Finite Group Databasebeta

← #500865

Subgroups — #500865

Order 512.

844 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic, complemented
2C2×#2226C211yesmaximal, characteristic, complemented
3C2×#2224C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic
5#13453C212yesmaximal, complemented
6#13365C212yesmaximal, complemented
7#13276C211yesmaximal, characteristic, complemented
8#13275C211yesmaximal, characteristic, complemented
9#12992C212yesmaximal
10#12921C212yesmaximal, complemented
11#12783C211yesmaximal, characteristic, complemented
12C2×#533—21no—
13#3160—24no—
14#3160—24no—
15C2×#500C2211yescharacteristic
16C2×#500C2211yescharacteristic
17C2×#499—21no—
18C2×#499C2212yes—
19C2×#498—21no—
20C2×#466C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×#427—21no—
23C2×#427C2212yes—
24C22×C4×C8C2211yescharacteristic
25C4×#121—21no—
26#2226—24no—
27#2224—24no—
28C2×#367C2212yes—
29C2×#366C2212yes—
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#357C2212yes—
33C2×#357C2212yes—
34C2×C22⋊C16C2212yes—
35#1970C2211yescharacteristic
36#1942C2211yescharacteristic
37#1934—21no—
38#1893—21no—
39#1808C2211yescharacteristic
40C4×#107C2211yescharacteristic
41C4×#102—21no—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45#1380C2212yes—
46#1374C2212yes—
47C22×D8C2311yescharacteristic
48C23×C8D411yescharacteristic
49C23×C8C2311yescharacteristic
50#533—28no—
51#533—44no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×#125—22no—
55C2×C22⋊Q8—41no—
56C2×C22⋊Q8—41no—
57C2×#121—21no—
58C2×C4×Q8—41no—
59C2×C4×Q8—41no—
60C2×C4×D4—21no—
61C2×C4×D4—41no—
62C2×#117—22no—
63C22×C42D411yescharacteristic
64C22×C16C2312yes—
65C22×C16C2312yes—
66#500—22no—
67#500—22no—
68#500D414yescomplemented
69#500D414yescomplemented
70#499—28no—
71#499—24no—
72#499—24no—
73#498—28no—
74#466—22no—
75#466Q814yescomplemented
76C4×D8—22no—
77C4×D8—22no—
78#427—28no—
79#427—24no—
80#427—24no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2312yes—
86C2×#107C2312yes—
87C2×#107C2311yescharacteristic
88C2×#106—22no—
89C2×#106—22no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×C4×C8—22no—
94C2×C4×C8—22no—
95C2×C4×C8D411yescharacteristic
96C2×C4×C8D411yescharacteristic
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#390—21no—
100#389—21no—
101#388—41no—
102#386—41no—
103#380—21no—
104C4×C4⋊C4—21no—
105C4×C22⋊C4—41no—
106#367—24no—
107#366—24no—
108C4⋊C16—22no—
109C4⋊C16—22no—
110#357—24no—
111#357—24no—
112C22⋊C16—24no—
113#340—21no—
114#340—21no—
115#340—21no—
116C22×Q8—41no—
117C22×Q8—41no—
118C22×D4—21no—
119C22×D4—41no—
120C23×C4—41no—
121C23×C4D811yescharacteristic
122C23×C4D811yescharacteristic
123C23×C4C2×D411yescharacteristic
124C2×D8—21no—
125C2×D8—21no—
126C2×D8—21no—
127C2×D8—21no—
128C2×D8—21no—
129C2×D8C2×Q812yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8C2×D411yescharacteristic
138C22×C8C2×D411yescharacteristic
139C22×C8C2×D411yescharacteristic
140C22×C8C2×Q812yes—
141C22×C8#4112yes—
142C22×C8C2411yescharacteristic
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#125—28no—
148#125—44no—
149C22⋊Q8—48no—
150C22⋊Q8—48no—
151#121—24no—
152#121—42no—
153C4×Q8—48no—
154C4×Q8—48no—
155C4×D4—42no—
156C4×D4—42no—
157C4×D4—84no—
158#117—28no—
159#117—44no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—42no—
179C2×C42—41no—
180C2×C42D811yescharacteristic
181C2×C42D811yescharacteristic
182C2×C42C2×D411yescharacteristic
183C2×C42C2×D411yescharacteristic
184C2×C16—22no—
185C2×C16—22no—
186C2×C16—22no—
187C2×C16—22no—
188C2×C16—22no—
189C2×C16—22no—
190#107—24no—
191#107—24no—
192#107—22no—
193#107—22no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107C2×D414yes—
