Finite Group Databasebeta

← #352503

Subgroups — #352503

Order 512.

588 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33297C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33067C211yesmaximal, characteristic
4Q8×#97C211yesmaximal, characteristic
5#9904C211yesmaximal, characteristic
6#9903C211yesmaximal, characteristic
7#9622C212yesmaximal
8#9619C212yesmaximal
9#8808C211yesmaximal, characteristic
10#8649C212yesmaximal
11#8647C212yesmaximal
12C2×#477—21no—
13C2×C8×Q8C2212yes—
14#2868C2211yescharacteristic
15#2859C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1993—22no—
18#1992—21no—
19#1992—21no—
20#1987—21no—
21#1948C2211yescharacteristic
22#1948C2211yescharacteristic
23#1947—21no—
24#1947C2212yes—
25#1919—21no—
26#1919—21no—
27#1865—21no—
28#1865—21no—
29#1842C2211yescharacteristic
30#1834C2212yes—
31#1834C2212yes—
32#1831C2211yescharacteristic
33#1803—21no—
34#1803C2211yescharacteristic
35#1803C2211yescharacteristic
36#1803C2212yes—
37#1770—22no—
38#1770C2212yes—
39#1767—21no—
40#1767C2211yescharacteristic
41C4×#97—21no—
42C4×#97C2211yescharacteristic
43C2×#335C2212yes—
44C2×#334C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#125—21no—
53C2×C4×Q8D411yescharacteristic
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8C2311yescharacteristic
56#477—44no—
57C8×Q8—24no—
58C8×Q8Q818yescomplemented
59C2×#107—22no—
60C2×#106—21no—
61C2×#106—21no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8C2311yescharacteristic
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2311yescharacteristic
68C2×C4⋊C8C2312yes—
69C2×#103—22no—
70C2×#103—21no—
71C2×#103—21no—
72C2×#97—22no—
73C2×#97—21no—
74C2×#97D411yescharacteristic
75C2×#97D411yescharacteristic
76C2×#97D412yes—
77C2×C4×C8—22no—
78C2×C4×C8C2312yes—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#384—21no—
84#384—21no—
85#384—21no—
86#384C2311yescharacteristic
87#384C2311yescharacteristic
88#384C2312yes—
89#382—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4D411yescharacteristic
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4C2311yescharacteristic
96C43—21no—
97C43C2311yescharacteristic
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#340—21no—
104#340—21no—
105#340—22no—
106#340—21no—
107#340—21no—
108#336—22no—
109#336—21no—
110#336—21no—
111#336—21no—
112#335—24no—
113#334—24no—
114C22×Q8—41no—
115C22×Q8C2×D411yescharacteristic
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8C2×D412yes—
121C22×C8C2×D412yes—
122C4⋊Q8—44no—
123C4⋊Q8—42no—
124C4⋊Q8—42no—
125C4⋊Q8—82no—
126#125—42no—
127#125—42no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8C2×Q814yes—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4C2×D411yescharacteristic
152C2×C4⋊C4C2×D411yescharacteristic
153C2×C4⋊C4C2×D411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42C2411yescharacteristic
167#107—44no—
168#106—42no—
169#106—42no—
170C4⋊C8—28no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8C2×Q814yes—
179#103—42no—
180#103—44no—
181#103—42no—
182#97—22no—
183#97—22no—
184#97—42no—
185#97—44no—
186#97—42no—
187C4×C8—28no—
188C4×C8C2×Q814yes—
189C4×C8#4114yes—
190#95—21no—
191#95—21no—
192#95—21no—
193#95—41no—
194#95—41no—
195C2×Q8—21no—
196C2×Q8—21no—
197C2×Q8—21no—
198C2×Q8—82no—
199C2×Q8—82no—
200C2×Q8—82no—
201C2×Q8C4⋊Q812yes—
202C2×Q8AGL(1,ℤ/8)12yes—
203C2×Q8#13712yes—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4C22×D411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C2×#4111yescharacteristic
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—22no—
222C2×C8—24no—
223C2×C8—42no—
224C2×C8—42no—
225C2×C8—42no—
226C2×C8—41no—
227C2×C8—41no—
228C2×C8—42no—
229C2×C8—41no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8C22⋊Q814yes—
235C2×C8C22⋊Q814yes—
236C4⋊C4—21no—
237C4⋊C4—24no—
238C4⋊C4—21no—
239C4⋊C4—21no—
240C4⋊C4—21no—
241C4⋊C4—21no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—81no—
256C4⋊C4—81no—
257C4⋊C4—82no—
258C4⋊C4—82no—
259C4⋊C4—82no—
260C4⋊C4—82no—
261C4⋊C4—82no—
262C4⋊C4—82no—
263C4⋊C4C4⋊Q812yes—
264C42—24no—
265C42—21no—
266C42—21no—
267C42—22no—
268C42—21no—
269C42—21no—
270C42—41no—
271C42—41no—
272C42—42no—
273C42—42no—
274C42—42no—
275C42—81no—
276C42—81no—
277C42#13512yes—
278C42#13712yes—
279C42C22×Q812yes—
280C42C2×#4112yes—
281C23C2×#12111yescharacteristic
282Q8—22no—
283Q8—22no—
284Q8—22no—
285Q8—164no—
286Q8#50114yes—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4—82no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4#46412yes—
321C2×C4#46412yes—
322C2×C4C2×C22⋊Q812yes—
323C2×C4C2×C4⋊Q811yescharacteristic
324C2×C4#53211yescharacteristic
325C2×C4#53312yes—
326C2×C4C2×#13511yescharacteristic
327C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
328C2×C4C2×#13711yescharacteristic
329C2×C4C2×#13711yescharacteristic
330C2×C4#57512yes—
331C2×C4#57511yescharacteristic
332C2×C4#57511yescharacteristic
333C8—24no—
334C8—24no—
335C8—44no—
336C8—82no—
337C8—81no—
338C8—81no—
339C22#285711yescharacteristic
340C22C2×#46411yescharacteristic
341C22#304011yescharacteristic
342C22#305511yescharacteristic
343C22#305611yescharacteristic
344C22#306811yescharacteristic
345C22#306911yescharacteristic
346C4—24no—
347C4—22no—
348C4—22no—
349C4—22no—
350C4—41no—
351C4—41no—
352C4—42no—
353C4—81no—
354C4—81no—
355C4—162no—
356C4—161no—
357C4—161no—
358C4#307612yes—
359C4#307812yes—
360C4C2×#50111yescharacteristic
361C4#316211yescharacteristic
362C4#318011yescharacteristic
363C4#318111yescharacteristic
364C2#1917611yescharacteristic
365C2#1917711yescharacteristic
366C2#1932411yescharacteristic
367C2#1932511yescharacteristic
368C2#3320211yescharacteristic
369C2#3321711yescharacteristic
370C2#3322311yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
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
58. C8×Q8Q8semidirect

Maximal subgroups

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

subgroupindex
#332972
#331982
#330672
Q8×#972
#99042
#99032
#96222
#96192
#88082
#86492
#86472