Finite Group Databasebeta

← #6389807

Subgroups — #6389807

Order 512.

1566 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3162C211yesmaximal, characteristic
2#60251C212yesmaximal
3#23172C218yesmaximal, complemented
4#22607C214yesmaximal
5#22606C214yesmaximal
6#22259C214yesmaximal, complemented
7#21844C212yesmaximal
8#21736C214yesmaximal, complemented
9#20830C211yesmaximal, characteristic
10#20479C211yesmaximal, characteristic
11C2×#533C2212yes—
12#3162C418yescomplemented
13#3162C2214yes—
14#3162C2212yes—
15#3162C2212yes—
16C2×#501C414yescomplemented
17C2×#501C414yescomplemented
18#2987—24no—
19#2985—24no—
20#2984—24no—
21#2983—24no—
22#2962C2218yes—
23#2961C2218yes—
24#2924C2218yes—
25#2920C2212yes—
26C2×#432C2211yescharacteristic
27C2×#427C2214yes—
28C22×#97C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1994C2218yes—
31#1986C2218yes—
32#1946C2218yes—
33#1945C2218yes—
34#1922C22116yes—
35#1922C22116yes—
36#1849C2212yes—
37#1838C2212yes—
38#1826C2214yes—
39#1826C2214yes—
40#1810C2214yes—
41#1778C2214yes—
42#1773C2218yes—
43#1772C2218yes—
44#1771C2214yes—
45#1752C2212yes—
46C8×C22⋊C4C2212yes—
47#1745C2212yes—
48C4×#97C2212yes—
49C23×C8C2312yes—
50#533—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533C2×C414yes—
55#533C2×C414yes—
56#533C2314yes—
57#533C2314yes—
58C2×C4⋊Q8C2×C412yes—
59C2×C4⋊Q8C2×C412yes—
60C2×#125C2×C414yes—
61C2×C22⋊Q8—22no—
62C2×C22⋊Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—22no—
65C2×#117—21no—
66C2×#117C2314yes—
67C22×C42C2311yescharacteristic
68#501C2×C418yes—
69#501C2×C418yes—
70#501C2×C418yes—
71#501C2×C418yes—
72#501C2×C418yes—
73#501C2×C418yes—
74#501C2×C418yes—
75#501C2×C418yes—
76#446—216no—
77#446—28no—
78#446—216no—
79#446—216no—
80C8×Q8—28no—
81#433—216no—
82#432—22no—
83#432—22no—
84#432—28no—
85#432D414yes—
86#432D414yes—
87#431—22no—
88#431—22no—
89#431—28no—
90C2×#108C2318yes—
91#427C2×C414yes—
92#427C2×C414yes—
93#427C2×C418yes—
94#427C2318yes—
95#427C2314yes—
96#427C2314yes—
97C2×#107C2×C414yes—
98C2×#107C2×C414yes—
99C2×#106C2×C418yes—
100#424C23116yes—
101#423C2314yes—
102C2×C4⋊C8C2312yes—
103#407C2314yes—
104C2×C22⋊C8C2312yes—
105#405C23116yes—
106C4×M16C2314yes—
107C2×#97C2×C414yes—
108C2×#97C2×C414yes—
109C2×#97C2314yes—
110C2×#97C2311yescharacteristic
111C2×#97C2311yescharacteristic
112C2×#97C2312yes—
113C2×#97C2312yes—
114C2×C4×C8C2314yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
117C4×C22⋊C4—21no—
118C4×C22⋊C4C2311yescharacteristic
119C4×C22⋊C4C2311yescharacteristic
120#376—21no—
121C43C2312yes—
122#341C2318yes—
123#339C23116yes—
124#339C23116yes—
125#336C2314yes—
126#336C2314yes—
127C22×Q8—22no—
128C22×Q8—22no—
129C23×C4#4111yescharacteristic
130C23×C4#4111yescharacteristic
131C23×C4C2411yescharacteristic
132C2×M16C2418yes—
133C22×C8—28no—
134C22×C8C22×C414yes—
135C22×C8C22×C414yes—
136C22×C8C2412yes—
137C22×C8C2414yes—
138C22×C8C2412yes—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8C22×C414yes—
144C4⋊Q8C22×C414yes—
145C4⋊Q8C22×C414yes—
146C4⋊Q8C22×C414yes—
147#125—24no—
148#125—24no—
149#125C22×C418yes—
150#125C22×C418yes—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—24no—
157C22⋊Q8—24no—
158C22⋊Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167#117—24no—
168#117—24no—
169#117—44no—
170#117C22×C418yes—
171#117C2418yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22×C414yes—
178C2×C4⋊C4C22×C414yes—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—21no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C22×C412yes—
