Finite Group Databasebeta

← #506340

Subgroups — #506340

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2229C212yesmaximal
2C4×#499C211yesmaximal, characteristic
3C4×#494C211yesmaximal, characteristic
4#13506C211yesmaximal, characteristic
5#13505C211yesmaximal, characteristic
6#13409C211yesmaximal, characteristic
7#13409C211yesmaximal, characteristic
8#13133C212yesmaximal
9#13132C212yesmaximal
10#12879C211yesmaximal, characteristic
11#12814C212yesmaximal
12C2×#499C2211yescharacteristic
13C2×#494C2211yescharacteristic
14C2×C4×Q16—21no—
15C2×C4×Q16C2212yes—
16C4×C4⋊Q8—21no—
17C4×#125—21no—
18#2229—28no—
19C2×#367C2212yes—
20C2×#366C2212yes—
21C2×C4⋊C16C2212yes—
22C2×#358C2212yes—
23C2×#358C2212yes—
24C2×#358C2212yes—
25C2×#358C2212yes—
26#1991—21no—
27#1969—21no—
28#1969C2212yes—
29#1956—21no—
30#1942C2211yescharacteristic
31#1942C2211yescharacteristic
32#1941—21no—
33#1846C2211yescharacteristic
34#1845C2211yescharacteristic
35C4×#107C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#106—21no—
38C42×C8C2211yescharacteristic
39#1380C2212yes—
40#1380C2212yes—
41#1374C2212yes—
42#1374C2211yescharacteristic
43#1374C2211yescharacteristic
44C22×Q16—21no—
45C22×Q16C2312yes—
46C2×C4⋊Q8—21no—
47C2×#125—21no—
48C2×C4×Q8—22no—
49C22×C16C2312yes—
50C22×C16C2312yes—
51#499—22no—
52#499D414yes—
53#494—24no—
54C4×Q16—24no—
55C4×Q16—24no—
56C4×Q16—44no—
57C2×#107—21no—
58C2×#107C2312yes—
59C2×#107C2311yescharacteristic
60C2×#107C2311yescharacteristic
61C2×#107C2311yescharacteristic
62C2×#107C2311yescharacteristic
63C2×#106—21no—
64C2×#106—21no—
65C2×#103—22no—
66C2×#103—22no—
67C2×C4×C8D411yescharacteristic
68C2×C4×C8D411yescharacteristic
69C2×C4×C8D411yescharacteristic
70C2×C4×C8C2311yescharacteristic
71C2×C4×C8C2311yescharacteristic
72C2×C4×C8C2311yescharacteristic
73#391—22no—
74#389—21no—
75#389—21no—
76#389—41no—
77#389—41no—
78#387—41no—
79#387—41no—
80#384—21no—
81#382—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—41no—
85C4×C4⋊C4—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C43D411yescharacteristic
89#367—24no—
90#366—24no—
91C4⋊C16—24no—
92#358—24no—
93#358—24no—
94#358—24no—
95#358—24no—
96#340—21no—
97#340—21no—
98#340—21no—
99#340—21no—
100C22×Q8—22no—
101C2×Q16—24no—
102C2×Q16—22no—
103C2×Q16—22no—
104C2×Q16—22no—
105C2×Q16—22no—
106C2×Q16—22no—
107C2×Q16—42no—
108C2×Q16—42no—
109C2×Q16#4114yes—
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2411yescharacteristic
114C4⋊Q8—24no—
115C4⋊Q8—42no—
116#125—24no—
117#125—42no—
118C4×Q8—44no—
119C4×Q8—44no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—41no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C42—42no—
135C2×C42—41no—
136C2×C42—41no—
137C2×C42D811yescharacteristic
138C2×C42D811yescharacteristic
139C2×C42D811yescharacteristic
140C2×C42D811yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C16—22no—
145C2×C16—22no—
146C2×C16—22no—
147C2×C16—22no—
148C2×C16—22no—
149C2×C16—22no—
150#107—21no—
151#107—21no—
152#107—22no—
153#107—22no—
154#107—22no—
