Finite Group Databasebeta

← #502335

Subgroups — #502335

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 331 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2209C211yesmaximal, characteristic
2C2×#2183C211yesmaximal, characteristic
3#28737C211yesmaximal, characteristic
4#28265C211yesmaximal, characteristic
5#13477C212yesmaximal
6#13450C212yesmaximal
7#13240C212yesmaximal
8#13203C211yesmaximal, characteristic
9#13177C211yesmaximal, characteristic
10#13150C212yesmaximal
11C4×#358C211yesmaximal, characteristic
12C22×Q32C2212yes—
13C2×#494C2211yescharacteristic
14C2×C8×Q8C2211yescharacteristic
15C2×C4×Q16C2211yescharacteristic
16#2582—21no—
17#2568—21no—
18#2209—24no—
19#2183—24no—
20C2×#367C2212yes—
21C2×C4⋊C16C2212yes—
22C2×C4⋊C16C2211yescharacteristic
23C2×#358C2212yes—
24C2×#358C2211yescharacteristic
25C2×#358C2211yescharacteristic
26C2×#358C2212yes—
27C2×#358C2212yes—
28C2×#346C2212yes—
29C2×C4×C16C2211yescharacteristic
30#2068—22no—
31#2056—22no—
32#1981—21no—
33#1969C2212yes—
34#1969C2212yes—
35#1969C2212yes—
36#1942C2212yes—
37#1845C2211yescharacteristic
38#1803—21no—
39C4×#107C2211yescharacteristic
40#1380C2211yescharacteristic
41#1380C2211yescharacteristic
42#1380C2212yes—
43C22×Q16—21no—
44C22×Q16C2312yes—
45C22×Q16C2311yescharacteristic
46C2×C4⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8D411yescharacteristic
49C2×Q32—24no—
50C2×Q32—24no—
51C2×Q32—42no—
52C22×C16C2311yescharacteristic
53C22×C16C2312yes—
54C22×C16C2311yescharacteristic
55#494—22no—
56#494—22no—
57C8×Q8—22no—
58C8×Q8—22no—
59C4×Q16—22no—
60C4×Q16—22no—
61C2×#107—21no—
62C2×#107C2312yes—
63C2×#107C2311yescharacteristic
64C2×#107C2311yescharacteristic
65C2×#107C2311yescharacteristic
66C2×C4⋊C8D412yes—
67C2×C4⋊C8D411yescharacteristic
68C2×#103—21no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—22no—
72C2×C4×C8C2312yes—
73C2×C4×C8C2311yescharacteristic
74#400—42no—
75#398—42no—
76#391—22no—
77#391—22no—
78#391—22no—
79#391—41no—
80#391—41no—
81#389—22no—
82#389—41no—
83#384—21no—
84#382—41no—
85C4×C4⋊C4—21no—
86#367—42no—
87C4⋊C16—24no—
88C4⋊C16D414yes—
89#358—22no—
90#358—24no—
91#358—22no—
92#358—24no—
93#358—42no—
94#346—24no—
95C4×C16—22no—
96#340—21no—
97#340—22no—
98C22×Q8—21no—
99C22×Q8—22no—
100C22×Q8D811yescharacteristic
101C2×Q16—21no—
102C2×Q16—22no—
103C2×Q16—21no—
104C2×Q16—21no—
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16—22no—
108C2×Q16—21no—
109C2×Q16—22no—
110C2×Q16—21no—
111C2×Q16—22no—
112C2×Q16—42no—
113C2×Q16—81no—
114C2×Q16—81no—
115C2×Q16#4112yes—
116C22×C8C2×D412yes—
117C22×C8C2×D411yescharacteristic
118C22×C8C2411yescharacteristic
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4×Q8—22no—
122C4×Q8—22no—
123C4×Q8—42no—
124C4×Q8—42no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4D811yescharacteristic
136C2×C4⋊C4D812yes—
137C2×C42—41no—
138C2×C42—41no—
139C2×C42C2×D412yes—
140C2×C42C2×D411yescharacteristic
141Q32—48no—
142C2×C16—21no—
143C2×C16—22no—
144C2×C16—22no—
145C2×C16—21no—
146C2×C16—21no—
147C2×C16—21no—
148C2×C16—21no—
149C2×C16C2×D414yes—
150C2×C16#4112yes—
151#107—21no—
152#107—21no—
153#107—22no—
154#107—21no—
155#107—21no—
156#107—22no—
