Finite Group Databasebeta

← #508568

Subgroups — #508568

Order 512.

684 conjugacy classes of subgroups.

The subgroup lattice has 411 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33209C211yesmaximal, characteristic, complemented
2#21224C211yesmaximal, characteristic
3#13531C211yesmaximal, characteristic, complemented
4#13526C211yesmaximal, characteristic, complemented
5#13476C212yesmaximal, complemented
6#13475C212yesmaximal, complemented
7#13364C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#12921C212yesmaximal, complemented
10#12783C211yesmaximal, characteristic, complemented
11#12733C212yesmaximal
12C22×#135C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#427C2211yescharacteristic
15C2×#416—21no—
16C22×C4×C8C2211yescharacteristic
17#2857—21no—
18#2297—21no—
19C2×#367C2212yes—
20C2×#366C2212yes—
21C2×C4⋊C16C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2212yes—
27#1990—21no—
28#1970C2211yescharacteristic
29#1970C2211yescharacteristic
30#1969C2211yescharacteristic
31#1969C2211yescharacteristic
32#1968—21no—
33#1968—21no—
34#1942—21no—
35#1942C2212yes—
36#1845C2211yescharacteristic
37#1808—21no—
38#1808C2212yes—
39#1380C2212yes—
40#1380C2212yes—
41#1374C2211yescharacteristic
42#1374C2211yescharacteristic
43C22×#41—21no—
44C2×#135—22no—
45C2×#135—42no—
46C2×#135—42no—
47C2×#135Q814yescomplemented
48C22×Q16C2311yescharacteristic
49C22×SD16—21no—
50C22×D8C2311yescharacteristic
51C23×C8D411yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4⋊Q8—21no—
54C2×C22⋊Q8—41no—
55C2×#117—21no—
56C22×C42D411yescharacteristic
57C22×C16C2312yes—
58C22×C16C2312yes—
59#500—42no—
60#427—42no—
61C2×#107—21no—
62C2×#107—21no—
63C2×#107C2312yes—
64C2×#107C2312yes—
65C2×#107C2311yescharacteristic
66C2×#107C2311yescharacteristic
67C2×#106—21no—
68#416—44no—
69C2×#103—21no—
70C2×#103—21no—
71C2×#102—21no—
72C2×#102—21no—
73C2×C4×C8—42no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—21no—
79#391—21no—
80#391—41no—
81#390—21no—
82#390—21no—
83#389—22no—
84#389—41no—
85#386—41no—
86#386—41no—
87#384—21no—
88#380—22no—
89#380—41no—
90C4×C22⋊C4—41no—
91C4×C22⋊C4—41no—
92#367—42no—
93#366—42no—
94C4⋊C16—42no—
95#358—42no—
96#358—42no—
97#357—42no—
98#357—42no—
99C22⋊C16—24no—
100#340—22no—
101#340—22no—
102C2×#41—24no—
103C2×#41—42no—
104C2×#41—42no—
105C2×#41—42no—
106C2×#41—42no—
107C2×#41—42no—
108C22×Q8—21no—
109C22×Q8—41no—
110C22×D4—21no—
111C22×D4—41no—
112C23×C4—41no—
113C23×C4D811yescharacteristic
114C23×C4D811yescharacteristic
115C23×C4C2×D411yescharacteristic
116#135—48no—
117#135—48no—
118C2×Q16—21no—
119C2×Q16—22no—
120C2×Q16—22no—
121C2×Q16C2×Q812yes—
122C2×SD16—24no—
123C2×SD16—42no—
124C2×SD16—42no—
125C2×D8—21no—
126C2×D8—22no—
127C2×D8—22no—
128C2×D8C2×Q812yes—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—41no—
134C22×C8C2×D411yescharacteristic
135C22×C8C2×D411yescharacteristic
136C22×C8C2×D411yescharacteristic
137C22×C8C2×Q812yes—
138C22×C8#4112yes—
139C22×C8C2411yescharacteristic
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C22⋊Q8—84no—
143#117—42no—
144#117—42no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—42no—
151C2×C4⋊C4—41no—
152C2×C4⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C22⋊C4—41no—
158C2×C22⋊C4—41no—
159C2×C42—41no—
160C2×C42—41no—
161C2×C42—41no—
162C2×C42D811yescharacteristic
163C2×C42D811yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C16—22no—
167C2×C16—22no—
168C2×C16—22no—
169C2×C16—22no—
170C2×C16—22no—
171C2×C16—22no—
172#107—22no—
173#107—22no—
174#107—22no—
175#107—22no—
176#107—22no—
177#107—22no—
178#107—42no—
179#107—42no—
180#106—42no—
181#103—42no—
182#103—42no—
183#102—42no—
184#102—42no—
