Finite Group Databasebeta

← #374559

Subgroups — #374559

Order 512.

1486 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic
3#21325C212yesmaximal
4C2×C4×#101C211yesmaximal, characteristic
5#11211C212yesmaximal
6#11210C212yesmaximal
7#10851C212yesmaximal
8#10750C212yesmaximal
9#10750C212yesmaximal
10C22×#125C2211yescharacteristic
11C2×#519C2212yes—
12C22×C4×Q8C2211yescharacteristic
13C22×#101C2212yes—
14C2×C4×M16C2212yes—
15#2565—24no—
16#2565C418yescomplemented
17C2×#398—22no—
18#2351—24no—
19Q8×C4⋊C4—24no—
20#2316—22no—
21#2316C2212yes—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#387—22no—
25#2285—22no—
26#2285C2212yes—
27C2×#384C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1992—24no—
31#1991—24no—
32C4×#101—24no—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C22×M16C2314yes—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—24no—
44C2×#125C2×C414yes—
45#519—24no—
46#519—24no—
47#519—44no—
48C2×C4×Q8—28no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—24no—
52C2×C4×Q8C2×C414yes—
53C2×C4×Q8D414yes—
54C2×#117—24no—
55C2×#117C2314yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4D412yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42D412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#101—24no—
69C2×#101—24no—
70C2×#101Q818yes—
71C4×M16—24no—
72C4×M16—24no—
73C2×#97—22no—
74C2×C4×C8—22no—
75#398—216no—
76#391—22no—
77#391—22no—
78#391C2×C414yes—
79#391C2×C414yes—
80#389—216no—
81#389—28no—
82#387—216no—
83#384—24no—
84#384—22no—
85#384C2×C414yes—
86#384D414yes—
87#382—24no—
88#382—22no—
89#382C2×C414yes—
90#382D414yes—
91#380—22no—
92#380—24no—
93#380—22no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C414yes—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104#343—24no—
105#340—28no—
106#340—28no—
107#339—44no—
108#339—44no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8C22×C412yes—
114C22×Q8C2×D412yes—
115C22×Q8C2×Q814yes—
116C22×Q8#4114yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16C2×Q818yes—
132C22×C8—24no—
133#125—28no—
134#125—28no—
135#125—44no—
136#125—44no—
137C4×Q8—28no—
138C4×Q8—28no—
139C4×Q8—28no—
140C4×Q8—28no—
141C4×Q8—416no—
142C4×Q8—416no—
143#117—24no—
144#117—24no—
145#117—24no—
146#117—24no—
147#117—48no—
148#117—48no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—41no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C418yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C4⋊C4C2×D418yes—
179C2×C4⋊C4C2×D412yes—
180C2×C22⋊C4—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—28no—
189C2×C42—24no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C22⋊C418yes—
193C2×C42C22×C412yes—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42#4114yes—
200#101—216no—
201#97—44no—
202C4×C8—44no—
203#95—28no—
204#95—24no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—24no—
209#95—24no—
210C24C22×D411yescharacteristic
211C2×Q8—24no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—28no—
216C2×Q8—28no—
217C2×Q8—28no—
218C2×Q8—28no—
219C2×Q8C4×Q818yes—
220C2×Q8C22⋊Q818yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—28no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—44no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C414yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4#12011yescharacteristic
254C22×C4#12012yes—
255C22×C4#12011yescharacteristic
256C22×C4#12312yes—
257C22×C4#12312yes—
258C22×C4#12414yes—
259C22×C4C4⋊Q814yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×Q812yes—
264C22×C4C2×#4112yes—
265C22×C4C2×#4112yes—
266C22×C4H(5,𝔽2)12yes—
267C22×C42−1+412yes—
268M16—28no—
269M16—28no—
270M16—44no—
271M16—44no—
272C2×C8—28no—
273C2×C8—44no—
274C2×C8—44no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—216no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—28no—
281C4⋊C4—28no—
282C4⋊C4—48no—
283C4⋊C4—48no—
284C4⋊C4—416no—
285C4⋊C4—416no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—84no—
292C4⋊C4—88no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C42—28no—
296C42—28no—
297C42—24no—
298C42—24no—
299C42—416no—
300C42—42no—
301C42—42no—
302C42—44no—
303C42—44no—
304C42C4×D414yes—
305C42C4×D414yes—
306C42#12114yes—
307C42#12114yes—
308C23—22no—
309C23—22no—
310C23C22×C22⋊C411yescharacteristic
311C23C2×#12011yescharacteristic
312C23C2×#12311yescharacteristic
313C23C2×#12411yescharacteristic
314C23C2×C4⋊Q811yescharacteristic
315C23#53511yescharacteristic
316C23#53611yescharacteristic
317Q8—216no—
318Q8—216no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—44no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—48no—
347C2×C4—48no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#38612yes—
358C2×C4#38612yes—
359C2×C4#38614yes—
360C2×C4#39118yes—
361C2×C4C2×C4×D412yes—
362C2×C4C2×C4×Q812yes—
363C2×C4#52012yes—
364C2×C4C2×#12112yes—
365C2×C4C2×C22⋊Q812yes—
366C2×C4#53712yes—
367C2×C4#54614yes—
368C2×C4D4×Q814yes—
369C2×C4#55214yes—
370C2×C4#55414yes—
371C8—48no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22C2×#38611yescharacteristic
381C22C2×#39111yescharacteristic
382C22#231111yescharacteristic
383C22#243411yescharacteristic
384C22#243511yescharacteristic
385C22#250211yescharacteristic
386C22#250311yescharacteristic
387C22#301812yes—
388C22#302012yes—
389C4—28no—
390C4—24no—
391C4—22no—
392C4—22no—
393C4—44no—
394C4—44no—
395C4—82no—
396C4—88no—
397C4—82no—
398C4—82no—
399C4#236014yes—
400C4#241714yes—
401C2—21no—
402C2—21no—
403C2—21no—
404C2—21no—
405C2#1413911yescharacteristic
406C2#2095211yescharacteristic
407C2#2095411yescharacteristic
408C2#2582712yes—
409C2#2587312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
16. #2565C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
#341822
#213252
C2×C4×#1012
#112112
#112102
#108512
#107502