Finite Group Databasebeta

← #215583

Subgroups — #215583

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9707C212yesmaximal, complemented
5#9704C211yesmaximal, characteristic, complemented
6#9702C211yesmaximal, characteristic, complemented
7#9701C212yesmaximal, complemented
8#9696C211yesmaximal, characteristic, complemented
9#9691C211yesmaximal, characteristic, complemented
10#8344C212yesmaximal
11#8343C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18#2858C2211yescharacteristic
19#2620—21no—
20#2620—21no—
21#2566—22no—
22#2545C414yes—
23#2500—22no—
24#2432C414yes—
25#2326—21no—
26#2326—21no—
27C4×#117C2211yescharacteristic
28#1924—22no—
29#1917—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
41C2×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41C2×C412yes—
50#533—42no—
51#533—42no—
52#532—42no—
53#532—42no—
54C2×C4⋊Q8C2×C414yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#125—21no—
60C2×#124—21no—
61C2×#124—21no—
62C2×#124C2×C412yes—
63C2×#124C2×C414yes—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8C2×C412yes—
66C2×#121—21no—
67C2×#121—21no—
68C2×C4×Q8—21no—
69C2×C4×Q8C2×C412yes—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C42C2311yescharacteristic
76#433—44no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89#396—24no—
90#396—24no—
91#394—24no—
92#394—24no—
93#391—21no—
94#391—21no—
95#391C2×C414yes—
96#390—22no—
97#390—22no—
98#390—21no—
99#390—21no—
100#390C2×C414yes—
101#388—24no—
102#388—24no—
103#385—24no—
104#385—24no—
105#384—22no—
106#384—21no—
107#384—21no—
108#382—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#333—42no—
124#333—42no—
125#333—42no—
126#333—42no—
127#332—42no—
128#332—42no—
129#331—42no—
130#331—42no—
131C2×#41—44no—
132C2×#41—84no—
133C22×Q8—21no—
134C22×Q8C22×C412yes—
135C22×Q8C22×C412yes—
136C22×D4—22no—
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C22×C412yes—
140C23×C4—22no—
141C23×C4C22×C411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—84no—
155#127—22no—
156#127—22no—
157#125—42no—
158#125—42no—
159#124—44no—
160#124—44no—
161#124—42no—
162#124—42no—
163#124—84no—
164C22⋊Q8—44no—
165C22⋊Q8—44no—
166#121—42no—
167#121—42no—
168#121—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—44no—
172C4×D4—42no—
173C4×D4—42no—
174#117—22no—
175#117—22no—
176#117—42no—
177#117—42no—
178#117—42no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C412yes—
205C2×C42C22×C412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2411yescharacteristic
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—44no—
224C4⋊C8—44no—
225C4⋊C8—82no—
226C22⋊C8—44no—
227C22⋊C8—82no—
228#97—42no—
229#97—82no—
230C4×C8—42no—
231#95—24no—
232#95—24no—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—22no—
238C24C2×C22⋊C411yescharacteristic
239#41—816no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—44no—
243C2×Q8—44no—
244C2×Q8—42no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—42no—
248C2×D4—24no—
249C2×D4—44no—
250C2×D4—44no—
251C2×D4—42no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—84no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—84no—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C412yes—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C412yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—48no—
317C22⋊C4—44no—
318C22⋊C4—48no—
319C22⋊C4—84no—
320C22⋊C4—84no—
321C42—22no—
322C42—22no—
323C42—21no—
324C42—21no—
325C42—21no—
326C42—21no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—42no—
334C42—42no—
335C42—44no—
336C42AGL(1,ℤ/8)12yes—
337C42#13712yes—
338C42H(5,𝔽2)12yes—
339C42H(5,𝔽2)12yes—
340C23—21no—
341C23—22no—
342C23—21no—
343C23—21no—
344C23—22no—
345C23—42no—
346C23—44no—
347C23—42no—
348C23—42no—
349C23—44no—
350C23C22×C22⋊C411yescharacteristic
351Q8—84no—
352Q8—84no—
353Q8—82no—
354Q8—82no—
355D4—84no—
356D4—88no—
357D4—82no—
358D4—82no—
359D4—81no—
360D4—81no—
361D4—82no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—88no—
395C2×C4#41714yes—
396C2×C4#51812yes—
397C2×C4#53512yes—
398C2×C4#53511yescharacteristic
399C2×C4#53511yescharacteristic
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13711yescharacteristic
402C2×C4#57512yes—
403C2×C4#57712yes—
404C2×C4#57812yes—
405C8—82no—
406C8—82no—
407C8—82no—
408C8—82no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—42no—
413C22—42no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—84no—
418C22—84no—
419C22#230911yescharacteristic
420C22#288812yes—
421C22C2×#41711yescharacteristic
422C22#289611yescharacteristic
423C22#289812yes—
424C4—22no—
425C4—24no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—82no—
433C4—82no—
434C4—82no—
435C4#321412yes—
436C4#321612yes—
437C4#322111yescharacteristic
438C4#322311yescharacteristic
439C4#322512yes—
440C2—81no—
441C2—162no—
442C2#1776412yes—
443C2#1776812yes—
444C2#2180111yescharacteristic
445C2#2180911yescharacteristic
446C2#2181411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. #20812C2semidirect
4. #9707C2semidirect
5. #9704C2semidirect
6. #9702C2semidirect
7. #9701C2semidirect
8. #9696C2semidirect
9. #9691C2semidirect
11. #8343C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#219262
#208122
#97072
#97042
#97022
#97012
#96962
#96912
#83442
#83432