Finite Group Databasebeta

← #380285

Subgroups — #380285

Order 512.

1446 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2316C211yesmaximal, characteristic
2#20846C212yesmaximal
3C2×#1821C211yesmaximal, characteristic
4#13924C211yesmaximal, characteristic
5#11342C214yesmaximal, complemented
6#11296C214yesmaximal, complemented
7#10733C212yesmaximal
8C22×#117C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10C2×#428C2214yes—
11C2×#407C2212yes—
12#2316—24no—
13#2316—22no—
14#2316C2214yes—
15C2×#389—22no—
16C2×#389—22no—
17C2×#380—22no—
18C2×#380C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1854—24no—
21#1853—22no—
22#1852—22no—
23#1821C418yescomplemented
24C2×#343C2214yes—
25#1736C2214yes—
26C2×#339C2218yes—
27#1445—22no—
28C4⋊#95C2212yes—
29C4⋊#95C2212yes—
30C22⋊#95—22no—
31C24×C4D412yes—
32C24×C4C2311yescharacteristic
33C22×M16C2314yes—
34C2×#117—22no—
35C2×#117—22no—
36C2×#117—48no—
37C2×#117D414yes—
38C2×#117C2314yes—
39C22×C4⋊C4—22no—
40C22×C4⋊C4—22no—
41C22×C4⋊C4—24no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4—22no—
44C22×C4⋊C4C2×C414yes—
45C22×C4⋊C4D412yes—
46C22×C4⋊C4D412yes—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4Q814yes—
49C22×C4⋊C4C2311yescharacteristic
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C22⋊C4—21no—
55C22×C42—22no—
56C22×C42C2312yes—
57#428—48no—
58C2×#107—24no—
59C2×#106—24no—
60#407—24no—
61#407C2×C418yes—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64#389—28no—
65#389—28no—
66#389—28no—
67#389—28no—
68#380—28no—
69#380—28no—
70#380—22no—
71#380—22no—
72C4×C22⋊C4—24no—
73C2×#95—24no—
74C2×#95—21no—
75C2×#95—21no—
76C2×#95—24no—
77C2×#95—24no—
78#343C2×C4116yescomplemented
79#340—24no—
80#340—24no—
81#339—48no—
82C25C2×D411yescharacteristic
83C23×C4—24no—
84C23×C4—24no—
85C23×C4—22no—
86C23×C4—22no—
87C23×C4—24no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—82no—
94C23×C4—82no—
95C23×C4C4⋊C418yes—
96C23×C4C22×C412yes—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D412yes—
99C23×C4C2×D412yes—
100C23×C4C2×D414yes—
101C23×C4C2×Q812yes—
102C23×C4#4114yes—
103C23×C4C2411yescharacteristic
104C2×M16—24no—
105C2×M16—28no—
106C2×M16—28no—
107C2×M16C22×C418yes—
108C22×C8—24no—
109#117—28no—
110#117—28no—
111#117—44no—
112#117—44no—
113#117—48no—
114#117—48no—
115C2×C4⋊C4—28no—
116C2×C4⋊C4—28no—
117C2×C4⋊C4—24no—
118C2×C4⋊C4—28no—
119C2×C4⋊C4—28no—
120C2×C4⋊C4—24no—
121C2×C4⋊C4—24no—
122C2×C4⋊C4—24no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—24no—
128C2×C4⋊C4—24no—
129C2×C4⋊C4—28no—
130C2×C4⋊C4—28no—
131C2×C4⋊C4—44no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—84no—
134C2×C4⋊C4—84no—
135C2×C4⋊C4C4⋊C418yes—
136C2×C4⋊C4C22×C414yes—
137C2×C4⋊C4C2×D414yes—
138C2×C4⋊C4C2×D414yes—
139C2×C4⋊C4C2×Q814yes—
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4—24no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—48no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—44no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—24no—
150C2×C42—24no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—24no—
154C2×C42—48no—
155#107—48no—
156#106—48no—
157C22⋊C8—28no—
158C22⋊C8—28no—
159#95—216no—
160#95—42no—
161#95—42no—
162#95—84no—
163#95—84no—
164C24—22no—
165C24—22no—
166C24—22no—
167C24—22no—
168C24—41no—
169C24—41no—
170C24C2×C4⋊C412yes—
171C24#12011yescharacteristic
172C24#12011yescharacteristic
173C24#12312yes—
174C24C22×D411yescharacteristic
