Finite Group Databasebeta

← #6389801

Subgroups — #6389801

Order 512.

1582 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#60250C212yesmaximal
3#23172C218yesmaximal, complemented
4#22608C218yesmaximal
5#22256C212yesmaximal, complemented
6#22256C212yesmaximal, complemented
7#21844C212yesmaximal
8#21734C212yesmaximal, complemented
9#21732C212yesmaximal, complemented
10#20831C211yesmaximal, characteristic
11#20472C211yesmaximal, characteristic
12C2×#533C2212yes—
13#3166C418yescomplemented
14#3165C414yescomplemented
15#3165C414yescomplemented
16#3162C2214yes—
17#3160C2212yes—
18#3160C2212yes—
19#2986—24no—
20#2985—22no—
21#2985—22no—
22#2984—24no—
23#2983—22no—
24#2982—22no—
25#2962C2218yes—
26#2960C2214yes—
27#2960C2214yes—
28#2924C2218yes—
29#2920C2212yes—
30C2×#431C2211yescharacteristic
31#2911C2214yes—
32C2×C4×M16C2211yescharacteristic
33C4×#117C2211yescharacteristic
34#1994C2214yes—
35#1994C2214yes—
36#1986C2218yes—
37#1946C2218yes—
38#1945C2218yes—
39#1922C22116yes—
40#1922C22116yes—
41#1848C2212yes—
42#1837C2212yes—
43#1826C2218yes—
44#1811C2214yes—
45#1778C2214yes—
46#1773C2218yes—
47#1772C2218yes—
48#1771C2214yes—
49#1753C2214yes—
50#1744C2211yescharacteristic
51#1744C2211yescharacteristic
52C4×#97C2211yescharacteristic
53C42×C8C2211yescharacteristic
54C22×M16C2312yes—
55#533—22no—
56#533—22no—
57#533—22no—
58#533—22no—
59#533C2×C414yes—
60#533C2×C414yes—
61#533C2314yes—
62#533C2314yes—
63C2×C4⋊Q8C2×C412yes—
64C2×C4⋊Q8C2×C412yes—
65C2×#125C2×C414yes—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—22no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×#117—21no—
71C2×#117C2314yes—
72C22×C42C2311yescharacteristic
73#501C2×C4116yes—
74#501C2×C4116yes—
75#500C2×C412yes—
76#500C2×C412yes—
77#500C2×C412yes—
78#500C2×C412yes—
79#499C2×C418yes—
80#499C2×C418yes—
81#498C2×C414yes—
82#498C2×C414yes—
83#446—28no—
84#446—28no—
85#446—28no—
86#446—24no—
87#446—24no—
88#446—24no—
89#446—24no—
90C8×Q8—28no—
91C8×Q8—24no—
92C8×Q8—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95#433—28no—
96#433—28no—
97#432—22no—
98#432—22no—
99#432—28no—
100#431—22no—
101#431—22no—
102#431—28no—
103#431D412yes—
104#431D412yes—
105#431D414yes—
106C2×#108C2318yes—
107#428C2×C4116yes—
108#428C2×C4116yes—
109#427C2318yes—
110#427C2318yes—
111#424C2318yes—
112#424C2318yes—
113#423C2314yes—
114C2×C4⋊C8C2312yes—
115#407C2314yes—
116C2×C22⋊C8C2312yes—
117#405C2318yes—
118#405C2318yes—
119C4×M16C2×C414yes—
120C4×M16C2×C414yes—
121C4×M16C2314yes—
122C2×#97C2314yes—
123C2×#97C2312yes—
124C2×C4×C8C2314yes—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131C4×C22⋊C4—21no—
132C4×C22⋊C4C2312yes—
133#376—21no—
134C43C2311yescharacteristic
135C43C2311yescharacteristic
136#341C2318yes—
137#339C23116yes—
138#339C23116yes—
139#336C2314yes—
140#336C2314yes—
141C22×Q8—22no—
142C22×Q8—22no—
143C23×C4#4111yescharacteristic
144C23×C4#4111yescharacteristic
145C23×C4C2411yescharacteristic
146C2×M16—28no—
147C2×M16C22×C418yes—
148C2×M16C2418yes—
149C22×C8C2412yes—
150C22×C8C2412yes—
151C22×C8C2412yes—
152C22×C8C2412yes—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8C22×C414yes—
158C4⋊Q8C22×C414yes—
159C4⋊Q8C22×C414yes—
160C4⋊Q8C22×C414yes—
161#125—24no—
162#125—24no—
163#125C22×C418yes—
164#125C22×C418yes—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177C4×Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181#117—24no—
182#117—24no—
183#117—44no—
184#117C22×C418yes—
185#117C2418yes—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C22×C414yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—22no—
