Finite Group Databasebeta

← #315754

Subgroups — #315754

Order 512.

894 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33175C211yesmaximal, characteristic
2#27906C212yesmaximal, complemented
3#27890C212yesmaximal, complemented
4#20800C211yesmaximal, characteristic, complemented
5#10683C212yesmaximal, complemented
6#10672C212yesmaximal, complemented
7#9837C211yesmaximal, characteristic, complemented
8#9824C211yesmaximal, characteristic, complemented
9#8819C211yesmaximal, characteristic, complemented
10#8813C211yesmaximal, characteristic, complemented
11#8146C211yesmaximal, characteristic, complemented
12C2×#493C2211yescharacteristic
13C2×#481—24no—
14C2×#469—22no—
15C2×#428—22no—
16C2×C4×M16C2211yescharacteristic
17#2856C2212yes—
18#2285C2211yescharacteristic
19C2×#384C2211yescharacteristic
20#2094—24no—
21#2034—24no—
22#1972—22no—
23#1957C2212yes—
24#1957C2212yes—
25#1957C2212yes—
26#1957C2212yes—
27#1957C2212yes—
28#1926—24no—
29#1847—22no—
30#1845C2211yescharacteristic
31#1845C2211yescharacteristic
32#1844C2212yes—
33#1844C2212yes—
34#1844C2211yescharacteristic
35#1844C2211yescharacteristic
36#1844C2211yescharacteristic
37#1844C2212yes—
38#1834C2212yes—
39C4×C22⋊C8C2212yes—
40C2×#322C2211yescharacteristic
41C2×C82C2211yescharacteristic
42#1273C2212yes—
43C22×D8—22no—
44C22×M16D412yes—
45C2×#127C2312yes—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121—22no—
49C2×#117—21no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—21no—
52C22×C42C2311yescharacteristic
53#493—22no—
54#493—22no—
55#481—416no—
56#469—48no—
57#428—48no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#106—24no—
61C2×#106—24no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106—22no—
65C2×#106—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—24no—
70C2×#102—24no—
71C2×C22⋊C8D414yes—
72C4×M16—22no—
73C4×M16—22no—
74C2×#97D412yes—
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78C2×C4×C8C2311yescharacteristic
79#390—22no—
80#390—22no—
81#390—22no—
82#384—22no—
83#384—21no—
84#384C2312yes—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2311yescharacteristic
89#382—21no—
90#380—22no—
91C4×C4⋊C4—22no—
92#340—24no—
93#336D414yes—
94#322—22no—
95C82—22no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—22no—
99C22×D4—42no—
100C23×C4—41no—
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C2×D8—44no—
104C2×D8—44no—
105C2×D8—44no—
106C2×M16—24no—
107C2×M16—24no—
108C2×M16SD1618yes—
109C22×C8C2×D414yes—
110C22×C8C2×D412yes—
111C22×C8C2×D414yes—
112C22×C8C2×D412yes—
113#127—22no—
114#127—22no—
115#127—22no—
116#127—22no—
117#121—48no—
118#121—48no—
119#121—84no—
120#117—44no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—44no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—41no—
136C2×C22⋊C4—41no—
137C2×C22⋊C4—42no—
138C2×C22⋊C4—42no—
139C2×C22⋊C4—42no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—22no—
143C2×C42—41no—
144C2×C42C2×D412yes—
145C2×C42C2×D411yescharacteristic
146C2×C42C2411yescharacteristic
147#107—44no—
148#107—44no—
149#106—44no—
150#106—48no—
151#106—44no—
152#106—44no—
153#106—48no—
154#106—84no—
155#102—48no—
156#102—48no—
157#102—48no—
158#102—48no—
159#102—84no—
160C22⋊C8—28no—
161#97—24no—
162C4×C8—22no—
163C4×C8—22no—
164C4×C8—21no—
165C4×C8—21no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8#4112yes—
169C4×C8#4112yes—
170#95—42no—
171C24—42no—
172C24#12711yescharacteristic
173C2×D4—42no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—42no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—44no—
181C2×D4—44no—
182C2×D4—44no—
183C2×D4—44no—
184C2×D4—44no—
185C2×D4—84no—
186C2×D4—84no—
187C2×D4—84no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—22no—
193C22×C4—24no—
194C22×C4—24no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—82no—
201C22×C4—84no—
202C22×C4#12712yes—
203C22×C4#12711yescharacteristic
204C22×C4C2×SD1614yes—
205C22×C4C22×D411yescharacteristic
206C22×C4C22×D412yes—
207D8—88no—
208M16—216no—
209C2×C8—22no—
210C2×C8—24no—
211C2×C8—22no—
212C2×C8—24no—
213C2×C8—24no—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8#12314yes—
217C2×C8C2×SD1614yes—
218C2×C8C2×SD1614yes—
219C2×C8AGL(1,ℤ/8)18yes—
220C2×C8AGL(1,ℤ/8)14yes—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—44no—
224C4⋊C4—44no—
225C4⋊C4—42no—
226C4⋊C4—44no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—44no—
231C4⋊C4—44no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C22⋊C4—84no—
235C22⋊C4—84no—
236C22⋊C4—82no—
237C22⋊C4—84no—
238C42—21no—
239C42—21no—
240C42—22no—
241C42—22no—
242C42—82no—
243C42C2×#4111yescharacteristic
244C42C2×#4111yescharacteristic
245C422−1+412yes—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—21no—
250C23—82no—
251C23—82no—
252C23—82no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23—82no—
257C23#49212yes—
258C23C2×#12711yescharacteristic
259D4—84no—
260D4—84no—
261D4—84no—
262D4—84no—
263D4—84no—
264D4—84no—
265D4—168no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—44no—
274C2×C4—44no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—164no—
287C2×C4#49212yes—
288C2×C4C2×#12311yescharacteristic
289C2×C4#53611yescharacteristic
290C2×C4#54712yes—
291C2×C4#54712yes—
292C2×C4C22×SD1612yes—
293C2×C4C2×AGL(1,ℤ/8)12yes—
294C2×C4C2×AGL(1,ℤ/8)14yes—
295C2×C4#57714yes—
296C8—24no—
297C8—28no—
298C8—44no—
299C8#48118yescomplemented
300C22—22no—
301C22—24no—
302C22—41no—
303C22—41no—
304C22—42no—
305C22—162no—
306C22—162no—
307C22—162no—
308C22—162no—
309C22—162no—
310C22—162no—
311C22—162no—
312C22#250111yescharacteristic
313C22C2×#49211yescharacteristic
314C22#315111yescharacteristic
315C22#315112yes—
316C22#315412yes—
317C4—22no—
318C4—42no—
319C4—161no—
320C4—161no—
321C4—161no—
322C4—161no—
323C4—162no—
324C4C2×#48112yes—
325C4#322712yes—
326C4#334514yes—
327C4#335414yes—
328C2—42no—
329C2—322no—
330C2#1962112yes—
331C2#1972111yescharacteristic
332C2#1974511yescharacteristic
333C2#2790611yescharacteristic
334C2#2794112yes—
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#33175C2
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
2. #27906C2semidirect
3. #27890C2semidirect
4. #20800C2semidirect
5. #10683C2semidirect
6. #10672C2semidirect
7. #9837C2semidirect
8. #9824C2semidirect
9. #8819C2semidirect
10. #8813C2semidirect
11. #8146C2semidirect
299. C8#481semidirect

Maximal subgroups

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

subgroupindex
#331752
#279062
#278902
#208002
#106832
#106722
#98372
#98242
#88192
#88132
#81462