Finite Group Databasebeta

← #318773

Subgroups — #318773

Order 512.

542 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#29257C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#10602C212yesmaximal
5#10594C212yesmaximal
6#9740C211yesmaximal, characteristic
7#9740C211yesmaximal, characteristic
8#9621C211yesmaximal, characteristic
9#9620C211yesmaximal, characteristic
10#8807C211yesmaximal, characteristic
11#8593C212yesmaximal
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14C4×#125C2211yescharacteristic
15#2097—22no—
16#2026—22no—
17#1990C2212yes—
18#1981C2212yes—
19#1981C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1916—21no—
27#1916—21no—
28#1865—22no—
29#1863—22no—
30#1845C2211yescharacteristic
31#1831C2212yes—
32#1831C2211yescharacteristic
33C4×#107—21no—
34C4×#107—21no—
35#1771—21no—
36#1771—21no—
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#335C2212yes—
41C2×#334C2211yescharacteristic
42C2×#334C2211yescharacteristic
43C2×#332C2212yes—
44C2×#332C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C2×C4⋊Q8C2312yes—
48C2×#125D411yescharacteristic
49C2×C4×Q8—22no—
50#500—22no—
51#500Q814yes—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8C2311yescharacteristic
61C2×C4⋊C8C2311yescharacteristic
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#97—21no—
69C2×#97—21no—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2311yescharacteristic
76#391—22no—
77#389—22no—
78#389—22no—
79#389D411yescharacteristic
80#389D411yescharacteristic
81#387—21no—
82#387—21no—
83#384C2312yes—
84#382D411yescharacteristic
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2311yescharacteristic
90C4×C4⋊C4C2311yescharacteristic
91C43C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—22no—
101#336—22no—
102#336—21no—
103#336—21no—
104#335—42no—
105#334—22no—
106#334—22no—
107#332—42no—
108#332—42no—
109C22×Q8—22no—
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8C2×Q814yes—
122#125—42no—
123C4×Q8—82no—
124C4×Q8—82no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—42no—
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42C2×D411yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42C2411yescharacteristic
149#107—24no—
150#107—24no—
151#107—42no—
152#107—42no—
153#107—82no—
154#106—82no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—21no—
162C4⋊C8—21no—
163C4⋊C8—42no—
164C4⋊C8—42no—
165#103—44no—
166#103—44no—
167#103—82no—
168#103—82no—
169#97—42no—
170#97—42no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—21no—
175C4×C8—21no—
176C4×C8—41no—
177C4×C8C2×Q812yes—
178C4×C8#4112yes—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—42no—
186#95—42no—
187C2×Q8—24no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4#12111yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C22×D411yescharacteristic
205C22×C4C2×#4111yescharacteristic
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—21no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8C22⋊Q812yes—
226C2×C8C22⋊Q812yes—
227C2×C8#13512yes—
228C2×C8#13712yes—
229C2×C8#13712yes—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—82no—
247C4⋊C4—84no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—22no—
253C42—22no—
254C42—22no—
255C42—41no—
256C42—41no—
257C42—41no—
258C42—41no—
259C42—41no—
260C42—84no—
261C42C22×Q811yescharacteristic
262C42C2×#4111yescharacteristic
263C42H(5,𝔽2)11yescharacteristic
264C422−1+411yescharacteristic
265C23C2×#12111yescharacteristic
266Q8—44no—
267Q8—44no—
268Q8—82no—
269Q8—82no—
270C2×C4—21no—
271C2×C4—21no—
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—21no—
282C2×C4—21no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
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#47312yes—
302C2×C4C2×C22⋊Q811yescharacteristic
303C2×C4#53711yescharacteristic
304C2×C4D4×Q811yescharacteristic
305C2×C4#55011yescharacteristic
306C2×C4#55411yescharacteristic
307C2×C4#55611yescharacteristic
308C2×C4C2×#13511yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4C2×#13711yescharacteristic
311C2×C4#57511yescharacteristic
312C2×C4#57512yes—
313C2×C4#57612yes—
314C8—22no—
315C8—22no—
316C8—41no—
317C8—82no—
318C8—82no—
319C8—81no—
320C8—81no—
321C8#47914yes—
322C22#253811yescharacteristic
323C22#303511yescharacteristic
324C22#303911yescharacteristic
325C22#305411yescharacteristic
326C22#305811yescharacteristic
327C22C2×#47311yescharacteristic
328C22#306711yescharacteristic
329C4—21no—
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—81no—
336C4—81no—
337C4—84no—
338C4—81no—
339C4—81no—
340C4—162no—
341C4C2×#47911yescharacteristic
342C4#307811yescharacteristic
343C4#324811yescharacteristic
344C4#325211yescharacteristic
345C4#338912yes—
346C4#340312yes—
347C2#1916111yescharacteristic
348C2#1922511yescharacteristic
349C2#1929811yescharacteristic
350C2#1935811yescharacteristic
351C2#2826911yescharacteristic
352C2#2828711yescharacteristic
353C2#2829711yescharacteristic
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

Maximal subgroups

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

subgroupindex
#292572
C4×#5002
#216732
#106022
#105942
#97402
#96212
#96202
#88072
#85932