Finite Group Databasebeta

← #497975

Subgroups — #497975

Order 512.

570 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2166C211yesmaximal, characteristic, complemented
2C22×#364C211yesmaximal, characteristic, complemented
3#20484C211yesmaximal, characteristic
4#13128C212yesmaximal, complemented
5#13107C212yesmaximal, complemented
6#12985C212yesmaximal
7#12881C211yesmaximal, characteristic, complemented
8#12790C211yesmaximal, characteristic, complemented
9#12788C211yesmaximal, characteristic, complemented
10#12660C211yesmaximal, characteristic, complemented
11#12616C212yesmaximal, complemented
12C22×M32C2212yes—
13C2×#435C2211yescharacteristic
14C2×C8×D4C411yescharacteristic
15C2×#405C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2290C411yescharacteristic
18#2166—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#1771C2211yescharacteristic
31C8×C4⋊C4C411yescharacteristic
32#1768C411yescharacteristic
33#1753C2211yescharacteristic
34#1752C2211yescharacteristic
35C8×C22⋊C4C411yescharacteristic
36C8×C22⋊C4C411yescharacteristic
37#1750C411yescharacteristic
38#1749C411yescharacteristic
39#1726C2211yescharacteristic
40#1377C2212yes—
41#1377C2212yes—
42#1376C2212yes—
43#1375C2212yes—
44#1375C2212yes—
45C22×M16C2311yescharacteristic
46C23×C8C2×C411yescharacteristic
47C23×C8C2×C411yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4×D4C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C22×C42C2×C411yescharacteristic
52C2×M32—22no—
53C2×M32—24no—
54C2×M32—42no—
55C2×M32C2312yes—
56C2×M32C2312yes—
57C2×M32C2312yes—
58C2×M32C2312yes—
59C22×C16—22no—
60#435—21no—
61#435—21no—
62#435—21no—
63#435—21no—
64C8×D4—22no—
65C8×D4—22no—
66C2×C4⋊C8C2×C411yescharacteristic
67C2×C4⋊C8C2×C411yescharacteristic
68C2×C22⋊C8C2×C411yescharacteristic
69C2×C22⋊C8C2×C411yescharacteristic
70C2×C22⋊C8C2×C411yescharacteristic
71C2×C22⋊C8C2×C411yescharacteristic
72#405—22no—
73#405—22no—
74C2×#97C2×C411yescharacteristic
75C2×#97C2×C411yescharacteristic
76C2×#97C2311yescharacteristic
77C2×#97C2311yescharacteristic
78C2×#97C2311yescharacteristic
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8C2×C411yescharacteristic
82C2×C4×C8C2×C411yescharacteristic
83C2×C4×C8C2×C411yescharacteristic
84C2×C4×C8C2×C411yescharacteristic
85C2×C4×C8C2×C411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#388—21no—
89#387—21no—
90#386—21no—
91#385—21no—
92C4×C4⋊C4C2×C411yescharacteristic
93C4×C22⋊C4C2×C411yescharacteristic
94C4×C22⋊C4C2×C411yescharacteristic
95#376C2×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×C4C22×C411yescharacteristic
116C23×C4C22×C411yescharacteristic
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—21no—
120C2×M16—21no—
121C2×M16—21no—
122C2×M16—21no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8M1612yes—
133C22×C8M1612yes—
134C22×C8C22×C411yescharacteristic
135C22×C8C22×C411yescharacteristic
136C22×C8C22×C411yescharacteristic
137C22×C8C22×C411yescharacteristic
138C22×C8C22×C411yescharacteristic
139C22×C8C22×C411yescharacteristic
140C22×C8C22×C411yescharacteristic
141C22×C8#4111yescharacteristic
142C22×C8#4111yescharacteristic
143C22×C8#4111yescharacteristic
144C22×C8#4111yescharacteristic
145C22×C8C2411yescharacteristic
146C4×D4—21no—
147C4×D4—21no—
148C4×D4—21no—
149C4×D4—21no—
150#117—22no—
151#117—41no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4M1611yescharacteristic
155C2×C22⋊C4—21no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—21no—
158C2×C22⋊C4M1611yescharacteristic
159C2×C22⋊C4M1611yescharacteristic
160C2×C42—22no—
161C2×C42—21no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C411yescharacteristic
168M32—22no—
169M32—22no—
170M32—22no—
171M32—22no—
172M32—22no—
173M32—22no—
174M32—22no—
175M32—44no—
176M32C2×D414yes—