157#107—21no—
158#107—21no—
159#107—81no—
160C4⋊C8—24no—
161C4⋊C8—22no—
162#103—42no—
163#103—82no—
164#103—82no—
165#103—82no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8C2×D412yes—
169C4×C8#4112yes—
170#95—41no—
171#95—42no—
172#95—42no—
173#95—42no—
174#95—42no—
175#95—42no—
176#95—42no—
177#95—42no—
178C2×Q8—22no—
179C2×Q8—22no—
180C2×Q8—22no—
181C2×Q8—22no—
182C2×Q8—41no—
183C2×Q8—41no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—41no—
187C2×Q8—42no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—41no—
191C2×Q8—42no—
192C2×Q8—42no—
193C22×C4—41no—
194C22×C4—41no—
195C22×C4—41no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4—41no—
199C22×C4C2×D812yes—
200C22×C4C2×D812yes—
201C22×C4C2×D811yescharacteristic
202C22×C4C2×D811yescharacteristic
203C22×C4C22×D411yescharacteristic
204Q16—22no—
205Q16—22no—
206Q16—42no—
207Q16—41no—
208Q16—42no—
209Q16—42no—
210Q16—41no—
211Q16—42no—
212Q16—84no—
213C2×C8—21no—
214C2×C8—21no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—22no—
218C2×C8#12112yes—
219C2×C8C22×D411yescharacteristic
220C2×C8C2×#4111yescharacteristic
221C2×C8C2×#4111yescharacteristic
222C2×C8C2×#4111yescharacteristic
223C2×C8H(5,𝔽2)11yescharacteristic
224C2×C82−1+411yescharacteristic
225C4⋊C4—22no—
226C4⋊C4—24no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—41no—
231C4⋊C4—41no—
232C4⋊C4—41no—
233C4⋊C4—42no—
234C4⋊C4—41no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—161no—
240C4⋊C4—162no—
241C42—22no—
242C42—22no—
243C42—82no—
244C42—82no—
245C42#12112yes—
246C42#13512yes—
247C16—24no—
248C16—22no—
249C16—41no—
250C23C22×D811yescharacteristic
251Q8—44no—
252Q8—42no—
253Q8—42no—
254Q8—81no—
255Q8—81no—
256Q8—82no—
257Q8—82no—
258Q8—82no—
259Q8—82no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—42no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—82no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—82no—
286C2×C4#46612yes—
287C2×C4#46612yes—
288C2×C4#51012yes—
289C2×C4#51012yes—
290C2×C4C2×#12111yescharacteristic
291C2×C4#54811yescharacteristic
292C2×C4#55011yescharacteristic
293C2×C4C2×#13511yescharacteristic
294C2×C4#57611yescharacteristic
295C2×C4#57711yescharacteristic
296C8—22no—
297C8—42no—
298C8#52911yescharacteristic
299C8#54211yescharacteristic
300C8#54611yescharacteristic
301C8#54811yescharacteristic
302C22C2×#46611yescharacteristic
303C22#335811yescharacteristic
304C22#339111yescharacteristic
305C22C2×#51011yescharacteristic
306C22C2×#51011yescharacteristic
307C22#341611yescharacteristic
308C22#341711yescharacteristic
309C4—22no—
310C4—42no—
311C4—42no—
312C4—82no—
313C4—161no—
314C4—161no—
315C4—161no—
316C4—162no—
317C4—162no—
318C4#221914yes—
319C4#242611yescharacteristic
320C4#306211yescharacteristic
321C4#336111yescharacteristic
322C4#339411yescharacteristic
323C2#2529211yescharacteristic
324C2C2×#221911yescharacteristic
325C2#3361311yescharacteristic
326C2#3393611yescharacteristic
327C2#3394111yescharacteristic
328C2#3396511yescharacteristic
329C2#3396711yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#22092
C2×#21832
#287372
#282652
#134772
#134502
#132402
#132032
#131772
#131502
C4×#3582