Finite Group Databasebeta

← #6389803

Subgroups — #6389803

Order 512.

1538 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#60663C211yesmaximal, characteristic
2#60251C212yesmaximal
3#23172C218yesmaximal, complemented
4#22608C218yesmaximal
5#22258C214yesmaximal, complemented
6#21845C212yesmaximal
7#21733C214yesmaximal, complemented
8#20831C211yesmaximal, characteristic
9#20477C211yesmaximal, characteristic
10C2×#533C2212yes—
11#3166C418yescomplemented
12#3165C414yescomplemented
13#3165C414yescomplemented
14#3162C2214yes—
15#3160C2212yes—
16#3160C2212yes—
17#2987—24no—
18#2985—24no—
19#2984—24no—
20#2983—24no—
21#2962C2218yes—
22#2961C2218yes—
23#2924C2218yes—
24#2920C2212yes—
25C2×#432C2211yescharacteristic
26#2911C2214yes—
27C2×C4×M16C2211yescharacteristic
28#2283C2211yescharacteristic
29#1995C2218yes—
30#1986C2218yes—
31#1946C2218yes—
32#1945C2218yes—
33#1922C22132yes—
34#1848C2212yes—
35#1837C2212yes—
36#1826C2218yes—
37#1811C2214yes—
38#1778C2214yes—
39#1773C2218yes—
40#1772C2218yes—
41#1771C2214yes—
42#1753C2214yes—
43#1746C2212yes—
44#1726C2212yes—
45C22×M16C2312yes—
46#533—22no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533C2×C414yes—
52#533C2314yes—
53#533C2314yes—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2×C412yes—
56C2×#125C2×C414yes—
57C2×C22⋊Q8—22no—
58C2×C22⋊Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×#117C2314yes—
62C22×C42C2311yescharacteristic
63#501C2×C4116yes—
64#501C2×C4116yes—
65#500C2×C412yes—
66#500C2×C412yes—
67#500C2×C412yes—
68#500C2×C412yes—
69#499C2×C418yes—
70#499C2×C418yes—
71#498C2×C414yes—
72#498C2×C414yes—
73#446—216no—
74#446—28no—
75#446—216no—
76#446—216no—
77C8×Q8—28no—
78#433—216no—
79#432—22no—
80#432—22no—
81#432—28no—
82#432D414yes—
83#432D414yes—
84#431—22no—
85#431—22no—
86#431—28no—
87C2×#108C2318yes—
88#428C2×C4116yes—
89#428C2×C4116yes—
90#427C2318yes—
91#427C2318yes—
92#424C23116yes—
93#423C2314yes—
94C2×C4⋊C8C2312yes—
95#407C2314yes—
96C2×C22⋊C8C2312yes—
97#405C23116yes—
98C4×M16C2×C414yes—
99C4×M16C2×C414yes—
100C4×M16C2314yes—
101C2×#97C2314yes—
102C2×#97C2312yes—
103C2×#97C2312yes—
104C2×C4×C8C2314yes—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#383—21no—
108#382—21no—
109#381—21no—
110#380—21no—
111C4×C4⋊C4C2312yes—
112C4×C22⋊C4C2312yes—
113#376C2312yes—
114#341C2318yes—
115#339C23116yes—
116#339C23116yes—
117#336C2318yes—
118C22×Q8—22no—
119C22×Q8—22no—
120C23×C4#4111yescharacteristic
121C23×C4#4111yescharacteristic
122C23×C4C2411yescharacteristic
123C2×M16—28no—
124C2×M16C22×C418yes—
125C2×M16C2418yes—
126C22×C8C2412yes—
127C22×C8C2414yes—
128C22×C8C2412yes—
129C4⋊Q8—22no—
130C4⋊Q8—22no—
131C4⋊Q8—22no—
132C4⋊Q8—22no—
133C4⋊Q8C22×C414yes—
134C4⋊Q8C22×C414yes—
135C4⋊Q8C22×C414yes—
136C4⋊Q8C22×C414yes—
137#125—24no—
138#125—24no—
139#125C22×C418yes—
140#125C22×C418yes—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—24no—
148C22⋊Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8—24no—
154C4×Q8—24no—
155C4×Q8—24no—
156C4×Q8—24no—
157#117—24no—
158#117—24no—
159#117C22×C418yes—
160#117C2418yes—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22×C414yes—
167C2×C4⋊C4C22×C414yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—22no—
170C2×C42—24no—
171C2×C42C22×C412yes—
172C2×C42C22×C412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42C2414yes—
178C2×C42C2411yescharacteristic
179C2×C42C2411yescharacteristic
180#108—216no—
