Finite Group Databasebeta

← #7234090

Subgroups — #7234090

Order 512.

1542 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#46333C212yesmaximal, complemented
2#33265C212yesmaximal, complemented
3#32862C211yesmaximal, characteristic, complemented
4#32858C211yesmaximal, characteristic, complemented
5#31723C213yesmaximal, complemented
6#31719C213yesmaximal, complemented
7#31670C216yesmaximal, complemented
8#27659C213yesmaximal, complemented
9#24777C216yesmaximal, complemented
10#24265C213yesmaximal, complemented
11C8×#127C211yesmaximal, characteristic, complemented
12C22×#135C2211yescharacteristic
13C2×#465C2216yes—
14C2×C8×D4C2213yes—
15C2×C8×D4C2213yes—
16C2×#427C2213yes—
17#2863C2212yes—
18#2860C2212yes—
19#2818—22no—
20#2803—26no—
21#2797—26no—
22#2485C2216yes—
23#2415C2216yes—
24#2414C2216yes—
25#2407C2216yes—
26#2383—22no—
27C4×#127C2211yescharacteristic
28#2087C2213yes—
29#2086C2213yes—
30#2076C2213yes—
31#2075C2213yes—
32#2068C2216yes—
33#2067C2216yes—
34#2056C2213yes—
35#2054C2213yes—
36#2036C2216yes—
37#2033C2216yes—
38#2001C2213yes—
39#1999C2213yes—
40#1968C2216yes—
41#1954C2213yes—
42#1954C2213yes—
43#1941C2216yes—
44#1934C2216yes—
45#1933C2216yes—
46#1875C2216yes—
47#1873C2216yes—
48#1842C2211yescharacteristic
49#1825C2216yes—
50#1807C2213yes—
51#1807C2213yes—
52#1788C2216yes—
53C42×C8C2211yescharacteristic
54C22×#41C2312yes—
55C2×#135—46no—
56C22×Q16C2311yescharacteristic
57C22×SD16C2312yes—
58C22×D8C2311yescharacteristic
59C23×C8C2311yescharacteristic
60C23×C8C2313yes—
61C2×#127D411yescharacteristic
62C2×#126—22no—
63C2×C22⋊Q8C2316yes—
64C2×#121C2316yes—
65C2×C4×D4C2313yes—
66C2×C4×D4C2313yes—
67C2×#117C2316yes—
68#465—412no—
69C8×D4—46no—
70C8×D4—46no—
71#427—46no—
72C2×#107C2313yes—
73C2×#107C2313yes—
74C2×#106C2316yes—
75C2×C4⋊C8C2313yes—
76C2×C4⋊C8C2313yes—
77C2×#103C2316yes—
78C2×#103C2316yes—
79C2×#102C2316yes—
80C2×#102C2316yes—
81C2×C22⋊C8C2313yes—
82C2×C22⋊C8C2313yes—
83C2×C22⋊C8C2316yes—
84C2×C4×C8C2313yes—
85C2×C4×C8C2313yes—
86#397—26no—
87#397—26no—
88#394—26no—
89#394—26no—
90#394—26no—
91#394—26no—
92#392—26no—
93#392—26no—
94#391C2316yes—
95#390—26no—
96#390D413yes—
97#390D413yes—
98#390C2316yes—
99#389—26no—
100#389C2316yes—
101#389C2316yes—
102#388—26no—
103#388—26no—
104#388—26no—
105#388—26no—
106#387—26no—
107#387—26no—
108#386—26no—
109#385—26no—
110#384D411yescharacteristic
111#383—22no—
112#380C2316yes—
113#380C2316yes—
114C43C2311yescharacteristic
115#340C2313yes—
116#340C2313yes—
117#340C2313yes—
118#340C2313yes—
119#340C2316yes—
120#340C2316yes—
121C2×#41—46no—
122C2×#41—46no—
123C22×Q8C2412yes—
124C22×D4—26no—
125C22×D4—22no—
126C22×D4C2×D413yes—
127C22×D4C2×D413yes—
128C22×D4C2412yes—
129C23×C4—22no—
130C23×C4C2413yes—
131C23×C4C2411yescharacteristic
132#135—416no—
133C2×Q16—43no—
134C2×SD16—212no—
135C2×D8—43no—
136C22×C8—26no—
137C22×C8—26no—
138C22×C8—212no—
139C22×C8C2411yescharacteristic
140C22×C8C2413yes—
141C22×C8C2413yes—
142C22×C8C2411yescharacteristic
143#127—24no—
144#126—48no—
145C22⋊Q8—46no—
146C22⋊Q8—46no—
147#121—46no—
148#121—46no—
149C4×D4—43no—
150C4×D4—43no—
151C4×D4—43no—
152C4×D4—43no—
153#117—46no—
154#117—46no—
155C2×C4⋊C4—26no—
156C2×C4⋊C4—26no—
157C2×C4⋊C4—26no—
158C2×C4⋊C4C2×D413yes—
159C2×C4⋊C4C2×D413yes—
160C2×C4⋊C4C2416yes—
161C2×C4⋊C4C2416yes—
162C2×C22⋊C4—26no—
