Finite Group Databasebeta

← #381136

Subgroups — #381136

Order 512.

1117 conjugacy classes of subgroups.

The subgroup lattice has 401 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35619C211yesmaximal, characteristic
2#20640C211yesmaximal, characteristic
3#20393C211yesmaximal, characteristic
4#11212C212yesmaximal, complemented
5#11148C212yesmaximal
6#11074C212yesmaximal, complemented
7#10985C212yesmaximal
8#10957C212yesmaximal, complemented
9#10888C212yesmaximal, complemented
10C2×#536C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#419C2214yes—
14C2×#413C2212yes—
15C2×#410C2212yes—
16#2503C412yescomplemented
17#2502C412yescomplemented
18#2499C412yescomplemented
19#2499C412yescomplemented
20#2447—22no—
21#2446—22no—
22#2419—24no—
23#2317C2211yescharacteristic
24#2310C2211yescharacteristic
25#2297C2211yescharacteristic
26C2×#391C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274C2212yes—
29#1801—22no—
30#1800—22no—
31C4×#99—24no—
32C2×#343C2212yes—
33#1740C2212yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40#536—44no—
41C2×C4⋊Q8—21no—
42C2×#124—21no—
43C2×#123—22no—
44C2×C22⋊Q8C2×C412yes—
45C2×C22⋊Q8C2×C412yes—
46#520—44no—
47C2×C4×Q8D412yes—
48C2×C4×D4D414yes—
49C2×C4×D4D412yes—
50C2×#117C2×C412yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C22×C4⋊C4C2×C412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#424—44no—
59#423—24no—
60#419—28no—
61#419—44no—
62C2×#103—44no—
63#413—44no—
64C2×#101—22no—
65C2×#100—22no—
66#410—28no—
67C2×#99—24no—
68#400—24no—
69#398—24no—
70#397—24no—
71#397—24no—
72#391—21no—
73#391—22no—
74#391—22no—
75#391C2×C412yes—
76#391C2×C412yes—
77#391C2×C412yes—
78#391C2×C412yes—
79#390—21no—
80#389—22no—
81#387—24no—
82#387—24no—
83#386—22no—
84#385—24no—
85#385—22no—
86#384C2×C412yes—
87#384C2×C411yescharacteristic
88#384C2×C411yescharacteristic
89#382C2×C412yes—
90#382C2×C411yescharacteristic
91#382C2×C411yescharacteristic
92#381—24no—
93#381—22no—
94#380—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—44no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4C2×C412yes—
100C2×#95—22no—
101#343—24no—
102#342—24no—
103#339—44no—
104#339—44no—
105H(3,ℤ/4)—28no—
106C2×#41—24no—
107C2×#41—24no—
108C2×#41—22no—
109C2×#41—22no—
110C22×Q8—21no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8C22×C412yes—
116C22×Q8C2×D411yescharacteristic
117C22×D4C2×D412yes—
118C22×D4C2×D411yescharacteristic
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C22⋊C414yes—
122C23×C4C22×C412yes—
123C23×C4C2×D412yes—
124C23×C4C2411yescharacteristic
125C2×M16—24no—
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C22×C8—24no—
132C4⋊Q8—44no—
133#124—44no—
134#123—48no—
135C22⋊Q8—44no—
136C22⋊Q8—44no—
137C4×Q8—44no—
138C4×D4—44no—
139C4×D4—48no—
140#117—24no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—44no—
147#117#4118yes—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—44no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C411yescharacteristic
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C411yescharacteristic
163C2×C4⋊C4C2×D412yes—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4C22⋊C414yes—
172C2×C22⋊C4C2×D412yes—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—44no—
179C2×C42C22⋊C414yes—
180C2×C42C22×C412yes—
181C2×C42C22×C411yescharacteristic
182C2×C42C22×C411yescharacteristic
183C2×C42C2×D412yes—
184C4⋊C8—48no—
185#103—416no—
186#101—44no—
187#100—44no—
188#99—216no—
189#95—24no—
190#95—24no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—44no—
199C24C2×C22⋊C412yes—
200C24C22×D411yescharacteristic
201#41—44no—
202#41—44no—
203#41—44no—
204#41—44no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—21no—
208C2×Q8—21no—
209C2×Q8—24no—
210C2×Q8—24no—
211C2×Q8—24no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—44no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×D4—24no—
221C2×D4—24no—
222C2×D4—24no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—21no—
226C2×D4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—44no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4—82no—
246C22×C4—84no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C2×#4114yes—
254C22×C4H(5,𝔽2)14yes—
255C22×C4H(5,𝔽2)12yes—
256C22×C42−1+412yes—
257M16—44no—
258M16—44no—
259M16—44no—
260M16—44no—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C4⋊C4—24no—
265C4⋊C4—22no—
266C4⋊C4—22no—
267C4⋊C4—24no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—28no—
271C4⋊C4—44no—
272C4⋊C4—44no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—44no—
277C4⋊C4—82no—
278C4⋊C4—84no—
279C22⋊C4—24no—
280C22⋊C4—28no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—82no—
289C42—24no—
290C42—24no—
291C42—22no—
292C42—22no—
293C42—28no—
294C42—44no—
295C42—42no—
296C42—41no—
297C42—41no—
298C42—44no—
299C42—164no—
300C23—22no—
301C23—22no—
302C23—22no—
303C23—41no—
304C23—41no—
305C23—42no—
306C23—42no—
307C23C22×C22⋊C411yescharacteristic
308C23#52512yes—
309C23#53412yes—
310C23#53712yes—
311Q8—42no—
312Q8—42no—
313Q8—48no—
314Q8—82no—
315Q8—82no—
316D4—48no—
317D4—42no—
318D4—42no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—24no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—48no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4—84no—
350C2×C4—84no—
351C2×C4—164no—
352C2×C4C4×#4112yes—
353C2×C4#51812yes—
354C2×C4#52012yes—
355C2×C4#53212yes—
356C2×C4#53412yes—
357C2×C4#53511yescharacteristic
358C2×C4#53611yescharacteristic
359C2×C4#53814yes—
360C2×C4#53914yes—
361C2×C4#54014yes—
362C2×C4#54214yes—
363C8—84no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—42no—
369C22—42no—
370C22—42no—
371C22—42no—
372C22—42no—
373C22—82no—
374C22—82no—
375C22#229711yescharacteristic
376C22#230811yescharacteristic
377C22#231111yescharacteristic
378C22#263014yes—
379C4—42no—
380C4—42no—
381C4—44no—
382C4—42no—
383C4—42no—
384C4—82no—
385C4—81no—
386C4—88no—
387C4—81no—
388C4—81no—
389C4—82no—
390C4#233514yes—
391C4#266112yes—
392C4#266312yes—
393C2—21no—
394C2—21no—
395C2—24no—
396C2—82no—
397C2#1428711yescharacteristic
398C2#2178311yescharacteristic
399C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
4. #11212C2semidirect
6. #11074C2semidirect
8. #10957C2semidirect
9. #10888C2semidirect
16. #2503C4semidirect
17. #2502C4semidirect
18. #2499C4semidirect
19. #2499C4semidirect

Maximal subgroups

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

subgroupindex
#356192
#206402
#203932
#112122
#111482
#110742
#109852
#109572
#108882