Finite Group Databasebeta

← #497976

Subgroups — #497976

Order 512.

547 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2166C211yesmaximal, characteristic, complemented
2C2×#2153C211yesmaximal, characteristic, complemented
3#20488C211yesmaximal, characteristic
4#13128C212yesmaximal, complemented
5#13107C212yesmaximal, complemented
6#12985C212yesmaximal
7#12882C211yesmaximal, characteristic, complemented
8#12790C211yesmaximal, characteristic, complemented
9#12789C211yesmaximal, characteristic, complemented
10#12659C211yesmaximal, characteristic, complemented
11#12616C212yesmaximal, complemented
12C22×M32C2212yes—
13C2×#435C411yescharacteristic
14C2×#435C2211yescharacteristic
15C2×#405C2211yescharacteristic
16C2×#405C2211yescharacteristic
17#2293C411yescharacteristic
18#2166—24no—
19#2153—24no—
20C2×#364C2212yes—
21C2×#364C2211yescharacteristic
22C2×#364C2211yescharacteristic
23C2×#364C2211yescharacteristic
24C2×#364C2211yescharacteristic
25C2×#349C2212yes—
26C2×#349C2212yes—
27#2112C2212yes—
28C2×#347C2212yes—
29C2×#347C2212yes—
30#1771C411yescharacteristic
31#1771C2211yescharacteristic
32#1767C411yescharacteristic
33#1753C411yescharacteristic
34#1753C2211yescharacteristic
35#1752C411yescharacteristic
36#1752C2211yescharacteristic
37#1750C411yescharacteristic
38#1750C411yescharacteristic
39C4×#97C2211yescharacteristic
40#1377C2212yes—
41#1377C2212yes—
42#1376C2212yes—
43#1375C2212yes—
44#1375C2212yes—
45C22×M16C2×C411yescharacteristic
46C22×M16C2311yescharacteristic
47C23×C8C2×C411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×M32—22no—
53C2×M32—24no—
54C2×M32—42no—
55C2×M32C2312yes—
56C2×M32C2312yes—
57C2×M32C2312yes—
58C2×M32C2312yes—
59C22×C16—22no—
60#435—24no—
61#435—22no—
62#435—22no—
63C2×C4⋊C8C2×C411yescharacteristic
64C2×C4⋊C8C2×C411yescharacteristic
65C2×C22⋊C8C2×C411yescharacteristic
66C2×C22⋊C8C2×C411yescharacteristic
67C2×C22⋊C8C2×C411yescharacteristic
68C2×C22⋊C8C2×C411yescharacteristic
69#405—22no—
70#405—22no—
71#405—22no—
72#405—22no—
73C2×#97C2×C411yescharacteristic
74C2×#97C2×C411yescharacteristic
75C2×#97C2×C411yescharacteristic
76C2×#97C2×C411yescharacteristic
77C2×#97C2×C411yescharacteristic
78C2×#97C2311yescharacteristic
79C2×#97C2311yescharacteristic
80C2×#97C2311yescharacteristic
81C2×C4×C8C2×C411yescharacteristic
82C2×C4×C8C2×C411yescharacteristic
83C2×C4×C8C2311yescharacteristic
84C2×C4×C8C2311yescharacteristic
85#390—21no—
86#389—21no—
87#388—21no—
88#388—21no—
89C4×C4⋊C4C2×C411yescharacteristic
90C4×C22⋊C4C2×C411yescharacteristic
91C4×C22⋊C4C2×C411yescharacteristic
92C43C2×C411yescharacteristic
93#364—24no—
94#364—22no—
95#364—22no—
96#364—22no—
97#364D414yescomplemented
98#349—24no—
99#349—42no—
100C22⋊C16—24no—
101#347—24no—
102#347—42no—
103#336C2×C411yescharacteristic
104#336C2×C411yescharacteristic
105#336C2×C411yescharacteristic
106#336C2×C411yescharacteristic
107#336C2×C411yescharacteristic
108#336C2×C411yescharacteristic
109C22×D4M1611yescharacteristic
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—21no—
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16M1614yes—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8C22×C411yescharacteristic
129C22×C8C22×C411yescharacteristic
130C22×C8C22×C411yescharacteristic
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C22×C411yescharacteristic
135C22×C8#4112yes—
136C22×C8#4112yes—
137C22×C8C2411yescharacteristic
138C4×D4—22no—
139C4×D4—22no—
140#117—22no—
141#117—22no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4M1611yescharacteristic
148C2×C22⋊C4—21no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4M1611yescharacteristic
