Finite Group Databasebeta

← #508120

Subgroups — #508120

Order 512.

787 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic, complemented
2C2×#2234C212yesmaximal
3#33129C211yesmaximal, characteristic
4#13453C212yesmaximal, complemented
5#13452C211yesmaximal, characteristic, complemented
6#13451C211yesmaximal, characteristic, complemented
7#13365C212yesmaximal, complemented
8#13363C211yesmaximal, characteristic, complemented
9#13363C211yesmaximal, characteristic, complemented
10#12868C212yesmaximal, complemented
11#12844C211yesmaximal, characteristic, complemented
12C2×#533—21no—
13#3160—28no—
14C2×#500C2211yescharacteristic
15C2×#500C2211yescharacteristic
16C2×#499—21no—
17C2×#499C2212yes—
18C2×#498—21no—
19C2×#466C2211yescharacteristic
20C2×#427—21no—
21C2×#427C2212yes—
22C22×C4×C8C2211yescharacteristic
23#2854—21no—
24#2234—28no—
25C2×#367C2212yes—
26C2×#366C2212yes—
27C2×#357C2212yes—
28C2×#357C2212yes—
29C2×C22⋊C16C2212yes—
30#1970C2211yescharacteristic
31#1970C2211yescharacteristic
32#1942C2211yescharacteristic
33#1942C2211yescharacteristic
34#1934—21no—
35#1934—21no—
36#1893—21no—
37#1808C2211yescharacteristic
38#1380C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44#1374C2212yes—
45C22×D8C2311yescharacteristic
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48#533—28no—
49#533—44no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—21no—
52C2×#125—22no—
53C2×#123—41no—
54C2×C22⋊Q8—41no—
55C2×C22⋊Q8—41no—
56C2×#121—21no—
57C2×C4×Q8—41no—
58C2×C4×Q8—41no—
59C2×#117—22no—
60C22×C42D411yescharacteristic
61C22×C16C2312yes—
62C22×C16C2312yes—
63#500—24no—
64#500—24no—
65#499—28no—
66#499—24no—
67#499—24no—
68#498—28no—
69#466—24no—
70#427—28no—
71#427—24no—
72#427D418yes—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107C2311yescharacteristic
76C2×#107C2311yescharacteristic
77C2×#107C2311yescharacteristic
78C2×#107C2312yes—
79C2×#107C2312yes—
80C2×#106—22no—
81C2×#106—22no—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×C4×C8—24no—
86C2×C4×C8D411yescharacteristic
87C2×C4×C8D411yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#390—21no—
91#390—21no—
92#389—21no—
93#389—21no—
94#388—41no—
95#388—41no—
96#386—41no—
97#380—21no—
98#367—24no—
99#366—24no—
100#357—24no—
101#357—24no—
102C22⋊C16—24no—
103#340—21no—
104#340—21no—
105#340—21no—
106C22×Q8—41no—
107C22×Q8—41no—
108C22×D4—21no—
109C22×D4—41no—
110C23×C4D811yescharacteristic
111C23×C4D811yescharacteristic
112C23×C4C2×D411yescharacteristic
113C2×D8—22no—
114C2×D8—22no—
115C2×D8—21no—
116C2×D8—21no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C2×D411yescharacteristic
124C22×C8C2×D414yes—
125C22×C8C2×D411yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2411yescharacteristic
128C4⋊Q8—24no—
129C4⋊Q8—24no—
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132#125—28no—
133#125—44no—
134#123—84no—
135C22⋊Q8—48no—
136C22⋊Q8—48no—
137#121—42no—
138#121—42no—
139C4×Q8—48no—
140C4×Q8—48no—
141#117—28no—
142#117—44no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C42—22no—
162C2×C42—22no—
163C2×C42—42no—
164C2×C42D811yescharacteristic
165C2×C42D811yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174#107—24no—
175#107—24no—
176#107—22no—
177#107—22no—
178#107—22no—
179#107—22no—
180#107—21no—
181#107—21no—
182#107C2×D414yes—
183#107C2×D414yes—
184#106—28no—
185#106—28no—
186#102—42no—
187#102—42no—
188#102—42no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—22no—
193#95—41no—
194#95—41no—
195#95—41no—
196C24—41no—
197C24C2×D811yescharacteristic
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—82no—
201C2×Q8—82no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—82no—
211C2×D4—82no—
212C2×D4—82no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4#12012yes—
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C2×D811yescharacteristic
233C22×C4C2×D811yescharacteristic
234C22×C4AGL(1,ℤ/8)12yes—
235C22×C4C22×D411yescharacteristic
236D8—42no—
237D8—42no—
238C2×C8—28no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—24no—
246C2×C8—24no—
247C2×C8#12012yes—
248C2×C8#13712yes—
249C2×C8#13712yes—
250C2×C8C22×D412yes—
251C2×C82−1+412yes—
252C2×C82−1+411yescharacteristic
253C2×C82−1+411yescharacteristic
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—48no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C22⋊C4—48no—
271C22⋊C4—82no—
272C22⋊C4—82no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—48no—
284C16—42no—
285C16—42no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—81no—
293C23—81no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23#44712yes—
300C23C22×D811yescharacteristic
301Q8—88no—
302Q8—88no—
303D4—81no—
304D4—81no—
305D4—81no—
306D4—81no—
307D4—164no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—44no—
327C2×C4—44no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4#44712yes—
342C2×C4#50812yes—
343C2×C4#50912yes—
344C2×C4C2×#12011yescharacteristic
345C2×C4#53611yescharacteristic
346C2×C4#53611yescharacteristic
347C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
348C2×C4C2×#13711yescharacteristic
349C2×C4C2×#13711yescharacteristic
350C8—22no—
351C8—22no—
352C8—22no—
353C8#53612yes—
354C8#53612yes—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—161no—
363C22—161no—
364C22—161no—
365C22—161no—
366C22—161no—
367C22—161no—
368C22—161no—
369C22C2×#44711yescharacteristic
370C22#322611yescharacteristic
371C22#322611yescharacteristic
372C22C2×#50811yescharacteristic
373C22C2×#50911yescharacteristic
374C22#341411yescharacteristic
375C22#341411yescharacteristic
376C4—22no—
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—84no—
382C4—84no—
383C4—161no—
384C4—161no—
385C4—161no—
386C4#261411yescharacteristic
387C4#300411yescharacteristic
388C4#323811yescharacteristic
389C4#323811yescharacteristic
390C2—42no—
391C2—321no—
392C2#2963811yescharacteristic
393C2#3351611yescharacteristic
394C2#3352011yescharacteristic
395C2#3379411yescharacteristic
396C2#3379911yescharacteristic
397C2#3380111yescharacteristic
398C2#3380611yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3160C2semidirect
4. #13453C2semidirect
5. #13452C2semidirect
6. #13451C2semidirect
7. #13365C2semidirect
8. #13363C2semidirect
9. #13363C2semidirect
10. #12868C2semidirect
11. #12844C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
C2×#22342
#331292
#134532
#134522
#134512
#133652
#133632
#128682
#128442