Finite Group Databasebeta

← #6389804

Subgroups — #6389804

Order 512.

1557 conjugacy classes of subgroups.

The subgroup lattice has 397 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60663C211yesmaximal, characteristic
2#60250C212yesmaximal
3#23171C214yesmaximal, complemented
4#23170C214yesmaximal, complemented
5#22608C218yesmaximal
6#22257C214yesmaximal, complemented
7#21845C212yesmaximal
8#21737C212yesmaximal, complemented
9#21735C212yesmaximal, complemented
10#20827C211yesmaximal, characteristic
11#20482C211yesmaximal, characteristic
12C2×#533C2212yes—
13#3166C418yescomplemented
14#3165C414yescomplemented
15#3165C414yescomplemented
16#3162C2212yes—
17#3162C2212yes—
18#3160C2214yes—
19#2986—22no—
20#2986—22no—
21#2985—24no—
22#2984—24no—
23#2983—22no—
24#2982—22no—
25#2962C2214yes—
26#2962C2214yes—
27#2960C2214yes—
28#2960C2214yes—
29#2924C2218yes—
30#2920C2212yes—
31C2×#431C2211yescharacteristic
32#2911C2214yes—
33C2×C4×M16C2211yescharacteristic
34#2283C2211yescharacteristic
35#1995C2214yes—
36#1995C2214yes—
37#1986C2218yes—
38#1946C2218yes—
39#1945C2218yes—
40#1922C22116yes—
41#1922C22116yes—
42#1849C2211yescharacteristic
43#1849C2211yescharacteristic
44#1838C2212yes—
45#1826C2218yes—
46#1811C2214yes—
47C4×#108C2214yes—
48#1773C2218yes—
49#1772C2218yes—
50#1770C2212yes—
51C8×C4⋊C4C2212yes—
52#1753C2214yes—
53#1746C2212yes—
54#1726C2212yes—
55C22×M16C2312yes—
56#533—22no—
57#533—22no—
58#533—22no—
59#533—22no—
60#533C2×C414yes—
61#533C2×C414yes—
62#533C2314yes—
63#533C2314yes—
64C2×C4⋊Q8C2×C412yes—
65C2×C4⋊Q8C2×C412yes—
66C2×#125C2×C414yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—22no—
69C2×C4×Q8—22no—
70C2×C4×Q8—22no—
71C2×#117C2314yes—
72C22×C42C2311yescharacteristic
73#501C2×C418yes—
74#501C2×C418yes—
75#501C2×C418yes—
76#501C2×C418yes—
77#500C2×C414yes—
78#500C2×C414yes—
79#499C2×C418yes—
80#499C2×C418yes—
81#498C2×C418yes—
82#446—24no—
83#446—24no—
84#446—28no—
85#446—28no—
86#446—24no—
87#446—24no—
88#446—24no—
89#446—24no—
90C8×Q8—24no—
91C8×Q8—24no—
92C8×Q8—24no—
93C8×Q8—24no—
94C8×Q8—24no—
95C8×Q8—24no—
96#433—28no—
97#433—28no—
98#432—22no—
99#432—22no—
100#432—28no—
101#431—22no—
102#431—22no—
103#431—28no—
104#431D412yes—
105#431D412yes—
106#431D414yes—
107C2×#108C2314yes—
108C2×#108C2314yes—
109#428C2×C4116yes—
110#428C2×C4116yes—
111#427C2318yes—
112#427C2318yes—
113#424C2318yes—
114#424C2318yes—
115#423C2312yes—
116#423C2312yes—
117C2×C4⋊C8C2311yescharacteristic
118C2×C4⋊C8C2311yescharacteristic
119#407C2314yes—
120C2×C22⋊C8C2312yes—
121#405C2318yes—
122#405C2318yes—
123C4×M16C2×C414yes—
124C4×M16C2×C414yes—
125C4×M16C2314yes—
126C2×#97C2314yes—
127C2×#97C2311yescharacteristic
128C2×#97C2311yescharacteristic
129C2×C4×C8C2314yes—
130C2×C4×C8C2312yes—
131C2×C4×C8C2312yes—
132#383—21no—
133#382—21no—
134#381—21no—
135#380—21no—
136C4×C4⋊C4C2312yes—
137C4×C22⋊C4C2312yes—
138#376C2312yes—
139#341C2318yes—
140#339C23116yes—
141#339C23116yes—
142#336C2314yes—
143#336C2314yes—
144C22×Q8—22no—
145C22×Q8—22no—
146C23×C4#4111yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4C2411yescharacteristic
149C2×M16—24no—
150C2×M16—24no—
151C2×M16C22×C418yes—
152C2×M16C2414yes—
153C2×M16C2414yes—
154C22×C8C2412yes—
155C22×C8C2412yes—
156C22×C8C2412yes—
157C22×C8C2412yes—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8C22×C414yes—
163C4⋊Q8C22×C414yes—
164C4⋊Q8C22×C414yes—
165C4⋊Q8C22×C414yes—
166#125—24no—
167#125—24no—
168#125C22×C418yes—
169#125C22×C418yes—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C22⋊Q8—24no—
174C22⋊Q8—24no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C4×Q8—24no—
179C4×Q8—24no—
180C4×Q8—24no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—24no—
184C4×Q8—24no—
185C4×Q8—24no—
186#117—24no—
