Finite Group Databasebeta

← #244472

Subgroups — #244472

Order 512.

434 conjugacy classes of subgroups.

The subgroup lattice has 413 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21679C211yesmaximal, characteristic, complemented
2#21663C211yesmaximal, characteristic
3#10689C211yesmaximal, characteristic, complemented
4#10689C211yesmaximal, characteristic, complemented
5#10630C211yesmaximal, characteristic, complemented
6#10629C211yesmaximal, characteristic
7#10547C211yesmaximal, characteristic
8#10547C211yesmaximal, characteristic
9#10373C211yesmaximal, characteristic, complemented
10#10372C211yesmaximal, characteristic
11#8256C211yesmaximal, characteristic, complemented
12#8255C211yesmaximal, characteristic
13#8252C211yesmaximal, characteristic, complemented
14#8251C211yesmaximal, characteristic
15#8138C211yesmaximal, characteristic, complemented
16#3135C2211yescharacteristic
17#2960C2211yescharacteristic
18#2931C2211yescharacteristic
19#2402C2211yescharacteristic
20#1995C2211yescharacteristic
21#1963C2211yescharacteristic
22#1940C2211yescharacteristic
23#1940C2211yescharacteristic
24#1937—21no—
25#1937—21no—
26#1916—21no—
27#1916—21no—
28#1838C2211yescharacteristic
29#1832C2211yescharacteristic
30#1771—21no—
31#1770—21no—
32#1770C2211yescharacteristic
33C8×C4⋊C4C2211yescharacteristic
34#1765—21no—
35#1765—21no—
36#1726C2211yescharacteristic
37#1542C2211yescharacteristic
38#1542C2211yescharacteristic
39#1540C2211yescharacteristic
40#1540C2211yescharacteristic
41#1503C2211yescharacteristic
42#1503C2211yescharacteristic
43#1502C2211yescharacteristic
44#1502C2211yescharacteristic
45#1450C2211yescharacteristic
46#1450C2211yescharacteristic
47#1303C2211yescharacteristic
48#1303C2211yescharacteristic
49#1303C2211yescharacteristic
50#1303C2211yescharacteristic
51#1301C2211yescharacteristic
52#1301C2211yescharacteristic
53#1301C2211yescharacteristic
54#1301C2211yescharacteristic
55#1275C2211yescharacteristic
56#1275C2211yescharacteristic
57#1274C2211yescharacteristic
58#1274C2211yescharacteristic
59C2×#131—21no—
60#533C2311yescharacteristic
61#532C2311yescharacteristic
62#491—21no—
63#490—21no—
64#489—21no—
65#489—21no—
66#446—21no—
67#446—21no—
68C8×Q8—21no—
69C8×Q8—21no—
70#436—21no—
71#436—21no—
72C8×D4—21no—
73C8×D4—21no—
74#431C2311yescharacteristic
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#424—21no—
79#423C2311yescharacteristic
80#423C2311yescharacteristic
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×#104—21no—
86C2×#104—21no—
87#416—21no—
88#416—21no—
89#408—21no—
90#405—21no—
91C2×#97—21no—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×#97C2311yescharacteristic
95C2×C4×C8—21no—
96C2×C4×C8D411yescharacteristic
97C2×C4×C8D411yescharacteristic
98C2×C4×C8C2311yescharacteristic
99#400—21no—
100#400—21no—
101#398—21no—
102#398—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2311yescharacteristic
107#376C2311yescharacteristic
108#341—21no—
109#341—21no—
110#339—21no—
111#339—21no—
112#336—21no—
113#336—21no—
114#336—21no—
115#336—21no—
116#336—21no—
117#336—21no—
118#333—21no—
119#333—21no—
120#332—21no—
121#332—21no—
122#332—21no—
123#332—21no—
124#331—21no—
125#331—21no—
126#328—21no—
127#328—21no—
128#328—21no—
129#328—21no—
130#327—21no—
131#327—21no—
132#327—21no—
133#327—21no—
134#322—21no—
135#322—21no—
136C82—21no—
137C82—21no—
138C2×#41—21no—
139#131—42no—
140#131—42no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—41no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—41no—
147C22×C8C2×D411yescharacteristic
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C4⋊Q8#4111yescharacteristic
152C4⋊Q8#4111yescharacteristic
153C4⋊Q8#4111yescharacteristic
154#127#4111yescharacteristic
155#125—21no—
156#124—21no—
157C22⋊Q8—21no—
158C22⋊Q8—21no—
159#121—21no—
