Finite Group Databasebeta

← #351636

Subgroups — #351636

Order 512.

553 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29939C211yesmaximal, characteristic
2#29936C211yesmaximal, characteristic
3#29844C211yesmaximal, characteristic
4#20537C211yesmaximal, characteristic
5#9737C212yesmaximal, complemented
6#9614C212yesmaximal, complemented
7#9576C212yesmaximal, complemented
8#9569C212yesmaximal, complemented
9#9140C211yesmaximal, characteristic
10#9136C211yesmaximal, characteristic
11#8599C211yesmaximal, characteristic
12C2×#487C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2857C2211yescharacteristic
16#2857C2211yescharacteristic
17#2285C2211yescharacteristic
18#2100—21no—
19#2099—21no—
20#2083—21no—
21#2083—21no—
22#2076—22no—
23#2075—22no—
24#2068—22no—
25#2036—22no—
26#1990C2211yescharacteristic
27#1988C2212yes—
28#1987C2212yes—
29#1956C2211yescharacteristic
30#1947C2212yes—
31#1947C2212yes—
32#1845C2211yescharacteristic
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1771—22no—
36#1770C2212yes—
37C8×C4⋊C4C2212yes—
38#1750—22no—
39C2×#335C2212yes—
40C2×#333C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2212yes—
45C2×C4⋊Q8C2311yescharacteristic
46C2×C4⋊Q8C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#487—24no—
52#432—24no—
53#431—24no—
54C2×#107—22no—
55C2×#107—22no—
56C2×#107—22no—
57C2×C4⋊C8D412yes—
58C2×C4⋊C8C2312yes—
59C2×C4⋊C8C2312yes—
60C2×C4⋊C8C2312yes—
61C2×#103—22no—
62C2×#103—22no—
63C2×C22⋊C8—21no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8D412yes—
67C2×#97—21no—
68C2×#97—22no—
69C2×C4×C8—22no—
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72#391—21no—
73#391—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#384D411yescharacteristic
79#384D411yescharacteristic
80#384C2311yescharacteristic
81#384C2311yescharacteristic
82#382—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380—21no—
87#380—21no—
88#380D412yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2312yes—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—22no—
99#336—22no—
100#336—21no—
101#336—21no—
102#335—24no—
103#333—24no—
104#332—24no—
105C22×Q8—21no—
106C22×Q8—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8C2×D412yes—
114C22×C8C2×D412yes—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C22⋊Q8—82no—
120C22⋊Q8—82no—
121#117—82no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4C2×D412yes—
139C2×C4⋊C4C2×D412yes—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C42—22no—
144C2×C42—22no—
145C2×C42—22no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42C2411yescharacteristic
150#107—82no—
151#107—82no—
152#107—82no—
153C4⋊C8—24no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8#4114yes—
160#103—44no—
161#103—82no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164C22⋊C8—81no—
165C22⋊C8—81no—
166#97—24no—
167#97—82no—
168C4×C8—22no—
169C4×C8—82no—
170C4×C8#4114yes—
171#95—22no—
172#95—22no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—41no—
177C24#12111yescharacteristic
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—81no—
184C2×Q8—81no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—21no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—41no—
202C22×C4#12111yescharacteristic
203C22×C4#12111yescharacteristic
204C22×C4#12111yescharacteristic
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D411yescharacteristic
207C22×C4C2×#4111yescharacteristic
208C2×C8—22no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—24no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8#12314yes—
223C4⋊C4—22no—
224C4⋊C4—24no—
225C4⋊C4—22no—
226C4⋊C4—41no—
227C4⋊C4—41no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—41no—
233C4⋊C4—41no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—41no—
237C4⋊C4—81no—
238C4⋊C4—81no—
239C4⋊C4—81no—
240C4⋊C4—81no—
241C4⋊C4—81no—
242C4⋊C4—81no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C22⋊C4—82no—
246C22⋊C4—82no—
247C22⋊C4—82no—
248C42—21no—
249C42—21no—
250C42—22no—
251C42—22no—
252C42—42no—
253C42—42no—
254C42C2×#4112yes—
255C42C2×#4112yes—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23C2×#12111yescharacteristic
264Q8—82no—
265Q8—82no—
266Q8—162no—
267C2×C4—21no—
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—21no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4C2×#12311yescharacteristic
302C2×C4#52511yescharacteristic
303C2×C4#52912yes—
304C2×C4#53212yes—
305C2×C4#57612yes—
306C2×C4#57611yescharacteristic
307C2×C4#57812yes—
308C2×C4#57812yes—
309C2×C4#57811yescharacteristic
310C8—24no—
311C8—42no—
312C8—44no—
313C8—82no—
314C8—82no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22#285411yescharacteristic
323C22#304111yescharacteristic
324C22#304411yescharacteristic
325C22#305911yescharacteristic
326C22#306111yescharacteristic
327C22#306811yescharacteristic
328C22#306911yescharacteristic
329C4—22no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4—82no—
335C4—161no—
336C4—161no—
337C4—161no—
338C4—161no—
339C4#309812yes—
340C4#312412yes—
341C4#317512yes—
342C4#317712yes—
343C2—81no—
344C2#1950711yescharacteristic
345C2#1951011yescharacteristic
346C2#1953111yescharacteristic
347C2#1953411yescharacteristic
348C2#3312011yescharacteristic
349C2#3312711yescharacteristic
350C2#3313911yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #9737C2semidirect
6. #9614C2semidirect
7. #9576C2semidirect
8. #9569C2semidirect

Maximal subgroups

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

subgroupindex
#299392
#299362
#298442
#205372
#97372
#96142
#95762
#95692
#91402
#91362
#85992