Finite Group Databasebeta

← #503039

Subgroups — #503039

Order 512.

928 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3169C211yesmaximal, characteristic
2C2×#2255C211yesmaximal, characteristic, complemented
3C2×#2244C211yesmaximal, characteristic, complemented
4#33208C211yesmaximal, characteristic, complemented
5#13381C212yesmaximal, complemented
6#13380C212yesmaximal, complemented
7#13283C211yesmaximal, characteristic, complemented
8#13206C211yesmaximal, characteristic, complemented
9#12937C212yesmaximal, complemented
10#12935C212yesmaximal, complemented
11#12599C211yesmaximal, characteristic, complemented
12C2×#532—21no—
13#3169—24no—
14#3169—24no—
15C2×#500C2211yescharacteristic
16C2×#499C2211yescharacteristic
17C2×#493C2211yescharacteristic
18C2×#486—21no—
19C2×#466—21no—
20C2×#466C2212yes—
21C2×C4×D8—21no—
22C2×C4×D8C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2851—21no—
25#2255—24no—
26#2244—24no—
27C2×#367C2212yes—
28C2×#366C2212yes—
29C2×#357C2212yes—
30C2×#357C2212yes—
31C2×#357C2212yes—
32C2×#357C2212yes—
33C2×C22⋊C16C2212yes—
34C2×#346C2211yescharacteristic
35C2×#346C2211yescharacteristic
36#1991—21no—
37#1942C2211yescharacteristic
38#1942C2211yescharacteristic
39#1941—21no—
40#1846C2211yescharacteristic
41#1808C2211yescharacteristic
42#1808C2211yescharacteristic
43#1807—21no—
44#1380C2212yes—
45#1380C2212yes—
46#1374C2212yes—
47C22×#41—41no—
48C22×D8—21no—
49C22×D8C2312yes—
50C23×C8D411yescharacteristic
51C23×C8C2311yescharacteristic
52#532—28no—
53#532—44no—
54C2×C4⋊Q8—21no—
55C2×#127—21no—
56C2×#125—21no—
57C2×#124—41no—
58C2×#121—22no—
59C2×#121—41no—
60C2×C4×D4—22no—
61C2×C4×D4—41no—
62C22×C42D411yescharacteristic
63C22×C16C2312yes—
64C22×C16C2312yes—
65#500—22no—
66#500D414yes—
67#499—22no—
68#499—22no—
69#493—22no—
70#493D414yes—
71#486—28no—
72#466—28no—
73#466—24no—
74#466—24no—
75C4×D8—28no—
76C4×D8—24no—
77C4×D8—24no—
78C2×#107—21no—
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2312yes—
84C2×#106—21no—
85C2×#106—21no—
86C2×#102—22no—
87C2×#102—22no—
88C2×C4×C8—22no—
89C2×C4×C8—22no—
90C2×C4×C8D411yescharacteristic
91C2×C4×C8D411yescharacteristic
92C2×C4×C8C2311yescharacteristic
93C2×C4×C8C2311yescharacteristic
94#389—21no—
95#389—21no—
96#387—41no—
97#387—41no—
98#385—41no—
99#385—41no—
100#382—21no—
101#380—21no—
102#380—21no—
103#367—24no—
104#366—24no—
105#357—24no—
106#357—24no—
107#357—24no—
108#357—24no—
109C22⋊C16—24no—
110#346—22no—
111#346—22no—
112#340—21no—
113#340—21no—
114#340—21no—
115#340—21no—
116C2×#41—48no—
117C2×#41—84no—
118C2×#41—84no—
119C22×Q8—41no—
120C22×D4—22no—
121C22×D4—42no—
122C22×D4—41no—
123C22×D4—41no—
124C23×C4—42no—
125C23×C4D811yescharacteristic
126C23×C4D811yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×D8—24no—
129C2×D8—22no—
130C2×D8—22no—
131C2×D8—22no—
132C2×D8—22no—
133C2×D8—22no—
134C2×D8—42no—
135C2×D8—42no—
136C2×D8C2×D414yes—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×D411yescharacteristic
147C22×C8C2411yescharacteristic
148C4⋊Q8—24no—
149C4⋊Q8—42no—
150#127—24no—
151#127—42no—
152#125—42no—
153#125—42no—
154#124—48no—
155#121—28no—
156#121—44no—
157#121—48no—
158C4×D4—28no—
159C4×D4—44no—
160C4×D4—48no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42D811yescharacteristic
178C2×C42D811yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—22no—
185C2×C16—22no—
186C2×C16#4114yescomplemented
187#107—24no—
188#107—22no—
189#107—21no—
190#107—21no—
191#107—21no—
