Finite Group Databasebeta

← #497977

Subgroups — #497977

Order 512.

592 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2165C211yesmaximal, characteristic, complemented
2C22×#364C211yesmaximal, characteristic, complemented
3C4×#435C211yesmaximal, characteristic
4#13128C212yesmaximal, complemented
5#13107C212yesmaximal, complemented
6#12985C212yesmaximal
7#12882C211yesmaximal, characteristic, complemented
8#12791C211yesmaximal, characteristic, complemented
9#12788C211yesmaximal, characteristic, complemented
10#12659C211yesmaximal, characteristic, complemented
11#12616C212yesmaximal, complemented
12C22×M32C2212yes—
13C2×#435C411yescharacteristic
14C2×#435C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C42×D4C411yescharacteristic
18#2165—24no—
19C2×#364—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
32C4×C4⋊C8C411yescharacteristic
33#1753C411yescharacteristic
34#1753C2211yescharacteristic
35#1752C411yescharacteristic
36#1752C2211yescharacteristic
37C4×C22⋊C8C411yescharacteristic
38C4×C22⋊C8C411yescharacteristic
39C4×#97C2211yescharacteristic
40#1377C2212yes—
41#1377C2212yes—
42#1376C2212yes—
43#1375C2212yes—
44#1375C2212yes—
45C22×M16C2×C411yescharacteristic
46C22×M16C2311yescharacteristic
47C23×C8C2×C411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4—21no—
50C2×C4×D4C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C22×C42C2×C411yescharacteristic
53C2×M32—22no—
54C2×M32—24no—
55C2×M32—42no—
56C2×M32C2312yes—
57C2×M32C2312yes—
58C2×M32C2312yes—
59C2×M32C2312yes—
60C22×C16—22no—
61#435—22no—
62#435—22no—
63#435—21no—
64#435—21no—
65#435—21no—
66#435—21no—
67C2×C4⋊C8C2×C411yescharacteristic
68C2×C4⋊C8C2×C411yescharacteristic
69C2×C22⋊C8C2×C411yescharacteristic
70C2×C22⋊C8C2×C411yescharacteristic
71C2×C22⋊C8C2×C411yescharacteristic
72C2×C22⋊C8C2×C411yescharacteristic
73C4×M16—22no—
74C4×M16—22no—
75C2×#97C2×C411yescharacteristic
76C2×#97C2×C411yescharacteristic
77C2×#97C2×C411yescharacteristic
78C2×#97C2311yescharacteristic
79C2×#97C2311yescharacteristic
80C2×#97C2311yescharacteristic
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2×C411yescharacteristic
86C2×C4×C8C2×C411yescharacteristic
87C2×C4×C8C2311yescharacteristic
88C2×C4×C8C2311yescharacteristic
89C4×C4⋊C4—21no—
90C4×C4⋊C4C2×C411yescharacteristic
91C4×C22⋊C4—21no—
92C4×C22⋊C4—21no—
93C4×C22⋊C4C2×C411yescharacteristic
94C4×C22⋊C4C2×C411yescharacteristic
95C43C2×C411yescharacteristic
96#364—24no—
97#364—22no—
98#364—22no—
99#364—22no—
100#364D414yescomplemented
101#349—24no—
102#349—42no—
103C22⋊C16—22no—
104C22⋊C16—22no—
105#347—24no—
106#347—42no—
107#336C2×C411yescharacteristic
108#336C2×C411yescharacteristic
109#336C2×C411yescharacteristic
110#336C2×C411yescharacteristic
111#336C2×C411yescharacteristic
112#336C2×C411yescharacteristic
113C22×D4M1611yescharacteristic
114C23×C4—21no—
115C23×C4—21no—
116C23×C4C22×C411yescharacteristic
117C23×C4C22×C411yescharacteristic
118C2×M16—22no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—21no—
122C2×M16—21no—
123C2×M16—21no—
124C2×M16—21no—
125C2×M16—21no—
126C2×M16—21no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8M1612yes—
134C22×C8M1612yes—
135C22×C8C22×C411yescharacteristic
136C22×C8C22×C411yescharacteristic
137C22×C8C22×C411yescharacteristic
138C22×C8C22×C411yescharacteristic
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8C22×C411yescharacteristic
142C22×C8#4111yescharacteristic
143C22×C8#4111yescharacteristic
144C22×C8#4111yescharacteristic
145C22×C8#4111yescharacteristic
146C22×C8C2411yescharacteristic
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150C4×D4—21no—
151C4×D4—44no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4M1611yescharacteristic
154C2×C22⋊C4—21no—
155C2×C22⋊C4—21no—
156C2×C22⋊C4M1611yescharacteristic
157C2×C22⋊C4M1611yescharacteristic
158C2×C42—22no—
159C2×C42—22no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—41no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171M32—22no—
