Finite Group Databasebeta

← #320233

Subgroups — #320233

Order 512.

680 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic
3#29141C211yesmaximal, characteristic
4#28748C211yesmaximal, characteristic
5#28717C211yesmaximal, characteristic
6C4×#500C211yesmaximal, characteristic
7#10675C212yesmaximal
8#10313C212yesmaximal
9#9840C212yesmaximal
10#9826C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#500C2211yescharacteristic
13C2×#487C2212yes—
14C2×#487C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2072—22no—
20#2058—22no—
21#2043—22no—
22#2024—22no—
23#1990—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1981—22no—
31#1956C2212yes—
32#1956C2212yes—
33#1948—22no—
34#1942—22no—
35#1845C2211yescharacteristic
36#1844C2212yes—
37#1834—22no—
38#1834C2212yes—
39#1803—22no—
40#1803—22no—
41C4×#107—22no—
42C4×C4⋊C8C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2312yes—
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—21no—
56C2×C4×Q8—21no—
57#500—22no—
58#500—22no—
59#487—24no—
60#487—24no—
61#487—42no—
62#487—42no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#106—24no—
66C2×C4⋊C8D412yes—
67C2×C4⋊C8D412yes—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—24no—
71C2×#103—24no—
72C2×C4×C8D412yes—
73C2×C4×C8D412yes—
74C2×C4×C8C2312yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#391—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384C2311yescharacteristic
92#384C2311yescharacteristic
93#384C2312yes—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C43C2311yescharacteristic
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#336D412yes—
107#336D412yes—
108#322—42no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—41no—
113C22×C8C2×D414yes—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123C4⋊Q8—22no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—22no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—82no—
130C4⋊Q8—82no—
131C4⋊Q8—82no—
132C4⋊Q8#4112yes—
133C4⋊Q8#4112yes—
134C4×Q8—44no—
135C4×Q8—44no—
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—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2411yescharacteristic
169#107—28no—
170#107—44no—
171#106—84no—
172C4⋊C8—24no—
173C4⋊C8—24no—
174#103—48no—
175#103—48no—
176#103—84no—
177#103—84no—
178C4×C8—24no—
179C4×C8—24no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—21no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—21no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192C2×Q8—22no—
193C2×Q8—22no—
194C2×Q8—41no—
195C2×Q8—41no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—42no—
203C2×Q8—42no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—82no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4#12711yescharacteristic
214C22×C4#12711yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4#12711yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C22×D411yescharacteristic
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—24no—
224C2×C8—24no—
225C2×C8—24no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—24no—
229C2×C8—24no—
230C2×C8—24no—
231C2×C8#13714yes—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—22no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C42—22no—
263C42—22no—
264C42—22no—
265C42—82no—
266C42—82no—
267C42#13514yes—
268C42#13714yes—
269C42#13714yes—
270C42C2×#4111yescharacteristic
271C42C2×#4111yescharacteristic
272C42H(5,𝔽2)11yescharacteristic
273C422−1+411yescharacteristic
274C23C2×#12711yescharacteristic
275Q8—44no—
276Q8—44no—
277Q8—82no—
278Q8—82no—
279Q8—84no—
280Q8—84no—
281Q8—164no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4#49512yes—
307C2×C4#49512yes—
308C2×C4#49712yes—
309C2×C4#49712yes—
310C2×C4#49712yes—
311C2×C4#49712yes—
312C2×C4#52511yescharacteristic
313C2×C4#53711yescharacteristic
314C2×C4#54611yescharacteristic
315C2×C4#54611yescharacteristic
316C2×C4#54811yescharacteristic
317C2×C4#54811yescharacteristic
318C2×C4C2×#13512yes—
319C2×C4C2×#13712yes—
320C2×C4C2×#13712yes—
321C2×C4C2×#13712yes—
322C2×C4#57612yes—
323C2×C4#57712yes—
324C8—24no—
325C8—44no—
326C8—42no—
327C8—44no—
328C22#254011yescharacteristic
329C22C2×#49511yescharacteristic
330C22C2×#49711yescharacteristic
331C22C2×#49711yescharacteristic
332C22#315211yescharacteristic
333C22#315411yescharacteristic
334C22#315511yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—82no—
339C4—82no—
340C4—162no—
341C4—161no—
342C4—161no—
343C4—162no—
344C4#171814yes—
345C4#311312yes—
346C4#325512yes—
347C4#330312yes—
348C4#333312yes—
349C4#336012yes—
350C4#337512yes—
351C2C2×#171811yescharacteristic
352C2#1963811yescharacteristic
353C2#1968011yescharacteristic
354C2#1970511yescharacteristic
355C2#2838511yescharacteristic
356C2#2839811yescharacteristic
357C2#2841811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#127
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
#332892
#292062
#291412
#287482
#287172
C4×#5002
#106752
#103132
#98402
#98262
#82612