Finite Group Databasebeta

← #380035

Subgroups — #380035

Order 512.

1490 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2566C211yesmaximal, characteristic
2#34058C211yesmaximal, characteristic
3#21314C212yesmaximal, complemented
4C2×#1800C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11210C212yesmaximal, complemented
7#11161C212yesmaximal
8#11161C212yesmaximal
9#10852C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#423C2212yes—
13C22×#100C2212yes—
14#2566—24no—
15#2566C418yescomplemented
16C2×#396—22no—
17#2319—24no—
18#2316C2212yes—
19#2312—22no—
20#2309C2212yes—
21#2293—24no—
22C2×#390C2211yescharacteristic
23C2×#390C2211yescharacteristic
24C2×#385—22no—
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#2280—22no—
28C4×#117C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30#1989—24no—
31#1989—24no—
32#1800—24no—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×D4C2311yescharacteristic
39C24×C4D412yes—
40C22×M16C2314yes—
41C2×#125—22no—
42C2×#125—22no—
43C2×#125—22no—
44C2×#125—24no—
45C2×#125C2×C414yes—
46C2×C4×D4—22no—
47C2×C4×D4—22no—
48C2×C4×D4—24no—
49C2×C4×D4C2×C414yes—
50C2×C4×D4D414yes—
51C2×#117—24no—
52C2×#117C2314yes—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#423—24no—
67#423—24no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×#100—24no—
71C2×#100—44no—
72#396—216no—
73#390—22no—
74#390—22no—
75#390—28no—
76#390C2×C414yes—
77#390C2×C414yes—
78#389—28no—
79#388—216no—
80#385—216no—
81#384—22no—
82#384C2×C414yes—
83#382—22no—
84#382C2×C414yes—
85#381—24no—
86#380—24no—
87#380—24no—
88#379—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2×C414yes—
91C4×C4⋊C4D414yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—24no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96#376—22no—
97C2×#95—22no—
98C2×#95—21no—
99C43—22no—
100#343—24no—
101#340—28no—
102#340—28no—
103#339—44no—
104#339—44no—
105C25C2×D412yes—
106C22×D4—22no—
107C22×D4—22no—
108C22×D4—22no—
109C22×D4—24no—
110C22×D4—22no—
111C22×D4C22×C412yes—
112C22×D4C2×D412yes—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—84no—
121C23×C4C22⋊C418yes—
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×M16—24no—
132C22×C8—24no—
133#125—28no—
134#125—28no—
135#125—44no—
136#125—44no—
137C4×D4—28no—
138C4×D4—28no—
139C4×D4—28no—
140C4×D4—28no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—24no—
145#117—88no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—84no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C2×D418yes—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—28no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—84no—
180C2×C22⋊C4C22⋊C418yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—24no—
190C2×C42C22×C412yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42#4114yes—
197C4⋊C8—44no—
198C4⋊C8—44no—
199#100—48no—
200#95—28no—
201#95—24no—
202#95—28no—
203#95—24no—
204#95—24no—
205C24—22no—
206C24—24no—
207C24—22no—
208C24—24no—
209C24—44no—
210C24—44no—
211C24C2×C22⋊C414yes—
212C24C22×D411yescharacteristic
213C2×D4—24no—
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—24no—
217C2×D4—22no—
218C2×D4—24no—
219C2×D4—22no—
220C2×D4—44no—
221C2×D4—44no—
222C2×D4—44no—
223C2×D4—44no—
224C22×C4—24no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—28no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—48no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—44no—
240C22×C4—44no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—84no—
250C22×C4—88no—
251C22×C4—88no—
252C22×C4C2×C22⋊C414yes—
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4#12011yescharacteristic
256C22×C4#12012yes—
257C22×C4#12011yescharacteristic
258C22×C4#12312yes—
259C22×C4#12312yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C42−1+414yes—
266C22×C42−1+412yes—
267M16—44no—
268M16—44no—
269M16—44no—
270M16—44no—
271C2×C8—44no—
272C2×C8—44no—
273C2×C8—44no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—48no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—88no—
287C4⋊C4—88no—
288C22⋊C4—216no—
289C22⋊C4—416no—
290C22⋊C4—44no—
291C22⋊C4—44no—
292C22⋊C4—88no—
293C22⋊C4—88no—
294C22⋊C4—88no—
295C42—24no—
296C42—24no—
297C42—48no—
298C42—41no—
299C42—41no—
300C42—41no—
301C42—41no—
302C42—44no—
303C42—44no—
304C42—84no—
305C42C4×D414yes—
306C42C4×D414yes—
307C42#12114yes—
308C42#12114yes—
309C23—22no—
310C23—22no—
311C23—24no—
312C23—24no—
313C23—24no—
314C23—44no—
315C23—44no—
316C23—416no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—82no—
322C23—82no—
323C23—88no—
324C23C22×C22⋊C411yescharacteristic
325C23C2×#12011yescharacteristic
326C23C2×#12311yescharacteristic
327C23#53612yes—
328C23#53712yes—
329D4—48no—
330D4—44no—
331D4—44no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—416no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—48no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—88no—
358C2×C4—82no—
359C2×C4—84no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#38614yes—
365C2×C4#38612yes—
366C2×C4#38612yes—
367C2×C4C2×C4×D412yes—
368C2×C4#51912yes—
369C2×C4#52012yes—
370C2×C4C2×#12112yes—
371C2×C4#53511yescharacteristic
372C2×C4#53614yes—
373C2×C4#53612yes—
374C2×C4#53611yescharacteristic
375C2×C4#53714yes—
376C2×C4#54114yes—
377C2×C4#54214yes—
378C8—84no—
379C22—24no—
380C22—24no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—48no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—84no—
390C22—88no—
391C22—88no—
392C22C2×#38611yescharacteristic
393C22#231011yescharacteristic
394C22#231111yescharacteristic
395C22#261612yes—
396C22#262512yes—
397C22#301812yes—
398C22#301912yes—
399C4—24no—
400C4—22no—
401C4—22no—
402C4—44no—
403C4—44no—
404C4—82no—
405C4—88no—
406C4—82no—
407C4—82no—
408C4#232314yes—
409C4#262012yes—
410C4#262112yes—
411C2—21no—
412C2—21no—
413C2—22no—
414C2—84no—
415C2#1425811yescharacteristic
416C2#2095211yescharacteristic
417C2#2095311yescharacteristic
418C2#2970714yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
3. #21314C2semidirect
5. #11211C2semidirect
6. #11210C2semidirect
9. #10852C2semidirect
15. #2566C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25662
#340582
#213142
C2×#18002
#112112
#112102
#111612
#108522