Finite Group Databasebeta

← #217078

Subgroups — #217078

Order 512.

1014 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35278C211yesmaximal, characteristic
2#21972C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9902C211yesmaximal, characteristic, complemented
5#9897C211yesmaximal, characteristic, complemented
6#9706C211yesmaximal, characteristic, complemented
7#9695C211yesmaximal, characteristic, complemented
8#9134C212yesmaximal, complemented
9#9133C212yesmaximal, complemented
10#8592C212yesmaximal, complemented
11#8588C212yesmaximal, complemented
12C2×#540—21no—
13C2×#431C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15#2867C2211yescharacteristic
16#2859C2211yescharacteristic
17#2856C2212yes—
18#2641—21no—
19#2619—21no—
20#2445—22no—
21#2445C414yes—
22#2428—22no—
23#2428C414yes—
24#2328—21no—
25C4×#127C2212yes—
26C4×#117C2211yescharacteristic
27#1939—22no—
28#1850—22no—
29#1842C2211yescharacteristic
30#1842C2211yescharacteristic
31#1840—21no—
32#1831C2212yes—
33#1831C2211yescharacteristic
34#1831C2211yescharacteristic
35#1818—21no—
36#1797C2212yes—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39#1750—21no—
40#1746—21no—
41C2×#331C2212yes—
42C2×#331C2212yes—
43C2×#331C2212yes—
44C2×#331C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2212yes—
48#1296C2212yes—
49C23×C8—21no—
50#540—48no—
51C2×#127—21no—
52C2×#127—21no—
53C2×#127—22no—
54C2×#127C2×C414yes—
55C2×#127C2312yes—
56C2×#125—21no—
57C2×#123—24no—
58C2×#121—22no—
59C2×#121—22no—
60C2×#121—24no—
61C2×#121C2×C414yes—
62C2×C4×D4—22no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C414yes—
65C2×#117D411yescharacteristic
66C2×#117D411yescharacteristic
67C22×C42C2311yescharacteristic
68#431—42no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—42no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2311yescharacteristic
78C2×C4⋊C8C2311yescharacteristic
79C2×C22⋊C8—22no—
80C2×C22⋊C8—21no—
81C2×#97—21no—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84#394—24no—
85#394—24no—
86#394—24no—
87#392—24no—
88#390—22no—
89#390—22no—
90#390—22no—
91#390—22no—
92#390C2×C414yes—
93#390C2×C414yes—
94#389—22no—
95#389C2×C414yes—
96#386—24no—
97#386—24no—
98#384C2312yes—
99#382—21no—
100C4×C4⋊C4C2×C412yes—
101C4×C4⋊C4D412yes—
102C4×C22⋊C4C2×C412yes—
103C4×C22⋊C4D412yes—
104#376D411yescharacteristic
105#376D411yescharacteristic
106C43C2311yescharacteristic
107C43C2311yescharacteristic
108#336—22no—
109#336—22no—
110#336—21no—
111#336—21no—
112#336—22no—
113#336—22no—
114#336—22no—
115#331—42no—
116#331—42no—
117#331—42no—
118#331—42no—
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—24no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4C22×C414yes—
125C23×C4—24no—
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—42no—
136C22×C8—82no—
137#127—22no—
138#127—22no—
139#127—22no—
140#127—22no—
141#127—44no—
142#127—42no—
143#127—42no—
144#127—82no—
145#127—82no—
146#127—84no—
147#125—44no—
148#123—416no—
149#121—416no—
150#121—44no—
151#121—44no—
152#121—82no—
153#121—82no—
154#121—84no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—82no—
158C4×D4—82no—
159C4×D4—84no—
160#117—24no—
161#117—22no—
162#117—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C414yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4C22⋊C412yes—
176C2×C22⋊C4C22⋊C412yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42C22×C411yescharacteristic
180C2×C42C22×C412yes—
181C2×C42C22×C412yes—
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—42no—
200C4⋊C8—82no—
201C4⋊C8—82no—
202C22⋊C8—42no—
203C22⋊C8—44no—
204#97—42no—
205C4×C8—42no—
206C4×C8—42no—
207#95—24no—
208#95C22⋊C412yes—
209#95C22⋊C412yes—
210#95C22⋊C412yes—
211#95C22⋊C412yes—
212C24—24no—
213C24C2×C22⋊C411yescharacteristic
214C2×D4—24no—
215C2×D4—24no—
216C2×D4—44no—
217C2×D4—44no—
218C2×D4—48no—
219C2×D4—48no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—48no—
233C2×D4—84no—
234C2×D4—84no—
235C2×D4—84no—
236C2×D4—84no—
237C2×D4—84no—
238C2×D4—84no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—48no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C412yes—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C412yes—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4AGL(1,ℤ/8)14yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C22×C4C22×D411yescharacteristic
263C2×C8—41no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—84no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—48no—
289C22⋊C4—48no—
290C22⋊C4—84no—
291C22⋊C4—84no—
292C22⋊C4—84no—
293C22⋊C4—84no—
294C42—22no—
295C42—21no—
296C42—21no—
297C42—22no—
298C42—21no—
299C42—21no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—42no—
307C42—42no—
308C42#13512yes—
309C42#13512yes—
310C42H(5,𝔽2)12yes—
311C422−1+412yes—
312C23—21no—
313C23—22no—
314C23—21no—
315C23—22no—
316C23—44no—
317C23—44no—
318C23—44no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23#41812yes—
324C23C22×C22⋊C411yescharacteristic
325D4—84no—
326D4—84no—
327D4—82no—
328D4—82no—
329D4—82no—
330D4—82no—
331D4—82no—
332D4—82no—
333D4—82no—
334D4—82no—
335D4—168no—
336D4—168no—
337D4—168no—
338C2×C4—21no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—82no—
365C2×C4—168no—
366C2×C4#41614yes—
367C2×C4#41812yes—
368C2×C4#52012yes—
369C2×C4#53511yescharacteristic
370C2×C4#53611yescharacteristic
371C2×C4#53712yes—
372C2×C4C2×#13511yescharacteristic
373C2×C4C2×#13511yescharacteristic
374C2×C4C2×AGL(1,ℤ/8)12yes—
375C2×C4#57612yes—
376C2×C4#57712yes—
377C8—81no—
378C8—81no—
379C8—81no—
380C8—81no—
381C8—82no—
382C8—82no—
383C22—24no—
384C22—22no—
385C22—42no—
386C22—41no—
387C22—41no—
388C22—84no—
389C22—84no—
390C22—84no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22#231111yescharacteristic
396C22#288812yes—
397C22C2×#41611yescharacteristic
398C22C2×#41811yescharacteristic
399C22#289712yes—
400C4—24no—
401C4—22no—
402C4—22no—
403C4—42no—
404C4—42no—
405C4—42no—
406C4—41no—
407C4—41no—
408C4—84no—
409C4#322011yescharacteristic
410C4#322111yescharacteristic
411C4#324112yes—
412C4#324712yes—
413C4#324712yes—
414C2—42no—
415C2—164no—
416C2#1772412yes—
417C2#1775812yes—
418C2#2187211yescharacteristic
419C2#2188211yescharacteristic
420C2#2188811yescharacteristic
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. #21972C2semidirect
3. #20786C2semidirect
4. #9902C2semidirect
5. #9897C2semidirect
6. #9706C2semidirect
7. #9695C2semidirect
8. #9134C2semidirect
9. #9133C2semidirect
10. #8592C2semidirect
11. #8588C2semidirect

Maximal subgroups

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

subgroupindex
#352782
#219722
#207862
#99022
#98972
#97062
#96952
#91342
#91332
#85922
#85882