Finite Group Databasebeta

← #501563

Subgroups — #501563

Order 512.

630 conjugacy classes of subgroups.

The subgroup lattice has 403 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×#2162C211yesmaximal, characteristic, complemented
3#33209C211yesmaximal, characteristic, complemented
4C4×#466C211yesmaximal, characteristic
5#13492C212yesmaximal, complemented
6#13380C212yesmaximal
7#13275C211yesmaximal, characteristic, complemented
8#13176C211yesmaximal, characteristic, complemented
9#13165C212yesmaximal, complemented
10#13147C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C2×#500C2211yescharacteristic
13C2×#466C2211yescharacteristic
14C2×C4×D8C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16#2857—21no—
17C4×#121—21no—
18#2224—24no—
19#2162—24no—
20C2×#366C2212yes—
21C2×#366C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×#357C2212yes—
25C2×#357C2212yes—
26C2×C22⋊C16C2212yes—
27C2×C4×C16C2212yes—
28#1990—21no—
29#1970C2211yescharacteristic
30#1942—21no—
31#1942C2211yescharacteristic
32#1942C2212yes—
33#1934—21no—
34#1893—21no—
35#1845C2211yescharacteristic
36#1808—21no—
37#1808C2211yescharacteristic
38#1808C2212yes—
39C4×#107C2211yescharacteristic
40C4×#102—21no—
41#1380C2212yes—
42#1380C2212yes—
43#1380C2212yes—
44#1380C2212yes—
45C22×D8C2311yescharacteristic
46C23×C8D411yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—21no—
49C2×C22⋊Q8—41no—
50C2×#121—21no—
51C2×C4×D4—21no—
52C2×C4×D4—41no—
53C22×C42D411yescharacteristic
54C22×C16C2312yes—
55C22×C16C2312yes—
56#500—22no—
57#500—22no—
58#466—22no—
59#466—22no—
60C4×D8—22no—
61C4×D8—22no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107C2312yes—
65C2×#107C2312yes—
66C2×#107C2311yescharacteristic
67C2×#107C2311yescharacteristic
68C2×#107C2311yescharacteristic
69C2×#102—21no—
70C2×#102—21no—
71C2×#102—21no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—41no—
79#390—21no—
80#389—22no—
81#389—21no—
82#389—41no—
83#388—41no—
84#386—41no—
85#384—21no—
86#380—22no—
87#380—21no—
88#380—41no—
89C4×C4⋊C4—21no—
90C4×C22⋊C4—41no—
91#366—24no—
92#366—24no—
93C4⋊C16—22no—
94C4⋊C16D414yescomplemented
95#357—24no—
96#357—24no—
97C22⋊C16—24no—
98C4×C16—24no—
99#340—22no—
100#340—21no—
101#340—22no—
102#340—21no—
103#340—21no—
104C22×Q8—41no—
105C22×D4—21no—
106C22×D4—41no—
107C23×C4—41no—
108C23×C4D811yescharacteristic
109C23×C4D811yescharacteristic
110C23×C4C2×D411yescharacteristic
111C2×D8—21no—
112C2×D8—21no—
113C2×D8—21no—
114C2×D8—21no—
115C2×D8—21no—
116C2×D8#4112yes—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—41no—
126C22×C8C2×D411yescharacteristic
127C22×C8C2×D411yescharacteristic
128C22×C8C2×D411yescharacteristic
129C22×C8C2411yescharacteristic
130C4⋊Q8—42no—
131C4⋊Q8—42no—
132C22⋊Q8—84no—
133#121—24no—
134#121—42no—
135C4×D4—42no—
136C4×D4—42no—
137C4×D4—84no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—41no—
152C2×C22⋊C4—41no—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42—41no—
157C2×C42D811yescharacteristic
158C2×C42D811yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C16—22no—
162C2×C16—22no—
163C2×C16—22no—
164C2×C16—22no—
165C2×C16—22no—
166C2×C16C2×D414yes—
167#107—21no—
168#107—22no—
169#107—21no—
170#107—22no—
171#107—22no—
172#107—21no—
173#107—22no—
174#107—21no—
175#107—21no—
176#107—42no—
177#107—42no—
178#107#4112yes—
179#102—24no—
180#102—42no—
181#102—42no—
182C4×C8—22no—
183C4×C8—21no—
184C4×C8—21no—
185C4×C8—22no—
186C4×C8C2×D412yes—
187C4×C8#4112yes—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—42no—
192#95—42no—
193C24—41no—
194C24C2×D811yescharacteristic
195C2×Q8—82no—
196C2×Q8—82no—
197C2×Q8—82no—
198C2×D4—22no—
