Finite Group Databasebeta

← #6389805

Subgroups — #6389805

Order 512.

1560 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3162C211yesmaximal, characteristic
2#60250C212yesmaximal
3#23172C218yesmaximal, complemented
4#22607C214yesmaximal
5#22606C214yesmaximal
6#22257C214yesmaximal, complemented
7#21845C212yesmaximal
8#21737C212yesmaximal, complemented
9#21735C212yesmaximal, complemented
10#20830C211yesmaximal, characteristic
11#20481C211yesmaximal, characteristic
12C2×#533C2212yes—
13#3162C414yescomplemented
14#3162C414yescomplemented
15#3162C2214yes—
16#3162C2211yescharacteristic
17#3162C2211yescharacteristic
18#3162C2212yes—
19C2×#501C414yescomplemented
20C2×#501C414yescomplemented
21#2986—24no—
22#2985—24no—
23#2984—22no—
24#2984—22no—
25#2983—22no—
26#2982—22no—
27#2962C2218yes—
28#2960C2214yes—
29#2960C2214yes—
30#2924C2214yes—
31#2924C2214yes—
32#2920C2212yes—
33C2×#431C2211yescharacteristic
34C2×#427C2214yes—
35C22×#97C2211yescharacteristic
36#2283C2211yescharacteristic
37#1995C2214yes—
38#1995C2214yes—
39#1986C2218yes—
40#1946C2218yes—
41#1945C2218yes—
42#1922C22116yes—
43#1922C22116yes—
44#1849C2212yes—
45#1838C2212yes—
46#1826C2218yes—
47#1810C2214yes—
48#1778C2214yes—
49#1773C2218yes—
50#1772C2218yes—
51#1771C2214yes—
52#1752C2212yes—
53C8×C22⋊C4C2212yes—
54#1746C2212yes—
55#1726C2212yes—
56C23×C8C2312yes—
57#533—22no—
58#533—22no—
59#533—22no—
60#533—22no—
61#533C2×C412yes—
62#533C2×C412yes—
63#533C2×C414yes—
64#533C2312yes—
65#533C2312yes—
66#533C2314yes—
67C2×C4⋊Q8C2×C412yes—
68C2×C4⋊Q8C2×C412yes—
69C2×#125C2×C414yes—
70C2×C22⋊Q8—22no—
71C2×C22⋊Q8—22no—
72C2×C4×Q8—22no—
73C2×C4×Q8—22no—
74C2×#117C2314yes—
75C22×C42C2311yescharacteristic
76#501C2×C418yes—
77#501C2×C418yes—
78#501C2×C418yes—
79#501C2×C418yes—
80#501C2×C414yes—
81#501C2×C414yes—
82#501C2×C414yes—
83#501C2×C414yes—
84#501C2×C414yes—
85#501C2×C414yes—
86#501C2×C414yes—
87#501C2×C414yes—
88#446—28no—
89#446—28no—
90#446—28no—
91#446—24no—
92#446—24no—
93#446—24no—
94#446—24no—
95C8×Q8—28no—
96C8×Q8—24no—
97C8×Q8—24no—
98C8×Q8—24no—
99C8×Q8—24no—
100#433—28no—
101#433—28no—
102#432—22no—
103#432—22no—
104#432—24no—
105#432—24no—
106#431—22no—
107#431—22no—
108#431—24no—
109#431—24no—
110#431D412yes—
111#431D412yes—
112#431D414yes—
113C2×#108C2318yes—
114#427C2×C414yes—
115#427C2×C414yes—
116#427C2×C418yes—
117#427C2318yes—
118#427C2314yes—
119#427C2314yes—
120C2×#107C2×C414yes—
121C2×#107C2×C414yes—
122C2×#106C2×C418yes—
123#424C2318yes—
124#424C2318yes—
125#423C2314yes—
126C2×C4⋊C8C2312yes—
127#407C2314yes—
128C2×C22⋊C8C2311yescharacteristic
129C2×C22⋊C8C2311yescharacteristic
130#405C2318yes—
131#405C2318yes—
132C4×M16C2314yes—
133C2×#97C2×C414yes—
134C2×#97C2×C414yes—
135C2×#97C2314yes—
136C2×#97C2311yescharacteristic
137C2×#97C2311yescharacteristic
138C2×#97C2312yes—
139C2×C4×C8C2314yes—
140C2×C4×C8C2312yes—
141#383—21no—
142#382—21no—
143#381—21no—
144#380—21no—
145C4×C4⋊C4C2312yes—
146C4×C22⋊C4C2312yes—
147#376C2312yes—
148#341C2318yes—
149#339C23116yes—
150#339C23116yes—
151#336C2314yes—
152#336C2314yes—
153C22×Q8—22no—
154C22×Q8—22no—
155C23×C4#4111yescharacteristic
156C23×C4#4111yescharacteristic
157C23×C4C2411yescharacteristic
158C2×M16C2418yes—
159C22×C8—28no—
160C22×C8C22×C414yes—
161C22×C8C22×C414yes—
162C22×C8C2412yes—
163C22×C8C2412yes—
164C22×C8C2412yes—
165C22×C8C2412yes—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8C22×C412yes—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C414yes—
175C4⋊Q8C22×C414yes—
176#125—24no—
177#125—24no—
178#125C22×C414yes—
179#125C22×C414yes—
180#125C22×C414yes—
181#125C22×C414yes—
182C22⋊Q8—22no—
183C22⋊Q8—22no—
184C22⋊Q8—24no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187C22⋊Q8—24no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—24no—
193C22⋊Q8—24no—
194C4×Q8—22no—