195C2×C42—24no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2412yes—
207C2×C42C2412yes—
208C2×C42C2411yescharacteristic
209C2×C42C2411yescharacteristic
210#108—216no—
211#107C22×C418yes—
212#107C22×C418yes—
213#107C22×C418yes—
214#107C22×C418yes—
215#106C22×C4116yes—
216#106C22×C4116yes—
217C4⋊C8—28no—
218C4⋊C8—28no—
219C4⋊C8—28no—
220C4⋊C8—28no—
221C4⋊C8—24no—
222C4⋊C8—24no—
223C4⋊C8C2×D414yes—
224C4⋊C8C2×D414yes—
225C22⋊C8—24no—
226C22⋊C8—24no—
227C22⋊C8C2×D414yes—
228C22⋊C8C2×D414yes—
229#97—28no—
230#97—28no—
231#97—28no—
232#97—24no—
233#97C22×C418yes—
234C4×C8—28no—
235C4×C8—28no—
236C4×C8—28no—
237C4×C8—24no—
238C4×C8C22×C412yes—
239C4×C8C22×C412yes—
240C4×C8C22×C412yes—
241C4×C8C22×C412yes—
242C4×C8C2×D411yescharacteristic
243C4×C8C2×D412yes—
244C4×C8C2×D412yes—
245C4×C8C2×D412yes—
246C4×C8C2×D411yescharacteristic
247#95—22no—
248#95—22no—
249C24C2×#4111yescharacteristic
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—24no—
254C2×Q8—48no—
255C2×Q8—84no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4C4×D412yes—
264C22×C4C4×D412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D412yes—
267C22×C4C23×C412yes—
268C22×C4C22×D412yes—
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4111yescharacteristic
271C22×C4C2×#4111yescharacteristic
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2511yescharacteristic
275M16—216no—
276M16—216no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—28no—
281C2×C8C23×C414yes—
282C2×C8C23×C414yes—
283C2×C8C22×D412yes—
284C2×C8C22×D414yes—
285C2×C8C22×D412yes—
286C2×C82−1+418yes—
287C4⋊C4—28no—
288C4⋊C4—28no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—44no—
298C4⋊C4C23×C418yes—
299C4⋊C4C23×C418yes—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—44no—
303C42—24no—
304C42—28no—
305C42—28no—
306C42—24no—
307C42—24no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42C4×D412yes—
312C42C4×D412yes—
313C42C4×D412yes—
314C42C4×D412yes—
315C42C23×C411yescharacteristic
316C42C23×C411yescharacteristic
317C42C23×C412yes—
318C42C22×D412yes—
319C42C22×D411yescharacteristic
320C42C22×D411yescharacteristic
321C23—22no—
322C23—41no—
323C23—41no—
324C23C2×C4×D412yes—
325C23C22×#4111yescharacteristic
326Q8—48no—
327Q8—48no—
328Q8—48no—
329Q8—84no—
330C2×C4—22no—
331C2×C4—24no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—84no—
351C2×C4C2×C4×D412yes—
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×C4×D411yescharacteristic
354C2×C4C2×C4×D412yes—
355C2×C4C2×C4×D412yes—
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C24×C411yescharacteristic
359C2×C4C23×D411yescharacteristic
360C2×C4C2×2−1+414yes—
361C8—24no—
362C8—22no—
363C8—22no—
364C8—24no—
365C8—48no—
366C8#51918yes—
367C22—21no—
368C22—21no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—82no—
373C22C22×C4×D411yescharacteristic
374C22#350912yes—
375C4—21no—
376C4—22no—
377C4—21no—
378C4—22no—
379C4—22no—
380C4—44no—
381C4—48no—
382C4—48no—
383C4—82no—
384C4#295512yes—
385C4#295612yes—
386C4C2×#51912yes—
387C4#349912yes—
388C2—22no—
389C2—42no—
390C2#3420911yescharacteristic
391C2C2×#295511yescharacteristic
392C2C2×#295611yescharacteristic
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×#431 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #23172C2semidirect
5. #22256C2semidirect
6. #22256C2semidirect
8. #21734C2semidirect
9. #21732C2semidirect
13. #3166C4semidirect
14. #3165C4semidirect
15. #3165C4semidirect

Maximal subgroups

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

subgroupindex
#606632
#602502
#231722
#226082
#222562
#218442
#217342
#217322
#208312
#204722