160#121—21no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×D4—21no—
164C4×D4—21no—
165#117—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×C42—21no—
173C2×C42—21no—
174C2×C42C2×D411yescharacteristic
175C2×C42C2×D411yescharacteristic
176C2×C42C2411yescharacteristic
177C4⋊C8—21no—
178C4⋊C8—21no—
179C4⋊C8—21no—
180C4⋊C8—21no—
181C4⋊C8—21no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—42no—
190C4⋊C8—42no—
191C4⋊C8—41no—
192C4⋊C8—41no—
193#104—42no—
194#104—42no—
195#103—21no—
196#103—21no—
197#103—21no—
198#103—21no—
199#102—21no—
200#102—21no—
201#102—21no—
202#102—21no—
203C22⋊C8—21no—
204C22⋊C8—21no—
205C22⋊C8—21no—
206C22⋊C8—21no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—22no—
210#97—22no—
211#97—21no—
212#97—21no—
213#97—41no—
214#97—42no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219C4×C8—41no—
220C4×C8—41no—
221C4×C8—41no—
222C4×C8—42no—
223C4×C8#4111yescharacteristic
224C4×C8#4111yescharacteristic
225C4×C8#4111yescharacteristic
226C4×C8#4111yescharacteristic
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—21no—
232#95—21no—
233#95#4111yescharacteristic
234#95#4111yescharacteristic
235#95#4111yescharacteristic
236#95#4111yescharacteristic
237#41—41no—
238#41—41no—
239#41—41no—
240#41—41no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257C22×C4C2×#4111yescharacteristic
258M16—41no—
259M16—41no—
260M16—41no—
261M16—41no—
262M16—82no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—82no—
284C2×C8#12411yescharacteristic
285C2×C8#12411yescharacteristic
286C2×C8#12411yescharacteristic
287C2×C8#12411yescharacteristic
288C2×C8#13512yes—
289C2×C8#13512yes—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—81no—
302C4⋊C4—81no—
303C4⋊C4—81no—
304C22⋊C4—41no—
305C22⋊C4—41no—
306C22⋊C4—41no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—41no—
312C42—41no—
313C42—41no—
314C42—41no—
315C42—41no—
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C42C2×#4111yescharacteristic
319C42C2×#4111yescharacteristic
320C23—41no—
321C23#52511yescharacteristic
322Q8—41no—
323Q8—41no—
324Q8—81no—
325Q8—81no—
326D4—41no—
327D4—41no—
328D4—81no—
329D4—81no—
330D4—81no—
331D4—81no—
332C2×C4—22no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4#52511yescharacteristic
358C2×C4C2×#12411yescharacteristic
359C2×C4#52911yescharacteristic
360C2×C4#52911yescharacteristic
361C2×C4#52911yescharacteristic
362C2×C4#52911yescharacteristic
363C2×C4C2×#13511yescharacteristic
364C2×C4C2×#13511yescharacteristic
365C2×C4#57511yescharacteristic
366C2×C4#57511yescharacteristic
367C8—21no—
368C8—21no—
369C8—41no—
370C8—41no—
371C8—81no—
372C8—81no—
373C8—81no—
374C8—81no—
375C8—81no—
376C8—81no—
377C8—81no—
378C8—81no—
379C8#48812yes—
380C8#48812yes—
381C22—21no—
382C22—21no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#239811yescharacteristic
387C22#310211yescharacteristic
388C22#310211yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—21no—
393C4—42no—
394C4—81no—
395C4—81no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4#310211yescharacteristic
403C4#310211yescharacteristic
404C4C2×#48811yescharacteristic
405C4#313511yescharacteristic
406C2—21no—
407C2—21no—
408C2—161no—
409C2#2381811yescharacteristic
410C2#2383311yescharacteristic
411C2#2383411yescharacteristic
subgroupG/H
centerC2×C4C2×#124
commutator, Frattini, cosocleC2×C42C24
socleC22#2398
prime radical#532C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21679C2semidirect
3. #10689C2semidirect
4. #10689C2semidirect
5. #10630C2semidirect
9. #10373C2semidirect
11. #8256C2semidirect
13. #8252C2semidirect
15. #8138C2semidirect

Maximal subgroups

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

subgroupindex
#216792
#216632
#106892
#106302
#106292
#105472
#103732
#103722
#82562
#82552
#82522
#82512
#81382