Finite Group Databasebeta

← #276561

Subgroups — #276561

Order 512.

567 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33198C211yesmaximal, characteristic
2#29141C211yesmaximal, characteristic
3#22825C211yesmaximal, characteristic
4C4×#475C211yesmaximal, characteristic
5#10626C212yesmaximal
6#10487C212yesmaximal
7#9898C211yesmaximal, characteristic
8#9619C212yesmaximal
9#9601C211yesmaximal, characteristic
10#9569C212yesmaximal
11#8586C211yesmaximal, characteristic
12C2×#477—21no—
13C2×#475C2211yescharacteristic
14#2868C2211yescharacteristic
15C4×C4⋊Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#2024—22no—
18#1988C2211yescharacteristic
19#1987—21no—
20#1984C2212yes—
21#1956C2212yes—
22#1948C2211yescharacteristic
23#1947—21no—
24#1947C2212yes—
25#1919—21no—
26#1917—22no—
27#1865—21no—
28#1842C2211yescharacteristic
29#1842C2211yescharacteristic
30#1834C2212yes—
31#1831—21no—
32#1803—21no—
33#1803—21no—
34#1803—21no—
35#1803C2211yescharacteristic
36#1803C2212yes—
37C4×#106—21no—
38#1770C2212yes—
39#1767—21no—
40C4×C4⋊C8C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103—21no—
43C2×#335C2212yes—
44C2×#332C2212yes—
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—44no—
58#475—22no—
59#475—22no—
60C2×#107—22no—
61C2×#106—21no—
62C2×#106—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8D411yescharacteristic
68C2×C4⋊C8D411yescharacteristic
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2311yescharacteristic
72C2×C4⋊C8C2311yescharacteristic
73C2×#103—21no—
74C2×#103—22no—
75C2×#103—21no—
76C2×#103—22no—
77C2×#103—22no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8C2312yes—
81#391—21no—
82#389—21no—
83#389—21no—
84#384—21no—
85#384—21no—
86#384—21no—
87#384C2311yescharacteristic
88#384C2312yes—
89C4×C4⋊C4—21no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93C4×C4⋊C4—21no—
94C4×C4⋊C4D411yescharacteristic
95C4×C4⋊C4D411yescharacteristic
96C4×C4⋊C4C2311yescharacteristic
97C43—21no—
98C43C2311yescharacteristic
99#340—22no—
100#340—22no—
101#340—22no—
102#340—21no—
103#340—22no—
104#340—21no—
105#340—21no—
106#340—21no—
107#336—22no—
108#336—22no—
109#336—22no—
110#336—21no—
111#336D412yes—
112#335—24no—
113#332—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—42no—
127C4⋊Q8—42no—
128C4⋊Q8—81no—
129C4⋊Q8—81no—
130C4⋊Q8—82no—
131C4⋊Q8#4112yes—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—21no—
136C4×Q8—21no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8—41no—
140C4×Q8—42no—
141C4×Q8—42no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4C2×D411yescharacteristic
158C2×C4⋊C4C2×D411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—41no—
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42C2411yescharacteristic
174#107—44no—
175#106—24no—
176#106—42no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—21no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—22no—
186C4⋊C8—42no—
187C4⋊C8—42no—
188C4⋊C8—42no—
189C4⋊C8—81no—
190C4⋊C8#4112yes—
191#103—24no—
192#103—44no—
193#103—42no—
194#103—82no—
195#103—82no—
196#97—82no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—42no—
200#95—21no—
201#95—21no—
202#95—21no—
203#95—41no—
204#95—41no—
205C2×Q8—21no—
206C2×Q8—21no—
207C2×Q8—21no—
208C2×Q8—22no—
209C2×Q8—21no—
210C2×Q8—21no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8—41no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×Q8—82no—
219C2×Q8#13712yes—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—42no—
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4C22×D411yescharacteristic
232C22×C4C22×D411yescharacteristic
233C22×C4C2×#4111yescharacteristic
234C2×C8—22no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—42no—
244C2×C8—41no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—41no—
248C2×C8—42no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—22no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—22no—
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—21no—
259C4⋊C4—21no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—81no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—41no—
286C42—42no—
287C42—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—42no—
292C42—42no—
293C42—81no—
294C42—82no—
295C42#13512yes—
296C42AGL(1,ℤ/8)12yes—
297C42#13712yes—
298C42#13712yes—
299C42C2×#4111yescharacteristic
300C42C2×#4111yescharacteristic
301C422−1+411yescharacteristic
302C422−1+411yescharacteristic
303C23C2×#12111yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308Q8—42no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312Q8—164no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—82no—
342C2×C4—81no—
343C2×C4#46412yes—
344C2×C4#46412yes—
345C2×C4#46812yes—
346C2×C4#46812yes—
347C2×C4#47312yes—
348C2×C4#52511yescharacteristic
349C2×C4#53611yescharacteristic
350C2×C4#54711yescharacteristic
351C2×C4#54811yescharacteristic
352C2×C4#55211yescharacteristic
353C2×C4#55611yescharacteristic
354C2×C4C2×#13511yescharacteristic
355C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
356C2×C4C2×#13711yescharacteristic
357C2×C4C2×#13711yescharacteristic
358C2×C4C2×#13711yescharacteristic
359C2×C4#57511yescharacteristic
360C2×C4#57611yescharacteristic
361C2×C4#57711yescharacteristic
362C8—42no—
363C8—42no—
364C8—42no—
365C8—82no—
366C8—82no—
367C8—81no—
368C22#241511yescharacteristic
369C22C2×#46411yescharacteristic
370C22#304511yescharacteristic
371C22C2×#46811yescharacteristic
372C22#305611yescharacteristic
373C22C2×#47311yescharacteristic
374C22#306811yescharacteristic
375C4—21no—
376C4—21no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—42no—
381C4—81no—
382C4—82no—
383C4—81no—
384C4—81no—
385C4—161no—
386C4—162no—
387C4#170014yes—
388C4#311311yescharacteristic
389C4#311511yescharacteristic
390C4#323711yescharacteristic
391C4#323911yescharacteristic
392C4#334011yescharacteristic
393C4#335711yescharacteristic
394C4#336111yescharacteristic
395C4#337211yescharacteristic
396C2C2×#170011yescharacteristic
397C2#1917611yescharacteristic
398C2#1944511yescharacteristic
399C2#1949711yescharacteristic
400C2#2467911yescharacteristic
401C2#2471011yescharacteristic
402C2#2472211yescharacteristic
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
#291412
#228252
C4×#4752
#106262
#104872
#98982
#96192
#96012
#95692
#85862