187#117—24no—
188#117C22×C418yes—
189#117C2418yes—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4C22×C414yes—
196C2×C4⋊C4C22×C414yes—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—22no—
199C2×C42—24no—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202C2×C42C2×D412yes—
203C2×C42C2×D412yes—
204C2×C42#4111yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42C2414yes—
207C2×C42C2411yescharacteristic
208C2×C42C2411yescharacteristic
209#108—28no—
210#108—28no—
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—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8C2×D411yescharacteristic
226C4⋊C8C2×D412yes—
227C4⋊C8C2×D411yescharacteristic
228C4⋊C8C2×D412yes—
229C4⋊C8C2×D412yes—
230C22⋊C8—24no—
231C22⋊C8—24no—
232C22⋊C8C2×D414yes—
233C22⋊C8C2×D414yes—
234#97—28no—
235#97—28no—
236#97—28no—
237#97—24no—
238#97C22×C414yes—
239#97C22×C414yes—
240C4×C8—28no—
241C4×C8—28no—
242C4×C8—28no—
243C4×C8—24no—
244C4×C8C22×C414yes—
245C4×C8C22×C414yes—
246C4×C8C2×D414yes—
247C4×C8C2×D414yes—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252C24C2×#4111yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—48no—
258C2×Q8—84no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—42no—
265C22×C4—81no—
266C22×C4C4×D412yes—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4C4×D412yes—
270C22×C4C23×C412yes—
271C22×C4C22×D412yes—
272C22×C4C2×#4111yescharacteristic
273C22×C4C2×#4111yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4111yescharacteristic
276C22×C4C2×#4111yescharacteristic
277C22×C4C2511yescharacteristic
278M16—216no—
279M16—28no—
280M16—28no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
286C2×C8—24no—
287C2×C8C23×C414yes—
288C2×C8C23×C414yes—
289C2×C8C22×D412yes—
290C2×C8C22×D414yes—
291C2×C8C22×D412yes—
292C2×C82−1+414yes—
293C2×C82−1+414yes—
294C4⋊C4—28no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—44no—
305C4⋊C4C23×C418yes—
306C4⋊C4C23×C418yes—
307C22⋊C4—28no—
308C22⋊C4—28no—
309C22⋊C4—82no—
310C42—28no—
311C42—28no—
312C42—24no—
313C42—24no—
314C42C4×D412yes—
315C42C4×D412yes—
316C42C4×D412yes—
317C42C4×D412yes—
318C42C23×C411yescharacteristic
319C42C23×C411yescharacteristic
320C42C23×C412yes—
321C42C22×D411yescharacteristic
322C42C22×D412yes—
323C42C22×D411yescharacteristic
324C23—22no—
325C23—41no—
326C23—41no—
327C23C2×C4×D412yes—
328C23C22×#4111yescharacteristic
329Q8—48no—
330Q8—48no—
331Q8—48no—
332Q8—84no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—42no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—48no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—84no—
352C2×C4C2×C4×D412yes—
353C2×C4C2×C4×D411yescharacteristic
354C2×C4C2×C4×D411yescharacteristic
355C2×C4C2×C4×D412yes—
356C2×C4C2×C4×D412yes—
357C2×C4C2×C4×D411yescharacteristic
358C2×C4C2×C4×D411yescharacteristic
359C2×C4C24×C411yescharacteristic
360C2×C4C23×D411yescharacteristic
361C2×C4C2×2−1+412yes—
362C2×C4C2×2−1+412yes—
363C8—24no—
364C8—22no—
365C8—22no—
366C8—24no—
367C8—44no—
368C8—44no—
369C8#51918yes—
370C22—21no—
371C22—21no—
372C22—22no—
373C22—22no—
374C22—22no—
375C22—82no—
376C22C22×C4×D411yescharacteristic
377C22#350912yes—
378C4—21no—
379C4—22no—
380C4—21no—
381C4—22no—
382C4—21no—
383C4—22no—
384C4—48no—
385C4—48no—
386C4—84no—
387C4#295512yes—
388C4C2×#51912yes—
389C4#349911yescharacteristic
390C4#349911yescharacteristic
391C2—22no—
392C2—42no—
393C2#3420911yescharacteristic
394C2C2×#295511yescharacteristic
395C2#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×#431 ◁ 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. #22257C2semidirect
8. #21737C2semidirect
9. #21735C2semidirect
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
#231712
#231702
#226082
#222572
#218452
#217372
#217352
#208272
#204822