Finite Group Databasebeta

← #317558

Subgroups — #317558

Order 512.

590 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29288C212yesmaximal
2C4×#494C211yesmaximal, characteristic
3C8×#125C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic
6#10538C211yesmaximal, characteristic
7#10538C211yesmaximal, characteristic
8#9562C212yesmaximal
9#9561C212yesmaximal
10#8784C211yesmaximal, characteristic
11#8591C212yesmaximal
12C2×#494C2211yescharacteristic
13C2×C4×Q16—21no—
14C2×C4×Q16—21no—
15C4×C4⋊Q8C2212yes—
16C4×#125C2211yescharacteristic
17#2088—22no—
18#2087—22no—
19#1989—22no—
20#1988C2212yes—
21#1988C2212yes—
22#1969—21no—
23#1969—21no—
24#1956C2212yes—
25#1949—22no—
26#1948C2212yes—
27#1947C2212yes—
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1916—21no—
31#1916—21no—
32#1845C2211yescharacteristic
33#1842C2212yes—
34#1831C2211yescharacteristic
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C8×C4⋊C4C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C22×Q16—21no—
48C22×Q16—21no—
49C2×C4⋊Q8C2312yes—
50C2×#125D411yescharacteristic
51C2×C4×Q8—22no—
52#494—24no—
53C4×Q16—44no—
54C4×Q16—44no—
55C2×#107—21no—
56C2×#107—21no—
57C2×#107—22no—
58C2×#106—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×#103—22no—
68C2×#103—22no—
69C2×C4×C8—21no—
70C2×C4×C8—21no—
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77#391—22no—
78#389—22no—
79#389—22no—
80#389D411yescharacteristic
81#389D411yescharacteristic
82#387—21no—
83#387—21no—
84#384C2312yes—
85#382D411yescharacteristic
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2311yescharacteristic
91C4×C4⋊C4C2311yescharacteristic
92C43C2311yescharacteristic
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#336—21no—
105#332—24no—
106#332—24no—
107C22×Q8—22no—
108C2×Q16—24no—
109C2×Q16—24no—
110C2×Q16—42no—
111C2×Q16—42no—
112C2×Q16—42no—
113C2×Q16—42no—
114C22×C8—21no—
115C22×C8—21no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8C2×D411yescharacteristic
119C22×C8C2×D411yescharacteristic
120C22×C8C2×D411yescharacteristic
121C22×C8C2×D411yescharacteristic
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8#4114yes—
126#125—22no—
127#125Q1614yescomplemented
128C4×Q8—44no—
129C4×Q8—44no—
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—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4C2×D411yescharacteristic
141C2×C4⋊C4C2×D411yescharacteristic
142C2×C4⋊C4C2×D411yescharacteristic
143C2×C4⋊C4C2×D411yescharacteristic
144C2×C42—21no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42C2411yescharacteristic
154#107—42no—
155#107—42no—
156#107—82no—
157#106—82no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8—44no—
167C4⋊C8—82no—
168#103—44no—
169#103—44no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—41no—
176C4×C8—42no—
177C4×C8—42no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—42no—
185#95—42no—
186C2×Q8—24no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205Q16—48no—
206Q16—48no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—21no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—41no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8—41no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8#13512yes—
229C2×C8AGL(1,ℤ/8)12yes—
230C2×C8AGL(1,ℤ/8)12yes—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—82no—
248C4⋊C4—84no—
249C4⋊C4C2×Q1612yes—
250C4⋊C4C2×Q1612yes—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—41no—
261C42—41no—
262C42—84no—
263C42C2×Q1612yes—
264C42#13512yes—
265C42C2×#4112yes—
266C422−1+412yes—
267C23C2×#12111yescharacteristic
268Q8—44no—
269Q8—44no—
270Q8—82no—
271Q8—82no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4#46712yes—
302C2×C4#46712yes—
303C2×C4#47312yes—
304C2×C4#52511yescharacteristic
305C2×C4#53611yescharacteristic
306C2×C4#54812yes—
307C2×C4#55212yes—
308C2×C4C22×Q1611yescharacteristic
309C2×C4C2×#13511yescharacteristic
310C2×C4C2×#13511yescharacteristic
311C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
312C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
313C2×C4#57511yescharacteristic
314C2×C4#57712yes—
315C8—22no—
316C8—22no—
317C8—22no—
318C8—41no—
319C8—81no—
320C8—82no—
321C8—82no—
322C8—81no—
323C22#253611yescharacteristic
324C22#303511yescharacteristic
325C22#303811yescharacteristic
326C22C2×#46711yescharacteristic
327C22#306011yescharacteristic
328C22C2×#47311yescharacteristic
329C22#306711yescharacteristic
330C4—22no—
331C4—22no—
332C4—22no—
333C4—22no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—81no—
338C4—84no—
339C4—81no—
340C4—162no—
341C4#310511yescharacteristic
342C4#311411yescharacteristic
343C4#322811yescharacteristic
344C4#323411yescharacteristic
345C4#336312yes—
346C4#336512yes—
347C2#1912611yescharacteristic
348C2#1922811yescharacteristic
349C2#1926811yescharacteristic
350C2#1935511yescharacteristic
351C2#2818711yescharacteristic
352C2#2820711yescharacteristic
353C2#2821911yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
127. #125Q16semidirect

Maximal subgroups

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

subgroupindex
#292882
C4×#4942
C8×#1252
#106522
#106502
#105382
#95622
#95612
#87842
#85912