Finite Group Databasebeta

← #276535

Subgroups — #276535

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#29218C211yesmaximal, characteristic
3#22828C211yesmaximal, characteristic
4#21616C211yesmaximal, characteristic
5#10624C212yesmaximal
6#10483C212yesmaximal
7#9903C211yesmaximal, characteristic
8#9616C212yesmaximal
9#9604C211yesmaximal, characteristic
10#9568C212yesmaximal
11#8590C211yesmaximal, characteristic
12C2×#477—21no—
13C2×#477C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#2100—21no—
18#2100—21no—
19#1990C2212yes—
20#1987—22no—
21#1987—21no—
22#1987C2211yescharacteristic
23#1984C2212yes—
24#1948C2211yescharacteristic
25#1947—21no—
26#1947C2212yes—
27#1918—21no—
28#1916—22no—
29#1865—21no—
30#1842C2211yescharacteristic
31#1839—21no—
32#1834C2212yes—
33#1831C2211yescharacteristic
34#1803—21no—
35#1803C2211yescharacteristic
36#1803C2212yes—
37#1776—21no—
38C8×C4⋊C4C2212yes—
39#1767C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1763—21no—
42#1744—21no—
43C2×#335C2212yes—
44C2×#333C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1277C2212yes—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8C2311yescharacteristic
53C2×C4×Q8—21no—
54C2×C4×Q8D411yescharacteristic
55C2×C4×Q8D411yescharacteristic
56C2×C4×Q8C2311yescharacteristic
57#477—22no—
58#477—22no—
59#477—44no—
60C2×#107—21no—
61C2×#107—22no—
62C2×#106—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2311yescharacteristic
69C2×C4⋊C8C2311yescharacteristic
70C2×#103—21no—
71C2×#103—22no—
72C2×#103—21no—
73C2×#97—21no—
74C2×C4×C8—22no—
75C2×C4×C8C2312yes—
76#391—21no—
77#389—21no—
78#389—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384C2311yescharacteristic
83#384C2312yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4D411yescharacteristic
90C4×C4⋊C4D411yescharacteristic
91C4×C4⋊C4C2311yescharacteristic
92C43—21no—
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—21no—
100#340—22no—
101#340—21no—
102#340—21no—
103#340—21no—
104#336—22no—
105#336—22no—
106#336—22no—
107#336—21no—
108#336—21no—
109#336—21no—
110#336D411yescharacteristic
111#336D411yescharacteristic
112#335—24no—
113#333—24no—
114C22×Q8—21no—
115C22×Q8—41no—
116C22×Q8C2×D411yescharacteristic
117C22×C8—21no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—22no—
121C22×C8C2×D412yes—
122C22×C8C2×D412yes—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—44no—
127C4⋊Q8—42no—
128C4⋊Q8—42no—
129C4⋊Q8—82no—
130C4⋊Q8#4112yes—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—41no—
138C4×Q8—82no—
139C4×Q8Q1614yes—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4C2×D411yescharacteristic
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—41no—
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42C2411yescharacteristic
172#107—24no—
173#107—44no—
174#106—42no—
175C4⋊C8—21no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—22no—
182C4⋊C8—42no—
183C4⋊C8—42no—
184C4⋊C8—42no—
185C4⋊C8#4112yes—
186#103—24no—
187#103—42no—
188#103—44no—
189#97—42no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—44no—
193#95—21no—
194#95—21no—
195#95—21no—
196#95—41no—
197#95—41no—
198C2×Q8—21no—
199C2×Q8—21no—
200C2×Q8—21no—
201C2×Q8—22no—
202C2×Q8—21no—
203C2×Q8—21no—
204C2×Q8—41no—
205C2×Q8—82no—
206C2×Q8—82no—
207C2×Q8—82no—
208C2×Q8—81no—
209C2×Q8—81no—
210C2×Q8C2×Q1612yes—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—42no—
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4C22×D411yescharacteristic
223C22×C4C22×D411yescharacteristic
224C22×C4C2×#4111yescharacteristic
225C2×C8—22no—
226C2×C8—22no—
227C2×C8—22no—
228C2×C8—22no—
229C2×C8—22no—
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—42no—
235C2×C8—41no—
236C2×C8—42no—
237C2×C8—41no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—41no—
242C2×C8—42no—
243C4⋊C4—21no—
244C4⋊C4—21no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—21no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—81no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C4⋊C4C2×Q1612yes—
268C4⋊C4AGL(1,ℤ/8)12yes—
269C42—21no—
270C42—22no—
271C42—21no—
272C42—22no—
273C42—24no—
274C42—22no—
275C42—41no—
276C42—42no—
277C42—41no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42—81no—
282C42—82no—
283C42C2×Q1612yes—
284C42#13512yes—
285C42C2×#4111yescharacteristic
286C42C2×#4111yescharacteristic
287C422−1+411yescharacteristic
288C422−1+411yescharacteristic
289C23C2×#12111yescharacteristic
290Q8—22no—
291Q8—22no—
292Q8—41no—
293Q8—41no—
294Q8—82no—
295Q8—82no—
296Q8—164no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4—82no—
324C2×C4—81no—
325C2×C4—82no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4#46712yes—
330C2×C4#46712yes—
331C2×C4#47112yes—
332C2×C4#52511yescharacteristic
333C2×C4#53611yescharacteristic
334C2×C4#54711yescharacteristic
335C2×C4#54811yescharacteristic
336C2×C4#55211yescharacteristic
337C2×C4#55611yescharacteristic
338C2×C4C22×Q1611yescharacteristic
339C2×C4C2×#13511yescharacteristic
340C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
341C2×C4#57512yes—
342C2×C4#57511yescharacteristic
343C2×C4#57812yes—
344C8—42no—
345C8—42no—
346C8—42no—
347C8—82no—
348C8—82no—
349C8—81no—
350C22#241511yescharacteristic
351C22#304011yescharacteristic
352C22#304411yescharacteristic
353C22C2×#46711yescharacteristic
354C22#305511yescharacteristic
355C22C2×#47111yescharacteristic
356C22#306811yescharacteristic
357C4—22no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—22no—
362C4—42no—
363C4—41no—
364C4—42no—
365C4—42no—
366C4—81no—
367C4—82no—
368C4—161no—
369C4—162no—
370C4#310411yescharacteristic
371C4#310611yescharacteristic
372C4#323011yescharacteristic
373C4#323311yescharacteristic
374C4#334912yes—
375C4#337712yes—
376C2#1927711yescharacteristic
377C2#1931611yescharacteristic
378C2#1940511yescharacteristic
379C2#1947411yescharacteristic
380C2#2469911yescharacteristic
381C2#2470211yescharacteristic
382C2#2472111yescharacteristic
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

Maximal subgroups

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

subgroupindex
#331982
#292182
#228282
#216162
#106242
#104832
#99032
#96162
#96042
#95682
#85902