Finite Group Databasebeta

← #217049

Subgroups — #217049

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35284C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic
3#20755C211yesmaximal, characteristic
4#9907C212yesmaximal, complemented
5#9705C212yesmaximal, complemented
6#9137C212yesmaximal
7#9129C212yesmaximal
8#8593C212yesmaximal, complemented
9#8589C212yesmaximal, complemented
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2647—21no—
16#2629—21no—
17#2507C414yes—
18#2497C414yes—
19#2446—22no—
20#2429—22no—
21#2330—21no—
22#2314—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—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8C2312yes—
49C2×#125—21no—
50C2×C22⋊Q8—22no—
51C2×C4×Q8—22no—
52C2×#117D411yescharacteristic
53C2×#117D411yescharacteristic
54C22×C42C2311yescharacteristic
55#423—41no—
56#423—41no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8—41no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C22⋊C8—22no—
67C2×#97—22no—
68#400—24no—
69#398—24no—
70#398—24no—
71#396—24no—
72#391—22no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—24no—
78#389C2×C414yes—
79#389C2×C414yes—
80#389C2×C414yes—
81#388—24no—
82#387—24no—
83#387—24no—
84#384—21no—
85#384—21no—
86#384C2×C414yes—
87#384C2312yes—
88#382—21no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C414yes—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4D412yes—
95C4×C22⋊C4C2×C412yes—
96C4×C22⋊C4D412yes—
97#376D411yescharacteristic
98#376D411yescharacteristic
99C43C2311yescharacteristic
100C43C2311yescharacteristic
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#332—42no—
109#332—42no—
110#332—42no—
111#332—42no—
112C22×Q8—22no—
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—42no—
124C22×C8—42no—
125C22×C8—42no—
126C4⋊Q8—22no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129C4⋊Q8—22no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132#125—42no—
133#125—42no—
134C22⋊Q8—82no—
135C22⋊Q8—82no—
136C4×Q8—82no—
137C4×Q8—82no—
138#117—22no—
139#117—22no—
140#117—41no—
141#117—41no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C412yes—
152C2×C4⋊C4C22×C414yes—
153C2×C4⋊C4C22×C414yes—
154C2×C22⋊C4—24no—
155C2×C22⋊C4C22⋊C412yes—
156C2×C22⋊C4C22⋊C412yes—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—24no—
162C2×C42C22×C411yescharacteristic
163C2×C42C22×C412yes—
164C2×C42C22×C412yes—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2411yescharacteristic
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—42no—
183C4⋊C8—42no—
184C4⋊C8—82no—
185C22⋊C8—82no—
186#97—82no—
187#95—24no—
188#95—24no—
189#95—24no—
190#95—24no—
191#95—24no—
192#95C22⋊C412yes—
193#95C22⋊C412yes—
194#95C22⋊C412yes—
195#95C22⋊C412yes—
196C24C2×C22⋊C411yescharacteristic
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—42no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4C2×C22⋊C411yescharacteristic
217C22×C4C2×C22⋊C412yes—
218C22×C4C2×C22⋊C412yes—
219C22×C4C2×C22⋊C412yes—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4#13712yes—
224C22×C4#13712yes—
225C22×C4C23×C411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D411yescharacteristic
228M16—84no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—84no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—42no—
246C4⋊C4—44no—
247C4⋊C4—48no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—84no—
255C4⋊C4—84no—
256C4⋊C4—84no—
257C4⋊C4—84no—
258C22⋊C4—24no—
259C22⋊C4—42no—
260C22⋊C4—84no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—42no—
280C42—42no—
281C42—84no—
282C42H(5,𝔽2)11yescharacteristic
283C42H(5,𝔽2)11yescharacteristic
284C422−1+411yescharacteristic
285C422−1+411yescharacteristic
286C23—21no—
287C23—22no—
288C23—21no—
289C23—22no—
290C23#41912yes—
291C23C22×C22⋊C411yescharacteristic
292Q8—82no—
293Q8—82no—
294Q8—82no—
295Q8—82no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
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—42no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4#41912yes—
331C2×C4#52011yescharacteristic
332C2×C4#52011yescharacteristic
333C2×C4#53511yescharacteristic
334C2×C4#53611yescharacteristic
335C2×C4#53711yescharacteristic
336C2×C4#53711yescharacteristic
337C2×C4C2×#13711yescharacteristic
338C2×C4C2×#13711yescharacteristic
339C2×C4#57511yescharacteristic
340C2×C4#57511yescharacteristic
341C2×C4#57612yes—
342C2×C4#57712yes—
343C8—82no—
344C8—82no—
345C8—82no—
346C8—82no—
347C22—24no—
348C22—22no—
349C22—42no—
350C22—41no—
351C22—41no—
352C22#231111yescharacteristic
353C22#288912yes—
354C22C2×#41911yescharacteristic
355C22#289611yescharacteristic
356C22#289712yes—
357C4—22no—
358C4—21no—
359C4—21no—
360C4—42no—
361C4—42no—
362C4—42no—
363C4—42no—
364C4—41no—
365C4—41no—
366C4—84no—
367C4—84no—
368C4#321511yescharacteristic
369C4#321511yescharacteristic
370C4#323611yescharacteristic
371C4#323611yescharacteristic
372C4#325712yes—
373C4#325911yescharacteristic
374C4#326011yescharacteristic
375C2—42no—
376C2#1776112yes—
377C2#1776912yes—
378C2#2186611yescharacteristic
379C2#2188311yescharacteristic
380C2#2189711yescharacteristic
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. #9705C2semidirect
8. #8593C2semidirect
9. #8589C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#219262
#207552
#99072
#97052
#91372
#91292
#85932
#85892