Finite Group Databasebeta

← #501539

Subgroups — #501539

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2227C211yesmaximal, characteristic
2C2×C4×Q32C211yesmaximal, characteristic
3#28531C211yesmaximal, characteristic
4#22808C211yesmaximal, characteristic
5#13481C212yesmaximal
6#13450C212yesmaximal
7#13282C211yesmaximal, characteristic
8#13176C212yesmaximal
9#13159C211yesmaximal, characteristic
10#13133C212yesmaximal
11#12812C211yesmaximal, characteristic
12C22×Q32C2211yescharacteristic
13C2×#500C2211yescharacteristic
14C2×#477—21no—
15C2×C4×Q16C2211yescharacteristic
16C2×C4×Q16C2211yescharacteristic
17#2567—21no—
18#2355—21no—
19#2227—24no—
20C4×Q32—24no—
21C2×#366C2212yes—
22C2×#366C2211yescharacteristic
23C2×C4⋊C16C2211yescharacteristic
24C2×C4⋊C16C2212yes—
25C2×#358C2212yes—
26C2×#358C2211yescharacteristic
27C2×#358C2212yes—
28C2×#358C2211yescharacteristic
29C2×C4×C16C2211yescharacteristic
30#2095—22no—
31#1969C2212yes—
32#1969C2212yes—
33#1928—22no—
34#1845C2211yescharacteristic
35#1845C2212yes—
36#1803—21no—
37C4×#107C2212yes—
38C8×C4⋊C4C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1380C2212yes—
42#1374C2212yes—
43C22×Q16C2311yescharacteristic
44C22×Q16C2311yescharacteristic
45C2×C4⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×Q32—22no—
49C2×Q32—22no—
50C2×Q32—22no—
51C22×C16C2311yescharacteristic
52C22×C16C2312yes—
53C22×C16C2311yescharacteristic
54#500—22no—
55#500—22no—
56#477—44no—
57C4×Q16—22no—
58C4×Q16—22no—
59C4×Q16—22no—
60C4×Q16D414yes—
61C2×#107—22no—
62C2×#107—21no—
63C2×#107—21no—
64C2×#107C2311yescharacteristic
65C2×#107C2312yes—
66C2×#107C2312yes—
67C2×#107C2311yescharacteristic
68C2×C4⋊C8D411yescharacteristic
69C2×#103—21no—
70C2×#103—22no—
71C2×#103—21no—
72C2×#103—22no—
73C2×C4×C8C2312yes—
74C2×C4×C8C2311yescharacteristic
75#395—42no—
76#391—22no—
77#391—22no—
78#389—41no—
79#389—41no—
80#387—42no—
81#384—22no—
82#384—21no—
83#382—41no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—41no—
86C4×C4⋊C4D411yescharacteristic
87#366—22no—
88#366—24no—
89C4⋊C16—24no—
90C4⋊C16—22no—
91#358—22no—
92#358—24no—
93#358—24no—
94#358—22no—
95C4×C16—22no—
96#340—22no—
97#336D412yes—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—41no—
101C2×Q16—21no—
102C2×Q16—21no—
103C2×Q16—21no—
104C2×Q16—21no—
105C2×Q16—21no—
106C2×Q16—21no—
107C2×Q16—21no—
108C2×Q16—21no—
109C2×Q16—21no—
110C2×Q16—21no—
111C2×Q16C2×D412yes—
112C2×Q16#4112yes—
113C22×C8C2×D412yes—
114C22×C8C2×D411yescharacteristic
115C22×C8C2411yescharacteristic
116C4⋊Q8—42no—
117C4⋊Q8—42no—
118C4×Q8—24no—
119C4×Q8—42no—
120C4×Q8—42no—
121C4×Q8—42no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4D811yescharacteristic
134C2×C4⋊C4D811yescharacteristic
135C2×C42—41no—
136C2×C42—41no—
137C2×C42—42no—
138C2×C42C2×D412yes—
139C2×C42C2×D411yescharacteristic
140Q32—28no—
141C2×C16—21no—
142C2×C16—21no—
143C2×C16—21no—
144C2×C16—22no—
145C2×C16—22no—
146C2×C16—22no—
147C2×C16—21no—
148C2×C16—21no—
149C2×C16#4112yes—
150#107—21no—
151#107—21no—
152#107—22no—
153#107—22no—
154#107—21no—
155#107—22no—
156#107—22no—
157#107—44no—
158#107—42no—
159#107—81no—
160#107C2×D412yes—
161C4⋊C8—22no—
162#103—24no—