172M32—22no—
173M32—22no—
174M32—22no—
175M32—22no—
176M32—22no—
177M32—22no—
178M32—44no—
179M32C2×D414yes—
180C2×C16—22no—
181C2×C16—22no—
182C2×C16—22no—
183C2×C16—22no—
184C2×C16—44no—
185C4⋊C8—21no—
186C4⋊C8—21no—
187C4⋊C8—22no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—21no—
192C22⋊C8—21no—
193C22⋊C8—21no—
194C22⋊C8—22no—
195#97—22no—
196#97—22no—
197#97—22no—
198#97—21no—
199#97—21no—
200#97—21no—
201#97—21no—
202#97—21no—
203#97—21no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—21no—
207C4×C8—22no—
208C4×C8—22no—
209C4×C8—41no—
210C4×C8C2×D412yes—
211C4×C8#4112yes—
212#95—21no—
213#95—21no—
214#95—21no—
215C24C2×M1611yescharacteristic
216C24C2×M1611yescharacteristic
217C2×D4—21no—
218C2×D4—21no—
219C2×D4—42no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—24no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4#11711yescharacteristic
241C22×C4#11711yescharacteristic
242C22×C4C2×M1611yescharacteristic
243C22×C4C2×M1611yescharacteristic
244C22×C4C2×M1611yescharacteristic
245C22×C4C2×M1611yescharacteristic
246C22×C4C2×M1611yescharacteristic
247C22×C4C2×M1611yescharacteristic
248C22×C4C23×C411yescharacteristic
249M16—41no—
250M16—41no—
251M16—41no—
252M16—41no—
253M16—44no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—21no—
264C2×C8—21no—
265C2×C8—21no—
266C2×C8—24no—
267C2×C8—24no—
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—21no—
277C2×C8—21no—
278C2×C8—21no—
279C2×C8—21no—
280C2×C8#11711yescharacteristic
281C2×C8#11711yescharacteristic
282C2×C8C4×D412yes—
283C2×C8C2×M1612yes—
284C2×C8C2×M1612yes—
285C2×C8#13112yes—
286C2×C8C22×D411yescharacteristic
287C2×C8C2×#4111yescharacteristic
288C2×C8C2×#4111yescharacteristic
289C2×C8C2×#4111yescharacteristic
290C2×C8H(5,𝔽2)11yescharacteristic
291C2×C82−1+411yescharacteristic
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—42no—
295C22⋊C4—21no—
296C22⋊C4—21no—
297C22⋊C4—21no—
298C22⋊C4—21no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—42no—
311C42—42no—
312C42—42no—
313C42—42no—
314C42C4×D412yes—
315C16—24no—
316C16—42no—
317C16—42no—
318C16—42no—
319C23—22no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23#43211yescharacteristic
329C23#43211yescharacteristic
330C23C22×M1611yescharacteristic
331D4—42no—
332D4—42no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—48no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—84no—
368C2×C4#43211yescharacteristic
369C2×C4#43211yescharacteristic
370C2×C4C2×#11711yescharacteristic
371C2×C4C2×C4×D411yescharacteristic
372C2×C4#52011yescharacteristic
373C2×C4C22×M1611yescharacteristic
374C2×C4C2×#13111yescharacteristic
375C2×C4#56811yescharacteristic
376C8—21no—
377C8—21no—
378C8—22no—
379C8—41no—
380C8—41no—
381C8—41no—
382C8—41no—
383C8#43514yes—
384C8#54611yescharacteristic
385C8#54811yescharacteristic
386C8#55211yescharacteristic
387C8#55311yescharacteristic
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—81no—
401C22—81no—
402C22#216512yescomplemented
403C22C2×#43211yescharacteristic
404C22D4×M1611yescharacteristic
405C22#299011yescharacteristic
406C4—24no—
407C4—21no—
408C4—21no—
409C4—42no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—81no—
414C4—81no—
415C4#235211yescharacteristic
416C4#292511yescharacteristic
417C4C2×#43511yescharacteristic
418C4#298911yescharacteristic
419C2—21no—
420C2—21no—
421C2—22no—
422C2—41no—
423C2—81no—
424C2#2275011yescharacteristic
425C2C2×#216511yescharacteristic
426C2#3342811yescharacteristic
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×#2165C2semidirect
2. C22×#364C2semidirect
4. #13128C2semidirect
5. #13107C2semidirect
7. #12882C2semidirect
8. #12791C2semidirect
9. #12788C2semidirect
10. #12659C2semidirect
11. #12616C2semidirect
100. #364D4semidirect
402. C22#2165semidirect

Maximal subgroups

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

subgroupindex
C2×#21652
C22×#3642
C4×#4352
#131282
#131072
#129852
#128822
#127912
#127882
#126592
#126162