Finite Group Databasebeta

← #330769

Subgroups — #330769

Order 512.

536 conjugacy classes of subgroups.

The subgroup lattice has 341 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33258C211yesmaximal, characteristic
2#33200C212yesmaximal
3#21339C211yesmaximal, characteristic
4#10656C212yesmaximal
5#10652C211yesmaximal, characteristic
6#10650C211yesmaximal, characteristic
7#10624C212yesmaximal
8#10618C211yesmaximal, characteristic
9#10618C211yesmaximal, characteristic
10#9900C212yesmaximal
11#8784C211yesmaximal, characteristic
12C2×#479—22no—
13#2859C2212yes—
14C42×Q8C2211yescharacteristic
15#1989—22no—
16#1984C2211yescharacteristic
17#1984C2211yescharacteristic
18#1949—22no—
19#1948C2212yes—
20#1947C2212yes—
21#1942—21no—
22#1942—21no—
23#1942—21no—
24#1942—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1916—21no—
28#1916—21no—
29#1845C2211yescharacteristic
30#1845C2211yescharacteristic
31#1842C2212yes—
32#1834C2212yes—
33#1805—22no—
34#1803C2212yes—
35#1803C2212yes—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2211yescharacteristic
38#1744—21no—
39C42×C8C2211yescharacteristic
40#1296C2212yes—
41#1296C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C2×#125—22no—
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8D411yescharacteristic
50C2×C4×Q8C2311yescharacteristic
51#479—48no—
52C2×#107—21no—
53C2×#107—21no—
54C2×#107—21no—
55C2×#107—21no—
56C2×#107—22no—
57C2×#106—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2311yescharacteristic
65C2×C4⋊C8C2311yescharacteristic
66C2×#103—22no—
67C2×#103—22no—
68C2×C4×C8—21no—
69C2×C4×C8D411yescharacteristic
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8D411yescharacteristic
73C2×C4×C8C2311yescharacteristic
74C2×C4×C8C2311yescharacteristic
75#389—22no—
76#389—22no—
77#389—22no—
78#389—22no—
79#384C2312yes—
80#384C2312yes—
81#382—22no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4D411yescharacteristic
87C4×C4⋊C4C2311yescharacteristic
88C43—21no—
89C43C2311yescharacteristic
90#340—22no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—21no—
97#336—21no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103C22×Q8C2×D411yescharacteristic
104C22×C8—21no—
105C22×C8—21no—
106C22×C8—22no—
107C22×C8—22no—
108C22×C8C2×D411yescharacteristic
109C22×C8C2×D411yescharacteristic
110C22×C8C2×D411yescharacteristic
111C22×C8C2×D411yescharacteristic
112#125—44no—
113#125—44no—
114C4×Q8—22no—
115C4×Q8—24no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8Q1614yes—
119C2×C4⋊C4—21no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4C2×D411yescharacteristic
131C2×C4⋊C4C2×D411yescharacteristic
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C42—21no—
134C2×C42—21no—
135C2×C42—21no—
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—21no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42C2411yescharacteristic
146#107—44no—
147#107—42no—
148#107—42no—
149#107—42no—
150#107—42no—
151#106—44no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—22no—
158C4⋊C8—42no—
159C4⋊C8—42no—
160C4⋊C8—44no—
161C4⋊C8—44no—
162#103—44no—
163#103—44no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—41no—
169C4×C8—41no—
170C4×C8—42no—
171#95—21no—
172#95—21no—
173#95—21no—
174#95—42no—
175#95—42no—
176C2×Q8—21no—
177C2×Q8—22no—
178C2×Q8—22no—
179C2×Q8C2×Q1612yes—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—42no—
185C22×C4—42no—
186C22×C4#12111yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4#12111yescharacteristic
189C22×C4#12111yescharacteristic
190C22×C4C22×D411yescharacteristic
191C22×C4C22×D411yescharacteristic
192C22×C4C2×#4111yescharacteristic
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—42no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8#13512yes—
216C2×C8#13712yes—
217C4⋊C4—21no—
218C4⋊C4—24no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—42no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—82no—
232C4⋊C4—82no—
233C4⋊C4—84no—
234C4⋊C4—84no—
235C4⋊C4C2×Q1612yes—
236C42—21no—
237C42—21no—
238C42—21no—
239C42—21no—
240C42—24no—
241C42—41no—
242C42—41no—
243C42—42no—
244C42—42no—
245C42—41no—
246C42—41no—
247C42—41no—
248C42—41no—
249C42—42no—
250C42C2×Q1612yes—
251C42#13512yes—
252C42AGL(1,ℤ/8)14yes—
253C422−1+412yes—
254C422−1+412yes—
255C23C2×#12111yescharacteristic
256Q8—24no—
257Q8—42no—
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—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—82no—
281C2×C4—82no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4#46712yes—
287C2×C4#46712yes—
288C2×C4#46912yes—
289C2×C4#46912yes—
290C2×C4#53612yes—
291C2×C4#53611yescharacteristic
292C2×C4#53611yescharacteristic
293C2×C4#54112yes—
294C2×C4C22×Q1611yescharacteristic
295C2×C4C2×#13511yescharacteristic
296C2×C4C2×#13511yescharacteristic
297C2×C4C2×AGL(1,ℤ/8)12yes—
298C2×C4C2×#13711yescharacteristic
299C2×C4#57511yescharacteristic
300C2×C4#57511yescharacteristic
301C8—22no—
302C8—22no—
303C8—41no—
304C8—41no—
305C8—82no—
306C8—82no—
307C8—81no—
308C8—81no—
309C22#262111yescharacteristic
310C22#303511yescharacteristic
311C22#303711yescharacteristic
312C22C2×#46711yescharacteristic
313C22C2×#46911yescharacteristic
314C22#306711yescharacteristic
315C22#306711yescharacteristic
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—22no—
321C4—42no—
322C4—42no—
323C4—81no—
324C4—81no—
325C4—162no—
326C4—162no—
327C4#322812yes—
328C4#323012yes—
329C4#323111yescharacteristic
330C4#323211yescharacteristic
331C4#323611yescharacteristic
332C4#324011yescharacteristic
333C2#1912611yescharacteristic
334C2#1915711yescharacteristic
335C2#1926211yescharacteristic
336C2#1928611yescharacteristic
337C2#2987111yescharacteristic
338C2#2987811yescharacteristic
339C2#2988811yescharacteristic
subgroupG/H
center, socle, prime radicalC23C2×#121
commutator, Frattini, cosocleC2×C42C24
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#332582
#332002
#213392
#106562
#106522
#106502
#106242
#106182
#99002
#87842