
380C2×C4—832no—
381C2×C4—816no—
382C2×C4#38612yes—
383C2×C4#38612yes—
384C2×C4C2×C4×D412yes—
385C2×C4C2×#12112yes—
386C8—84no—
387C22—216no—
388C22—216no—
389C22—22no—
390C22—22no—
391C22—28no—
392C22—28no—
393C22—28no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—81no—
401C22—84no—
402C22—81no—
403C22—81no—
404C22#189714yes—
405C22#189914yes—
406C22C2×#38611yescharacteristic
407C22C2×#41012yes—
408C22#289612yes—
409C22C2×#42112yes—
410C22#300512yes—
411C22#309312yes—
412C4—22no—
413C4—22no—
414C4—22no—
415C4—22no—
416C4—44no—
417C4—44no—
418C4—88no—
419C4—81no—
420C4—84no—
421C4—81no—
422C4—81no—
423C2—22no—
424C2—22no—
425C2—24no—
426C2—24no—
427C2—161no—
428C2#1759314yes—
429C2#2091011yescharacteristic
430C2C2×#189711yescharacteristic
431C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
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
2. C22×#383C2semidirect
3. #20682C2semidirect
5. #9157C2semidirect
6. #9153C2semidirect
7. #8020C2semidirect
11. #2855C4semidirect
13. #2854C4semidirect
20. #2297C4semidirect
21. #2296C4semidirect
68. #391C2×C4semidirect
69. #390C2×C4semidirect
70. #389C2×C4semidirect
71. #389C2×C4semidirect

Maximal subgroups

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

subgroupindex
#366262
C22×#3832
#206822
C2×#15042
#91572
#91532
#80202