Finite Group Databasebeta

← #319031

Subgroups — #319031

Order 512.

1001 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33170C211yesmaximal, characteristic
2#27948C211yesmaximal, characteristic, complemented
3#27906C211yesmaximal, characteristic
4#25811C211yesmaximal, characteristic, complemented
5#25718C211yesmaximal, characteristic
6C2×#1990C211yesmaximal, characteristic, complemented
7#10685C212yesmaximal, complemented
8#10315C212yesmaximal, complemented
9#9820C212yesmaximal, complemented
10#9819C212yesmaximal, complemented
11C8×C22⋊C8C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2212yes—
14C2×#492C2212yes—
15C2×#492C2211yescharacteristic
16C2×#481—22no—
17C2×#465—22no—
18C2×#448—22no—
19C22×C4×C8C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2094—22no—
24#2084—22no—
25#2058—22no—
26#2013—22no—
27#1990—22no—
28#1990C2212yes—
29#1990C2212yes—
30#1990C2212yes—
31#1990C2212yes—
32#1968—22no—
33#1957C2212yes—
34#1957C2211yescharacteristic
35#1956C2211yescharacteristic
36#1956C2212yes—
37#1853—22no—
38#1853—22no—
39#1844C2212yes—
40#1844C2212yes—
41#1844C2211yescharacteristic
42C4×C22⋊C8C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C2×#340—22no—
45C2×C82C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C22×SD16—22no—
49C22×SD16—24no—
50C23×C8D412yes—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2312yes—
58C2×#127C2311yescharacteristic
59C2×C22⋊Q8—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C42C2311yescharacteristic
69#498—24no—
70#498—42no—
71#492—24no—
72#492—42no—
73#492—42no—
74#481—48no—
75#465—48no—
76#448—48no—
77C2×#106—24no—
78C2×#106—24no—
79C2×#106—22no—
80C2×#106—24no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#102—24no—
84C2×#102—22no—
85C2×C22⋊C8D412yes—
86C2×C22⋊C8D412yes—
87C2×C4×C8—42no—
88C2×C4×C8D412yes—
89C2×C4×C8C2311yescharacteristic
90C2×C4×C8C2312yes—
91C2×C4×C8C2312yes—
92C2×C4×C8C2311yescharacteristic
93#391—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#389—21no—
98#389—21no—
99#384—21no—
100#384—21no—
101#384—21no—
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#384C2312yes—
105#380—21no—
106#380—21no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#336D412yes—
112#336D412yes—
113C82—42no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—44no—
117C22×Q8—41no—
118C22×Q8—44no—
119C22×Q8—82no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—41no—
124C23×C4—41no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×SD16—48no—
130C2×SD16—44no—
131C2×SD16—44no—
132C2×SD16—48no—
133C2×SD16—44no—
134C2×SD16—84no—
135C22×C8—42no—
136C22×C8—42no—
137C22×C8SD1618yescomplemented
138C22×C8C2×D412yes—
139C22×C8C2×D412yes—
140C22×C8C2×D414yes—
141C22×C8C2×D414yes—
142C4⋊Q8—24no—
143C4⋊Q8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—44no—
149C4⋊Q8—44no—
150C4⋊Q8—42no—
151C4⋊Q8#4114yes—
152#127—22no—
153#127—22no—
154C22⋊Q8—44no—
155#121—44no—
156#121—44no—
157#121—44no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—82no—
183C2×C22⋊C4—41no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—41no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2411yescharacteristic
193#106—48no—
194#106—44no—
195#106—48no—
196#106—84no—
197#103—48no—
198#103—44no—
199#102—44no—
200#102—48no—
201C22⋊C8SD1618yescomplemented
202C22⋊C8SD1618yescomplemented
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—22no—
206C4×C8—22no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—42no—
210#95—41no—
211#95—41no—
212C24—41no—
213C24#12711yescharacteristic
214C2×Q8—24no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—44no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—816no—
224C2×Q8—82no—
225C2×Q8—82no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—82no—
253C22×C4—82no—
254C22×C4—82no—
255C22×C4—82no—
256C22×C4—82no—
257C22×C4#12711yescharacteristic
258C22×C4#12711yescharacteristic
259C22×C4#12711yescharacteristic
260C22×C4C2×SD1614yes—
261C22×C4C2×SD1614yes—
262C22×C4C2×SD1614yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266SD16—816no—
267SD16—88no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—48no—
277C2×C8C2×SD1618yes—
278C2×C8C2×SD1618yes—
279C2×C8C2×SD1614yes—
280C2×C8C2×SD1614yes—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—44no—
297C4⋊C4—42no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—82no—
301C4⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C22⋊C4—82no—
305C22⋊C4—82no—
306C42—21no—
307C42—21no—
308C42—22no—
309C42—22no—
310C42C2×#4112yes—
311C42H(5,𝔽2)12yes—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—81no—
320C23—81no—
321C23—81no—
322C23—81no—
323C23#49212yes—
324C23#49212yes—
325C23#49212yes—
326C23C2×#12711yescharacteristic
327Q8—48no—
328Q8—88no—
329Q8—84no—
330Q8—82no—
331Q8—82no—
332D4—82no—
333D4—82no—
334D4—84no—
335D4—84no—
336D4—164no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—44no—
348C2×C4—88no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—164no—
364C2×C4#49212yes—
365C2×C4#49212yes—
366C2×C4#49212yes—
367C2×C4#52511yescharacteristic
368C2×C4#53411yescharacteristic
369C2×C4#54612yes—
370C2×C4#54612yes—
371C2×C4C22×SD1612yes—
372C2×C4C22×SD1612yes—
373C2×C4C22×SD1612yes—
374C2×C4#57512yes—
375C2×C4#57612yes—
376C2×C4#57812yes—
377C8—24no—
378C8—24no—
379C8—28no—
380C8—28no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—161no—
388C22—161no—
389C22—161no—
390C22—161no—
391C22—161no—
392C22—161no—
393C22—161no—
394C22#170514yes—
395C22#253911yescharacteristic
396C22C2×#49211yescharacteristic
397C22C2×#49211yescharacteristic
398C22C2×#49211yescharacteristic
399C22#315311yescharacteristic
400C22#315511yescharacteristic
401C22#315611yescharacteristic
402C4—22no—
403C4—22no—
404C4—22no—
405C4—84no—
406C4—161no—
407C4—161no—
408C4—162no—
409C4—161no—
410C4#310312yes—
411C4#318712yes—
412C4#329514yes—
413C4#329814yes—
414C2—24no—
415C2—321no—
416C2C2×#170511yescharacteristic
417C2#1962311yescharacteristic
418C2#1962411yescharacteristic
419C2#1965511yescharacteristic
420C2#2830511yescharacteristic
421C2#2833211yescharacteristic
422C2#2833611yescharacteristic
subgroupG/H
center, socleC23C2×#127
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #27948C2semidirect
4. #25811C2semidirect
6. C2×#1990C2semidirect
7. #10685C2semidirect
8. #10315C2semidirect
9. #9820C2semidirect
10. #9819C2semidirect
11. C8×C22⋊C8C2semidirect
137. C22×C8SD16semidirect
201. C22⋊C8SD16semidirect
202. C22⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#331702
#279482
#279062
#258112
#257182
C2×#19902
#106852
#103152
#98202
#98192
C8×C22⋊C82