186C2×C42C22×C412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42#4111yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42C2412yes—
192C2×C42C2412yes—
193C2×C42C2411yescharacteristic
194C2×C42C2411yescharacteristic
195#108—216no—
196#107C22×C418yes—
197#107C22×C418yes—
198#107C22×C414yes—
199#107C22×C414yes—
200#107C22×C414yes—
201#107C22×C414yes—
202#106C22×C418yes—
203#106C22×C418yes—
204#106C22×C418yes—
205#106C22×C418yes—
206C4⋊C8—216no—
207C4⋊C8—216no—
208C4⋊C8—24no—
209C4⋊C8—24no—
210C4⋊C8C2×D418yes—
211C22⋊C8—24no—
212C22⋊C8—24no—
213C22⋊C8C2×D414yes—
214C22⋊C8C2×D414yes—
215#97—216no—
216#97—28no—
217#97—24no—
218#97C22×C414yes—
219#97C22×C418yes—
220#97C22×C414yes—
221#97C2×D418yes—
222C4×C8—216no—
223C4×C8—28no—
224C4×C8—24no—
225#95—22no—
226#95—21no—
227#95—21no—
228C24C2×#4111yescharacteristic
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—84no—
234C2×Q8—84no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—42no—
241C22×C4—81no—
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4C4×D412yes—
245C22×C4C4×D412yes—
246C22×C4C23×C411yescharacteristic
247C22×C4C23×C411yescharacteristic
248C22×C4C22×D411yescharacteristic
249C22×C4C22×D411yescharacteristic
250C22×C4C2×#4111yescharacteristic
251C22×C4C2×#4111yescharacteristic
252C22×C4C2×#4111yescharacteristic
253C22×C4C2×#4111yescharacteristic
254C22×C4C2×#4111yescharacteristic
255C22×C4C2511yescharacteristic
256M16—216no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—28no—
261C2×C8—28no—
262C2×C8—28no—
263C2×C8C23×C414yes—
264C2×C8C23×C414yes—
265C2×C8C22×D418yes—
266C2×C82−1+418yes—
267C4⋊C4—28no—
268C4⋊C4—28no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4C23×C418yes—
279C4⋊C4C23×C418yes—
280C22⋊C4—24no—
281C22⋊C4—28no—
282C22⋊C4—28no—
283C22⋊C4—42no—
284C42—28no—
285C42—28no—
286C42—24no—
287C42—24no—
288C42—44no—
289C42—82no—
290C42C4×D412yes—
291C42C4×D412yes—
292C42C4×D412yes—
293C42C4×D412yes—
294C42C23×C412yes—
295C42C23×C412yes—
296C42C22×D412yes—
297C42C22×D412yes—
298C23—22no—
299C23—22no—
300C23—41no—
301C23C2×C4×D411yescharacteristic
302C23C2×C4×D411yescharacteristic
303C23C22×#4111yescharacteristic
304Q8—48no—
305Q8—84no—
306Q8—84no—
307Q8—84no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—28no—
316C2×C4—28no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—42no—
325C2×C4—48no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—84no—
331C2×C4C2×C4×D411yescharacteristic
332C2×C4C2×C4×D411yescharacteristic
333C2×C4C2×C4×D412yes—
334C2×C4C2×C4×D412yes—
335C2×C4C2×C4×D412yes—
336C2×C4C2×C4×D412yes—
337C2×C4C24×C411yescharacteristic
338C2×C4C23×D411yescharacteristic
339C2×C4C2×2−1+414yes—
340C8—24no—
341C8—28no—
342C8—48no—
343C8#51918yes—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22#295612yes—
353C22C22×C4×D411yescharacteristic
354C22#350912yes—
355C4—22no—
356C4—22no—
357C4—22no—
358C4—22no—
359C4—21no—
360C4—21no—
361C4—44no—
362C4—48no—
363C4—48no—
364C4—82no—
365C4C2×#51912yes—
366C4#349912yes—
367C2—22no—
368C2—22no—
369C2—41no—
370C2#3420911yescharacteristic
371C2C2×#295611yescharacteristic
372C2#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
6. #22259C2semidirect
8. #21736C2semidirect
12. #3162C4semidirect
16. C2×#501C4semidirect
17. C2×#501C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31622
#602512
#231722
#226072
#226062
#222592
#218442
#217362
#208302
#204792