Finite Group Databasebeta

← #327990

Subgroups — #327990

Order 512.

779 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2571C211yesmaximal, characteristic
2#33149C211yesmaximal, characteristic
3#33148C211yesmaximal, characteristic
4#27972C211yesmaximal, characteristic
5#10578C214yesmaximal, complemented
6#10352C214yesmaximal, complemented
7#8724C211yesmaximal, characteristic
8#8707C211yesmaximal, characteristic
9#8706C211yesmaximal, characteristic
10#2857C2211yescharacteristic
11#2571—22no—
12#2571—22no—
13#2571—22no—
14#2571C2214yes—
15C2×#400—22no—
16C2×#389—21no—
17C2×#389—21no—
18C2×#384C2211yescharacteristic
19C2×#382—21no—
20C2×#382—21no—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2072—22no—
24#1987C2212yes—
25#1987C2212yes—
26#1956C2212yes—
27#1948C2212yes—
28#1947C2212yes—
29#1921—22no—
30#1920—22no—
31#1872—22no—
32#1853—21no—
33#1852—21no—
34#1834C2212yes—
35#1824—21no—
36#1823—21no—
37#1797C2211yescharacteristic
38#1797C2211yescharacteristic
39C4×C22⋊C8C2211yescharacteristic
40#1296C2214yes—
41#1279C2214yes—
42#1279C2214yes—
43C2×C4⋊Q8C2311yescharacteristic
44C2×C22⋊Q8—21no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D412yes—
53C22×C42—21no—
54C22×C42—21no—
55C22×C42C2311yescharacteristic
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×#103—24no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—21no—
63C2×C22⋊C8—21no—
64C2×C4×C8C2312yes—
65#400—28no—
66#400—28no—
67#391—21no—
68#389—24no—
69#389—24no—
70#389—24no—
71#389—24no—
72#389—21no—
73#389—21no—
74#384—21no—
75#384—21no—
76#384D412yes—
77#384C2311yescharacteristic
78#384C2312yes—
79#382—24no—
80#382—24no—
81#382—24no—
82#382—24no—
83#380—21no—
84#380—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4C2312yes—
92C4×C4⋊C4C2312yes—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98#340—24no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#336—21no—
106#336—21no—
107#336—21no—
108C22×Q8—21no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C4⋊Q8—24no—
121C22⋊Q8—44no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—44no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—82no—
148C2×C4⋊C4—84no—
149C2×C4⋊C4C2×D412yes—
150C2×C4⋊C4C2×D412yes—
151C2×C4⋊C4C2×D412yes—
152C2×C4⋊C4C2×D412yes—
153C2×C4⋊C4C2×D414yes—
154C2×C22⋊C4—41no—
155C2×C42—24no—
156C2×C42—22no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—44no—
161C2×C42—42no—
162C2×C42—42no—
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168#106—44no—
169C4⋊C8—24no—
170C4⋊C8—24no—
171#103—84no—
172C22⋊C8—42no—
173C22⋊C8—42no—
174C22⋊C8—42no—
175C4×C8—24no—
176#95—28no—
177#95—28no—
178#95—28no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—41no—
184#95—41no—
185C24#12011yescharacteristic
186C2×Q8—42no—
187C2×Q8—42no—
188C2×Q8—42no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—41no—
201C22×C4—42no—
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—42no—
212C22×C4—42no—
213C22×C4—44no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#13512yes—
222C22×C4#13512yes—
223C22×C4#13712yes—
224C22×C4#13712yes—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D411yescharacteristic
227C22×C4C22×D411yescharacteristic
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—44no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—88no—
263C22⋊C4—82no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—44no—
268C42—42no—
269C42—42no—
270C42—44no—
271C42—44no—
272C422−1+414yes—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23#45212yes—
278C23#45212yes—
279C23#45412yes—
280C23C2×#12011yescharacteristic
281Q8—84no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—44no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—88no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4#45212yes—
323C2×C4#45212yes—
324C2×C4#53612yes—
325C2×C4#53612yes—
326C2×C4#53612yes—
327C2×C4C2×#13511yescharacteristic
328C2×C4C2×#13511yescharacteristic
329C2×C4C2×#13711yescharacteristic
330C2×C4C2×#13711yescharacteristic
331C2×C4#57512yes—
332C2×C4#57512yes—
333C8—82no—
334C8—82no—
335C8—82no—
336C22—22no—
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22#163814yes—
343C22#261411yescharacteristic
344C22C2×#45211yescharacteristic
345C22C2×#45211yescharacteristic
346C22#300511yescharacteristic
347C22#300511yescharacteristic
348C22C2×#45411yescharacteristic
349C22#301911yescharacteristic
350C4—22no—
351C4—22no—
352C4—22no—
353C4—41no—
354C4—82no—
355C4—84no—
356C4—82no—
357C4—82no—
358C4—82no—
359C4—161no—
360C4—161no—
361C4#323212yes—
362C4#323212yes—
363C4#323612yes—
364C4#323612yes—
365C2—24no—
366C2C2×#163811yescharacteristic
367C2#1853611yescharacteristic
368C2#1875111yescharacteristic
369C2#1877311yescharacteristic
370C2#2964311yescharacteristic
371C2#2964311yescharacteristic
372C2#2965111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10578C2semidirect
6. #10352C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25712
#331492
#331482
#279722
#105782
#103522
#87242
#87072
#87062