Finite Group Databasebeta

← #507429

Subgroups — #507429

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33207C211yesmaximal, characteristic, complemented
2#21544C211yesmaximal, characteristic
3#13543C211yesmaximal, characteristic, complemented
4#13533C211yesmaximal, characteristic, complemented
5#13457C211yesmaximal, characteristic, complemented
6#13371C211yesmaximal, characteristic, complemented
7#13116C212yesmaximal, complemented
8#13065C212yesmaximal
9#12950C212yesmaximal, complemented
10#12831C212yesmaximal, complemented
11C2×#1376C211yesmaximal, characteristic, complemented
12C22×M32C2212yes—
13C2×#498C2211yescharacteristic
14#3050C2211yescharacteristic
15C2×#443C2211yescharacteristic
16#2929—21no—
17C22×C4×C8C2211yescharacteristic
18#2857—21no—
19C2×#365C2212yes—
20C2×#365C2212yes—
21C2×#360C2212yes—
22C2×#360C2212yes—
23#1990—21no—
24#1974C2211yescharacteristic
25#1945C2211yescharacteristic
26#1941—21no—
27#1941C2212yes—
28#1937—21no—
29#1896—21no—
30#1844C2211yescharacteristic
31#1810C2211yescharacteristic
32#1807—21no—
33#1807C2212yes—
34#1772C2211yescharacteristic
35#1764—21no—
36#1386C2212yes—
37#1386C2212yes—
38#1383C2212yes—
39#1383C2212yes—
40#1376C2211yescharacteristic
41#1376C2212yes—
42#1376C2211yescharacteristic
43C2×#135C2311yescharacteristic
44C2×#131—21no—
45C23×C8D411yescharacteristic
46C23×C8C2311yescharacteristic
47C2×C4⋊Q8—21no—
48#525—21no—
49C2×C22⋊Q8—41no—
50C22×C42D411yescharacteristic
51C2×M32—22no—
52C2×M32—24no—
53C2×M32—42no—
54C2×M32C2312yes—
55C2×M32C2312yes—
56C22×C16—22no—
57#498—42no—
58#467Q811yescharacteristic, complemented
59#467Q811yescharacteristic, complemented
60#466Q811yescharacteristic, complemented
61#466Q811yescharacteristic, complemented
62#465—21no—
63#465—21no—
64#443—42no—
65#435—41no—
66#435—41no—
67C8×D4—41no—
68C8×D4—41no—
69C2×#108C2311yescharacteristic
70C2×#108C2311yescharacteristic
71#427C2311yescharacteristic
72C2×#106—21no—
73C2×#106—21no—
74C2×#106C2312yes—
75C2×#106C2312yes—
76#424—21no—
77C2×#104—21no—
78C2×#104—21no—
79#416—21no—
80#408—21no—
81#407—21no—
82#405—21no—
83C2×C4×C8—42no—
84C2×C4×C8D411yescharacteristic
85C2×C4×C8D411yescharacteristic
86C2×C4×C8C2311yescharacteristic
87C2×C4×C8C2311yescharacteristic
88#391—41no—
89#389—22no—
90#389—41no—
91#384—21no—
92#380—22no—
93#380—41no—
94#365—24no—
95#365—42no—
96#360—24no—
97#360—42no—
98#341—21no—
99#340—22no—
100#340—22no—
101#339—21no—
102#339—21no—
103C2×#41—21no—
104C22×Q8—41no—
105C23×C4—21no—
106C23×C4C2×D411yescharacteristic
107#135—22no—
108#135—22no—
109C2×Q16C2×Q811yescharacteristic
110C2×SD16—21no—
111C2×D8C2×Q811yescharacteristic
112#131—42no—
113#131—42no—
114C2×M16—21no—
115C2×M16—21no—
116C2×M16—41no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—41no—
120C22×C8—41no—
121C22×C8—41no—
122C22×C8SD1611yescharacteristic
123C22×C8SD1611yescharacteristic
124C22×C8SD1611yescharacteristic
125C22×C8SD1611yescharacteristic
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2×Q811yescharacteristic
130C22×C8C2×Q811yescharacteristic
131C22×C8#4111yescharacteristic
132C22×C8#4111yescharacteristic
133C22×C8C2411yescharacteristic
134C4⋊Q8—42no—
135C4⋊Q8—42no—
136#123—41no—
137C22⋊Q8—21no—
138C22⋊Q8—21no—
139C22⋊Q8—84no—
140#121—21no—
141#121—21no—
142#120—41no—
143C4×D4—41no—
144C4×D4—41no—
145#117—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4—41no—
151C2×C4⋊C4—41no—
152C2×C22⋊C4—42no—
153C2×C42—22no—
154C2×C42—21no—
155C2×C42—41no—
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158M32—22no—
159M32—22no—
160M32—22no—
161M32—44no—
162M32#4114yes—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—44no—
166#108—22no—
167#108—22no—