153C2×C22⋊C4M1611yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161M32—22no—
162M32—22no—
163M32—22no—
164M32—22no—
165M32—22no—
166M32—22no—
167M32—22no—
168M32—44no—
169M32C2×D414yes—
170C2×C16—22no—
171C2×C16—22no—
172C2×C16—22no—
173C2×C16—22no—
174C2×C16—44no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8—22no—
180C22⋊C8—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—21no—
185#97—21no—
186#97—21no—
187#97—21no—
188#97—21no—
189#97—21no—
190#97—22no—
191#97—22no—
192C4×C8—22no—
193C4×C8—21no—
194C4×C8—21no—
195C4×C8—41no—
196C4×C8C2×D412yes—
197C4×C8#4112yes—
198#95—21no—
199#95—21no—
200#95—21no—
201C24C2×M1611yescharacteristic
202C24C2×M1611yescharacteristic
203C2×D4—22no—
204C2×D4—81no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4#11712yes—
220C22×C4C2×M1611yescharacteristic
221C22×C4C2×M1612yes—
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C2×M1611yescharacteristic
225C22×C4C23×C411yescharacteristic
226M16—28no—
227M16—41no—
228M16—41no—
229M16—41no—
230M16—41no—
231C2×C8—21no—
232C2×C8—21no—
233C2×C8—21no—
234C2×C8—21no—
235C2×C8—22no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—21no—
247C2×C8—21no—
248C2×C8—21no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—44no—
252C2×C8#11712yes—
253C2×C8C4×D412yes—
254C2×C8C2×M1612yes—
255C2×C8C2×M1612yes—
256C2×C8#13112yes—
257C2×C8C22×D411yescharacteristic
258C2×C8C2×#4111yescharacteristic
259C2×C8C2×#4111yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8H(5,𝔽2)11yescharacteristic
262C2×C82−1+411yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—81no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—81no—
272C22⋊C4—81no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—81no—
283C42—81no—
284C42C4×D412yes—
285C16—24no—
286C16—42no—
287C16—42no—
288C16—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#43212yes—
299C23C22×M1611yescharacteristic
300D4—82no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4#43212yes—
331C2×C4C2×#11711yescharacteristic
332C2×C4C2×C4×D411yescharacteristic
333C2×C4#52011yescharacteristic
334C2×C4C22×M1611yescharacteristic
335C2×C4C2×#13111yescharacteristic
336C2×C4#56811yescharacteristic
337C8—22no—
338C8—22no—
339C8—41no—
340C8—41no—
341C8—41no—
342C8—41no—
343C8#43514yes—
344C8#54611yescharacteristic
345C8#54811yescharacteristic
346C8#55211yescharacteristic
347C8#55311yescharacteristic
348C22—21no—
349C22—21no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22—81no—
360C22C2×#43211yescharacteristic
361C22D4×M1611yescharacteristic
362C22#299011yescharacteristic
363C4—22no—
364C4—42no—
365C4—41no—
366C4—41no—
367C4—41no—
368C4—81no—
369C4—81no—
370C4—81no—
371C4#235211yescharacteristic
372C4#292511yescharacteristic
373C4C2×#43511yescharacteristic
374C4#298911yescharacteristic
375C2—21no—
376C2—21no—
377C2—42no—
378C2—81no—
379C2#2275011yescharacteristic
380C2#3342611yescharacteristic
381C2#3342711yescharacteristic
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC22×C8C24
socleC22C2×#432
prime radicalC22×D4M16
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #1752 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2166C2semidirect
2. C2×#2153C2semidirect
4. #13128C2semidirect
5. #13107C2semidirect
7. #12882C2semidirect
8. #12790C2semidirect
9. #12789C2semidirect
10. #12659C2semidirect
11. #12616C2semidirect
97. #364D4semidirect

Maximal subgroups

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

subgroupindex
C2×#21662
C2×#21532
#204882
#131282
#131072
#129852
#128822
#127902
#127892
#126592
#126162