Finite Group Databasebeta

← #350981

Subgroups — #350981

Order 512.

660 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33225C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4C4×#446C211yesmaximal, characteristic
5#9896C211yesmaximal, characteristic
6#9694C211yesmaximal, characteristic
7#9569C214yesmaximal
8#9228C214yesmaximal
9#8784C211yesmaximal, characteristic
10C2×#487—22no—
11C2×#477—21no—
12C2×#475C2212yes—
13C2×#446C2212yes—
14#2868C2211yescharacteristic
15#2868C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1988C2212yes—
18#1987—21no—
19#1956—21no—
20#1956—21no—
21#1956C2214yes—
22#1948—21no—
23#1948—21no—
24#1947—21no—
25#1947C2212yes—
26#1842C2211yescharacteristic
27#1842C2211yescharacteristic
28#1803—21no—
29#1803—21no—
30#1803—21no—
31#1803C2212yes—
32#1771—22no—
33#1770C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2211yescharacteristic
36C4×#97—21no—
37C42×C8C2211yescharacteristic
38C2×#332C2214yes—
39#1296C2214yes—
40#1277C2214yes—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8C2312yes—
47C2×C4×Q8D411yescharacteristic
48C2×C4×Q8D411yescharacteristic
49C2×C4×Q8C2311yescharacteristic
50#487—48no—
51#477—44no—
52#475—28no—
53#446—24no—
54#446—24no—
55C2×#107—22no—
56C2×#106—22no—
57C2×C4⋊C8—22no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#97—22no—
70C2×#97—22no—
71C2×C4×C8—21no—
72C2×C4×C8D412yes—
73C2×C4×C8D411yescharacteristic
74C2×C4×C8D411yescharacteristic
75C2×C4×C8C2312yes—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384—21no—
81#384—21no—
82#384—21no—
83#384C2312yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4D411yescharacteristic
88C4×C4⋊C4D411yescharacteristic
89C4×C4⋊C4C2311yescharacteristic
90C43—21no—
91C43C2311yescharacteristic
92#340—22no—
93#340—22no—
94#340—22no—
95#340—22no—
96#336—22no—
97#336—21no—
98#332—28no—
99C22×Q8—22no—
100C22×Q8—41no—
101C22×Q8C2×D411yescharacteristic
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—22no—
105C22×C8C2×D412yes—
106C22×C8C2×D412yes—
107C4⋊Q8—24no—
108C4⋊Q8—24no—
109C4⋊Q8—44no—
110C4⋊Q8—44no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—42no—
114C4⋊Q8—42no—
115C4⋊Q8—82no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8#4114yes—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C4⋊C4C2×D411yescharacteristic
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42—21no—
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42#4111yescharacteristic
152C2×C42C2411yescharacteristic
153#107—44no—
154#106—44no—
155C4⋊C8—28no—
156C4⋊C8—22no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—42no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8—42no—
163C4⋊C8#4114yes—
164#103—48no—
165#103—44no—
166#103—44no—
167#103—48no—
168#97—28no—
169#97—44no—
170C4×C8—24no—
171C4×C8—22no—
172C4×C8—22no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—42no—
176#95—21no—
177#95—21no—
178#95—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—21no—
183C2×Q8—21no—
184C2×Q8—44no—
185C2×Q8—42no—
186C2×Q8—44no—
187C2×Q8—42no—
188C2×Q8—82no—
189C2×Q8—82no—
190C2×Q8—82no—
191C2×Q8#12412yes—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—42no—
197C22×C4—42no—
198C22×C4#12111yescharacteristic
199C22×C4#12111yescharacteristic
200C22×C4#12111yescharacteristic
201C22×C4#12111yescharacteristic
202C22×C4C22×D411yescharacteristic
203C22×C4C22×D411yescharacteristic
204C22×C4C2×#4111yescharacteristic
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—22no—
208C2×C8—24no—
209C2×C8—22no—
210C2×C8—22no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8—42no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8AGL(1,ℤ/8)14yes—
220C4⋊C4—24no—
221C4⋊C4—21no—
222C4⋊C4—21no—
223C4⋊C4—21no—
224C4⋊C4—21no—
225C4⋊C4—21no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—44no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4—82no—
240C4⋊C4—82no—
241C4⋊C4—82no—
242C4⋊C4#12412yes—
243C42—24no—
244C42—22no—
245C42—22no—
246C42—22no—
247C42—42no—
248C42—42no—
249C42—42no—
250C42—42no—
251C42—42no—
252C42#13512yes—
253C42AGL(1,ℤ/8)12yes—
254C42#13712yes—
255C42#13712yes—
256C42#13712yes—
257C42#13712yes—
258C42C2×#4112yes—
259C42C2×#4112yes—
260C23C2×#12111yescharacteristic
261Q8—22no—
262Q8—22no—
263Q8—41no—
264Q8—41no—
265Q8—84no—
266Q8—84no—
267Q8—164no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—82no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4#46412yes—
293C2×C4#46412yes—
294C2×C4#46812yes—
295C2×C4#46812yes—
296C2×C4#47012yes—
297C2×C4#47012yes—
298C2×C4#47112yes—
299C2×C4#47312yes—
300C2×C4#52512yes—
301C2×C4C2×#12411yescharacteristic
302C2×C4#52912yes—
303C2×C4#53211yescharacteristic
304C2×C4C2×#13511yescharacteristic
305C2×C4C2×AGL(1,ℤ/8)12yes—
306C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
307C2×C4C2×#13711yescharacteristic
308C2×C4C2×#13711yescharacteristic
309C2×C4C2×#13711yescharacteristic
310C2×C4C2×#13711yescharacteristic
311C8—24no—
312C8—42no—
313C8—44no—
314C8—82no—
315C8—82no—
316C22#285011yescharacteristic
317C22C2×#46411yescharacteristic
318C22#303811yescharacteristic
319C22C2×#46811yescharacteristic
320C22C2×#47011yescharacteristic
321C22C2×#47111yescharacteristic
322C22C2×#47311yescharacteristic
323C4—22no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—42no—
328C4—81no—
329C4—81no—
330C4—162no—
331C4—162no—
332C4#170014yes—
333C4#170414yes—
334C4#310612yes—
335C4#311212yes—
336C4#313911yescharacteristic
337C4#314111yescharacteristic
338C4#318211yescharacteristic
339C4#318511yescharacteristic
340C2C2×#170011yescharacteristic
341C2C2×#170411yescharacteristic
342C2#1929111yescharacteristic
343C2#1929311yescharacteristic
344C2#3303111yescharacteristic
345C2#3304911yescharacteristic
346C2#3306111yescharacteristic
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
#332252
#331982
#331962
C4×#4462
#98962
#96942
#95692
#92282
#87842