185C4×C8—22no—
186C4×C8—22no—
187C4×C8—41no—
188C4×C8—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—42no—
194#95—42no—
195C24—41no—
196C24C2×D811yescharacteristic
197#41—48no—
198#41—48no—
199#41—84no—
200#41—84no—
201C2×Q8—22no—
202C2×Q8—41no—
203C2×Q8—41no—
204C2×Q8—41no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—82no—
208C2×Q8—82no—
209C2×Q8—82no—
210C2×D4—22no—
211C2×D4—44no—
212C2×D4—41no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—82no—
218C2×D4—82no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—82no—
236C22×C4—82no—
237C22×C4C22⋊Q812yes—
238C22×C4C2×D811yescharacteristic
239C22×C4C2×D811yescharacteristic
240C22×C4C2×D811yescharacteristic
241C22×C4C2×D811yescharacteristic
242C22×C4C2×D811yescharacteristic
243C22×C4#13712yes—
244C22×C4C22×D411yescharacteristic
245Q16—24no—
246Q16—42no—
247SD16—48no—
248D8—24no—
249D8—42no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—44no—
260C2×C8C22⋊Q812yes—
261C2×C8AGL(1,ℤ/8)12yes—
262C2×C8C22×Q811yescharacteristic
263C2×C8C2×#4111yescharacteristic
264C2×C8H(5,𝔽2)12yes—
265C2×C8H(5,𝔽2)12yes—
266C4⋊C4—42no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—42no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—84no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C22⋊C4—82no—
279C22⋊C4—82no—
280C22⋊C4—84no—
281C22⋊C4—82no—
282C22⋊C4—82no—
283C22⋊C4—82no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—41no—
289C42—41no—
290C42—82no—
291C16—42no—
292C16—42no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—81no—
301C23—81no—
302C23—81no—
303C23—81no—
304C23—81no—
305C23—81no—
306C23#47412yes—
307C23C22×D811yescharacteristic
308Q8—42no—
309Q8—42no—
310Q8—81no—
311Q8—81no—
312Q8—164no—
313D4—42no—
314D4—42no—
315D4—88no—
316D4—81no—
317D4—81no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—42no—
336C2×C4—88no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4—81no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—82no—
351C2×C4#47412yes—
352C2×C4#50812yes—
353C2×C4#50912yes—
354C2×C4#51012yes—
355C2×C4C2×C22⋊Q811yescharacteristic
356C2×C4#53411yescharacteristic
357C2×C4#53511yescharacteristic
358C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
359C2×C4C2×#13711yescharacteristic
360C2×C4#57511yescharacteristic
361C8—22no—
362C8—22no—
363C8—41no—
364C8#53812yes—
365C8#54312yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—82no—
374C22—82no—
375C22—82no—
376C22—161no—
377C22—161no—
378C22—161no—
379C22—161no—
380C22C2×#47411yescharacteristic
381C22#318611yescharacteristic
382C22#320511yescharacteristic
383C22C2×#50811yescharacteristic
384C22C2×#50911yescharacteristic
385C22C2×#51011yescharacteristic
386C22#341411yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—22no—
391C4—41no—
392C4—82no—
393C4—162no—
394C4—161no—
395C4—162no—
396C4—161no—
397C4#262611yescharacteristic
398C4#307711yescharacteristic
399C4#319311yescharacteristic
400C4#321511yescharacteristic
401C2—42no—
402C2—162no—
403C2#2995011yescharacteristic
404C2#3362611yescharacteristic
405C2#3362911yescharacteristic
406C2#3375611yescharacteristic
407C2#3376011yescharacteristic
408C2#3377311yescharacteristic
409C2#3378211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ 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. #33209C2semidirect
3. #13531C2semidirect
4. #13526C2semidirect
5. #13476C2semidirect
6. #13475C2semidirect
7. #13364C2semidirect
8. #13363C2semidirect
9. #12921C2semidirect
10. #12783C2semidirect
47. C2×#135Q8semidirect

Maximal subgroups

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

subgroupindex
#332092
#212242
#135312
#135262
#134762
#134752
#133642
#133632
#129212
#127832
#127332