163#103—44no—
164#103—42no—
165#103—82no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8C2×D412yes—
169C4×C8#4112yes—
170#95—42no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95D812yes—
176C2×Q8—22no—
177C2×Q8—22no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—41no—
185C2×Q8—41no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—82no—
189C2×Q8—161no—
190C2×Q8—161no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4—42no—
195C22×C4—42no—
196C22×C4—41no—
197C22×C4C2×D812yes—
198C22×C4C2×D811yescharacteristic
199C22×C4C2×D812yes—
200C22×C4C2×D811yescharacteristic
201C22×C4C22×D411yescharacteristic
202Q16—22no—
203Q16—22no—
204Q16—22no—
205Q16—22no—
206Q16—41no—
207Q16—41no—
208Q16—41no—
209Q16—41no—
210C2×C8—21no—
211C2×C8—21no—
212C2×C8—22no—
213C2×C8—22no—
214C2×C8—22no—
215C2×C8#12112yes—
216C2×C8C22×D411yescharacteristic
217C2×C8C2×#4111yescharacteristic
218C2×C8C2×#4111yescharacteristic
219C2×C8C2×#4111yescharacteristic
220C2×C82−1+411yescharacteristic
221C2×C82−1+411yescharacteristic
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—41no—
225C4⋊C4—44no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—42no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C4⋊C4—82no—
235C4⋊C4—162no—
236C4⋊C4—161no—
237C4⋊C4—162no—
238C4⋊C4Q3214yescomplemented
239C42—22no—
240C42—22no—
241C42—44no—
242C42—82no—
243C42—162no—
244C42#12112yes—
245C42AGL(1,ℤ/8)12yes—
246C16—22no—
247C16—42no—
248C16—41no—
249C23C22×D811yescharacteristic
250Q8—42no—
251Q8—42no—
252Q8—42no—
253Q8—42no—
254Q8—81no—
255Q8—81no—
256Q8—81no—
257Q8—81no—
258Q8—164no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4—82no—
280C2×C4—81no—
281C2×C4—82no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4#45912yes—
285C2×C4#45912yes—
286C2×C4C2×Q3214yes—
287C2×C4C2×Q3212yes—
288C2×C4#51012yes—
289C2×C4C2×#12111yescharacteristic
290C2×C4#54711yescharacteristic
291C2×C4#54811yescharacteristic
292C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
293C2×C4#57711yescharacteristic
294C2×C4#57811yescharacteristic
295C8—22no—
296C8—42no—
297C8#52511yescharacteristic
298C8#53611yescharacteristic
299C8#55211yescharacteristic
300C8#55611yescharacteristic
301C22C2×#45911yescharacteristic
302C22#333411yescharacteristic
303C22#335811yescharacteristic
304C22C22×Q3211yescharacteristic
305C22C2×#51011yescharacteristic
306C22#341511yescharacteristic
307C22#341611yescharacteristic
308C4—22no—
309C4—24no—
310C4—42no—
311C4—82no—
312C4—82no—
313C4—82no—
314C4—161no—
315C4—162no—
316C4—162no—
317C4#220914yes—
318C4#241511yescharacteristic
319C4#304611yescharacteristic
320C4#335411yescharacteristic
321C4#336711yescharacteristic
322C2#2467611yescharacteristic
323C2C2×#220911yescharacteristic
324C2#3358911yescharacteristic
325C2#3390511yescharacteristic
326C2#3392311yescharacteristic
327C2#3393011yescharacteristic
328C2#3394211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
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
238. C4⋊C4Q32semidirect

Maximal subgroups

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

subgroupindex
C2×#22272
C2×C4×Q322
#285312
#228082
#134812
#134502
#132822
#131762
#131592
#131332
#128122