Finite Group Databasebeta

← #319168

Subgroups — #319168

Order 512.

957 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33042C211yesmaximal, characteristic
2#27909C211yesmaximal, characteristic
3#27905C211yesmaximal, characteristic, complemented
4#25903C211yesmaximal, characteristic
5#25811C211yesmaximal, characteristic, complemented
6C2×#1956C211yesmaximal, characteristic, complemented
7#10680C212yesmaximal, complemented
8#10312C212yesmaximal, complemented
9#9841C212yesmaximal, complemented
10#9827C212yesmaximal, complemented
11#8145C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C2×#498C2212yes—
14C2×#486C2212yes—
15C2×#486C2211yescharacteristic
16C2×#483—22no—
17C2×#480—22no—
18C22×#103—22no—
19C22×C4×C8C2211yescharacteristic
20#2857C2211yescharacteristic
21#2856C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2096—22no—
24#2093—22no—
25#2058—22no—
26#2023—22no—
27#1990C2211yescharacteristic
28#1990C2212yes—
29#1990C2212yes—
30#1957C2212yes—
31#1957C2211yescharacteristic
32#1956—22no—
33#1956C2212yes—
34#1956C2212yes—
35#1956C2212yes—
36#1934—22no—
37#1893—22no—
38#1874—22no—
39#1853—22no—
40#1845C2212yes—
41#1844C2212yes—
42#1834C2211yescharacteristic
43C4×C22⋊C8C2211yescharacteristic
44C4×C22⋊C8C2211yescharacteristic
45C2×#322C2212yes—
46#1273C2212yes—
47C23×Q8—21no—
48C23×C8D412yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—42no—
54C2×C4⋊Q8C2311yescharacteristic
55C2×C4⋊Q8C2312yes—
56C2×#127C2311yescharacteristic
57C2×C22⋊Q8—21no—
58C2×#121—21no—
59C2×#121—21no—
60C2×#121—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C42C2311yescharacteristic
67#498—24no—
68#498—42no—
69#486—24no—
70#486—42no—
71#486—42no—
72#483—48no—
73#480—48no—
74C2×#107—24no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—48no—
81C2×#102—24no—
82C2×#102—24no—
83C2×#102—22no—
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—22no—
109#340—24no—
110#340—22no—
111#336D412yes—
112#336D412yes—
113#322—42no—
114C22×Q8—24no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—44no—
119C22×Q8—41no—
120C22×Q8—44no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—41no—
125C23×C4—41no—
126C23×C4—41no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C22×C8—24no—
131C22×C8—42no—
132C22×C8—42no—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D414yes—
136C22×C8C2×D414yes—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—42no—
146C4⋊Q8#4114yes—
147#127—22no—
148#127—22no—
149C22⋊Q8—44no—
150#121—44no—
151#121—44no—
152#121—44no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C22⋊C4—41no—
179C2×C22⋊C4—41no—
180C2×C22⋊C4—41no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42C2×D411yescharacteristic
184C2×C42C2×D411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2411yescharacteristic
187#107—48no—
188#106—48no—
189#106—48no—
190#103—48no—
191#103—48no—
192#103—48no—
193#102—48no—
194#102—44no—
195#102—44no—
196#102—48no—
197C22⋊C8—24no—
198C22⋊C8SD1618yes—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—42no—
206#95—41no—
207#95—41no—
208C24—41no—
209C24#12711yescharacteristic
210C2×Q8—24no—
211C2×Q8—44no—
212C2×Q8—416no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—44no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—42no—
219C2×Q8—82no—
220C2×Q8—88no—
221C2×Q8—82no—
222C2×D4—41no—
223C2×D4—41no—
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—82no—
233C2×D4—82no—
234C2×D4—82no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—44no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4—82no—
252C22×C4—82no—
253C22×C4#12711yescharacteristic
254C22×C4#12711yescharacteristic
255C22×C4#12711yescharacteristic
256C22×C4C2×SD1614yes—
257C22×C4AGL(1,ℤ/8)14yes—
258C22×C4AGL(1,ℤ/8)14yes—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D411yescharacteristic
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—22no—
265C2×C8—22no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—24no—
273C2×C8—48no—
274C2×C8C2×SD1618yes—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—42no—
284C4⋊C4—44no—
285C4⋊C4—42no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—82no—
296C4⋊C4—82no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C22⋊C4—82no—
300C22⋊C4—82no—
301C42—21no—
302C42—21no—
303C42—22no—
304C42—22no—
305C42C2×#4112yes—
306C42H(5,𝔽2)12yes—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23—81no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23#49212yes—
319C23#49612yes—
320C23#49612yes—
321C23C2×#12711yescharacteristic
322Q8—48no—
323Q8—88no—
324Q8—84no—
325Q8—84no—
326D4—82no—
327D4—82no—
328D4—84no—
329D4—84no—
330D4—164no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—44no—
342C2×C4—88no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—164no—
358C2×C4#49212yes—
359C2×C4#49612yes—
360C2×C4#49612yes—
361C2×C4#52511yescharacteristic
362C2×C4#53411yescharacteristic
363C2×C4#54612yes—
364C2×C4#54612yes—
365C2×C4C22×SD1612yes—
366C2×C4C2×AGL(1,ℤ/8)12yes—
367C2×C4C2×AGL(1,ℤ/8)12yes—
368C2×C4#57512yes—
369C2×C4#57812yes—
370C2×C4#57812yes—
371C8—28no—
372C8—42no—
373C8—42no—
374C8—44no—
375C22—22no—
376C22—22no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—161no—
382C22—161no—
383C22—161no—
384C22—161no—
385C22—161no—
386C22—161no—
387C22—161no—
388C22#171514yes—
389C22#253911yescharacteristic
390C22C2×#49211yescharacteristic
391C22C2×#49611yescharacteristic
392C22C2×#49611yescharacteristic
393C22#315311yescharacteristic
394C22#315611yescharacteristic
395C22#315611yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—84no—
400C4—162no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4#311412yes—
405C4#319412yes—
406C4#329814yes—
407C4#330814yes—
408C2—24no—
409C2—321no—
410C2C2×#171511yescharacteristic
411C2#1962411yescharacteristic
412C2#1967411yescharacteristic
413C2#1969911yescharacteristic
414C2#2831211yescharacteristic
415C2#2833611yescharacteristic
416C2#2834611yescharacteristic
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
3. #27905C2semidirect
5. #25811C2semidirect
6. C2×#1956C2semidirect
7. #10680C2semidirect
8. #10312C2semidirect
9. #9841C2semidirect
10. #9827C2semidirect
11. #8145C2semidirect

Maximal subgroups

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

subgroupindex
#330422
#279092
#279052
#259032
#258112
C2×#19562
#106802
#103122
#98412
#98272
#81452