Finite Group Databasebeta

← #380038

Subgroups — #380038

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2565C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic
3#21314C212yesmaximal
4C2×#1801C211yesmaximal, characteristic
5#11211C212yesmaximal
6#11210C212yesmaximal
7#11162C212yesmaximal
8#11162C212yesmaximal
9#10852C212yesmaximal
10C22×#125C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#423C2212yes—
13C22×#101C2212yes—
14#2565—24no—
15#2565C418yescomplemented
16C2×#398—22no—
17#2320—24no—
18#2316C2212yes—
19#2313—22no—
20#2310C2212yes—
21#2295—24no—
22C2×#391C2211yescharacteristic
23C2×#391C2211yescharacteristic
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26#2283—22no—
27C2×#382C2211yescharacteristic
28C4×#117C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1989—24no—
31#1989—24no—
32#1801—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—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yes—
49C2×C4×Q8D414yes—
50C2×#117C2314yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42D412yes—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C22×C42C2311yescharacteristic
63#423—24no—
64#423—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×#101—24no—
68C2×#101—44no—
69#398—216no—
70#391—22no—
71#391—22no—
72#391—28no—
73#391C2×C414yes—
74#391C2×C414yes—
75#389—28no—
76#387—216no—
77#387—216no—
78#384—22no—
79#384C2×C414yes—
80#383—24no—
81#382—24no—
82#382—22no—
83#382C2×C414yes—
84#381—24no—
85#380—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4C2×C414yes—
89C4×C4⋊C4D414yes—
90C4×C4⋊C4D414yes—
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93#376—24no—
94#376—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C43—22no—
100#343—24no—
101#340—28no—
102#340—28no—
103#339—44no—
104#339—44no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—24no—
109C22×Q8—22no—
110C22×Q8C22×C412yes—
111C22×Q8C2×D412yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2411yescharacteristic
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C22×C8—24no—
128#125—28no—
129#125—28no—
130#125—44no—
131#125—44no—
132C4×Q8—28no—
133C4×Q8—28no—
134C4×Q8—28no—
135C4×Q8—28no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—42no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4C22⋊C418yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C4⋊C4C2×D418yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—24no—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—24no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42C22⋊C418yes—
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42#4114yes—
189C4⋊C8—44no—
190C4⋊C8—44no—
191#101—48no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—24no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—24no—
200#95—24no—
201C24C22×D411yescharacteristic
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—22no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—42no—
231C22×C4—44no—
232C22×C4—42no—
233C22×C4—44no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4C2×C22⋊C414yes—
240C22×C4C2×C22⋊C414yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C412yes—
243C22×C4#12011yescharacteristic
244C22×C4#12012yes—
245C22×C4#12011yescharacteristic
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4112yes—
252C22×C4H(5,𝔽2)12yes—
253C22×C42−1+414yes—
254C22×C42−1+412yes—
255M16—44no—
256M16—44no—
257M16—44no—
258M16—44no—
259C2×C8—44no—
260C2×C8—44no—
261C2×C8—44no—
262C4⋊C4—24no—
263C4⋊C4—24no—
264C4⋊C4—216no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—48no—
270C4⋊C4—416no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—48no—
276C4⋊C4—88no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—28no—
280C42—28no—
281C42—24no—
282C42—24no—
283C42—48no—
284C42—41no—
285C42—41no—
286C42—41no—
287C42—41no—
288C42—44no—
289C42—44no—
290C42—84no—
291C42C4×D414yes—
292C42C4×D414yes—
293C42#12114yes—
294C42#12114yes—
295C23—22no—
296C23—22no—
297C23C22×C22⋊C411yescharacteristic
298C23C2×#12011yescharacteristic
299C23C2×#12311yescharacteristic
300C23#53612yes—
301C23#53712yes—
302Q8—48no—
303Q8—44no—
304Q8—44no—
305C2×C4—24no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—48no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—44no—
321C2×C4—44no—
322C2×C4—44no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—44no—
326C2×C4—42no—
327C2×C4—44no—
328C2×C4—42no—
329C2×C4—48no—
330C2×C4—48no—
331C2×C4—48no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—84no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4#38612yes—
341C2×C4#38612yes—
342C2×C4#38614yes—
343C2×C4C2×C4×D412yes—
344C2×C4#51912yes—
345C2×C4#52012yes—
346C2×C4C2×#12112yes—
347C2×C4#53511yescharacteristic
348C2×C4#53614yes—
349C2×C4#53612yes—
350C2×C4#53611yescharacteristic
351C2×C4#53714yes—
352C2×C4#54114yes—
353C2×C4#54214yes—
354C8—84no—
355C22—24no—
356C22—24no—
357C22—21no—
358C22—21no—
359C22—21no—
360C22—21no—
361C22C2×#38611yescharacteristic
362C22#231011yescharacteristic
363C22#231111yescharacteristic
364C22#261612yes—
365C22#262512yes—
366C22#301812yes—
367C22#302012yes—
368C4—24no—
369C4—22no—
370C4—22no—
371C4—44no—
372C4—44no—
373C4—44no—
374C4—82no—
375C4—88no—
376C4—82no—
377C4—82no—
378C4#232314yes—
379C4#262012yes—
380C4#262112yes—
381C2—21no—
382C2—21no—
383C2—22no—
384C2#1425811yescharacteristic
385C2#2095211yescharacteristic
386C2#2095411yescharacteristic
387C2#2970814yes—
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
15. #2565C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25652
#340592
#213142
C2×#18012
#112112
#112102
#111622
#108522