163C2×C22⋊C4—26no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—26no—
166C2×C22⋊C4—26no—
167C2×C22⋊C4—26no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4C2×D413yes—
170C2×C22⋊C4C2×D413yes—
171C2×C22⋊C4C2×D416yes—
172C2×C42—26no—
173C2×C42C2×D413yes—
174C2×C42C2×D411yescharacteristic
175C2×C42C2413yes—
176#107—43no—
177#107—43no—
178#106—212no—
179C4⋊C8—43no—
180C4⋊C8—43no—
181#103—46no—
182#103—46no—
183#102—46no—
184#102—46no—
185C22⋊C8—46no—
186C22⋊C8—43no—
187C22⋊C8—43no—
188C4×C8—26no—
189C4×C8—43no—
190#95—26no—
191#95—26no—
192#95—26no—
193#95—26no—
194#95—26no—
195#95—26no—
196C24—22no—
197C24C22×D411yescharacteristic
198C24C22×D413yes—
199#41—48no—
200#41—812no—
201C2×Q8—26no—
202C2×Q8—26no—
203C2×D4—26no—
204C2×D4—26no—
205C2×D4—26no—
206C2×D4—26no—
207C2×D4—43no—
208C2×D4—43no—
209C2×D4—43no—
210C2×D4—43no—
211C2×D4—86no—
212C2×D4—86no—
213C2×D4—86no—
214C2×D4—86no—
215C22×C4—23no—
216C22×C4—26no—
217C22×C4—23no—
218C22×C4—23no—
219C22×C4—26no—
220C22×C4—23no—
221C22×C4—26no—
222C22×C4—22no—
223C22×C4—26no—
224C22×C4—86no—
225C22×C4C22×D411yescharacteristic
226C22×C4C22×D413yes—
227C22×C4C22×D413yes—
228C22×C4C22×D413yes—
229C22×C4C2511yescharacteristic
230Q16—44no—
231SD16—48no—
232D8—44no—
233C2×C8—23no—
234C2×C8—26no—
235C2×C8—23no—
236C2×C8—23no—
237C2×C8—26no—
238C2×C8—412no—
239C2×C8—44no—
240C2×C8H(5,𝔽2)16yes—
241C4⋊C4—26no—
242C4⋊C4—26no—
243C4⋊C4—26no—
244C4⋊C4—26no—
245C4⋊C4—43no—
246C4⋊C4—43no—
247C4⋊C4—412no—
248C4⋊C4—86no—
249C4⋊C4—86no—
250C22⋊C4—43no—
251C22⋊C4—46no—
252C22⋊C4—43no—
253C22⋊C4—412no—
254C22⋊C4—86no—
255C22⋊C4—86no—
256C22⋊C4—86no—
257C22⋊C4—86no—
258C22⋊C4—86no—
259C22⋊C4—86no—
260C42—23no—
261C42—23no—
262C42—43no—
263C42—86no—
264C42#13514yescomplemented
265C23—23no—
266C23—26no—
267C23—23no—
268C23—23no—
269C23—26no—
270C23—23no—
271C23—21no—
272C23—23no—
273C23—46no—
274C23—46no—
275C23—42no—
276C23C23×D411yescharacteristic
277Q8—42no—
278Q8—46no—
279D4—412no—
280D4—412no—
281D4—42no—
282D4—46no—
283D4—88no—
284D4—1612no—
285C2×C4—23no—
286C2×C4—23no—
287C2×C4—26no—
288C2×C4—23no—
289C2×C4—23no—
290C2×C4—26no—
291C2×C4—23no—
292C2×C4—43no—
293C2×C4—41no—
294C2×C4—43no—
295C2×C4—46no—
296C2×C4—43no—
297C2×C4—46no—
298C2×C4—46no—
299C2×C4—46no—
300C2×C4—46no—
301C2×C4—46no—
302C2×C4—46no—
303C2×C4—46no—
304C2×C4—88no—
305C2×C4C2×#13516yes—
306C2×C4C2×H(5,𝔽2)13yes—
307C2×C4#58513yes—
308C8—43no—
309C8—41no—
310C8—43no—
311C8#56014yes—
312C22—43no—
313C22—46no—
314C22—43no—
315C22—46no—
316C22—43no—
317C22—43no—
318C22—41no—
319C22—43no—
320C22—86no—
321C22—86no—
322C22—82no—
323C22#349613yes—
324C22C22×#13511yescharacteristic
325C22#358413yes—
326C4—26no—
327C4—43no—
328C4—43no—
329C4—41no—
330C4—86no—
331C4—86no—
332C4—82no—
333C4C2×#56011yescharacteristic
334C4#355113yes—
335C2—83no—
336C2—81no—
337C2—162no—
338C2#4632811yescharacteristic
339C2#6224813yes—
340C2#6232713yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #46333C2semidirect
2. #33265C2semidirect
3. #32862C2semidirect
4. #32858C2semidirect
5. #31723C2semidirect
6. #31719C2semidirect
7. #31670C2semidirect
8. #27659C2semidirect
9. #24777C2semidirect
10. #24265C2semidirect
11. C8×#127C2semidirect
264. C42#135semidirect

Maximal subgroups

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

subgroupindex
#463332
#332652
#328622
#328582
#317232
#317192
#316702
#276592
#247772
#242652
C8×#1272