192#107—21no—
193#107—22no—
194#107—22no—
195#107C2×D414yes—
196#106—42no—
197#106—42no—
198#102—28no—
199#102—28no—
200C4×C8—22no—
201C4×C8—21no—
202C4×C8—21no—
203C4×C8—22no—
204C4×C8C2×D412yes—
205C4×C8C2×D412yes—
206#95—41no—
207#95—41no—
208#95—41no—
209#95—41no—
210#95—41no—
211#95—41no—
212C24—42no—
213C24C2×D811yescharacteristic
214#41—832no—
215C2×Q8—44no—
216C2×Q8—82no—
217C2×Q8—82no—
218C2×D4—24no—
219C2×D4—48no—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—42no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—44no—
226C2×D4—44no—
227C2×D4—82no—
228C2×D4—82no—
229C2×D4—82no—
230C2×D4—82no—
231C2×D4—84no—
232C2×D4—84no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—48no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—41no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4C2×D811yescharacteristic
258C22×C4C22×D411yescharacteristic
259D8—24no—
260D8—24no—
261D8—42no—
262D8—42no—
263D8—48no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—44no—
273C2×C8#12012yes—
274C2×C8#13512yes—
275C2×C8#13712yes—
276C2×C8C22×D411yescharacteristic
277C2×C8C22×D411yescharacteristic
278C2×C8C2×#4112yes—
279C2×C82−1+412yes—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C22⋊C4—48no—
296C22⋊C4—48no—
297C22⋊C4—82no—
298C22⋊C4—82no—
299C42—21no—
300C42—21no—
301C42—22no—
302C42—22no—
303C42#12012yes—
304C42AGL(1,ℤ/8)12yes—
305C16—24no—
306C16—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—44no—
314C23—82no—
315C23—82no—
316C23—82no—
317C23—82no—
318C23—82no—
319C23—82no—
320C23#50912yes—
321C23C22×D811yescharacteristic
322Q8—88no—
323D4—44no—
324D4—44no—
325D4—88no—
326D4—88no—
327D4—816no—
328D4—82no—
329D4—82no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—816no—
350C2×C4—82no—
351C2×C4—82no—
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#44712yes—
365C2×C4#44712yes—
366C2×C4#50912yes—
367C2×C4C2×#12011yescharacteristic
368C2×C4#53211yescharacteristic
369C2×C4#53611yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C8—41no—
374C8—41no—
375C8#45214yes—
376C8#54712yes—
377C8#54812yes—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—84no—
386C22—84no—
387C22—84no—
388C22—162no—
389C22—162no—
390C22—162no—
391C22—162no—
392C22C2×#44711yescharacteristic
393C22#316911yescharacteristic
394C22#322611yescharacteristic
395C22C2×#50911yescharacteristic
396C22C2×#50911yescharacteristic
397C22#341411yescharacteristic
398C22#341411yescharacteristic
399C4—22no—
400C4—22no—
401C4—22no—
402C4—41no—
403C4—41no—
404C4—84no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4#243211yescharacteristic
410C4C2×#45211yescharacteristic
411C4#318211yescharacteristic
412C4#322811yescharacteristic
413C2—42no—
414C2—164no—
415C2#2567311yescharacteristic
416C2#3351711yescharacteristic
417C2#3351911yescharacteristic
418C2#3373711yescharacteristic
419C2#3374611yescharacteristic
420C2#3380011yescharacteristic
421C2#3380111yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#3169C2
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
2. C2×#2255C2semidirect
3. C2×#2244C2semidirect
4. #33208C2semidirect
5. #13381C2semidirect
6. #13380C2semidirect
7. #13283C2semidirect
8. #13206C2semidirect
9. #12937C2semidirect
10. #12935C2semidirect
11. #12599C2semidirect
186. C2×C16#41semidirect

Maximal subgroups

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

subgroupindex
C2×#31692
C2×#22552
C2×#22442
#332082
#133812
#133802
#132832
#132062
#129372
#129352
#125992