199C2×D4—44no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—82no—
206C2×D4—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—41no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—82no—
224C22×C4—82no—
225C22×C4—82no—
226C22×C4D1614yes—
227C22×C4C2×D811yescharacteristic
228C22×C4C2×D811yescharacteristic
229C22×C4C2×D811yescharacteristic
230C22×C4C2×D811yescharacteristic
231C22×C4C2×D811yescharacteristic
232C22×C4C22×D411yescharacteristic
233D8—22no—
234D8—22no—
235D8—41no—
236D8—41no—
237C2×C8—21no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8—44no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8#12112yes—
250C2×C8#13712yes—
251C2×C8C22×D411yescharacteristic
252C2×C8C2×#4111yescharacteristic
253C2×C8C2×#4111yescharacteristic
254C2×C8C2×#4111yescharacteristic
255C2×C82−1+411yescharacteristic
256C2×C82−1+411yescharacteristic
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C4⋊C4—82no—
270C4⋊C4—82no—
271C4⋊C4—84no—
272C22⋊C4—44no—
273C22⋊C4—82no—
274C22⋊C4—84no—
275C22⋊C4—82no—
276C22⋊C4—82no—
277C22⋊C4—82no—
278C42—21no—
279C42—21no—
280C42—22no—
281C42—22no—
282C42—82no—
283C42#12112yes—
284C42#13512yes—
285C16—24no—
286C16—42no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—42no—
294C23—81no—
295C23—81no—
296C23—81no—
297C23—81no—
298C23—81no—
299C23—81no—
300C23C2×D1612yes—
301C23C22×D811yescharacteristic
302Q8—164no—
303D4—42no—
304D4—42no—
305D4—88no—
306D4—81no—
307D4—81no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—28no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—44no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—81no—
325C2×C4—81no—
326C2×C4—82no—
327C2×C4—81no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4—82no—
331C2×C4—81no—
332C2×C4—82no—
333C2×C4—81no—
334C2×C4—82no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—82no—
338C2×C4—164no—
339C2×C4#46612yes—
340C2×C4#46612yes—
341C2×C4C2×D1612yes—
342C2×C4C2×D1612yes—
343C2×C4#50812yes—
344C2×C4#51012yes—
345C2×C4C2×#12111yescharacteristic
346C2×C4#52511yescharacteristic
347C2×C4#53611yescharacteristic
348C2×C4C2×#13511yescharacteristic
349C2×C4C2×#13711yescharacteristic
350C2×C4#57511yescharacteristic
351C8—22no—
352C8—22no—
353C8—41no—
354C8#54711yescharacteristic
355C8#54811yescharacteristic
356C8#55211yescharacteristic
357C8#55611yescharacteristic
358C22—22no—
359C22—22no—
360C22—22no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—161no—
369C22—161no—
370C22—161no—
371C22—161no—
372C22C2×#46611yescharacteristic
373C22#310011yescharacteristic
374C22#322611yescharacteristic
375C22C22×D1611yescharacteristic
376C22C2×#50811yescharacteristic
377C22C2×#51011yescharacteristic
378C22#341411yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—82no—
384C4—161no—
385C4—161no—
386C4—162no—
387C4—162no—
388C4#221514yes—
389C4#241511yescharacteristic
390C4#305611yescharacteristic
391C4#310711yescharacteristic
392C4#323311yescharacteristic
393C2—42no—
394C2—162no—
395C2#2470011yescharacteristic
396C2C2×#221511yescharacteristic
397C2#3360611yescharacteristic
398C2#3366011yescharacteristic
399C2#3366711yescharacteristic
400C2#3379211yescharacteristic
401C2#3379911yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC2×#466C22
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. C2×#2224C2semidirect
2. C2×#2162C2semidirect
3. #33209C2semidirect
5. #13492C2semidirect
7. #13275C2semidirect
8. #13176C2semidirect
9. #13165C2semidirect
10. #13147C2semidirect
11. #12811C2semidirect
94. C4⋊C16D4semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×#21622
#332092
C4×#4662
#134922
#133802
#132752
#131762
#131652
#131472
#128112