Finite Group Databasebeta

← #352487

Subgroups — #352487

Order 512.

544 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33258C211yesmaximal, characteristic
2#29997C212yesmaximal
3#20516C211yesmaximal, characteristic
4#9906C212yesmaximal
5#9621C211yesmaximal, characteristic
6#9620C211yesmaximal, characteristic
7#9613C212yesmaximal
8#8805C211yesmaximal, characteristic
9#8653C211yesmaximal, characteristic
10#8653C211yesmaximal, characteristic
11#8647C212yesmaximal
12C2×#446C2211yescharacteristic
13C2×C8×Q8C2211yescharacteristic
14#2859C2212yes—
15#2295C2211yescharacteristic
16#2088—22no—
17#2086—22no—
18#2068—22no—
19#2066—22no—
20#1948C2212yes—
21#1947C2212yes—
22#1942—21no—
23#1942—21no—
24#1942—21no—
25#1942—21no—
26#1845C2211yescharacteristic
27#1845C2211yescharacteristic
28#1842C2212yes—
29#1834C2212yes—
30#1803C2212yes—
31#1803C2212yes—
32#1771—21no—
33#1770C2211yescharacteristic
34C8×C4⋊C4—21no—
35C8×C4⋊C4C2211yescharacteristic
36#1768—21no—
37#1767C2211yescharacteristic
38#1726—21no—
39C42×C8C2211yescharacteristic
40C2×#334C2212yes—
41C2×#334C2211yescharacteristic
42C2×#334C2211yescharacteristic
43#1296C2212yes—
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47C2×#125—22no—
48C2×C4×Q8C2311yescharacteristic
49#446—22no—
50#446Q814yes—
51C8×Q8—22no—
52C8×Q8Q814yes—
53C2×#107—21no—
54C2×#107—21no—
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⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2311yescharacteristic
67C2×#103—22no—
68C2×#103—22no—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8D411yescharacteristic
77C2×C4×C8D411yescharacteristic
78C2×C4×C8C2311yescharacteristic
79C2×C4×C8C2311yescharacteristic
80#391D411yescharacteristic
81#391D411yescharacteristic
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#389D411yescharacteristic
87#389D411yescharacteristic
88#387—21no—
89#387—21no—
90#384C2312yes—
91#384C2312yes—
92#382—22no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4C2311yescharacteristic
95#376—21no—
96C43C2311yescharacteristic
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—21no—
104#336—21no—
105#336—21no—
106#334—24no—
107#334—22no—
108#334—22no—
109C22×Q8C2×D411yescharacteristic
110C22×C8—21no—
111C22×C8—21no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8C2×D411yescharacteristic
115C22×C8C2×D411yescharacteristic
116C22×C8C2×D411yescharacteristic
117C22×C8C2×D411yescharacteristic
118#125—84no—
119C4×Q8—21no—
120C4×Q8—21no—
121C4×Q8C2×Q812yes—
122C4×Q8C2×Q812yes—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42C2411yescharacteristic
146#107—42no—
147#107—42no—
148#107—42no—
149#107—42no—
150#107—82no—
151#106—82no—
152C4⋊C8—24no—
153C4⋊C8—24no—
154C4⋊C8—21no—
155C4⋊C8—21no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—82no—
161C4⋊C8—82no—
162C4⋊C8C2×Q812yes—
163C4⋊C8C2×Q812yes—
164#103—82no—
165#103—82no—
166#97—24no—
167#97—42no—
168#97—42no—
169C4×C8—24no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—41no—
173C4×C8—41no—
174C4×C8—42no—
175C4×C8C2×Q812yes—
176C4×C8C2×Q812yes—
177C4×C8#4112yes—
178C4×C8#4112yes—
179#95—21no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95—21no—
184#95—21no—
185#95—21no—
186#95—42no—
187#95—42no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8C4⋊Q812yes—
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
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—21no—
211C2×C8—22no—
212C2×C8—22no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8—42no—
223C2×C8C22⋊Q812yes—
224C2×C8C22⋊Q812yes—
225C2×C8C22⋊Q812yes—
226C2×C8C22⋊Q812yes—
227C2×C8#13512yes—
228C2×C8#13712yes—
229C4⋊C4—21no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
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—42no—
243C4⋊C4—42no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—84no—
247C4⋊C4—84no—
248C4⋊C4C4⋊Q812yes—
249C42—22no—
250C42—22no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42—22no—
259C42—41no—
260C42—41no—
261C42C22×Q811yescharacteristic
262C42C22×Q811yescharacteristic
263C42C2×#4111yescharacteristic
264C42C2×#4111yescharacteristic
265C23C2×#12111yescharacteristic
266Q8—24no—
267Q8—42no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#47212yes—
299C2×C4#47312yes—
300C2×C4C2×C22⋊Q811yescharacteristic
301C2×C4C2×C22⋊Q811yescharacteristic
302C2×C4C2×C4⋊Q811yescharacteristic
303C2×C4#53211yescharacteristic
304C2×C4#53311yescharacteristic
305C2×C4#53311yescharacteristic
306C2×C4C2×#13511yescharacteristic
307C2×C4C2×#13711yescharacteristic
308C2×C4#57511yescharacteristic
309C2×C4#57511yescharacteristic
310C2×C4#57712yes—
311C2×C4#57712yes—
312C8—22no—
313C8—22no—
314C8—22no—
315C8—42no—
316C8—42no—
317C8—82no—
318C8—82no—
319C8#47914yes—
320C22#285711yescharacteristic
321C22#303511yescharacteristic
322C22#303711yescharacteristic
323C22#306011yescharacteristic
324C22#306011yescharacteristic
325C22C2×#47211yescharacteristic
326C22C2×#47311yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—42no—
334C4—42no—
335C4—81no—
336C4—81no—
337C4—162no—
338C4—162no—
339C4C2×#47911yescharacteristic
340C4#307611yescharacteristic
341C4#307811yescharacteristic
342C4#308011yescharacteristic
343C4#316312yes—
344C4#317412yes—
345C2#1922711yescharacteristic
346C2#1922811yescharacteristic
347C2#1934911yescharacteristic
348C2#1935011yescharacteristic
349C2#3320011yescharacteristic
350C2#3321411yescharacteristic
351C2#3322211yescharacteristic
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
#332582
#299972
#205162
#99062
#96212
#96202
#96132
#88052
#86532
#86472