Finite Group Databasebeta

← #375691

Subgroups — #375691

Order 512.

1219 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35530C211yesmaximal, characteristic
2#20397C211yesmaximal, characteristic, complemented
3#20393C211yesmaximal, characteristic, complemented
4#11148C212yesmaximal, complemented
5#10986C212yesmaximal
6#10959C212yesmaximal, complemented
7#10949C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10889C212yesmaximal, complemented
10C2×#537C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#421C2214yes—
13C2×#413C2212yes—
14C2×#410C2212yes—
15#2873C2212yes—
16#2502C412yescomplemented
17#2501C412yescomplemented
18#2500C412yescomplemented
19#2499C412yescomplemented
20#2443—24no—
21#2423—22no—
22#2422—22no—
23#2317C2211yescharacteristic
24#2311C2211yescharacteristic
25#2297C2211yescharacteristic
26#2297C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#2274C2212yes—
29C4×#101—22no—
30C4×#100—22no—
31C4×#99—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1736C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×#41C2311yescharacteristic
39C22×M16C2314yes—
40#537—44no—
41C2×#123—24no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8C2×C412yes—
44C2×#121—21no—
45C2×#121—22no—
46C2×#121C2×C412yes—
47#520—44no—
48C2×C4×Q8D412yes—
49C2×C4×D4D414yes—
50C2×C4×D4D412yes—
51C2×#117C2×C412yes—
52C2×#117C2314yes—
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C412yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42C2311yescharacteristic
60#421—28no—
61#421—28no—
62C2×#104—44no—
63#413—28no—
64C2×#101—22no—
65C2×#100—22no—
66#410—28no—
67C2×#99—24no—
68#405—44no—
69C4×M16—24no—
70#399—24no—
71#398—24no—
72#397—24no—
73#394—24no—
74#391—21no—
75#391C2×C412yes—
76#391C2×C411yescharacteristic
77#391C2×C411yescharacteristic
78#390—21no—
79#390C2×C412yes—
80#390C2×C411yescharacteristic
81#390C2×C411yescharacteristic
82#389—22no—
83#388—24no—
84#387—24no—
85#386—22no—
86#386—22no—
87#386—22no—
88#385—24no—
89#385—22no—
90#384—22no—
91#384C2×C412yes—
92#384C2×C412yes—
93#384C2×C412yes—
94#382C2×C412yes—
95#380—22no—
96C4×C4⋊C4—44no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—24no—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4C2×C412yes—
101#343—24no—
102#342—24no—
103#340—44no—
104#339—28no—
105H(3,ℤ/4)—28no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—24no—
109C2×#41—24no—
110C2×#41—24no—
111C2×#41—24no—
112C22×Q8C22×C411yescharacteristic
113C22×Q8C2×D411yescharacteristic
114C22×D4—21no—
115C22×D4—22no—
116C22×D4C22×C411yescharacteristic
117C22×D4C2×D412yes—
118C22×D4C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22⋊C414yes—
123C23×C4C22×C412yes—
124C23×C4C2×D412yes—
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16#4118yes—
132C22×C8—24no—
133#123—88no—
134C22⋊Q8—44no—
135C22⋊Q8—44no—
136#121—44no—
137#121—48no—
138#121—44no—
139C4×Q8—44no—
140C4×D4—44no—
141C4×D4—48no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—44no—
148#117#4118yes—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—84no—
160C2×C4⋊C4C22⋊C414yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C411yescharacteristic
165C2×C4⋊C4C2×D412yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C22⋊C414yes—
176C2×C22⋊C4C2×D412yes—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—21no—
180C2×C42—44no—
181C2×C42C22⋊C414yes—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D412yes—
186#104—416no—
187#101—28no—
188#100—28no—
189#99—216no—
190#97—44no—
191C4×C8—44no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—22no—
196#95—24no—
197#95—24no—
198#95—44no—
199C24—21no—
200C24C2×C22⋊C412yes—
201C24C22×D411yescharacteristic
202#41—44no—
203#41—44no—
204#41—44no—
205#41—44no—
206#41—44no—
207#41—44no—
208#41—44no—
209#41—44no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×D4—24no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—44no—
229C2×D4—84no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C414yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C2×#4114yes—
258C22×C4C2×#4114yes—
259C22×C4H(5,𝔽2)14yes—
260M16—28no—
261M16—28no—
262M16—44no—
263M16—44no—
264C2×C8—28no—
265C2×C8—44no—
266C2×C8—44no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—28no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—44no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—44no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C22⋊C4—24no—
282C22⋊C4—28no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—42no—
291C22⋊C4—44no—
292C22⋊C4—84no—
293C42—24no—
294C42—22no—
295C42—22no—
296C42—24no—
297C42—28no—
298C42—44no—
299C42—42no—
300C42—44no—
301C42—164no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—41no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23C22×C22⋊C411yescharacteristic
315C23#52512yes—
316C23#53212yes—
317C23#53412yes—
318Q8—44no—
319Q8—44no—
320D4—48no—
321D4—44no—
322D4—44no—
323D4—84no—
324D4—164no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—48no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—48no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—42no—
346C2×C4—41no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—84no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—84no—
355C2×C4—164no—
356C2×C4—162no—
357C2×C4C4×#4112yes—
358C2×C4C4×#4112yes—
359C2×C4#51812yes—
360C2×C4#52512yes—
361C2×C4#53212yes—
362C2×C4#53412yes—
363C2×C4#55114yes—
364C2×C4#55314yes—
365C2×C4#55314yes—
366C2×C4#55914yes—
367C8—48no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—22no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—82no—
380C22—82no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22#229711yescharacteristic
385C22#229711yescharacteristic
386C22#230811yescharacteristic
387C22#260214yes—
388C4—42no—
389C4—42no—
390C4—44no—
391C4—42no—
392C4—42no—
393C4—42no—
394C4—82no—
395C4—88no—
396C4—82no—
397C4—81no—
398C4#237414yes—
399C4#260214yes—
400C2—21no—
401C2—21no—
402C2—24no—
403C2—82no—
404C2—161no—
405C2#1416711yescharacteristic
406C2#2178211yescharacteristic
407C2#2178311yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#537C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20397C2semidirect
3. #20393C2semidirect
4. #11148C2semidirect
6. #10959C2semidirect
7. #10949C2semidirect
8. #10891C2semidirect
9. #10889C2semidirect
16. #2502C4semidirect
17. #2501C4semidirect
18. #2500C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#355302
#203972
#203932
#111482
#109862
#109592
#109492
#108912
#108892