155#107—22no—
156#107—42no—
157#107C2×D412yes—
158#107C2×D412yes—
159#106—24no—
160#106—42no—
161#103—44no—
162#103—44no—
163C4×C8—22no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—41no—
168C4×C8C2×D412yes—
169C4×C8#4112yes—
170#95—41no—
171#95—41no—
172#95—41no—
173#95—41no—
174#95—41no—
175#95—41no—
176#95—42no—
177#95—42no—
178C2×Q8—24no—
179C2×Q8—42no—
180C2×Q8—42no—
181C2×Q8—42no—
182C2×Q8—42no—
183C2×Q8—42no—
184C22×C4—41no—
185C22×C4—41no—
186C22×C4—42no—
187C22×C4—41no—
188C22×C4—41no—
189C22×C4—42no—
190C22×C4C2×D811yescharacteristic
191C22×C4C2×D811yescharacteristic
192C22×C4C2×D811yescharacteristic
193C22×C4C2×D811yescharacteristic
194C22×C4C2×D811yescharacteristic
195C22×C4C2×D811yescharacteristic
196C22×C4C22×D411yescharacteristic
197Q16—24no—
198Q16—24no—
199Q16—48no—
200Q16—42no—
201Q16—42no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8#12112yes—
210C2×C8#13712yes—
211C2×C8C22×D411yescharacteristic
212C2×C8C2×#4112yes—
213C2×C8C2×#4111yescharacteristic
214C2×C82−1+412yes—
215C4⋊C4—22no—
216C4⋊C4—22no—
217C4⋊C4—44no—
218C4⋊C4—44no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—82no—
230C4⋊C4—84no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—82no—
234C42—21no—
235C42—21no—
236C42—21no—
237C42—21no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—41no—
242C42—82no—
243C42—84no—
244C42—82no—
245C42#12112yes—
246C42AGL(1,ℤ/8)12yes—
247C16—42no—
248C16—42no—
249C23C22×D811yescharacteristic
250Q8—44no—
251Q8—44no—
252Q8—82no—
253Q8—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—82no—
275C2×C4—82no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4#45912yes—
286C2×C4#45912yes—
287C2×C4#50812yes—
288C2×C4#51012yes—
289C2×C4#51012yes—
290C2×C4C2×#12111yescharacteristic
291C2×C4#52511yescharacteristic
292C2×C4#53611yescharacteristic
293C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
294C2×C4C2×#13711yescharacteristic
295C2×C4#57511yescharacteristic
296C8—22no—
297C8—22no—
298C8—41no—
299C8#54812yes—
300C8#55212yes—
301C22C2×#45911yescharacteristic
302C22#310011yescharacteristic
303C22#322611yescharacteristic
304C22C2×#50811yescharacteristic
305C22C2×#51011yescharacteristic
306C22C2×#51011yescharacteristic
307C22#341411yescharacteristic
308C4—22no—
309C4—22no—
310C4—22no—
311C4—22no—
312C4—22no—
313C4—41no—
314C4—84no—
315C4—82no—
316C4—82no—
317C4—162no—
318C4—161no—
319C4—161no—
320C4#253611yescharacteristic
321C4#303711yescharacteristic
322C4#311011yescharacteristic
323C4#323511yescharacteristic
324C2#2820111yescharacteristic
325C2#3357911yescharacteristic
326C2#3358311yescharacteristic
327C2#3366811yescharacteristic
328C2#3367711yescharacteristic
329C2#3379711yescharacteristic
330C2#3380211yescharacteristic
subgroupG/H
center, socle, prime radicalC23C22×D8
commutator, Frattini, cosocleC22×C8C24
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ 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×#22292
C4×#4992
C4×#4942
#135062
#135052
#134092
#131332
#131322
#128792
#128142