181#107C22×C418yes—
182#107C22×C418yes—
183#107C22×C418yes—
184#107C22×C418yes—
185#106C22×C4116yes—
186#106C22×C4116yes—
187C4⋊C8—216no—
188C4⋊C8—216no—
189C4⋊C8—24no—
190C4⋊C8—24no—
191C4⋊C8C2×D418yes—
192C22⋊C8—24no—
193C22⋊C8—24no—
194C22⋊C8C2×D418yes—
195#97—216no—
196#97—28no—
197#97—24no—
198#97C22×C418yes—
199#97C2×D414yes—
200#97C2×D414yes—
201C4×C8—216no—
202C4×C8—28no—
203C4×C8—24no—
204C4×C8C22×C412yes—
205C4×C8C22×C412yes—
206C4×C8C22×C412yes—
207C4×C8C22×C412yes—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212C24C2×#4111yescharacteristic
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—24no—
217C2×Q8—84no—
218C2×Q8—84no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—81no—
224C22×C4—81no—
225C22×C4C4×D412yes—
226C22×C4C4×D412yes—
227C22×C4C4×D412yes—
228C22×C4C4×D412yes—
229C22×C4C23×C412yes—
230C22×C4C22×D412yes—
231C22×C4C2×#4111yescharacteristic
232C22×C4C2×#4111yescharacteristic
233C22×C4C2×#4111yescharacteristic
234C22×C4C2×#4111yescharacteristic
235C22×C4C2×#4111yescharacteristic
236C22×C4C2511yescharacteristic
237M16—216no—
238M16—216no—
239C2×C8—24no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—28no—
243C2×C8C23×C414yes—
244C2×C8C23×C414yes—
245C2×C8C22×D418yes—
246C2×C82−1+418yes—
247C4⋊C4—28no—
248C4⋊C4—28no—
249C4⋊C4—28no—
250C4⋊C4—28no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—24no—
257C4⋊C4—82no—
258C4⋊C4C23×C418yes—
259C4⋊C4C23×C418yes—
260C22⋊C4—28no—
261C22⋊C4—28no—
262C22⋊C4—82no—
263C42—28no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C4×D412yes—
268C42C4×D412yes—
269C42C4×D412yes—
270C42C4×D412yes—
271C42C23×C411yescharacteristic
272C42C23×C411yescharacteristic
273C42C23×C412yes—
274C42C22×D412yes—
275C42C22×D411yescharacteristic
276C42C22×D411yescharacteristic
277C23—22no—
278C23—41no—
279C23—41no—
280C23C2×C4×D412yes—
281C23C22×#4111yescharacteristic
282Q8—48no—
283Q8—84no—
284Q8—84no—
285Q8—84no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—24no—
289C2×C4—28no—
290C2×C4—28no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—48no—
300C2×C4—44no—
301C2×C4—44no—
302C2×C4—44no—
303C2×C4—84no—
304C2×C4C2×C4×D412yes—
305C2×C4C2×C4×D411yescharacteristic
306C2×C4C2×C4×D411yescharacteristic
307C2×C4C2×C4×D412yes—
308C2×C4C2×C4×D412yes—
309C2×C4C2×C4×D411yescharacteristic
310C2×C4C2×C4×D411yescharacteristic
311C2×C4C24×C411yescharacteristic
312C2×C4C23×D411yescharacteristic
313C2×C4C2×2−1+414yes—
314C8—24no—
315C8—28no—
316C8—48no—
317C8#51918yes—
318C22—21no—
319C22—21no—
320C22—22no—
321C22—22no—
322C22—22no—
323C22—82no—
324C22C22×C4×D411yescharacteristic
325C22#350912yes—
326C4—21no—
327C4—21no—
328C4—22no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—22no—
333C4—48no—
334C4—48no—
335C4—84no—
336C4C2×#51912yes—
337C4#349912yes—
338C2—22no—
339C2—42no—
340C2#3420911yescharacteristic
341C2#6018211yescharacteristic
342C2#6018211yescharacteristic
subgroupG/H
centerC2×C4C23×D4
commutatorC2×C4C24×C4
Frattini, cosocleC22×C4C25
socleC22C22×C4×D4
prime radicalC24C2×#41
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×#432 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #23172C2semidirect
5. #22258C2semidirect
7. #21733C2semidirect
11. #3166C4semidirect
12. #3165C4semidirect
13. #3165C4semidirect

Maximal subgroups

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

subgroupindex
#606632
#602512
#231722
#226082
#222582
#218452
#217332
#208312
#204772