Finite Group Databasebeta

← #501533

Subgroups — #501533

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2224C211yesmaximal, characteristic, complemented
2C2×C4×D16C211yesmaximal, characteristic
3#28527C211yesmaximal, characteristic, complemented
4#22807C211yesmaximal, characteristic, complemented
5#13480C212yesmaximal, complemented
6#13448C212yesmaximal
7#13282C211yesmaximal, characteristic, complemented
8#13176C212yesmaximal, complemented
9#13159C211yesmaximal, characteristic, complemented
10#13130C212yesmaximal, complemented
11#12811C211yesmaximal, characteristic, complemented
12C22×D16C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#474—21no—
15C2×C4×D8C2211yescharacteristic
16C2×C4×D8C2211yescharacteristic
17#2542—21no—
18#2354—21no—
19#2224—24no—
20C4×D16—24no—
21C2×#366C2212yes—
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2212yes—
25C2×#357C2212yes—
26C2×#357C2211yescharacteristic
27C2×#357C2212yes—
28C2×#357C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#2093—22no—
31#1970C2212yes—
32#1970C2212yes—
33#1925—22no—
34#1845C2211yescharacteristic
35#1845C2212yes—
36#1802—21no—
37C4×#107C2212yes—
38C8×C4⋊C4C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×D8C2311yescharacteristic
44C22×D8C2311yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C22⋊Q8—41no—
47C2×C22⋊Q8—41no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×#117—41no—
51C2×D16—22no—
52C2×D16—22no—
53C2×D16—22no—
54C22×C16C2311yescharacteristic
55C22×C16C2312yes—
56C22×C16C2311yescharacteristic
57#500—22no—
58#500—22no—
59#474—44no—
60C4×D8—22no—
61C4×D8—22no—
62C4×D8—22no—
63C4×D8D414yes—
64C2×#107—21no—
65C2×#107—22no—
66C2×#107—21no—
67C2×#107C2311yescharacteristic
68C2×#107C2312yes—
69C2×#107C2312yes—
70C2×#107C2311yescharacteristic
71C2×C4⋊C8D411yescharacteristic
72C2×#102—21no—
73C2×#102—21no—
74C2×#102—22no—
75C2×#102—22no—
76C2×C4×C8C2312yes—
77C2×C4×C8C2311yescharacteristic
78#399—42no—
79#390—22no—
80#390—22no—
81#388—42no—
82#384—21no—
83#384—22no—
84#380—41no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4D411yescharacteristic
87#366—22no—
88#366—24no—
89C4⋊C16—24no—
90C4⋊C16—22no—
91#357—22no—
92#357—24no—
93#357—24no—
94#357—22no—
95C4×C16—22no—
96#340—22no—
97#336D412yes—
98C22×Q8—41no—
99C22×D4—21no—
100C22×D4—21no—
101C23×C4—41no—
102C23×C4—41no—
103C2×D8—21no—
104C2×D8—21no—
105C2×D8—21no—
106C2×D8—21no—
107C2×D8—21no—
108C2×D8—21no—
109C2×D8—21no—
110C2×D8—21no—
111C2×D8—21no—
112C2×D8—21no—
113C2×D8C2×D412yes—
114C2×D8#4112yes—
115C22×C8C2×D412yes—
116C22×C8C2×D411yescharacteristic
117C22×C8C2411yescharacteristic
118C4⋊Q8—42no—
119C4⋊Q8—42no—
120C22⋊Q8—84no—
121C22⋊Q8—84no—
122C4×D4—24no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126#117—84no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4D811yescharacteristic
137C2×C4⋊C4D811yescharacteristic
138C2×C22⋊C4—41no—
139C2×C22⋊C4—42no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—42no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C42—42no—
145C2×C42C2×D412yes—
146C2×C42C2×D411yescharacteristic
147D16—28no—
148C2×C16—21no—
149C2×C16—21no—
150C2×C16—21no—
151C2×C16—22no—
152C2×C16—22no—
153C2×C16—22no—
154C2×C16—21no—
155C2×C16—21no—
156C2×C16#4112yes—
157#107—21no—
158#107—21no—
159#107—22no—
160#107—22no—
161#107—21no—
162#107—22no—
163#107—22no—
164#107—44no—
165#107—42no—
166#107—81no—
167#107C2×D412yes—
168C4⋊C8—22no—
169#102—24no—
170#102—44no—
171#102—42no—
172#102—82no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8C2×D412yes—
176C4×C8#4112yes—
177#95—42no—
178#95D812yes—
179C24—41no—
180C24—41no—
181C2×Q8—82no—
182C2×Q8—82no—
183C2×Q8—82no—
184C2×D4—22no—
