Finite Group Databasebeta

← #6389806

Subgroups — #6389806

Order 512.

1563 conjugacy classes of subgroups.

The subgroup lattice has 373 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#60251C212yesmaximal
3#23171C214yesmaximal, complemented
4#23170C214yesmaximal, complemented
5#22608C218yesmaximal
6#22259C214yesmaximal, complemented
7#21844C212yesmaximal
8#21736C214yesmaximal, complemented
9#20827C211yesmaximal, characteristic
10#20480C211yesmaximal, characteristic
11C2×#533C2212yes—
12#3166C418yescomplemented
13#3165C414yescomplemented
14#3165C414yescomplemented
15#3162C2212yes—
16#3162C2212yes—
17#3160C2214yes—
18#2987—24no—
19#2985—24no—
20#2984—24no—
21#2983—24no—
22#2962C2218yes—
23#2961C2218yes—
24#2924C2218yes—
25#2920C2212yes—
26C2×#432C2211yescharacteristic
27#2911C2214yes—
28C2×C4×M16C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1994C2218yes—
31#1986C2214yes—
32#1986C2214yes—
33#1946C2218yes—
34#1945C2218yes—
35#1922C22116yes—
36#1922C22116yes—
37#1849C2212yes—
38#1838C2212yes—
39#1826C2218yes—
40#1811C2214yes—
41C4×#108C2214yes—
42#1773C2218yes—
43#1772C2218yes—
44#1770C2212yes—
45C8×C4⋊C4C2212yes—
46#1753C2214yes—
47#1745C2212yes—
48C4×#97C2212yes—
49C22×M16C2312yes—
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#500C2×C414yes—
73#500C2×C414yes—
74#499C2×C418yes—
75#499C2×C418yes—
76#498C2×C418yes—
77#446—216no—
78#446—28no—
79#446—216no—
80#446—216no—
81C8×Q8—28no—
82#433—216no—
83#432—22no—
84#432—22no—
85#432—28no—
86#432D414yes—
87#432D414yes—
88#431—22no—
89#431—22no—
90#431—28no—
91C2×#108C2314yes—
92C2×#108C2314yes—
93#428C2×C4116yes—
94#428C2×C4116yes—
95#427C2318yes—
96#427C2318yes—
97#424C23116yes—
98#423C2314yes—
99C2×C4⋊C8C2312yes—
100#407C2314yes—
101C2×C22⋊C8C2312yes—
102#405C23116yes—
103C4×M16C2×C414yes—
104C4×M16C2×C414yes—
105C4×M16C2314yes—
106C2×#97C2314yes—
107C2×#97C2311yescharacteristic
108C2×#97C2311yescharacteristic
109C2×#97C2312yes—
110C2×C4×C8C2314yes—
111C2×C4×C8C2312yes—
112C4×C4⋊C4—21no—
113C4×C4⋊C4C2311yescharacteristic
114C4×C4⋊C4C2311yescharacteristic
115C4×C22⋊C4—21no—
116C4×C22⋊C4C2312yes—
117#376—21no—
118C43C2312yes—
119#341C2318yes—
120#339C23116yes—
121#339C23116yes—
122#336C2314yes—
123#336C2314yes—
124C22×Q8—22no—
125C22×Q8—22no—
126C23×C4#4111yescharacteristic
127C23×C4#4111yescharacteristic
128C23×C4C2411yescharacteristic
129C2×M16—24no—
130C2×M16—24no—
131C2×M16C22×C418yes—
132C2×M16C2418yes—
133C22×C8C2412yes—
134C22×C8C2414yes—
135C22×C8C2412yes—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8C22×C414yes—
141C4⋊Q8C22×C414yes—
142C4⋊Q8C22×C414yes—
143C4⋊Q8C22×C414yes—
144#125—24no—
145#125—24no—
146#125C22×C418yes—
147#125C22×C418yes—
148C22⋊Q8—24no—
149C22⋊Q8—24no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C22⋊Q8—24no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C4×Q8—24no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164#117—24no—
165#117—24no—
166#117—44no—
167#117C22×C418yes—
168#117C2418yes—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4C22×C414yes—
176C2×C4⋊C4C22×C414yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—22no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42C22×C412yes—
183C2×C42C22×C412yes—
184C2×C42C2×D412yes—
185C2×C42C2×D412yes—
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42C2412yes—
189C2×C42C2412yes—
190C2×C42C2411yescharacteristic
191C2×C42C2411yescharacteristic
192#108—28no—
