Finite Group Databasebeta

← #217046

Subgroups — #217046

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9703C212yesmaximal, complemented
6#9139C212yesmaximal
7#9131C212yesmaximal
8#8593C212yesmaximal, complemented
9#8591C212yesmaximal, complemented
10C2×#541—21no—
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2868C2211yescharacteristic
14#2859C2211yescharacteristic
15#2857C2212yes—
16#2642—21no—
17#2621—21no—
18#2506—22no—
19#2506C414yes—
20#2496—22no—
21#2496C414yes—
22#2327—21no—
23C4×C4⋊Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1842C2211yescharacteristic
28#1832—21no—
29#1832—21no—
30#1831C2212yes—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1767C2212yes—
35#1729—21no—
36#1729—21no—
37C2×#332C2212yes—
38C2×#332C2212yes—
39C2×#332C2212yes—
40C2×#332C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×M16—21no—
46C23×C8—21no—
47#541—44no—
48#541—44no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2312yes—
52C2×#126—24no—
53C2×#125—24no—
54C2×#125—21no—
55C2×C22⋊Q8—22no—
56C2×C4×Q8—22no—
57C2×#117D411yescharacteristic
58C2×#117D411yescharacteristic
59C22×C42C2311yescharacteristic
60#423—41no—
61#423—41no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—41no—
66C2×C4⋊C8—41no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—22no—
72C2×#97—22no—
73#400—24no—
74#400—24no—
75#398—24no—
76#397—24no—
77#391—22no—
78#391—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389C2×C414yes—
83#389C2×C414yes—
84#389C2×C414yes—
85#387—24no—
86#387—24no—
87#384—22no—
88#384C2×C414yes—
89#384C2312yes—
90#382—21no—
91#380—22no—
92#380C2×C414yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4D412yes—
97C4×C22⋊C4C2×C412yes—
98C4×C22⋊C4D412yes—
99#376D411yescharacteristic
100#376D411yescharacteristic
101C43C2311yescharacteristic
102C43C2311yescharacteristic
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#332—42no—
111#332—42no—
112#332—42no—
113#332—42no—
114C22×Q8—22no—
115C23×C4C22×C411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—42no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—42no—
126C22×C8—42no—
127C22×C8—42no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—42no—
133C4⋊Q8—42no—
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136#126—416no—
137#125—42no—
138#125—42no—
139#125—416no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142C4×Q8—44no—
143C4×Q8—44no—
144#117—22no—
145#117—22no—
146#117—41no—
147#117—41no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22×C414yes—
159C2×C4⋊C4C22×C414yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—24no—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D411yescharacteristic
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D411yescharacteristic
179C2×C42C2411yescharacteristic
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—42no—
189C4⋊C8—42no—
190C4⋊C8—82no—
191C22⋊C8—82no—
192#97—82no—
193#95—24no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202C24C2×C22⋊C411yescharacteristic
203C2×Q8—42no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—42no—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C2×C22⋊C412yes—
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4AGL(1,ℤ/8)12yes—
230C22×C4AGL(1,ℤ/8)12yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C22×D411yescharacteristic
234M16—84no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—84no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C4⋊C4—48no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—48no—
264C22⋊C4—24no—
265C22⋊C4—42no—
266C22⋊C4—48no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—42no—
286C42—42no—
287C42—48no—
288C42H(5,𝔽2)11yescharacteristic
289C42H(5,𝔽2)11yescharacteristic
290C422−1+411yescharacteristic
291C422−1+411yescharacteristic
292C23—21no—
293C23—22no—
294C23—21no—
295C23—22no—
296C23#41812yes—
297C23C22×C22⋊C411yescharacteristic
298Q8—82no—
299Q8—82no—
300Q8—82no—
301Q8—82no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—44no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4#41812yes—
337C2×C4#52011yescharacteristic
338C2×C4#52011yescharacteristic
339C2×C4#53511yescharacteristic
340C2×C4#53611yescharacteristic
341C2×C4#53711yescharacteristic
342C2×C4#53711yescharacteristic
343C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
344C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C4#57511yescharacteristic
346C2×C4#57511yescharacteristic
347C2×C4#57712yes—
348C2×C4#57812yes—
349C8—82no—
350C8—82no—
351C8—82no—
352C8—82no—
353C22—24no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22#231111yescharacteristic
359C22#288912yes—
360C22C2×#41811yescharacteristic
361C22#289611yescharacteristic
362C22#289812yes—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—41no—
371C4—41no—
372C4—84no—
373C4—84no—
374C4#321311yescharacteristic
375C4#321311yescharacteristic
376C4#323411yescharacteristic
377C4#323411yescharacteristic
378C4#325812yes—
379C4#326011yescharacteristic
380C4#326111yescharacteristic
381C2—42no—
382C2#1775912yes—
383C2#1776712yes—
384C2#2186511yescharacteristic
385C2#2188311yescharacteristic
386C2#2189811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9907C2semidirect
5. #9703C2semidirect
8. #8593C2semidirect
9. #8591C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#219262
#207552
#99072
#97032
#91392
#91312
#85932
#85912