185C2×D4—22no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—41no—
194C2×D4—41no—
195C2×D4—41no—
196C22×C4—41no—
197C22×C4—44no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—41no—
201C22×C4—82no—
202C22×C4—82no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4—82no—
206C22×C4C2×D812yes—
207C22×C4C2×D811yescharacteristic
208C22×C4C2×D812yes—
209C22×C4C2×D811yescharacteristic
210C22×C4C22×D411yescharacteristic
211D8—22no—
212D8—22no—
213D8—22no—
214D8—22no—
215D8—41no—
216D8—41no—
217D8—41no—
218D8—41no—
219C2×C8—21no—
220C2×C8—21no—
221C2×C8—22no—
222C2×C8—22no—
223C2×C8—22no—
224C2×C8#12112yes—
225C2×C8C22×D411yescharacteristic
226C2×C8C2×#4111yescharacteristic
227C2×C8C2×#4111yescharacteristic
228C2×C8C2×#4111yescharacteristic
229C2×C82−1+411yescharacteristic
230C2×C82−1+411yescharacteristic
231C4⋊C4—22no—
232C4⋊C4—22no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—41no—
239C4⋊C4—42no—
240C4⋊C4—84no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—84no—
244C4⋊C4—161no—
245C4⋊C4D1614yescomplemented
246C22⋊C4—44no—
247C22⋊C4—84no—
248C22⋊C4—82no—
249C22⋊C4—84no—
250C22⋊C4—84no—
251C22⋊C4—162no—
252C42—22no—
253C42—22no—
254C42—84no—
255C42#12112yes—
256C42AGL(1,ℤ/8)12yes—
257C16—22no—
258C16—42no—
259C16—41no—
260C23—42no—
261C23—42no—
262C23—81no—
263C23—81no—
264C23—81no—
265C23—81no—
266C23—81no—
267C23—81no—
268C23—81no—
269C23—81no—
270C23—81no—
271C23—81no—
272C23—81no—
273C23—81no—
274C23C22×D811yescharacteristic
275Q8—164no—
276D4—42no—
277D4—42no—
278D4—42no—
279D4—42no—
280D4—81no—
281D4—81no—
282D4—81no—
283D4—81no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—42no—
292C2×C4—81no—
293C2×C4—81no—
294C2×C4—88no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—82no—
302C2×C4—81no—
303C2×C4—82no—
304C2×C4—164no—
305C2×C4#45912yes—
306C2×C4#45912yes—
307C2×C4C2×D1614yes—
308C2×C4C2×D1612yes—
309C2×C4#50912yes—
310C2×C4C2×#12111yescharacteristic
311C2×C4#54711yescharacteristic
312C2×C4#54811yescharacteristic
313C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
314C2×C4#57711yescharacteristic
315C2×C4#57811yescharacteristic
316C8—22no—
317C8—42no—
318C8#52511yescharacteristic
319C8#53611yescharacteristic
320C8#55211yescharacteristic
321C8#55611yescharacteristic
322C22—82no—
323C22—82no—
324C22—82no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—161no—
329C22—161no—
330C22—161no—
331C22—161no—
332C22—161no—
333C22—161no—
334C22—161no—
335C22—161no—
336C22C2×#45911yescharacteristic
337C22#333411yescharacteristic
338C22#335811yescharacteristic
339C22C22×D1611yescharacteristic
340C22C2×#50911yescharacteristic
341C22#341611yescharacteristic
342C22#341711yescharacteristic
343C4—22no—
344C4—24no—
345C4—42no—
346C4—82no—
347C4—161no—
348C4—162no—
349C4—162no—
350C4#220614yes—
351C4#241511yescharacteristic
352C4#304611yescharacteristic
353C4#335411yescharacteristic
354C4#336711yescharacteristic
355C2—162no—
356C2—162no—
357C2#2467611yescharacteristic
358C2C2×#220611yescharacteristic
359C2#3359011yescharacteristic
360C2#3390111yescharacteristic
361C2#3392111yescharacteristic
362C2#3392511yescharacteristic
363C2#3393411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ C8×C4⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2224C2semidirect
3. #28527C2semidirect
4. #22807C2semidirect
5. #13480C2semidirect
7. #13282C2semidirect
8. #13176C2semidirect
9. #13159C2semidirect
10. #13130C2semidirect
11. #12811C2semidirect
245. C4⋊C4D16semidirect

Maximal subgroups

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

subgroupindex
C2×#22242
C2×C4×D162
#285272
#228072
#134802
#134482
#132822
#131762
#131592
#131302
#128112