Finite Group Databasebeta

← #244487

Subgroups — #244487

Order 512.

428 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#22310C211yesmaximal, characteristic, complemented
2#22235C211yesmaximal, characteristic
3#10688C211yesmaximal, characteristic, complemented
4#10687C211yesmaximal, characteristic, complemented
5#10630C211yesmaximal, characteristic, complemented
6#10629C211yesmaximal, characteristic
7#10547C211yesmaximal, characteristic
8#10547C211yesmaximal, characteristic
9#10371C211yesmaximal, characteristic, complemented
10#10370C211yesmaximal, characteristic
11#8257C211yesmaximal, characteristic
12#8255C211yesmaximal, characteristic
13#8254C211yesmaximal, characteristic, complemented
14#8252C211yesmaximal, characteristic, complemented
15#8142C211yesmaximal, characteristic, complemented
16#3135C2211yescharacteristic
17#2960C2211yescharacteristic
18#2931C2211yescharacteristic
19#2399C2211yescharacteristic
20#1995C2211yescharacteristic
21#1985—21no—
22#1984—21no—
23#1963C2211yescharacteristic
24#1940C2211yescharacteristic
25#1940C2211yescharacteristic
26#1929—21no—
27#1929—21no—
28#1917—21no—
29#1916—21no—
30#1866—21no—
31#1866—21no—
32#1849C2211yescharacteristic
33#1832C2211yescharacteristic
34#1770C2211yescharacteristic
35C8×C4⋊C4C2211yescharacteristic
36#1746C2211yescharacteristic
37#1542C2211yescharacteristic
38#1542C2211yescharacteristic
39#1540C2211yescharacteristic
40#1540C2211yescharacteristic
41#1503C2211yescharacteristic
42#1503C2211yescharacteristic
43#1502C2211yescharacteristic
44#1502C2211yescharacteristic
45#1452C2211yescharacteristic
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
62C2×C4×Q8—21no—
63#491—21no—
64#490—21no—
65#489—21no—
66#489—21no—
67#446—21no—
68#446—21no—
69C8×Q8—21no—
70C8×Q8—21no—
71#436—21no—
72#436—21no—
73C8×D4—21no—
74C8×D4—21no—
75#431C2311yescharacteristic
76#431C2311yescharacteristic
77#431C2311yescharacteristic
78#431C2311yescharacteristic
79C2×#108—21no—
80C2×#108—21no—
81#424—21no—
82#423C2311yescharacteristic
83#423C2311yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#104—21no—
89C2×#104—21no—
90#416—21no—
91#416—21no—
92#408—21no—
93#405—21no—
94C2×#97—21no—
95C2×#97C2311yescharacteristic
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98#398—21no—
99#398—21no—
100#395—21no—
101#395—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4C2311yescharacteristic
104C4×C4⋊C4C2311yescharacteristic
105#376C2311yescharacteristic
106#339—21no—
107#339—21no—
108#336—21no—
109#336—21no—
110#336—21no—
111#336—21no—
112#336—21no—
113#336—21no—
114#336D411yescharacteristic
115#336D411yescharacteristic
116#336D411yescharacteristic
117#336D411yescharacteristic
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—
136#322—21no—
137C82—21no—
138C2×#41—21no—
139C22×Q8—21no—
140#131—81no—
141#131—81no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—41no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C22×C8C2×D411yescharacteristic
149C22×C8C2×D411yescharacteristic
150C22×C8C2×D411yescharacteristic
151C22×C8C2×D411yescharacteristic
152C4⋊Q8#4111yescharacteristic
153C4⋊Q8#4111yescharacteristic
154C4⋊Q8#4111yescharacteristic
155#127#4111yescharacteristic
156#125—21no—
157#124—21no—
158C22⋊Q8—21no—
159C22⋊Q8—21no—
160#121—21no—
161#121—21no—
162C4×Q8—21no—
163C4×Q8—21no—
164C4×Q8—81no—
165C4×Q8—81no—
166C4×D4—21no—
167C4×D4—21no—
168#117—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C42C2411yescharacteristic
181#108—81no—
182#108—81no—
183C4⋊C8—21no—
184C4⋊C8—21no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—41no—
196C4⋊C8—41no—
197C4⋊C8—81no—
198C4⋊C8—81no—
199#104—81no—
200#104—81no—
201#103—21no—
202#103—21no—
203#103—21no—
204#103—21no—
205#102—21no—
206#102—21no—
207#102—21no—
208#102—21no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215#97—21no—
216#97—21no—
217#97—41no—
218#97—81no—
219C4×C8—21no—
220C4×C8—21no—
221C4×C8—21no—
222C4×C8—21no—
223C4×C8—41no—
224C4×C8—81no—
225C4×C8#4111yescharacteristic
226C4×C8#4111yescharacteristic
227C4×C8#4111yescharacteristic
228C4×C8#4111yescharacteristic
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×Q8—42no—
245C2×Q8—81no—
246C2×Q8—81no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—21no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—41no—
258C22×C4C22×D411yescharacteristic
259C22×C4C2×#4111yescharacteristic
260C22×C4C2×#4111yescharacteristic
261M16—41no—
262M16—41no—
263M16—41no—
264M16—41no—
265M16—82no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—21no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—82no—
289C2×C8#12411yescharacteristic
290C2×C8#12411yescharacteristic
291C2×C8#12411yescharacteristic
292C2×C8#12411yescharacteristic
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—81no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—41no—
316C42—41no—
317C42—41no—
318C42—41no—
319C42—41no—
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—41no—
325C23#52511yescharacteristic
326Q8—41no—
327Q8—41no—
328Q8—81no—
329Q8—81no—
330Q8—84no—
331D4—41no—
332D4—41no—
333D4—81no—
334D4—81no—
335D4—81no—
336D4—81no—
337C2×C4—22no—
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—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4#52511yescharacteristic
363C2×C4C2×#12411yescharacteristic
364C2×C4#52911yescharacteristic
365C2×C4#52911yescharacteristic
366C2×C4#52911yescharacteristic
367C2×C4#52911yescharacteristic
368C2×C4#57611yescharacteristic
369C2×C4#57711yescharacteristic
370C2×C4#57711yescharacteristic
371C2×C4#57811yescharacteristic
372C8—21no—
373C8—21no—
374C8—21no—
375C8—21no—
376C8—41no—
377C8—41no—
378C8—81no—
379C8—81no—
380C8—81no—
381C8—81no—
382C8—81no—
383C8—81no—
384C8—81no—
385C8—81no—
386C22—21no—
387C22—21no—
388C22—81no—
389C22—81no—
390C22—81no—
391C22#239811yescharacteristic
392C22#311811yescharacteristic
393C22#312111yescharacteristic
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—42no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4—81no—
407C4#312411yescharacteristic
408C4#312511yescharacteristic
409C4#313611yescharacteristic
410C4#313711yescharacteristic
411C2—21no—
412C2—21no—
413C2—161no—
414C2#2382711yescharacteristic
415C2#2383311yescharacteristic
416C2#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. #22310C2semidirect
3. #10688C2semidirect
4. #10687C2semidirect
5. #10630C2semidirect
9. #10371C2semidirect
13. #8254C2semidirect
14. #8252C2semidirect
15. #8142C2semidirect

Maximal subgroups

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

subgroupindex
#223102
#222352
#106882
#106872
#106302
#106292
#105472
#103712
#103702
#82572
#82552
#82542
#82522
#81422