Finite Group Databasebeta

← #208322

Subgroups — #208322

Order 512.

1051 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#20812C211yesmaximal, characteristic, complemented
3#20549C211yesmaximal, characteristic, complemented
4#9900C212yesmaximal, complemented
5#9695C212yesmaximal, complemented
6#9142C212yesmaximal, complemented
7#9141C212yesmaximal, complemented
8#8234C212yesmaximal, complemented
9#8231C212yesmaximal, complemented
10C2×#540—21no—
11C2×#433—21no—
12C2×#432C2211yescharacteristic
13C2×#423C2211yescharacteristic
14#2867C2211yescharacteristic
15#2859C2211yescharacteristic
16#2856C2212yes—
17#2641—21no—
18#2619—21no—
19#2445—22no—
20#2445C414yes—
21#2428—22no—
22#2428C414yes—
23#2328—21no—
24C4×#127C2212yes—
25C4×#117C2211yescharacteristic
26#1842C2212yes—
27#1832—21no—
28#1798—21no—
29#1797C2212yes—
30#1771—22no—
31#1768—21no—
32#1767C2211yescharacteristic
33#1767C2212yes—
34C4×C4⋊C8C2211yescharacteristic
35C4×C4⋊C8C2212yes—
36#1753—22no—
37C2×#331C2212yes—
38C2×#331C2212yes—
39C2×#331C2212yes—
40C2×#331C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45C22×M16—21no—
46#540—48no—
47C2×#127—21no—
48C2×#127—21no—
49C2×#127—22no—
50C2×#127C2×C414yes—
51C2×#127C2312yes—
52C2×#125—21no—
53C2×#123—24no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—24no—
57C2×#121C2×C414yes—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4C2×C414yes—
61C2×#117D411yescharacteristic
62C2×#117D411yescharacteristic
63C22×C42C2311yescharacteristic
64#433—44no—
65#432—24no—
66#423—24no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C22⋊C8—21no—
75C2×C22⋊C8—22no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—22no—
79C2×C4×C8—21no—
80C2×C4×C8—22no—
81#394—24no—
82#394—24no—
83#394—24no—
84#392—24no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390—22no—
89#390C2×C414yes—
90#390C2×C414yes—
91#389—22no—
92#389C2×C414yes—
93#386—24no—
94#386—24no—
95#384C2312yes—
96#382—21no—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4D412yes—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101#376D411yescharacteristic
102#376D411yescharacteristic
103C43C2311yescharacteristic
104C43C2311yescharacteristic
105#336—22no—
106#336—21no—
107#336—21no—
108#336—22no—
109#336—22no—
110#331—24no—
111#331—24no—
112#331—24no—
113#331—24no—
114C22×D4—24no—
115C22×D4—24no—
116C22×D4—24no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4C22×C414yes—
120C23×C4—24no—
121C23×C4C22×C411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×M16—24no—
125C2×M16—82no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130#127—22no—
131#127—22no—
132#127—22no—
133#127—22no—
134#127—42no—
135#127—44no—
136#127—42no—
137#127—82no—
138#127—82no—
139#127—84no—
140#125—44no—
141#123—416no—
142#121—416no—
143#121—44no—
144#121—44no—
145#121—82no—
146#121—82no—
147#121—84no—
148C4×D4—44no—
149C4×D4—44no—
150C4×D4—82no—
151C4×D4—82no—
152C4×D4—84no—
153#117—24no—
154#117—22no—
155#117—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4C22⋊C412yes—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22×C414yes—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C22⋊C412yes—
169C2×C22⋊C4C22⋊C412yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C22×C411yescharacteristic
173C2×C42C22×C412yes—
174C2×C42C22×C412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2411yescharacteristic
184C4⋊C8—24no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189C4⋊C8—22no—
190C4⋊C8—22no—
191C4⋊C8—44no—
192C4⋊C8—44no—
193C4⋊C8#4114yes—
194C4⋊C8#4114yes—
195C22⋊C8—24no—
196C22⋊C8—44no—
197#97—24no—
198#97—42no—
199#97—82no—
200C4×C8—42no—
201C4×C8—82no—
202#95—24no—
203#95C22⋊C412yes—
204#95C22⋊C412yes—
205#95C22⋊C412yes—
206#95C22⋊C412yes—
207C24—24no—
208C24C2×C22⋊C411yescharacteristic
209C2×D4—24no—
210C2×D4—24no—
211C2×D4—44no—
212C2×D4—44no—
213C2×D4—48no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—48no—
227C2×D4—48no—
228C2×D4—84no—
229C2×D4—84no—
230C2×D4—84no—
231C2×D4—84no—
232C2×D4—84no—
233C2×D4—84no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C412yes—
254C22×C4C2×C22⋊C412yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258M16—84no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C4⋊C4—42no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—42no—
278C4⋊C4—48no—
279C22⋊C4—24no—
280C22⋊C4—24no—
281C22⋊C4—48no—
282C22⋊C4—48no—
283C22⋊C4—84no—
284C22⋊C4—84no—
285C22⋊C4—84no—
286C22⋊C4—84no—
287C42—22no—
288C42—21no—
289C42—21no—
290C42—22no—
291C42—21no—
292C42—22no—
293C42—21no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—42no—
299C42—42no—
300C42AGL(1,ℤ/8)12yes—
301C42AGL(1,ℤ/8)14yes—
302C42#13712yes—
303C42C2×#4112yes—
304C42C2×#4112yes—
305C23—22no—
306C23—21no—
307C23—21no—
308C23—22no—
309C23—21no—
310C23—44no—
311C23—44no—
312C23—44no—
313C23—44no—
314C23—44no—
315C23—44no—
316C23—44no—
317C23C22×C22⋊C411yescharacteristic
318D4—84no—
319D4—84no—
320D4—82no—
321D4—82no—
322D4—82no—
323D4—82no—
324D4—82no—
325D4—82no—
326D4—82no—
327D4—82no—
328D4—168no—
329D4—168no—
330D4—168no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—168no—
359C2×C4#41714yes—
360C2×C4#41814yes—
361C2×C4C4×#4112yes—
362C2×C4#52511yescharacteristic
363C2×C4#52511yescharacteristic
364C2×C4#53212yes—
365C2×C4C2×AGL(1,ℤ/8)12yes—
366C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C2×#13711yescharacteristic
368C2×C4#57612yes—
369C2×C4#57712yes—
370C8—44no—
371C8—44no—
372C8—82no—
373C8—82no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22—84no—
383C22—84no—
384C22—84no—
385C22—84no—
386C22#229711yescharacteristic
387C22#288812yes—
388C22C2×#41811yescharacteristic
389C22C2×#41711yescharacteristic
390C22#289712yes—
391C4—24no—
392C4—24no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—41no—
398C4—41no—
399C4—84no—
400C4#311912yes—
401C4#312211yescharacteristic
402C4#312611yescharacteristic
403C4#318212yes—
404C4#318412yes—
405C2—81no—
406C2—164no—
407C2#1775812yes—
408C2#1776212yes—
409C2#2139911yescharacteristic
410C2#2141011yescharacteristic
411C2#2142811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35278C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20812C2semidirect
3. #20549C2semidirect
4. #9900C2semidirect
5. #9695C2semidirect
6. #9142C2semidirect
7. #9141C2semidirect
8. #8234C2semidirect
9. #8231C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#208122
#205492
#99002
#96952
#91422
#91412
#82342
#82312