200#107C2×D414yes—
201#107C2×Q812yes—
202#106—28no—
203#106—28no—
204#102—24no—
205#102—42no—
206#102—42no—
207C4×C8—22no—
208C4×C8—21no—
209C4×C8—21no—
210C4×C8—22no—
211C4×C8C2×D412yes—
212C4×C8C2×D412yes—
213#95—41no—
214#95—41no—
215#95—41no—
216C24—41no—
217C24C2×D811yescharacteristic
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—82no—
221C2×Q8—82no—
222C2×Q8—82no—
223C2×Q8—82no—
224C2×D4—22no—
225C2×D4—44no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—82no—
232C2×D4—82no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4—82no—
251C22×C4C22⋊Q812yes—
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×D811yescharacteristic
255C22×C4C2×D811yescharacteristic
256C22×C4C2×D811yescharacteristic
257C22×C4#13712yes—
258C22×C4C22×D411yescharacteristic
259D8—22no—
260D8—22no—
261D8—41no—
262D8—41no—
263C2×C8—28no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—22no—
271C2×C8—22no—
272C2×C8—22no—
273C2×C8—22no—
274C2×C8#12012yes—
275C2×C8C22⋊Q812yes—
276C2×C8#13712yes—
277C2×C8C22×D411yescharacteristic
278C2×C8C22×D411yescharacteristic
279C2×C8C22×Q811yescharacteristic
280C2×C8C2×#4111yescharacteristic
281C2×C82−1+411yescharacteristic
282C2×C82−1+411yescharacteristic
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—22no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—82no—
298C22⋊C4—48no—
299C22⋊C4—44no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C22⋊C4—82no—
303C22⋊C4—82no—
304C42—21no—
305C42—21no—
306C42—22no—
307C42—22no—
308C42—48no—
309C42—82no—
310C42#12012yes—
311C42AGL(1,ℤ/8)12yes—
312C16—42no—
313C16—42no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—42no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23#47412yes—
328C23C22×D811yescharacteristic
329Q8—88no—
330Q8—88no—
331D4—42no—
332D4—42no—
333D4—88no—
334D4—81no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—82no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—164no—
368C2×C4#44712yes—
369C2×C4#44712yes—
370C2×C4#47412yes—
371C2×C4#50812yes—
372C2×C4#50912yes—
373C2×C4C2×#12011yescharacteristic
374C2×C4C2×C22⋊Q811yescharacteristic
375C2×C4#53611yescharacteristic
376C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
377C2×C4C2×#13711yescharacteristic
378C2×C4C2×#13711yescharacteristic
379C8—22no—
380C8—22no—
381C8—22no—
382C8#54811yescharacteristic
383C8#54811yescharacteristic
384C8D4×Q811yescharacteristic
385C8D4×Q811yescharacteristic
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22—161no—
397C22—161no—
398C22—161no—
399C22—161no—
400C22C2×#44711yescharacteristic
401C22C2×#47411yescharacteristic
402C22#322611yescharacteristic
403C22C2×#50811yescharacteristic
404C22C2×#50911yescharacteristic
405C22#341411yescharacteristic
406C22#341411yescharacteristic
407C4—22no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—82no—
412C4—84no—
413C4—84no—
414C4—161no—
415C4—161no—
416C4#218614yes—
417C4#240911yescharacteristic
418C4#300411yescharacteristic
419C4#307911yescharacteristic
420C4#323911yescharacteristic
421C2—42no—
422C2—162no—
423C2#2429811yescharacteristic
424C2C2×#218611yescharacteristic
425C2#3351911yescharacteristic
426C2#3362611yescharacteristic
427C2#3362711yescharacteristic
428C2#3379911yescharacteristic
429C2#3380111yescharacteristic
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
2. C2×#2226C2semidirect
3. C2×#2224C2semidirect
5. #13453C2semidirect
6. #13365C2semidirect
7. #13276C2semidirect
8. #13275C2semidirect
10. #12921C2semidirect
11. #12783C2semidirect
68. #500D4semidirect
69. #500D4semidirect
75. #466Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
C2×#22262
C2×#22242
C4×#4662
#134532
#133652
#132762
#132752
#129922
#129212
#127832