Finite Group Databasebeta

← #317681

Subgroups — #317681

Order 512.

562 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28287C212yesmaximal
2C4×#500C211yesmaximal, characteristic
3#21673C211yesmaximal, characteristic
4#10652C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic
6#10545C211yesmaximal, characteristic
7#10545C211yesmaximal, characteristic
8#9574C212yesmaximal
9#9573C212yesmaximal
10#8784C211yesmaximal, characteristic
11#8589C212yesmaximal
12C2×#500C2211yescharacteristic
13C2×#479—22no—
14C4×C4⋊Q8C2212yes—
15C4×#125C2211yescharacteristic
16#2068—22no—
17#2066—22no—
18#1990C2212yes—
19#1981C2212yes—
20#1981C2212yes—
21#1948C2212yes—
22#1947C2212yes—
23#1942—21no—
24#1942—21no—
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1916—21no—
28#1916—21no—
29#1845C2211yescharacteristic
30#1842C2212yes—
31#1831C2211yescharacteristic
32#1805—22no—
33C4×#107—21no—
34C4×#107—21no—
35#1771—21no—
36#1771—21no—
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39C42×C8C2211yescharacteristic
40C2×#332C2212yes—
41C2×#332C2212yes—
42#1296C2212yes—
43#1296C2212yes—
44#1277C2211yescharacteristic
45#1277C2211yescharacteristic
46#1277C2212yes—
47C2×C4⋊Q8C2312yes—
48C2×#125D411yescharacteristic
49C2×C4×Q8—22no—
50#500—24no—
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⋊C8C2311yescharacteristic
63C2×C4⋊C8C2311yescharacteristic
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8D411yescharacteristic
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8D411yescharacteristic
76C2×C4×C8C2311yescharacteristic
77C2×C4×C8C2311yescharacteristic
78#391—22no—
79#389—22no—
80#389—22no—
81#389D411yescharacteristic
82#389D411yescharacteristic
83#387—21no—
84#387—21no—
85#384C2312yes—
86#382D411yescharacteristic
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4C2311yescharacteristic
92C4×C4⋊C4C2311yescharacteristic
93C43C2311yescharacteristic
94#340—22no—
95#340—22no—
96#340—22no—
97#340—22no—
98#336—21no—
99#336—21no—
100#336—21no—
101#336—21no—
102#336—21no—
103#336—21no—
104#332—24no—
105#332—24no—
106C22×Q8—22no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C2×D411yescharacteristic
112C22×C8C2×D411yescharacteristic
113C22×C8C2×D411yescharacteristic
114C22×C8C2×D411yescharacteristic
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—22no—
118C4⋊Q8#4114yes—
119#125—24no—
120C4×Q8—44no—
121C4×Q8—44no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C2×D411yescharacteristic
133C2×C4⋊C4C2×D411yescharacteristic
134C2×C4⋊C4C2×D411yescharacteristic
135C2×C4⋊C4C2×D411yescharacteristic
136C2×C42—21no—
137C2×C42—21no—
138C2×C42—42no—
139C2×C42C2×D411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42#4111yescharacteristic
144C2×C42#4111yescharacteristic
145C2×C42C2411yescharacteristic
146#107—24no—
147#107—24no—
148#107—44no—
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—82no—
162#103—44no—
163#103—44no—
164#103—82no—
165#103—82no—
166#97—42no—
167#97—42no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—41no—
174#95—21no—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—21no—
180#95—42no—
181#95—42no—
182C2×Q8—24no—
183C2×Q8—42no—
184C2×Q8—42no—
185C2×Q8—42no—
186C2×Q8—42no—
187C2×Q8—42no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—42no—
193C22×C4—42no—
194C22×C4#12111yescharacteristic
195C22×C4#12111yescharacteristic
196C22×C4#12111yescharacteristic
197C22×C4#12111yescharacteristic
198C22×C4C22×D411yescharacteristic
199C22×C4C22×D411yescharacteristic
200C22×C4C2×#4111yescharacteristic
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—21no—
206C2×C8—21no—
207C2×C8—21no—
208C2×C8—21no—
209C2×C8—21no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—41no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—42no—
221C2×C8—42no—
222C2×C8#13512yes—
223C2×C8#13712yes—
224C2×C8#13712yes—
225C4⋊C4—24no—
226C4⋊C4—24no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—21no—
230C4⋊C4—21no—
231C4⋊C4—21no—
232C4⋊C4—21no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—82no—
244C4⋊C4—84no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C42—21no—
250C42—21no—
251C42—22no—
252C42—41no—
253C42—41no—
254C42—41no—
255C42—41no—
256C42—41no—
257C42—84no—
258C42AGL(1,ℤ/8)14yes—
259C42C2×#4112yes—
260C422−1+412yes—
261C23C2×#12111yescharacteristic
262Q8—44no—
263Q8—44no—
264Q8—82no—
265Q8—82no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4#46912yes—
296C2×C4#46912yes—
297C2×C4#47312yes—
298C2×C4#52511yescharacteristic
299C2×C4#53611yescharacteristic
300C2×C4#54812yes—
301C2×C4#55212yes—
302C2×C4C2×#13511yescharacteristic
303C2×C4C2×AGL(1,ℤ/8)12yes—
304C2×C4C2×#13711yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4#57511yescharacteristic
307C2×C4#57611yescharacteristic
308C2×C4#57811yescharacteristic
309C8—22no—
310C8—22no—
311C8—22no—
312C8—41no—
313C8—81no—
314C8—82no—
315C8—82no—
316C8—81no—
317C22#253611yescharacteristic
318C22#303511yescharacteristic
319C22#303911yescharacteristic
320C22C2×#46911yescharacteristic
321C22#306111yescharacteristic
322C22C2×#47311yescharacteristic
323C22#306711yescharacteristic
324C4—22no—
325C4—22no—
326C4—22no—
327C4—22no—
328C4—41no—
329C4—81no—
330C4—84no—
331C4—81no—
332C4—81no—
333C4—81no—
334C4—162no—
335C4#310811yescharacteristic
336C4#311711yescharacteristic
337C4#323111yescharacteristic
338C4#323611yescharacteristic
339C4#336612yes—
340C4#336712yes—
341C2#1915711yescharacteristic
342C2#1922311yescharacteristic
343C2#1929611yescharacteristic
344C2#1936111yescharacteristic
345C2#2819411yescharacteristic
346C2#2820911yescharacteristic
347C2#2821911yescharacteristic
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
#282872
C4×#5002
#216732
#106522
#106502
#105452
#95742
#95732
#87842
#85892