Finite Group Databasebeta

← #217059

Subgroups — #217059

Order 512.

991 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20815C211yesmaximal, characteristic, complemented
4#9907C212yesmaximal, complemented
5#9706C212yesmaximal, complemented
6#9142C212yesmaximal, complemented
7#9134C212yesmaximal, complemented
8#8588C212yesmaximal, complemented
9#8585C212yesmaximal, complemented
10C2×#540—21no—
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2867C2211yescharacteristic
14#2859C2211yescharacteristic
15#2856C2212yes—
16#2641—21no—
17#2619—21no—
18#2445—22no—
19#2445C414yes—
20#2428—22no—
21#2428C414yes—
22#2328—21no—
23C4×#127C2212yes—
24C4×#117C2211yescharacteristic
25#1924—22no—
26#1917—22no—
27#1842C2212yes—
28#1842C2211yescharacteristic
29#1831C2212yes—
30#1831C2211yescharacteristic
31#1798—21no—
32#1798—21no—
33#1797C2212yes—
34#1768—21no—
35#1768—21no—
36#1767C2212yes—
37C2×#331C2212yes—
38C2×#331C2212yes—
39C2×#331C2212yes—
40C2×#331C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1296C2212yes—
45#540—48no—
46C2×#127—21no—
47C2×#127—21no—
48C2×#127—22no—
49C2×#127C2×C414yes—
50C2×#127C2312yes—
51C2×#125—21no—
52C2×#123—24no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121—24no—
56C2×#121C2×C414yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C414yes—
60C2×#117D411yescharacteristic
61C2×#117D411yescharacteristic
62C22×C42C2311yescharacteristic
63#432—41no—
64#432—41no—
65#431—41no—
66#431—41no—
67C2×C4⋊C8—22no—
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—21no—
76C2×C22⋊C8—22no—
77C2×#97—21no—
78C2×#97—22no—
79C2×C4×C8—21no—
80#394—24no—
81#394—24no—
82#394—24no—
83#392—24no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390C2×C414yes—
89#390C2×C414yes—
90#389—22no—
91#389C2×C414yes—
92#386—24no—
93#386—24no—
94#384C2312yes—
95#382—21no—
96C4×C4⋊C4C2×C412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4C2×C412yes—
99C4×C22⋊C4D412yes—
100#376D411yescharacteristic
101#376D411yescharacteristic
102C43C2311yescharacteristic
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#331—42no—
110#331—42no—
111#331—42no—
112#331—42no—
113C22×D4—24no—
114C22×D4—24no—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4C22×C414yes—
119C23×C4—24no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127#127—22no—
128#127—22no—
129#127—22no—
130#127—22no—
131#127—44no—
132#127—44no—
133#127—82no—
134#127—82no—
135#127—84no—
136#125—44no—
137#123—416no—
138#121—416no—
139#121—48no—
140#121—82no—
141#121—82no—
142#121—84no—
143C4×D4—48no—
144C4×D4—82no—
145C4×D4—82no—
146C4×D4—84no—
147#117—24no—
148#117—42no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C414yes—
155C2×C22⋊C4—24no—
156C2×C22⋊C4—24no—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4C22⋊C412yes—
162C2×C22⋊C4C22⋊C412yes—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C412yes—
169C2×C42C22×C412yes—
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×D411yescharacteristic
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2×D411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2411yescharacteristic
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—44no—
188C4⋊C8—44no—
189C4⋊C8—82no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192C22⋊C8—82no—
193#97—42no—
194#97—82no—
195C4×C8—42no—
196#95—24no—
197#95C22⋊C412yes—
198#95C22⋊C412yes—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201C24—24no—
202C24C2×C22⋊C411yescharacteristic
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—48no—
209C2×D4—48no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—84no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C2×D4—84no—
225C2×D4—84no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—42no—
238C22×C4—48no—
239C22×C4—84no—
240C22×C4—84no—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C411yescharacteristic
243C22×C4C2×C22⋊C411yescharacteristic
244C22×C4C2×C22⋊C412yes—
245C22×C4C2×C22⋊C412yes—
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C412yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C22×D411yescharacteristic
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C4⋊C4—24no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—48no—
267C4⋊C4—44no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C22⋊C4—24no—
271C22⋊C4—42no—
272C22⋊C4—48no—
273C22⋊C4—48no—
274C22⋊C4—84no—
275C22⋊C4—84no—
276C22⋊C4—84no—
277C22⋊C4—84no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—22no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—42no—
292C42—42no—
293C42AGL(1,ℤ/8)14yes—
294C42H(5,𝔽2)11yescharacteristic
295C42H(5,𝔽2)11yescharacteristic
296C422−1+411yescharacteristic
297C422−1+411yescharacteristic
298C23—21no—
299C23—22no—
300C23—21no—
301C23—21no—
302C23—22no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23C22×C22⋊C411yescharacteristic
311D4—84no—
312D4—84no—
313D4—84no—
314D4—84no—
315D4—82no—
316D4—82no—
317D4—82no—
318D4—82no—
319D4—168no—
320D4—168no—
321D4—168no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—168no—
352C2×C4#41814yes—
353C2×C4#52011yescharacteristic
354C2×C4#52011yescharacteristic
355C2×C4#53511yescharacteristic
356C2×C4#53611yescharacteristic
357C2×C4#53711yescharacteristic
358C2×C4#53711yescharacteristic
359C2×C4C2×AGL(1,ℤ/8)12yes—
360C2×C4#57512yes—
361C2×C4#57612yes—
362C2×C4#57712yes—
363C8—82no—
364C8—82no—
365C8—82no—
366C8—82no—
367C22—42no—
368C22—42no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—84no—
373C22—84no—
374C22—84no—
375C22—84no—
376C22—84no—
377C22—84no—
378C22—84no—
379C22#231111yescharacteristic
380C22#288812yes—
381C22C2×#41811yescharacteristic
382C22#289611yescharacteristic
383C22#289712yes—
384C4—24no—
385C4—21no—
386C4—21no—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—84no—
394C4#321412yes—
395C4#323512yes—
396C4#325712yes—
397C4#325911yescharacteristic
398C4#326011yescharacteristic
399C2—81no—
400C2—164no—
401C2#1775812yes—
402C2#1776612yes—
403C2#2186911yescharacteristic
404C2#2188211yescharacteristic
405C2#2189711yescharacteristic
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. #21926C2semidirect
3. #20815C2semidirect
4. #9907C2semidirect
5. #9706C2semidirect
6. #9142C2semidirect
7. #9134C2semidirect
8. #8588C2semidirect
9. #8585C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#219262
#208152
#99072
#97062
#91422
#91342
#85882
#85852