Finite Group Databasebeta

← #6849746

Subgroups — #6849746

Order 512.

1518 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36181C212yesmaximal
2#33263C211yesmaximal, characteristic, complemented
3#32516C214yesmaximal, complemented
4#32469C214yesmaximal
5#29877C212yesmaximal, complemented
6#29179C212yesmaximal, complemented
7#27950C212yesmaximal, complemented
8#27875C212yesmaximal, complemented
9#27475C218yesmaximal, complemented
10#21982C212yesmaximal, complemented
11#21727C211yesmaximal, characteristic, complemented
12#21265C211yesmaximal, characteristic
13C2×#552—22no—
14C2×#498C2211yescharacteristic
15C2×#431C2212yes—
16#2859C2212yes—
17#2857C2212yes—
18#2857C2212yes—
19#2565—22no—
20#2506—22no—
21#2506C2214yes—
22#2503—22no—
23#2499—22no—
24#2496—22no—
25#2496C2214yes—
26#2486—22no—
27#2486C2214yes—
28#2344—22no—
29#2344C2214yes—
30C4×C4⋊Q8C2212yes—
31#2285C2212yes—
32C2×#384C2212yes—
33C42×D4C2211yescharacteristic
34#2088C2218yes—
35#2087C2218yes—
36#2083C2218yes—
37#2077C2218yes—
38#2077C2218yes—
39#1990C2212yes—
40#1990C2212yes—
41#1990C2212yes—
42#1955C2212yes—
43#1947C2218yes—
44#1944C2214yes—
45#1944C2214yes—
46#1943C2214yes—
47#1940C2212yes—
48#1924C2214yes—
49#1921C2218yes—
50#1920C2218yes—
51#1852C2218yes—
52#1852C2218yes—
53#1845C2211yescharacteristic
54#1845C2212yes—
55#1845C2211yescharacteristic
56#1844C2211yescharacteristic
57#1834C2212yes—
58#1834C2212yes—
59#1776C2214yes—
60C4×C4⋊C8C2211yescharacteristic
61C4×C22⋊C8C2212yes—
62C4×#97C2211yescharacteristic
63#552—416no—
64C2×C4⋊Q8C2312yes—
65C2×#126—24no—
66C2×#125—24no—
67C2×#125—22no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×C22⋊Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×D4D412yes—
73C2×C4×D4C2311yescharacteristic
74C2×#117—22no—
75C2×#117—24no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4C2314yes—
79C22×C42C2312yes—
80#498—24no—
81#431—28no—
82C2×#107C2314yes—
83C2×#107C2318yes—
84C2×#107C2314yes—
85C2×#106C2314yes—
86C2×#106C2314yes—
87C2×C4⋊C8C2314yes—
88C2×C4⋊C8C2312yes—
89C2×C22⋊C8C2314yes—
90C2×C22⋊C8C2314yes—
91C2×#97C2312yes—
92C2×#97C2312yes—
93C2×C4×C8C2312yes—
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#400—24no—
97#400—24no—
98#398—24no—
99#398—24no—
100#397—24no—
101#397—24no—
102#391—22no—
103#391—22no—
104#391—22no—
105#389—22no—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#389C2314yes—
115#387—24no—
116#387—24no—
117#387—24no—
118#387—24no—
119#384—22no—
120#384—22no—
121#384C2312yes—
122#384C2314yes—
123#384C2312yes—
124#384C2312yes—
125#382—22no—
126#382—22no—
127#381—24no—
128#381—24no—
129#380—22no—
130#380—22no—
131#380—22no—
132#380C2314yes—
133#380C2314yes—
134#380C2314yes—
135C4×C4⋊C4—22no—
136C4×C4⋊C4—22no—
137C4×C4⋊C4D412yes—
138C4×C4⋊C4C2314yes—
139C4×C4⋊C4C2314yes—
140C4×C4⋊C4C2311yescharacteristic
141C4×C22⋊C4D412yes—
142C4×C22⋊C4D412yes—
143C4×C22⋊C4C2312yes—
144C43C2311yescharacteristic
145#340C2318yes—
146#340C2318yes—
147#340C2318yes—
148#340C2318yes—
149#340C2318yes—
150#336C2314yes—
151#336C2312yes—
152C22×Q8—22no—
153C22×D4C2×D411yescharacteristic
154C23×C4—24no—
155C23×C4C2×D412yes—
156C23×C4C2×D412yes—
157C23×C4C2412yes—
158C22×C8C2414yes—
159C22×C8C2412yes—
160C22×C8C2412yes—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163#126—416no—
164#125—48no—
165#125—416no—
166#123—416no—
167C22⋊Q8—48no—
168C22⋊Q8—48no—
169C4×Q8—48no—
170C4×D4—28no—
171C4×D4—42no—
172#117—416no—
173#117—48no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4—48no—
