Finite Group Databasebeta

← #215935

Subgroups — #215935

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35167C211yesmaximal, characteristic
2#21916C211yesmaximal, characteristic
3#13950C211yesmaximal, characteristic
4#9912C212yesmaximal
5#9909C212yesmaximal
6#9889C212yesmaximal
7#9888C212yesmaximal
8#8717C212yesmaximal
9#8717C212yesmaximal
10C2×#544—21no—
11#2857C2212yes—
12#2857C2212yes—
13#2659—21no—
14#2659—21no—
15#2552C414yescomplemented
16#2551C414yescomplemented
17#2550—24no—
18#2316—21no—
19#2285C2212yes—
20C2×C4×C4⋊C4C2211yescharacteristic
21#1917—24no—
22#1817C2212yes—
23#1797C2212yes—
24#1749C2212yes—
25#1445—21no—
26#1443—21no—
27#1436—21no—
28#1434—21no—
29#1434C2211yescharacteristic
30#1434C2211yescharacteristic
31#1434C2211yescharacteristic
32#1434C2211yescharacteristic
33#1434C2211yescharacteristic
34C4×#95C2211yescharacteristic
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39#544—44no—
40#544—44no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8C2×C412yes—
43C2×C4⋊Q8C2×C412yes—
44C2×#125—24no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×#117—22no—
48C22×C4⋊C4D411yescharacteristic
49C22×C4⋊C4D411yescharacteristic
50C22×C42—21no—
51C22×C42C2311yescharacteristic
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C2×C4⋊C8—22no—
55C2×C22⋊C8C2314yes—
56C2×#97—22no—
57#398—24no—
58#398—24no—
59#395—24no—
60#395—24no—
61#391—22no—
62#391—22no—
63#389—24no—
64#389—24no—
65#389—22no—
66#389—22no—
67#389C2×C414yes—
68#389C2×C414yes—
69#384C2×C412yes—
70#384C2×C412yes—
71#384C2×C412yes—
72#384C2×C412yes—
73#382—22no—
74#380—22no—
75#380—22no—
76#380—22no—
77#380C2314yes—
78C4×C4⋊C4—22no—
79C4×C4⋊C4C2×C414yes—
80C4×C4⋊C4C2×C414yes—
81C4×C4⋊C4D414yes—
82C2×#95—21no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—21no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95C2×#95C2311yescharacteristic
96C2×#95C2311yescharacteristic
97C2×#95C2311yescharacteristic
98C2×#95C2311yescharacteristic
99#336—22no—
100#336—24no—
101#336—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4C2×D411yescharacteristic
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2411yescharacteristic
115C22×C8—24no—
116C4⋊Q8—416no—
117C4⋊Q8—82no—
118C4⋊Q8—82no—
119#125—416no—
120C22⋊Q8—44no—
121C22⋊Q8—44no—
122C22⋊Q8—44no—
123C22⋊Q8—44no—
124#117—44no—
125#117—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4C22⋊C414yes—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22×C414yes—
145C2×C4⋊C4C22×C414yes—
146C2×C22⋊C4—24no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—24no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—81no—
153C2×C42—81no—
154C2×C42—81no—
155C2×C42C22×C412yes—
156C2×C42C22×C412yes—
157C2×C42C22×C412yes—
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D412yes—
161C4⋊C8—82no—
162C22⋊C8—24no—
163C22⋊C8—24no—
164#97—82no—
165#95—24no—
166#95—24no—
167#95—24no—
168#95—24no—
169#95—22no—
170#95—22no—
171#95—22no—
172#95—22no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—42no—
180#95—42no—
181#95—81no—
182#95—81no—
183#95—81no—
184#95C22⋊C414yes—
185#95C22⋊C414yes—
186C24C22×D411yescharacteristic
187C2×Q8—44no—
188C2×Q8—44no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8—82no—
192C2×Q8—82no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4C2×C22⋊C412yes—
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C412yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4AGL(1,ℤ/8)12yes—
254C22×C4AGL(1,ℤ/8)12yes—
255C22×C4#13712yes—
256C22×C4#13712yes—
257C22×C4C23×C411yescharacteristic
258C22×C4C22×D411yescharacteristic
259C22×C42−1+412yes—
260C22×C42−1+412yes—
261C2×C8—44no—
262C2×C8—44no—
263C2×C8—44no—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—44no—
267C4⋊C4—44no—
268C4⋊C4—44no—
269C4⋊C4—44no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—82no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C22⋊C4—44no—
281C22⋊C4—44no—
282C42—24no—
283C42—24no—
284C42—42no—
285C42—42no—
286C42—42no—
287C42—42no—
288C42—48no—
289C42—84no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23C22×C22⋊C411yescharacteristic
295C23#53611yescharacteristic
296C23#53611yescharacteristic
297C23C2×AGL(1,ℤ/8)11yescharacteristic
298C23C2×AGL(1,ℤ/8)11yescharacteristic
299C23C2×#13711yescharacteristic
300C23C2×#13711yescharacteristic
301Q8—164no—
302Q8—164no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
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—44no—
342C2×C4—44no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4#41814yes—
374C2×C4#41914yes—
375C2×C4#51912yes—
376C2×C4#53612yes—
377C8—84no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22#231011yescharacteristic
391C22#288311yescharacteristic
392C22#288411yescharacteristic
393C22C2×#41811yescharacteristic
394C22C2×#41911yescharacteristic
395C22#290111yescharacteristic
396C22#290111yescharacteristic
397C22#323911yescharacteristic
398C22#323911yescharacteristic
399C22#323911yescharacteristic
400C22#323911yescharacteristic
401C4—22no—
402C4—22no—
403C4—24no—
404C4—24no—
405C4—42no—
406C4—42no—
407C4—42no—
408C4—42no—
409C4—82no—
410C4—82no—
411C4—84no—
412C4—84no—
413C4—161no—
414C4—161no—
415C2—21no—
416C2—21no—
417C2—21no—
418C2—21no—
419C2#1762011yescharacteristic
420C2#1762711yescharacteristic
421C2#1763111yescharacteristic
422C2#1763211yescharacteristic
423C2#2182811yescharacteristic
424C2#2184111yescharacteristic
425C2#2184611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #2552C4semidirect
16. #2551C4semidirect

Maximal subgroups

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

subgroupindex
#351672
#219162
#139502
#99122
#99092
#98892
#98882
#87172