Finite Group Databasebeta

← #217048

Subgroups — #217048

Order 512.

750 conjugacy classes of subgroups.

The subgroup lattice has 375 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#9702C212yesmaximal, complemented
6#9135C212yesmaximal
7#9127C212yesmaximal
8#8594C212yesmaximal, complemented
9#8590C212yesmaximal, complemented
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12#2868C2211yescharacteristic
13#2859C2211yescharacteristic
14#2857C2212yes—
15#2647—21no—
16#2629—21no—
17#2507—22no—
18#2497—22no—
19#2446C414yes—
20#2429C414yes—
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×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2212yes—
40C2×#333C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×M16—21no—
46C23×C8—21no—
47C2×C4⋊Q8C2×C414yes—
48C2×C4⋊Q8C2312yes—
49C2×#125—21no—
50C2×C22⋊Q8C2×C414yes—
51C2×C4×Q8C2×C414yes—
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#391C2×C414yes—
73#391C2×C414yes—
74#389—22no—
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#389—24no—
80#389C2×C414yes—
81#388—24no—
82#387—24no—
83#387—24no—
84#384—21no—
85#384—21no—
86#384—22no—
87#384C2312yes—
88#382—21no—
89#380—22no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
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#333—42no—
109#333—42no—
110#333—42no—
111#333—42no—
112C22×Q8C22×C414yes—
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—84no—
131#125—42no—
132#125—42no—
133C22⋊Q8—84no—
134C4×Q8—84no—
135#117—22no—
136#117—22no—
137#117—41no—
138#117—41no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4C22⋊C412yes—
149C2×C4⋊C4C22⋊C412yes—
150C2×C4⋊C4C22×C414yes—
151C2×C22⋊C4—24no—
152C2×C22⋊C4C22⋊C412yes—
153C2×C22⋊C4C22⋊C412yes—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—24no—
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C412yes—
161C2×C42C22×C412yes—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D411yescharacteristic
170C2×C42C2411yescharacteristic
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—82no—
182C22⋊C8—82no—
183#97—82no—
184#95—24no—
185#95—24no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95C22⋊C412yes—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192#95C22⋊C412yes—
193C24C2×C22⋊C411yescharacteristic
194C2×Q8—24no—
195C2×Q8—24no—
196C2×Q8—44no—
197C2×Q8—44no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C2×C22⋊C412yes—
213C22×C4C2×C22⋊C412yes—
214C22×C4C2×C22⋊C412yes—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C22⋊C412yes—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4#13712yes—
219C22×C4#13712yes—
220C22×C4C23×C411yescharacteristic
221C22×C4C22×D411yescharacteristic
222C22×C4C22×D411yescharacteristic
223M16—84no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—84no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—48no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—84no—
250C4⋊C4—84no—
251C4⋊C4—84no—
252C4⋊C4—84no—
253C22⋊C4—24no—
254C22⋊C4—42no—
255C22⋊C4—84no—
256C42—21no—
257C42—21no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—42no—
275C42—42no—
276C42—84no—
277C42H(5,𝔽2)11yescharacteristic
278C42H(5,𝔽2)11yescharacteristic
279C422−1+411yescharacteristic
280C422−1+411yescharacteristic
281C23—21no—
282C23—22no—
283C23—21no—
284C23—22no—
285C23#41912yes—
286C23C22×C22⋊C411yescharacteristic
287Q8—84no—
288Q8—84no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—42no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—44no—
316C2×C4—44no—
317C2×C4—44no—
318C2×C4—44no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4#41912yes—
324C2×C4#52011yescharacteristic
325C2×C4#52011yescharacteristic
326C2×C4#53511yescharacteristic
327C2×C4#53611yescharacteristic
328C2×C4#53711yescharacteristic
329C2×C4#53711yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C2×C4C2×#13711yescharacteristic
332C2×C4#57511yescharacteristic
333C2×C4#57511yescharacteristic
334C2×C4#57712yes—
335C2×C4#57812yes—
336C8—82no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—24no—
341C22—22no—
342C22—42no—
343C22—41no—
344C22—41no—
345C22#231111yescharacteristic
346C22#288812yes—
347C22C2×#41911yescharacteristic
348C22#289611yescharacteristic
349C22#289812yes—
350C4—22no—
351C4—21no—
352C4—21no—
353C4—42no—
354C4—42no—
355C4—42no—
356C4—42no—
357C4—41no—
358C4—41no—
359C4—84no—
360C4—84no—
361C4#321511yescharacteristic
362C4#321511yescharacteristic
363C4#323611yescharacteristic
364C4#323611yescharacteristic
365C4#325812yes—
366C4#326011yescharacteristic
367C4#326111yescharacteristic
368C2—42no—
369C2#1776012yes—
370C2#1776812yes—
371C2#2186611yescharacteristic
372C2#2188211yescharacteristic
373C2#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. #9702C2semidirect
8. #8594C2semidirect
9. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#352842
#219262
#207552
#99072
#97022
#91352
#91272
#85942
#85902