195C4×Q8—22no—
196C4×Q8—24no—
197C4×Q8—22no—
198C4×Q8—22no—
199C4×Q8—24no—
200C4×Q8—24no—
201C4×Q8—22no—
202C4×Q8—22no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—24no—
206#117—24no—
207#117—24no—
208#117C22×C414yes—
209#117C22×C414yes—
210#117C2414yes—
211#117C2414yes—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4C22×C414yes—
218C2×C4⋊C4C22×C414yes—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—22no—
221C2×C42—24no—
222C2×C42C22×C412yes—
223C2×C42C22×C412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42C2414yes—
229C2×C42C2411yescharacteristic
230C2×C42C2411yescharacteristic
231#108—216no—
232#107C22×C418yes—
233#107C22×C418yes—
234#107C22×C414yes—
235#107C22×C414yes—
236#107C22×C414yes—
237#107C22×C414yes—
238#106C22×C418yes—
239#106C22×C418yes—
240#106C22×C418yes—
241#106C22×C418yes—
242C4⋊C8—28no—
243C4⋊C8—28no—
244C4⋊C8—28no—
245C4⋊C8—28no—
246C4⋊C8—24no—
247C4⋊C8—24no—
248C4⋊C8C2×D414yes—
249C4⋊C8C2×D414yes—
250C22⋊C8—24no—
251C22⋊C8—24no—
252C22⋊C8C2×D411yescharacteristic
253C22⋊C8C2×D412yes—
254C22⋊C8C2×D411yescharacteristic
255C22⋊C8C2×D412yes—
256C22⋊C8C2×D412yes—
257#97—28no—
258#97—28no—
259#97—28no—
260#97—24no—
261#97C22×C414yes—
262#97C22×C418yes—
263#97C22×C414yes—
264C4×C8—28no—
265C4×C8—28no—
266C4×C8—28no—
267C4×C8—24no—
268C4×C8C2×D414yes—
269C4×C8C2×D414yes—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274C24C2×#4111yescharacteristic
275C2×Q8—24no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—24no—
281C2×Q8—48no—
282C2×Q8—84no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—81no—
289C22×C4C4×D412yes—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2511yescharacteristic
303M16—216no—
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—28no—
308C2×C8—28no—
309C2×C8—28no—
310C2×C8C23×C414yes—
311C2×C8C23×C414yes—
312C2×C8C22×D412yes—
313C2×C8C22×D414yes—
314C2×C8C22×D412yes—
315C2×C82−1+418yes—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—82no—
333C4⋊C4C23×C414yes—
334C4⋊C4C23×C414yes—
335C4⋊C4C23×C414yes—
336C4⋊C4C23×C414yes—
337C22⋊C4—24no—
338C22⋊C4—24no—
339C22⋊C4—24no—
340C22⋊C4—24no—
341C22⋊C4—44no—
342C42—24no—
343C42—24no—
344C42—24no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42C4×D412yes—
349C42C4×D412yes—
350C42C4×D412yes—
351C42C4×D412yes—
352C42C23×C412yes—
353C42C23×C412yes—
354C42C22×D412yes—
355C42C22×D412yes—
356C23—22no—
357C23—22no—
358C23—41no—
359C23C2×C4×D411yescharacteristic
360C23C2×C4×D411yescharacteristic
361C23C22×#4111yescharacteristic
362Q8—48no—
363Q8—44no—
364Q8—44no—
365Q8—44no—
366Q8—44no—
367Q8—84no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—48no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—84no—
390C2×C4C2×C4×D411yescharacteristic
391C2×C4C2×C4×D411yescharacteristic
392C2×C4C2×C4×D412yes—
393C2×C4C2×C4×D412yes—
394C2×C4C2×C4×D412yes—
395C2×C4C2×C4×D412yes—
396C2×C4C24×C411yescharacteristic
397C2×C4C23×D411yescharacteristic
398C2×C4C2×2−1+414yes—
399C8—24no—
400C8—22no—
401C8—22no—
402C8—24no—
403C8—48no—
404C8#51918yes—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—44no—
413C22#295512yes—
414C22C22×C4×D411yescharacteristic
415C22#350912yes—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—21no—
421C4—21no—
422C4—44no—
423C4—44no—
424C4—44no—
425C4—44no—
426C4—84no—
427C4C2×#51912yes—
428C4#349912yes—
429C2—22no—
430C2—22no—
431C2—41no—
432C2#3420911yescharacteristic
433C2C2×#295511yescharacteristic
434C2#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. #23172C2semidirect
6. #22257C2semidirect
8. #21737C2semidirect
9. #21735C2semidirect
13. #3162C4semidirect
14. #3162C4semidirect
19. C2×#501C4semidirect
20. C2×#501C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31622
#602502
#231722
#226072
#226062
#222572
#218452
#217372
#217352
#208302
#204812