Finite Group Databasebeta

← #325134

Subgroups — #325134

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33291C211yesmaximal, characteristic
2#29313C211yesmaximal, characteristic
3#29312C211yesmaximal, characteristic
4C8×C4⋊Q8C211yesmaximal, characteristic
5#10656C212yesmaximal
6#10655C212yesmaximal
7#9900C211yesmaximal, characteristic
8#9900C211yesmaximal, characteristic
9#9613C212yesmaximal
10#9613C212yesmaximal
11C4×#334C211yesmaximal, characteristic
12C2×C8×Q8—21no—
13#2859C2211yescharacteristic
14#2859C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16#2084—21no—
17#2083—21no—
18#2058—21no—
19#2057—21no—
20#1991—21no—
21#1991—21no—
22#1991—21no—
23#1991—21no—
24#1984—21no—
25#1948C2212yes—
26#1948C2212yes—
27#1947C2212yes—
28#1947C2212yes—
29#1942—22no—
30#1941—22no—
31#1939—21no—
32#1939C2212yes—
33#1842C2211yescharacteristic
34#1834C2211yescharacteristic
35#1834C2211yescharacteristic
36#1834C2211yescharacteristic
37#1834C2211yescharacteristic
38C8×C4⋊C4—21no—
39C8×C4⋊C4C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C42×C8C2211yescharacteristic
43C2×#334C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×#125—21no—
51C2×#125—21no—
52C2×C4×Q8—21no—
53C8×Q8—44no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#106—22no—
57C2×#106—22no—
58C2×C4⋊C8—22no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4×C8—22no—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389D412yes—
84#384D411yescharacteristic
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#382—21no—
90#382—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4C2312yes—
93C43C2311yescharacteristic
94#340—21no—
95#340—21no—
96#340—21no—
97#340—21no—
98#340—21no—
99#340—21no—
100#340—21no—
101#340—21no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#336—22no—
107#336—22no—
108#336—21no—
109#336—21no—
110#334Q818yescomplemented
111C22×Q8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C22×C8C2×D411yescharacteristic
119C4⋊Q8—24no—
120#125—42no—
121#125—42no—
122#125—42no—
123#125—42no—
124C4×Q8—42no—
125C4×Q8—42no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C2×D412yes—
147C2×C4⋊C4C2×D412yes—
148C2×C42—22no—
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42C2411yescharacteristic
156#107—44no—
157#107—82no—
158#106—44no—
159#106—82no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—24no—
163C4⋊C8—44no—
164C4⋊C8—42no—
165C4⋊C8—42no—
166C4⋊C8C2×Q814yes—
167C4⋊C8C2×Q814yes—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—44no—
174C4×C8—81no—
175C4×C8—81no—
176C4×C8C2×Q814yes—
177#95—22no—
178#95—22no—
179#95—41no—
180#95—41no—
181#95—41no—
182#95—41no—
183C2×Q8—24no—
184C2×Q8—42no—
185C2×Q8—42no—
186C22×C4—22no—
187C22×C4—22no—
188C22×C4—41no—
189C22×C4—41no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4C22×D411yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C2×#4111yescharacteristic
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—24no—
206C2×C8—24no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8C4⋊Q812yes—
219C2×C8C4⋊Q812yes—
220C4⋊C4—24no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—44no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—81no—
244C4⋊C4—81no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4#13514yes—
250C42—22no—
251C42—21no—
252C42—21no—
253C42—41no—
254C42—41no—
255C42—42no—
256C42—42no—
257C42#13512yes—
258C42#13512yes—
259C42AGL(1,ℤ/8)12yes—
260C42#13712yes—
261C42C22×Q812yes—
262C422−1+412yes—
263C23C2×#12111yescharacteristic
264Q8—48no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—81no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4#46312yes—
294C2×C4#46312yes—
295C2×C4#46412yes—
296C2×C4#46412yes—
297C2×C4C2×C4⋊Q811yescharacteristic
298C2×C4#53611yescharacteristic
299C2×C4D4×Q812yes—
300C2×C4#55712yes—
301C2×C4C2×#13512yes—
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×#13511yescharacteristic
304C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4#57711yescharacteristic
307C2×C4#57711yescharacteristic
308C8—22no—
309C8—22no—
310C8—44no—
311C8—44no—
312C8—41no—
313C8—41no—
314C8—82no—
315C22#254611yescharacteristic
316C22C2×#46311yescharacteristic
317C22C2×#46411yescharacteristic
318C22#305011yescharacteristic
319C22#306011yescharacteristic
320C22#306711yescharacteristic
321C22#306711yescharacteristic
322C4—22no—
323C4—22no—
324C4—22no—
325C4—22no—
326C4—22no—
327C4—44no—
328C4—82no—
329C4—161no—
330C4—161no—
331C4—161no—
332C4—161no—
333C4#316011yescharacteristic
334C4#316511yescharacteristic
335C4#322811yescharacteristic
336C4#323111yescharacteristic
337C4#338112yes—
338C4#338112yes—
339C2#1911011yescharacteristic
340C2#1912311yescharacteristic
341C2#1920011yescharacteristic
342C2#1921611yescharacteristic
343C2#2867811yescharacteristic
344C2#2868811yescharacteristic
345C2#2870211yescharacteristic
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
110. #334Q8semidirect

Maximal subgroups

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

subgroupindex
#332912
#293132
#293122
C8×C4⋊Q82
#106562
#106552
#99002
#96132
C4×#3342