Finite Group Databasebeta

← #223254

Subgroups — #223254

Order 512.

951 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36211C211yesmaximal, characteristic
2#22648C212yesmaximal, complemented
3#9906C212yesmaximal, complemented
4#9141C212yesmaximal, complemented
5#9133C212yesmaximal, complemented
6#8727C212yesmaximal, complemented
7#8585C211yesmaximal, characteristic, complemented
8#8231C211yesmaximal, characteristic, complemented
9#8045C212yesmaximal, complemented
10C2×#435—22no—
11C2×#423C2212yes—
12C22×C4⋊C8C2212yes—
13#2859C2211yescharacteristic
14#2856C2212yes—
15#2535C412yes—
16#2513—22no—
17#2490C414yes—
18#2420—22no—
19#2414—22no—
20#2363—22no—
21#2357C412yes—
22C4×#127C2211yescharacteristic
23#2285C2212yes—
24C42×D4C2211yescharacteristic
25#1940—22no—
26#1851—22no—
27#1850—22no—
28#1842C2212yes—
29#1797C2212yes—
30#1797C2212yes—
31#1767C2212yes—
32C2×#331C2212yes—
33C2×#331C2211yescharacteristic
34C2×#331C2211yescharacteristic
35#1296C2211yescharacteristic
36#1296C2212yes—
37#1296C2212yes—
38#1296C2212yes—
39#1296C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42C22×M16—22no—
43C23×C8—22no—
44C2×#127C2311yescharacteristic
45C2×#125C2×C412yes—
46C2×#121—22no—
47C2×#121—22no—
48C2×#121C2×C414yes—
49C2×C4×D4—22no—
50C2×C4×D4C2×C412yes—
51C2×C4×D4C2×C411yescharacteristic
52C2×C4×D4D412yes—
53C2×#117—22no—
54C22×C42C2312yes—
55#435—48no—
56#423—24no—
57#423—42no—
58C2×C4⋊C8—24no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—42no—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×C4⋊C8C2312yes—
67C2×C4⋊C8C2312yes—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71#397—22no—
72#397—24no—
73#394—24no—
74#392—22no—
75#390—24no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390C2×C412yes—
80#390C2×C414yes—
81#389—22no—
82#389—22no—
83#388—24no—
84#388—24no—
85#386—24no—
86#384C2311yescharacteristic
87#384C2312yes—
88#382—22no—
89#382C2×C412yes—
90#381—22no—
91#380—22no—
92#380C2×C414yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4C2×C411yescharacteristic
96C4×C4⋊C4D412yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D412yes—
100C4×C22⋊C4D412yes—
101C43C2311yescharacteristic
102#336—22no—
103#336—22no—
104#336—22no—
105#331—24no—
106#331—41no—
107#331—41no—
108C22×D4—24no—
109C22×D4—22no—
110C22×D4C22⋊C411yescharacteristic
111C22×D4C22×C412yes—
112C23×C4—22no—
113C23×C4C22×C412yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C2×M16—44no—
117C2×M16—44no—
118C2×M16—82no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—44no—
124C22×C8—44no—
125C22×C8—82no—
126#127—22no—
127#127—21no—
128#127—21no—
129#125—82no—
130#121—44no—
131#121—44no—
132#121—82no—
133#121—82no—
134#121—84no—
135C4×D4—24no—
136C4×D4—24no—
137C4×D4—42no—
138C4×D4—44no—
139C4×D4—82no—
140C4×D4—82no—
141#117—82no—
142#117—82no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4C22⋊C412yes—
148C2×C4⋊C4C22⋊C411yescharacteristic
149C2×C4⋊C4C22×C412yes—
150C2×C4⋊C4C22×C414yes—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4C22⋊C412yes—
158C2×C22⋊C4C22⋊C412yes—
159C2×C22⋊C4C22⋊C412yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—24no—
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C412yes—
167C2×C42C22×C411yescharacteristic
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2411yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—44no—
181C4⋊C8—44no—
182C4⋊C8—44no—
183C4⋊C8—82no—
184C4⋊C8#4114yes—
185C22⋊C8—44no—
186C22⋊C8—44no—
187#97—44no—
188#95—22no—
189#95—24no—
190#95—24no—
191#95—22no—
192#95C22⋊C411yescharacteristic
193#95C22⋊C412yes—
194#95C22⋊C412yes—
195#95C22⋊C411yescharacteristic
196C24—22no—
197C24C2×C22⋊C412yes—
198C2×D4—24no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—41no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—48no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—84no—
212C2×D4—84no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—84no—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4AGL(1,ℤ/8)14yes—
238C22×C4#13714yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241M16—88no—
242C2×C8—24no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—88no—
255C4⋊C4—24no—
256C4⋊C4—41no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—84no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—24no—
269C22⋊C4—44no—
270C22⋊C4—48no—
271C22⋊C4—84no—
272C22⋊C4—84no—
273C22⋊C4—84no—
274C22⋊C4—84no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—22no—
285C42—22no—
286C42—41no—
287C42—84no—
288C42C2×#4112yes—
289C42H(5,𝔽2)11yescharacteristic
290C422−1+411yescharacteristic
291C23—22no—
292C23—22no—
293C23—22no—
294C23—22no—
295C23—22no—
296C23—22no—
297C23—42no—
298C23—42no—
299C23—42no—
300C23—44no—
301C23—44no—
302C23#41714yes—
303C23C22×C22⋊C411yescharacteristic
304D4—44no—
305D4—82no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—82no—
311D4—168no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—164no—
344C2×C4#41312yes—
345C2×C4#41714yes—
346C2×C4C4×#4111yescharacteristic
347C2×C4#52011yescharacteristic
348C2×C4#54612yes—
349C2×C4#54712yes—
350C2×C4C2×AGL(1,ℤ/8)12yes—
351C2×C4C2×#13712yes—
352C2×C4#57612yes—
353C2×C4#57712yes—
354C8—44no—
355C8—82no—
356C8—82no—
357C8—82no—
358C22—24no—
359C22—24no—
360C22—24no—
361C22—42no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—82no—
366C22—84no—
367C22—82no—
368C22—82no—
369C22—84no—
370C22#233911yescharacteristic
371C22C2×#41311yescharacteristic
372C22#288811yescharacteristic
373C22C2×#41712yes—
374C22#289712yes—
375C4—22no—
376C4—21no—
377C4—21no—
378C4—24no—
379C4—41no—
380C4—42no—
381C4—42no—
382C4—42no—
383C4—84no—
384C4—82no—
385C4#331812yes—
386C4#332012yes—
387C4#334512yes—
388C4#334612yes—
389C2—44no—
390C2—162no—
391C2#1775211yescharacteristic
392C2#1776212yes—
393C2#1777111yescharacteristic
394C2#2248311yescharacteristic
395C2#2249312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36211C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22648C2semidirect
3. #9906C2semidirect
4. #9141C2semidirect
5. #9133C2semidirect
6. #8727C2semidirect
7. #8585C2semidirect
8. #8231C2semidirect
9. #8045C2semidirect

Maximal subgroups

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

subgroupindex
#362112
#226482
#99062
#91412
#91332
#87272
#85852
#82312
#80452