191C2×C4⋊C4—48no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
197C2×C4⋊C4C2414yes—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4C2×D412yes—
202C2×C22⋊C4C2×D412yes—
203C2×C22⋊C4C2×D412yes—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2411yescharacteristic
213C2×C42C2411yescharacteristic
214C2×C42C2412yes—
215C2×C42C2411yescharacteristic
216#107—44no—
217#107—44no—
218#107—48no—
219#106—28no—
220#106—28no—
221C4⋊C8—28no—
222C4⋊C8—42no—
223C22⋊C8—28no—
224C22⋊C8—28no—
225#97—24no—
226#97—42no—
227C4×C8—24no—
228C4×C8—22no—
229C4×C8#4114yes—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95C2×D411yescharacteristic
241#95C2×D411yescharacteristic
242#95C2×D412yes—
243#95C2×D412yes—
244C24C22×D412yes—
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×D4—24no—
249C2×D4—41no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—48no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4AGL(1,ℤ/8)18yes—
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D412yes—
269C22×C4C22×D412yes—
270C22×C4C22×D412yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C2511yescharacteristic
275C2×C8—22no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—22no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8C2×#4114yes—
283C2×C82−1+414yes—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—24no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—41no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—48no—
294C4⋊C4—48no—
295C4⋊C4—48no—
296C4⋊C4—48no—
297C4⋊C4—44no—
298C4⋊C4—48no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—44no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—42no—
309C22⋊C4—48no—
310C22⋊C4—48no—
311C22⋊C4—48no—
312C42—24no—
313C42—24no—
314C42—24no—
315C42—24no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—48no—
321C42—48no—
322C42C2×#4112yes—
323C422−1+412yes—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—22no—
328C23—22no—
329C23—22no—
330C23C2×AGL(1,ℤ/8)14yes—
331C23C23×D411yescharacteristic
332Q8—88no—
333D4—44no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—216no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—48no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—816no—
366C2×C4#54714yes—
367C2×C4C2×AGL(1,ℤ/8)14yes—
368C2×C4C2×AGL(1,ℤ/8)14yes—
369C2×C4#57814yes—
370C2×C4C22×#4111yescharacteristic
371C2×C4C2×2−1+412yes—
372C2×C4C2×2−1+411yescharacteristic
373C2×C4#58512yes—
374C8—24no—
375C8—24no—
376C8—44no—
377C22—24no—
378C22—24no—
379C22—24no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22#344911yescharacteristic
385C22C2×#54711yescharacteristic
386C22#349911yescharacteristic
387C22C22×AGL(1,ℤ/8)12yes—
388C22C2×#57812yes—
389C4—24no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—42no—
394C4—41no—
395C4—84no—
396C4—84no—
397C4—84no—
398C4—84no—
399C4#334318yes—
400C4#350712yes—
401C4#356312yes—
402C2—44no—
403C2#3948411yescharacteristic
404C2#6096811yescharacteristic
405C2#6101311yescharacteristic
406C2C2×#334312yes—
407C2#6247912yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33263C2semidirect
3. #32516C2semidirect
5. #29877C2semidirect
6. #29179C2semidirect
7. #27950C2semidirect
8. #27875C2semidirect
9. #27475C2semidirect
10. #21982C2semidirect
11. #21727C2semidirect

Maximal subgroups

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

subgroupindex
#361812
#332632
#325162
#324692
#298772
#291792
#279502
#278752
#274752
#219822
#217272
#212652