168#107C2×Q811yescharacteristic
169#107C2×Q811yescharacteristic
170#106—21no—
171#106—22no—
172#106—22no—
173#106—22no—
174#106—22no—
175#106—42no—
176#106—42no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179#104—42no—
180#104—42no—
181#103—21no—
182#103—21no—
183#102—21no—
184#102—21no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189#97—41no—
190C4×C8—22no—
191C4×C8—22no—
192C4×C8—41no—
193C4×C8—41no—
194C4×C8—41no—
195#95—42no—
196#95—42no—
197C24AGL(1,ℤ/8)11yescharacteristic
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8—21no—
203C2×Q8—82no—
204C2×Q8—82no—
205C2×Q8—82no—
206C2×D4—21no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—21no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C2×SD1611yescharacteristic
225C22×C4C2×SD1611yescharacteristic
226C22×C4AGL(1,ℤ/8)11yescharacteristic
227C22×C4C22×D411yescharacteristic
228Q16—22no—
229SD16—42no—
230D8—22no—
231M16—41no—
232M16—41no—
233M16—41no—
234M16—41no—
235M16—82no—
236C2×C8—21no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—82no—
253C2×C8C22⋊Q811yescharacteristic
254C2×C8C22⋊Q811yescharacteristic
255C2×C8C2×SD1611yescharacteristic
256C2×C8C2×SD1611yescharacteristic
257C2×C8C2×SD1611yescharacteristic
258C2×C8C2×SD1611yescharacteristic
259C2×C8C22×Q811yescharacteristic
260C2×C8C2×#4111yescharacteristic
261C2×C8C2×#4112yes—
262C2×C8H(5,𝔽2)12yes—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—82no—
272C4⋊C4—82no—
273C4⋊C4—84no—
274C4⋊C4—82no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—84no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—41no—
286C42#13512yes—
287C16—24no—
288C16—42no—
289C23—21no—
290C23—21no—
291C23—22no—
292C23—41no—
293C23—41no—
294C23#47611yescharacteristic
295C23#47611yescharacteristic
296C23C2×AGL(1,ℤ/8)11yescharacteristic
297Q8—41no—
298Q8—41no—
299Q8—164no—
300D4—41no—
301D4—41no—
302D4—82no—
303D4—81no—
304D4—81no—
305D4—82no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—21no—
309C2×C4—22no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—48no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—82no—
334C2×C4#47611yescharacteristic
335C2×C4#47611yescharacteristic
336C2×C4C2×C22⋊Q811yescharacteristic
337C2×C4#53211yescharacteristic
338C2×C4#53411yescharacteristic
339C2×C4C22×SD1611yescharacteristic
340C2×C4C2×#13511yescharacteristic
341C2×C4#57511yescharacteristic
342C8—21no—
343C8—21no—
344C8—21no—
345C8—21no—
346C8—21no—
347C8—21no—
348C8—81no—
349C8—81no—
350C8#55112yes—
351C8#55412yes—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—42no—
360C22—42no—
361C22—81no—
362C22—81no—
363C22—81no—
364C22#223012yes—
365C22C2×#47611yescharacteristic
366C22#318011yescharacteristic
367C22#318811yescharacteristic
368C4—24no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—162no—
376C4—162no—
377C4#256411yescharacteristic
378C4#307611yescharacteristic
379C4#316711yescharacteristic
380C4#318711yescharacteristic
381C2—21no—
382C2—21no—
383C2—22no—
384C2—41no—
385C2—161no—
386C2#2915611yescharacteristic
387C2C2×#223011yescharacteristic
388C2#3363711yescharacteristic
subgroupG/H
center, socleC22C2×#476
commutator, Frattini, cosocleC22×C8C24
prime radical#3050C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #33207C2semidirect
3. #13543C2semidirect
4. #13533C2semidirect
5. #13457C2semidirect
6. #13371C2semidirect
7. #13116C2semidirect
9. #12950C2semidirect
10. #12831C2semidirect
11. C2×#1376C2semidirect
58. #467Q8semidirect
59. #467Q8semidirect
60. #466Q8semidirect
61. #466Q8semidirect

Maximal subgroups

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

subgroupindex
#332072
#215442
#135432
#135332
#134572
#133712
#131162
#130652
#129502
#128312
C2×#13762