Finite Group Databasebeta

← #318712

Subgroups — #318712

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29275C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3C8×#125C211yesmaximal, characteristic
4#10597C212yesmaximal
5#10589C212yesmaximal
6#9730C211yesmaximal, characteristic
7#9730C211yesmaximal, characteristic
8#9615C211yesmaximal, characteristic
9#9614C211yesmaximal, characteristic
10#8807C211yesmaximal, characteristic
11#8594C212yesmaximal
12C2×#500C2211yescharacteristic
13C4×C4⋊Q8C2212yes—
14C4×#125C2211yescharacteristic
15#2101—22no—
16#2059—22no—
17#1990C2212yes—
18#1987C2212yes—
19#1987C2212yes—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1920—22no—
27#1918—22no—
28#1916—21no—
29#1916—21no—
30#1845C2211yescharacteristic
31#1831C2212yes—
32#1831C2211yescharacteristic
33C4×#107—21no—
34C4×#107—21no—
35C8×C4⋊C4—21no—
36C8×C4⋊C4—21no—
37C8×C4⋊C4C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#335C2211yescharacteristic
41C2×#335C2211yescharacteristic
42C2×#334C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
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×C4×C8—21no—
65C2×C4×C8—21no—
66C2×C4×C8D411yescharacteristic
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—22no—
73#389—22no—
74#389—22no—
75#389D411yescharacteristic
76#389D411yescharacteristic
77#387—21no—
78#387—21no—
79#384C2312yes—
80#382D411yescharacteristic
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4C2311yescharacteristic
86C4×C4⋊C4C2311yescharacteristic
87C43C2311yescharacteristic
88#340—22no—
89#340—22no—
90#340—22no—
91#340—22no—
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—21no—
102#336—21no—
103#336—22no—
104#335—22no—
105#335—22no—
106#334—42no—
107#333—42no—
108#333—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—22no—
123#125—22no—
124C4×Q8—84no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
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—44no—
154#106—44no—
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—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—21no—
169C4×C8—21no—
170C4×C8—41no—
171C4×C8—42no—
172C4×C8—42no—
173C4×C8C2×Q812yes—
174C4×C8#4112yes—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—21no—
181#95—42no—
182#95—42no—
183C2×Q8—24no—
184C2×Q8—42no—
185C2×Q8—82no—
186C2×Q8—82no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—42no—
192C22×C4—42no—
193C22×C4#12111yescharacteristic
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4C22×D411yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C2×#4111yescharacteristic
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8C22⋊Q812yes—
220C2×C8C22⋊Q812yes—
221C2×C8#13512yes—
222C2×C8#13712yes—
223C2×C8#13712yes—
224C4⋊C4—24no—
225C4⋊C4—24no—
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—21no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C4⋊C4#13512yes—
245C4⋊C4#13512yes—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
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—84no—
267Q8—84no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—22no—
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—41no—
288C2×C4—41no—
289C2×C4—44no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#47212yes—
300C2×C4C2×C22⋊Q811yescharacteristic
301C2×C4#53711yescharacteristic
302C2×C4D4×Q811yescharacteristic
303C2×C4#55011yescharacteristic
304C2×C4#55411yescharacteristic
305C2×C4#55611yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4C2×#13511yescharacteristic
308C2×C4C2×#13511yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4C2×#13711yescharacteristic
311C2×C4#57511yescharacteristic
312C2×C4#57811yescharacteristic
313C2×C4#57811yescharacteristic
314C8—22no—
315C8—22no—
316C8—41no—
317C8—82no—
318C8—82no—
319C8—81no—
320C8—81no—
321C8#47914yes—
322C22#253811yescharacteristic
323C22#303511yescharacteristic
324C22#303911yescharacteristic
325C22#305011yescharacteristic
326C22#305911yescharacteristic
327C22C2×#47211yescharacteristic
328C22#306711yescharacteristic
329C4—21no—
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—42no—
335C4—41no—
336C4—42no—
337C4—81no—
338C4—84no—
339C4—81no—
340C4—162no—
341C4C2×#47911yescharacteristic
342C4#307811yescharacteristic
343C4#324811yescharacteristic
344C4#325211yescharacteristic
345C4#338311yescharacteristic
346C4#338311yescharacteristic
347C4#339311yescharacteristic
348C4#339311yescharacteristic
349C2#1912811yescharacteristic
350C2#1922011yescharacteristic
351C2#1927311yescharacteristic
352C2#1935911yescharacteristic
353C2#2826811yescharacteristic
354C2#2828711yescharacteristic
355C2#2829511yescharacteristic
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
#292752
C4×#5002
C8×#1252
#105972
#105892
#97302
#96152
#96142
#88072
#85942