Finite Group Databasebeta

← #320304

Subgroups — #320304

Order 512.

672 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29141C211yesmaximal, characteristic
4#28748C211yesmaximal, characteristic
5#28689C211yesmaximal, characteristic
6C4×#487C211yesmaximal, characteristic
7#10682C212yesmaximal
8#10313C212yesmaximal
9#9840C212yesmaximal
10#9831C212yesmaximal
11#8261C211yesmaximal, characteristic
12C2×#500C2212yes—
13C2×#487C2211yescharacteristic
14C2×#487C2212yes—
15#2868C2211yescharacteristic
16#2868C2211yescharacteristic
17C4×C4⋊Q8C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2100—22no—
20#2083—22no—
21#2043—22no—
22#2024—22no—
23#1990—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2212yes—
27#1990C2212yes—
28#1990C2211yescharacteristic
29#1987—22no—
30#1981—22no—
31#1956C2212yes—
32#1956C2212yes—
33#1956C2211yescharacteristic
34#1947—22no—
35#1933—22no—
36#1834—22no—
37#1834C2212yes—
38#1834C2212yes—
39#1834C2211yescharacteristic
40#1803—22no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43C4×#103—22no—
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—24no—
58#500—42no—
59#487—22no—
60#487—22no—
61#487—24no—
62#487—42no—
63C2×#107—24no—
64C2×C4⋊C8D412yes—
65C2×C4⋊C8D412yes—
66C2×#103—22no—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—24no—
70C2×#103—24no—
71C2×C4×C8D412yes—
72C2×C4×C8D412yes—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75C2×C4×C8C2312yes—
76C2×C4×C8C2311yescharacteristic
77#391—21no—
78#391—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384C2311yescharacteristic
91#384C2311yescharacteristic
92#384C2312yes—
93C4×C4⋊C4—21no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—21no—
97C43C2311yescharacteristic
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—22no—
104#340—24no—
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—44no—
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—48no—
170C4⋊C8—24no—
171C4⋊C8—24no—
172#103—28no—
173#103—48no—
174#103—44no—
175#103—84no—
176#103—84no—
177C4×C8—24no—
178C4×C8—21no—
179C4×C8—21no—
180C4×C8—22no—
181C4×C8—22no—
182C4×C8—21no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—21no—
186C4×C8—42no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191C2×Q8—22no—
192C2×Q8—22no—
193C2×Q8—41no—
194C2×Q8—41no—
195C2×Q8—42no—
196C2×Q8—42no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—44no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—82no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4#12711yescharacteristic
213C22×C4#12711yescharacteristic
214C22×C4#12711yescharacteristic
215C22×C4#12711yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8—24no—
224C2×C8—22no—
225C2×C8—22no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—24no—
230C2×C8—24no—
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—22no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—44no—
246C4⋊C4—42no—
247C4⋊C4—44no—
248C4⋊C4—42no—
249C4⋊C4—44no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C4⋊C4—82no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C42—22no—
262C42—22no—
263C42—22no—
264C42—82no—
265C42—82no—
266C42#13514yes—
267C42#13714yes—
268C42#13714yes—
269C42C2×#4111yescharacteristic
270C42C2×#4111yescharacteristic
271C42H(5,𝔽2)11yescharacteristic
272C422−1+411yescharacteristic
273C23C2×#12711yescharacteristic
274Q8—44no—
275Q8—44no—
276Q8—82no—
277Q8—82no—
278Q8—84no—
279Q8—84no—
280Q8—164no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—82no—
295C2×C4—81no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—81no—
304C2×C4—82no—
305C2×C4#49512yes—
306C2×C4#49512yes—
307C2×C4#49712yes—
308C2×C4#49712yes—
309C2×C4#49712yes—
310C2×C4#49712yes—
311C2×C4#52511yescharacteristic
312C2×C4#53711yescharacteristic
313C2×C4#54611yescharacteristic
314C2×C4#54611yescharacteristic
315C2×C4#54811yescharacteristic
316C2×C4#54811yescharacteristic
317C2×C4C2×#13512yes—
318C2×C4C2×#13712yes—
319C2×C4C2×#13712yes—
320C2×C4#57512yes—
321C2×C4#57612yes—
322C2×C4#57812yes—
323C8—44no—
324C8—42no—
325C8—42no—
326C8—44no—
327C22#254011yescharacteristic
328C22C2×#49511yescharacteristic
329C22C2×#49711yescharacteristic
330C22C2×#49711yescharacteristic
331C22#315311yescharacteristic
332C22#315511yescharacteristic
333C22#315611yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—82no—
338C4—82no—
339C4—161no—
340C4—162no—
341C4—162no—
342C4—161no—
343C4#171814yes—
344C4#311612yes—
345C4#325212yes—
346C4#330512yes—
347C4#332812yes—
348C4#336112yes—
349C4#337412yes—
350C2C2×#171811yescharacteristic
351C2#1962811yescharacteristic
352C2#1968211yescharacteristic
353C2#1970411yescharacteristic
354C2#2838411yescharacteristic
355C2#2839911yescharacteristic
356C2#2841711yescharacteristic
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
#292182
#291412
#287482
#286892
C4×#4872
#106822
#103132
#98402
#98312
#82612