Finite Group Databasebeta

← #352697

Subgroups — #352697

Order 512.

730 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33316C211yesmaximal, characteristic
2#33314C211yesmaximal, characteristic
3#33278C211yesmaximal, characteristic
4#33225C211yesmaximal, characteristic
5#30230C211yesmaximal, characteristic
6#30205C211yesmaximal, characteristic
7#9841C212yesmaximal
8#9836C212yesmaximal
9#9823C212yesmaximal
10#9819C212yesmaximal
11C8×#97C211yesmaximal, characteristic
12C2×#501—22no—
13C2×#500C2212yes—
14C2×#498C2212yes—
15C2×#487—22no—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18#2862C2211yescharacteristic
19#2859C2211yescharacteristic
20#2100—22no—
21#2100—22no—
22#2072—22no—
23#2024—22no—
24#1990C2212yes—
25#1990C2212yes—
26#1990C2211yescharacteristic
27#1990C2212yes—
28#1990C2211yescharacteristic
29#1956—22no—
30#1956C2212yes—
31#1956C2212yes—
32#1956C2211yescharacteristic
33#1956C2211yescharacteristic
34#1956C2212yes—
35#1933—22no—
36#1933—22no—
37#1847—22no—
38#1847—22no—
39#1847—22no—
40#1845C2212yes—
41#1844C2212yes—
42#1744C2211yescharacteristic
43C4×#97C2211yescharacteristic
44C42×C8C2211yescharacteristic
45C2×#322C2212yes—
46C2×C82C2212yes—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4⋊Q8C2311yescharacteristic
54C2×C4⋊Q8C2312yes—
55C2×#125—21no—
56#501—48no—
57#500—24no—
58#500—24no—
59#498—24no—
60#498—24no—
61#487—48no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#97D412yes—
72C2×#97D412yes—
73C2×C4×C8D412yes—
74C2×C4×C8D412yes—
75C2×C4×C8C2311yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2312yes—
78C2×C4×C8C2312yes—
79#391—21no—
80#391—21no—
81#391—21no—
82#391—21no—
83#389—21no—
84#389—21no—
85#389—21no—
86#389—21no—
87#384—21no—
88#384—21no—
89#384—21no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384—21no—
94#384C2311yescharacteristic
95#384C2311yescharacteristic
96#384C2312yes—
97#382—21no—
98#382—21no—
99#382—21no—
100C43C2311yescharacteristic
101#340—24no—
102#340—24no—
103#340—24no—
104#340—22no—
105#340—22no—
106#336D412yes—
107#336D412yes—
108#322—24no—
109C82—24no—
110C22×Q8—22no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—41no—
114C22×C8C2×D414yes—
115C22×C8C2×D412yes—
116C22×C8C2×D412yes—
117C22×C8C2×D414yes—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—44no—
127C4⋊Q8—44no—
128C4⋊Q8—44no—
129C4⋊Q8—44no—
130C4⋊Q8—82no—
131#125—44no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2411yescharacteristic
163#107—48no—
164#107—48no—
165#106—48no—
166#106—48no—
167#103—44no—
168#103—48no—
169#103—44no—
170#103—48no—
171#103—84no—
172#97SD1618yes—
173#97Q1618yes—
174C4×C8—24no—
175C4×C8—22no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—42no—
180C4×C8#4114yes—
181C4×C8#4114yes—
182#95—41no—
183#95—41no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C2×Q8—42no—
189C2×Q8—41no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—41no—
193C2×Q8—44no—
194C2×Q8—42no—
195C2×Q8—42no—
196C2×Q8—44no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8—82no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4#12711yescharacteristic
209C22×C4#12711yescharacteristic
210C22×C4#12711yescharacteristic
211C22×C4#12711yescharacteristic
212C22×C4C22×D411yescharacteristic
213C22×C4C22×D411yescharacteristic
214C22×C4C22×D411yescharacteristic
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8—24no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—24no—
222C2×C8—24no—
223C2×C8#12418yes—
224C2×C8C2×SD1618yes—
225C2×C8C2×Q1618yes—
226C2×C8AGL(1,ℤ/8)14yes—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—41no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—41no—
236C4⋊C4—42no—
237C4⋊C4—44no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—44no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C42—22no—
255C42—22no—
256C42—22no—
257C42—22no—
258C42C2×SD1614yes—
259C42C2×Q1614yes—
260C42C2×#4112yes—
261C42C2×#4112yes—
262C23C2×#12711yescharacteristic
263Q8—84no—
264Q8—82no—
265Q8—84no—
266Q8—84no—
267Q8—82no—
268Q8—164no—
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—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—82no—
291C2×C4—81no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—81no—
295C2×C4#49212yes—
296C2×C4#49212yes—
297C2×C4#49412yes—
298C2×C4#49412yes—
299C2×C4C2×#12412yes—
300C2×C4#53211yescharacteristic
301C2×C4#53212yes—
302C2×C4#53211yescharacteristic
303C2×C4C22×SD1612yes—
304C2×C4C22×Q1612yes—
305C2×C4C2×AGL(1,ℤ/8)12yes—
306C2×C4#57512yes—
307C2×C4#57712yes—
308C2×C4#57812yes—
309C8—24no—
310C8—42no—
311C8—44no—
312C8#487116yescomplemented
313C22#285811yescharacteristic
314C22C2×#49211yescharacteristic
315C22C2×#49411yescharacteristic
316C22#315111yescharacteristic
317C22#315311yescharacteristic
318C22#315411yescharacteristic
319C22#315611yescharacteristic
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—161no—
325C4—162no—
326C4—161no—
327C4—161no—
328C4—162no—
329C4—161no—
330C4C2×#48714yes—
331C4#317112yes—
332C4#317814yes—
333C4#318012yes—
334C2#1962111yescharacteristic
335C2#1962411yescharacteristic
336C2#1963211yescharacteristic
337C2#1963411yescharacteristic
338C2#3322511yescharacteristic
339C2#3324411yescharacteristic
340C2#3325111yescharacteristic
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

Complements

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

NH ≅ G/Ntype
312. C8#487semidirect

Maximal subgroups

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

subgroupindex
#333162
#333142
#332782
#332252
#302302
#302052
#98412
#98362
#98232
#98192
C8×#972