193#108—28no—
194#107C22×C418yes—
195#107C22×C418yes—
196#107C22×C418yes—
197#107C22×C418yes—
198#106C22×C4116yes—
199#106C22×C4116yes—
200C4⋊C8—216no—
201C4⋊C8—216no—
202C4⋊C8—24no—
203C4⋊C8—24no—
204C4⋊C8C2×D414yes—
205C4⋊C8C2×D414yes—
206C22⋊C8—24no—
207C22⋊C8—24no—
208C22⋊C8C2×D418yes—
209#97—216no—
210#97—28no—
211#97—24no—
212#97C22×C414yes—
213#97C22×C414yes—
214#97C2×D418yes—
215C4×C8—216no—
216C4×C8—28no—
217C4×C8—24no—
218C4×C8C22×C414yes—
219C4×C8C22×C414yes—
220#95—21no—
221#95—21no—
222#95—22no—
223C24C2×#4111yescharacteristic
224C2×Q8—24no—
225C2×Q8—24no—
226C2×Q8—24no—
227C2×Q8—24no—
228C2×Q8—84no—
229C2×Q8—84no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—42no—
237C22×C4—81no—
238C22×C4C4×D412yes—
239C22×C4C4×D412yes—
240C22×C4C4×D412yes—
241C22×C4C4×D412yes—
242C22×C4C23×C412yes—
243C22×C4C22×D412yes—
244C22×C4C2×#4111yescharacteristic
245C22×C4C2×#4111yescharacteristic
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4111yescharacteristic
249C22×C4C2511yescharacteristic
250M16—216no—
251M16—28no—
252M16—28no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—28no—
258C2×C8C23×C414yes—
259C2×C8C23×C414yes—
260C2×C8C22×D418yes—
261C2×C82−1+414yes—
262C2×C82−1+414yes—
263C4⋊C4—24no—
264C4⋊C4—28no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—28no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—24no—
274C4⋊C4—42no—
275C4⋊C4C23×C418yes—
276C4⋊C4C23×C418yes—
277C22⋊C4—28no—
278C22⋊C4—28no—
279C22⋊C4—44no—
280C42—28no—
281C42—28no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—82no—
286C42C4×D412yes—
287C42C4×D412yes—
288C42C4×D412yes—
289C42C4×D412yes—
290C42C23×C411yescharacteristic
291C42C23×C411yescharacteristic
292C42C23×C412yes—
293C42C22×D411yescharacteristic
294C42C22×D411yescharacteristic
295C42C22×D412yes—
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23C2×C4×D412yes—
301C23C22×#4111yescharacteristic
302Q8—48no—
303Q8—84no—
304Q8—84no—
305Q8—84no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—24no—
311C2×C4—24no—
312C2×C4—24no—
313C2×C4—28no—
314C2×C4—28no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—48no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—84no—
328C2×C4C2×C4×D412yes—
329C2×C4C2×C4×D411yescharacteristic
330C2×C4C2×C4×D411yescharacteristic
331C2×C4C2×C4×D412yes—
332C2×C4C2×C4×D412yes—
333C2×C4C2×C4×D411yescharacteristic
334C2×C4C2×C4×D411yescharacteristic
335C2×C4C24×C411yescharacteristic
336C2×C4C23×D411yescharacteristic
337C2×C4C2×2−1+412yes—
338C2×C4C2×2−1+412yes—
339C8—24no—
340C8—28no—
341C8—48no—
342C8#51918yes—
343C22—21no—
344C22—21no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—81no—
349C22—81no—
350C22C22×C4×D411yescharacteristic
351C22#350911yescharacteristic
352C22#350911yescharacteristic
353C4—21no—
354C4—22no—
355C4—21no—
356C4—22no—
357C4—21no—
358C4—22no—
359C4—44no—
360C4—48no—
361C4—48no—
362C4—82no—
363C4#295612yes—
364C4C2×#51912yes—
365C4#349912yes—
366C2—21no—
367C2—21no—
368C2—42no—
369C2#3420911yescharacteristic
370C2C2×#295611yescharacteristic
371C2#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. #23171C2semidirect
4. #23170C2semidirect
6. #22259C2semidirect
8. #21736C2semidirect
12. #3166C4semidirect
13. #3165C4semidirect
14. #3165C4semidirect

Maximal subgroups

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

subgroupindex
#606632
#602512
#231712
#231702
#226082
#222592
#218442
#217362
#208272
#204802