175C22×C4—22no—
176C22×C4—22no—
177C22×C4—28no—
178C22×C4—24no—
179C22×C4—24no—
180C22×C4—24no—
181C22×C4—24no—
182C22×C4—216no—
183C22×C4—28no—
184C22×C4—24no—
185C22×C4—28no—
186C22×C4—24no—
187C22×C4—28no—
188C22×C4—24no—
189C22×C4—24no—
190C22×C4—24no—
191C22×C4—44no—
192C22×C4—44no—
193C22×C4—44no—
194C22×C4—44no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—44no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—44no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—44no—
206C22×C4—44no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—88no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—88no—
219C22×C4C2×C4⋊C412yes—
220C22×C4C2×C4⋊C418yes—
221C22×C4C4×D414yes—
222C22×C4C4×D414yes—
223C22×C4#12012yes—
224C22×C4C22⋊Q814yes—
225C22×C4C22⋊Q814yes—
226C22×C4#12312yes—
227C22×C4C23×C411yescharacteristic
228C22×C4C22×D412yes—
229C22×C4C22×Q811yescharacteristic
230C22×C4C2×#4112yes—
231C22×C4H(5,𝔽2)14yes—
232C22×C42−1+414yes—
233M16—216no—
234M16—48no—
235C2×C8—28no—
236C2×C8—44no—
237C2×C8—44no—
238C4⋊C4—28no—
239C4⋊C4—28no—
240C4⋊C4—216no—
241C4⋊C4—24no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—88no—
250C4⋊C4—88no—
251C22⋊C4—48no—
252C22⋊C4—48no—
253C22⋊C4—48no—
254C22⋊C4—82no—
255C22⋊C4—82no—
256C42—48no—
257C42—48no—
258C42—48no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—24no—
268C23—21no—
269C23—21no—
270C23—24no—
271C23—24no—
272C23—24no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—44no—
281C23—44no—
282C23—44no—
283C23#38512yes—
284C23#38512yes—
285C23#45412yes—
286C23#45612yes—
287C23C22×C4⋊C411yescharacteristic
288C23C2×#12011yescharacteristic
289C23C2×#12311yescharacteristic
290C23#53512yes—
291C23#53612yes—
292C2×C4—24no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—28no—
297C2×C4—28no—
298C2×C4—28no—
299C2×C4—28no—
300C2×C4—28no—
301C2×C4—416no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—48no—
305C2×C4—44no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—84no—
315C2×C4—88no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—88no—
320C2×C4—88no—
321C2×C4—82no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—81no—
330C2×C4#38514yes—
331C2×C4C2×C4×D412yes—
332C2×C4#52014yes—
333C2×C4C2×C22⋊Q812yes—
334C2×C4#53414yes—
335C2×C4#53614yes—
336C2×C4#53918yes—
337C2×C4#54414yes—
338C8—48no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—21no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—22no—
355C22—22no—
356C22—48no—
357C22—48no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22#185714yes—
363C22C2×#38511yescharacteristic
364C22#231712yes—
365C22#261311yescharacteristic
366C22#261411yescharacteristic
367C22#262312yes—
368C22C2×#45411yescharacteristic
369C22C2×#45611yescharacteristic
370C22#301911yescharacteristic
371C22#301911yescharacteristic
372C4—24no—
373C4—48no—
374C4—84no—
375C4—84no—
376C4—84no—
377C4—161no—
378C4—161no—
379C4#232214yes—
380C4#262714yes—
381C2—22no—
382C2—22no—
383C2—24no—
384C2—42no—
385C2#1426311yescharacteristic
386C2C2×#185711yescharacteristic
387C2#2085611yescharacteristic
388C2#2961711yescharacteristic
389C2#2961911yescharacteristic
390C2#2965111yescharacteristic
391C2#2965311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11342C2semidirect
6. #11296C2semidirect
23. #1821C4semidirect
78. #343C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23162
#208462
C2×#18212
#139242
#113422
#112962
#107332