Finite Group Databasebeta

← #302884

Subgroups — #302884

Order 512.

845 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#59832C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#29208C211yesmaximal, characteristic
4#29206C211yesmaximal, characteristic
5#10575C214yesmaximal
6#9865C211yesmaximal, characteristic
7#9864C211yesmaximal, characteristic
8#8926C214yesmaximal
9C4×#332C211yesmaximal, characteristic
10C2×Q82—21no—
11#2868C2211yescharacteristic
12#2734—22no—
13#2582—21no—
14#2551—21no—
15#2551C2212yes—
16#2546—22no—
17#2451—22no—
18#2451—22no—
19#2365—21no—
20#2365C2212yes—
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23C4×C4⋊Q8C2211yescharacteristic
24#2100—22no—
25#2058—22no—
26#2057—22no—
27#1990—21no—
28#1990—21no—
29#1990C2212yes—
30#1987—22no—
31#1948C2214yes—
32#1947C2214yes—
33#1845—21no—
34#1844—21no—
35#1834C2212yes—
36C4×C4⋊C8C2211yescharacteristic
37C4×C4⋊C8C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C2×#332C2212yes—
40#1296C2212yes—
41#1279C2214yes—
42#1279C2214yes—
43Q82—44no—
44Q82—44no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—24no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8D411yescharacteristic
53C2×C4⋊Q8D411yescharacteristic
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×#107—24no—
61C2×#106—24no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4×C8—21no—
66C2×C4×C8—21no—
67C2×C4×C8C2312yes—
68#398—24no—
69#398—24no—
70#395—24no—
71#395—24no—
72#391—22no—
73#391—24no—
74#391—22no—
75#391—21no—
76#391—21no—
77#391—21no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—22no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389D412yes—
86#389D412yes—
87#389D412yes—
88#387—24no—
89#387—24no—
90#384—21no—
91#384—21no—
92#384—21no—
93#384D411yescharacteristic
94#384D411yescharacteristic
95#384C2312yes—
96#384C2311yescharacteristic
97C4×C4⋊C4—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4D412yes—
102C4×C4⋊C4C2312yes—
103C4×C4⋊C4C2312yes—
104C43C2311yescharacteristic
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—21no—
110#336—21no—
111#336—21no—
112#332Q818yescomplemented
113C22×Q8—24no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—41no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126C4⋊Q8—48no—
127C4⋊Q8—416no—
128C4⋊Q8—44no—
129C4⋊Q8—82no—
130C4⋊Q8—82no—
131C4⋊Q8—84no—
132C4⋊Q8SD1614yes—
133C4⋊Q8Q1614yes—
134C4⋊Q8C2×Q814yes—
135C4×Q8—44no—
136C4×Q8—48no—
137C4×Q8—44no—
138C4×Q8—42no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—42no—
142C4×Q8—82no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4C2×D412yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4C2×D412yes—
170C2×C42—24no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2411yescharacteristic
181#107—84no—
182#106—84no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—42no—
186C4⋊C8C2×Q814yes—
187C4⋊C8C2×Q814yes—
188C4×C8—24no—
189C4×C8—42no—
190C4×C8—42no—
191#95—24no—
192#95—24no—
193#95—24no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—22no—
200C2×Q8—24no—
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—48no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—42no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—81no—
210C2×Q8—81no—
211C2×Q8—82no—
212C2×Q8—82no—
213C2×Q8—82no—
214C2×Q8—84no—
215C2×Q8—84no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—42no—
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4C22×D411yescharacteristic
228C22×C4C22×D411yescharacteristic
229C22×C4C22×D411yescharacteristic
230C2×C8—24no—
231C2×C8—24no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—42no—
248C4⋊C4—48no—
249C4⋊C4—48no—
250C4⋊C4—44no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—42no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—81no—
261C4⋊C4—81no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C4⋊C4—82no—
267C4⋊C4C4⋊Q814yes—
268C4⋊C4C2×SD1614yes—
269C4⋊C4C2×Q1614yes—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—48no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—44no—
279C42C2×SD1612yes—
280C42C2×Q1612yes—
281C42AGL(1,ℤ/8)12yes—
282C42#13712yes—
283C42C22×Q812yes—
284C422−1+412yes—
285C23C2×#12011yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—48no—
289Q8—84no—
290Q8—168no—
291Q8—164no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#45012yes—
323C2×C4#45012yes—
324C2×C4#45112yes—
325C2×C4#45112yes—
326C2×C4#45412yes—
327C2×C4C2×C4⋊Q811yescharacteristic
328C2×C4#53611yescharacteristic
329C2×C4D4×Q812yes—
330C2×C4D4×Q812yes—
331C2×C4C22×SD1611yescharacteristic
332C2×C4C22×Q1611yescharacteristic
333C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
334C2×C4C2×#13711yescharacteristic
335C2×C4#57612yes—
336C2×C4#57712yes—
337C8—44no—
338C8—44no—
339C8—82no—
340C22#243411yescharacteristic
341C22C2×#45011yescharacteristic
342C22C2×#45111yescharacteristic
343C22#300811yescharacteristic
344C22#301011yescharacteristic
345C22C2×#45411yescharacteristic
346C22#302211yescharacteristic
347C4—22no—
348C4—21no—
349C4—21no—
350C4—22no—
351C4—41no—
352C4—44no—
353C4—44no—
354C4—44no—
355C4—82no—
356C4—82no—
357C4—82no—
358C4—162no—
359C4#163714yes—
360C4Q8×SD1612yes—
361C4Q8×Q1612yes—
362C4#338412yes—
363C4#338512yes—
364C2C2×#163711yescharacteristic
365C2#1872911yescharacteristic
366C2#1890311yescharacteristic
367C2#1892511yescharacteristic
368C2#2582611yescharacteristic
369C2#2583411yescharacteristic
370C2#2584111yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#120
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
112. #332Q8semidirect

Maximal subgroups

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

subgroupindex
#598322
#292182
#292082
#292062
#105752
#98652
#98642
#89262
C4×#3322