177C2×C16—22no—
178C2×C16—22no—
179C2×C16—22no—
180C2×C16—22no—
181C2×C16—44no—
182C4⋊C8—21no—
183C4⋊C8—21no—
184C4⋊C8—22no—
185C22⋊C8—21no—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—21no—
189C22⋊C8—21no—
190C22⋊C8—21no—
191C22⋊C8—22no—
192#97—21no—
193#97—21no—
194#97—21no—
195#97—21no—
196#97—21no—
197#97—21no—
198#97—22no—
199#97—41no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—22no—
203C4×C8—22no—
204C4×C8—21no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8C2×D412yes—
208C4×C8#4112yes—
209#95—21no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214C24C2×M1611yescharacteristic
215C24C2×M1611yescharacteristic
216C2×D4—21no—
217C2×D4—21no—
218C2×D4—42no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4#11711yescharacteristic
237C22×C4#11711yescharacteristic
238C22×C4C2×M1611yescharacteristic
239C22×C4C2×M1611yescharacteristic
240C22×C4C2×M1611yescharacteristic
241C22×C4C2×M1611yescharacteristic
242C22×C4C2×M1611yescharacteristic
243C22×C4C2×M1611yescharacteristic
244C22×C4C23×C411yescharacteristic
245M16—41no—
246M16—41no—
247M16—41no—
248M16—41no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—24no—
262C2×C8—24no—
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—21no—
272C2×C8—21no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—44no—
276C2×C8#11711yescharacteristic
277C2×C8#11711yescharacteristic
278C2×C8C4×D412yes—
279C2×C8C2×M1612yes—
280C2×C8C2×M1612yes—
281C2×C8#13112yes—
282C2×C8C22×D411yescharacteristic
283C2×C8C2×#4111yescharacteristic
284C2×C8C2×#4111yescharacteristic
285C2×C8C2×#4111yescharacteristic
286C2×C8H(5,𝔽2)11yescharacteristic
287C2×C82−1+411yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—21no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—42no—
309C42C4×D412yes—
310C16—24no—
311C16—42no—
312C16—42no—
313C16—42no—
314C23—22no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23#43211yescharacteristic
324C23#43211yescharacteristic
325C23C22×M1611yescharacteristic
326D4—42no—
327D4—42no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—48no—
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—41no—
361C2×C4—41no—
362C2×C4#43211yescharacteristic
363C2×C4#43211yescharacteristic
364C2×C4C2×#11711yescharacteristic
365C2×C4C2×C4×D411yescharacteristic
366C2×C4#52011yescharacteristic
367C2×C4C22×M1611yescharacteristic
368C2×C4C2×#13111yescharacteristic
369C2×C4#56811yescharacteristic
370C8—21no—
371C8—21no—
372C8—22no—
373C8—41no—
374C8—41no—
375C8—41no—
376C8—41no—
377C8#43514yes—
378C8#54611yescharacteristic
379C8#54811yescharacteristic
380C8#55211yescharacteristic
381C8#55311yescharacteristic
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—42no—
390C22—42no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—81no—
395C22—81no—
396C22#216612yescomplemented
397C22C2×#43211yescharacteristic
398C22D4×M1611yescharacteristic
399C22#299011yescharacteristic
400C4—21no—
401C4—21no—
402C4—42no—
403C4—42no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—81no—
408C4—81no—
409C4#235211yescharacteristic
410C4#292511yescharacteristic
411C4C2×#43511yescharacteristic
412C4#298911yescharacteristic
413C2—21no—
414C2—21no—
415C2—22no—
416C2—41no—
417C2—81no—
418C2#2275011yescharacteristic
419C2C2×#216611yescharacteristic
420C2#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×#2166C2semidirect
2. C22×#364C2semidirect
4. #13128C2semidirect
5. #13107C2semidirect
7. #12881C2semidirect
8. #12790C2semidirect
9. #12788C2semidirect
10. #12660C2semidirect
11. #12616C2semidirect
100. #364D4semidirect
396. C22#2166semidirect

Maximal subgroups

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

subgroupindex
C2×#21662
C22×#3642
#204842
#131282
#131072
#129852
#128812
#127902
#127882
#126602
#126162