Finite Group Databasebeta

← #350987

Subgroups — #350987

Order 512.

624 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33289C211yesmaximal, characteristic
2#33198C211yesmaximal, characteristic
3#33196C211yesmaximal, characteristic
4C4×#446C211yesmaximal, characteristic
5#9898C211yesmaximal, characteristic
6#9691C211yesmaximal, characteristic
7#9575C214yesmaximal
8#9224C214yesmaximal
9#8784C211yesmaximal, characteristic
10C2×#477C2212yes—
11C2×#475—21no—
12C2×#446C2212yes—
13#2868C2211yescharacteristic
14#2868C2211yescharacteristic
15C42×Q8C2211yescharacteristic
16#1990—21no—
17#1990—21no—
18#1990C2214yes—
19#1988—21no—
20#1987C2212yes—
21#1948—21no—
22#1948C2212yes—
23#1947—21no—
24#1947—21no—
25#1842C2211yescharacteristic
26#1842C2211yescharacteristic
27#1834—22no—
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×#333C2214yes—
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#477—28no—
51#475—44no—
52#446—24no—
53#446—24no—
54C2×#107—22no—
55C2×#106—22no—
56C2×C4⋊C8—22no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—21no—
60C2×C4⋊C8—21no—
61C2×C4⋊C8C2312yes—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×#103—22no—
65C2×#103—22no—
66C2×#97—22no—
67C2×#97—22no—
68C2×C4×C8—21no—
69C2×C4×C8D412yes—
70C2×C4×C8D411yescharacteristic
71C2×C4×C8D411yescharacteristic
72C2×C4×C8C2312yes—
73#384—21no—
74#384—21no—
75#384—21no—
76#384—21no—
77#384—21no—
78#384—21no—
79#384—21no—
80#384C2312yes—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4D411yescharacteristic
85C4×C4⋊C4D411yescharacteristic
86C4×C4⋊C4C2311yescharacteristic
87C43—21no—
88C43C2311yescharacteristic
89#340—24no—
90#340—24no—
91#340—22no—
92#340—22no—
93#340—22no—
94#340—22no—
95#336—22no—
96#336—21no—
97#333—28no—
98C22×Q8—22no—
99C22×Q8—41no—
100C22×Q8C2×D411yescharacteristic
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8C2×D412yes—
105C22×C8C2×D412yes—
106C4⋊Q8—24no—
107C4⋊Q8—24no—
108C4⋊Q8—42no—
109C4⋊Q8—42no—
110C4⋊Q8—42no—
111C4⋊Q8—42no—
112C4⋊Q8—82no—
113C4⋊Q8—82no—
114C4⋊Q8—82no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8#4114yes—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
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—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4C2×D411yescharacteristic
137C2×C4⋊C4C2×D411yescharacteristic
138C2×C4⋊C4C2×D411yescharacteristic
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—21no—
145C2×C42C2×D411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C2×C42C2411yescharacteristic
152#107—44no—
153#106—44no—
154C4⋊C8—28no—
155C4⋊C8—22no—
156C4⋊C8—24no—
157C4⋊C8—24no—
158C4⋊C8—42no—
159C4⋊C8—42no—
160C4⋊C8—42no—
161C4⋊C8—42no—
162C4⋊C8#4114yes—
163#103—44no—
164#103—44no—
165#97—28no—
166#97—44no—
167C4×C8—24no—
168C4×C8—22no—
169C4×C8—22no—
170C4×C8—22no—
171C4×C8—22no—
172C4×C8—42no—
173#95—21no—
174#95—21no—
175#95—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—21no—
180C2×Q8—21no—
181C2×Q8—44no—
182C2×Q8—82no—
183C2×Q8—82no—
184C2×Q8—82no—
185C2×Q8—82no—
186C2×Q8—82no—
187C2×Q8#12412yes—
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—22no—
202C2×C8—22no—
203C2×C8—22no—
204C2×C8—24no—
205C2×C8—22no—
206C2×C8—22no—
207C2×C8—42no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—42no—
214C2×C8—42no—
215C2×C8#13714yes—
216C4⋊C4—24no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—44no—
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—82no—
234C4⋊C4—82no—
235C4⋊C4—82no—
236C4⋊C4—82no—
237C4⋊C4—82no—
238C4⋊C4—82no—
239C4⋊C4#12412yes—
240C42—24no—
241C42—22no—
242C42—22no—
243C42—22no—
244C42—42no—
245C42—42no—
246C42—42no—
247C42—42no—
248C42—42no—
249C42#13512yes—
250C42AGL(1,ℤ/8)12yes—
251C42#13712yes—
252C42#13712yes—
253C42#13712yes—
254C42#13712yes—
255C42C2×#4112yes—
256C42C2×#4112yes—
257C23C2×#12111yescharacteristic
258Q8—22no—
259Q8—22no—
260Q8—41no—
261Q8—41no—
262Q8—164no—
263Q8—164no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—22no—
271C2×C4—22no—
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—41no—
281C2×C4—41no—
282C2×C4—82no—
283C2×C4—82no—
284C2×C4—82no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4#46412yes—
289C2×C4#46412yes—
290C2×C4#46812yes—
291C2×C4#46812yes—
292C2×C4#47012yes—
293C2×C4#47012yes—
294C2×C4#47112yes—
295C2×C4#47212yes—
296C2×C4#52512yes—
297C2×C4C2×#12411yescharacteristic
298C2×C4#52912yes—
299C2×C4#53211yescharacteristic
300C2×C4C2×#13511yescharacteristic
301C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
302C2×C4C2×#13712yes—
303C2×C4C2×#13711yescharacteristic
304C2×C4C2×#13711yescharacteristic
305C2×C4C2×#13711yescharacteristic
306C2×C4C2×#13711yescharacteristic
307C8—24no—
308C8—42no—
309C8—44no—
310C8—82no—
311C8—82no—
312C22#285011yescharacteristic
313C22C2×#46411yescharacteristic
314C22#303911yescharacteristic
315C22C2×#46811yescharacteristic
316C22C2×#47011yescharacteristic
317C22C2×#47111yescharacteristic
318C22C2×#47211yescharacteristic
319C4—22no—
320C4—22no—
321C4—22no—
322C4—42no—
323C4—42no—
324C4—81no—
325C4—81no—
326C4—162no—
327C4—162no—
328C4#169814yes—
329C4#170414yes—
330C4#310712yes—
331C4#311312yes—
332C4#313911yescharacteristic
333C4#314111yescharacteristic
334C4#318211yescharacteristic
335C4#318511yescharacteristic
336C2C2×#169811yescharacteristic
337C2C2×#170411yescharacteristic
338C2#1929511yescharacteristic
339C2#1929711yescharacteristic
340C2#3303311yescharacteristic
341C2#3304911yescharacteristic
342C2#3306011yescharacteristic
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
#332892
#331982
#331962
C4×#4462
#98982
#96912
#95752
#92242
#87842