Finite Group Databasebeta

← #6480128

Subgroups — #6480128

Order 512.

1768 conjugacy classes of subgroups.

The subgroup lattice has 439 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#33143C214yesmaximal, complemented
3#30298C211yesmaximal, characteristic, complemented
4#30247C211yesmaximal, characteristic
5#30197C212yesmaximal, complemented
6#29580C214yesmaximal, complemented
7#29558C214yesmaximal, complemented
8#29513C214yesmaximal, complemented
9#29493C214yesmaximal, complemented
10#22187C214yesmaximal, complemented
11#20364C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×#469C2214yes—
14C2×C4×M16C2212yes—
15#2861—22no—
16#2860—22no—
17#2860C2214yes—
18#2859C2212yes—
19#2858C2212yes—
20#2857C2212yes—
21#2856C2212yes—
22#2854—22no—
23#2854C2214yes—
24#2850—22no—
25C4×C4⋊Q8C2212yes—
26C4×#127C2212yes—
27C4×#124—22no—
28C4×#121—22no—
29C4×#121C2214yes—
30#2293—22no—
31#2293C2214yes—
32C2×C43C2211yescharacteristic
33#2100C2214yes—
34#2094C2214yes—
35#2083C2214yes—
36#2071C2214yes—
37#2058C2214yes—
38#2041C2214yes—
39#2023C2214yes—
40#2011C2214yes—
41#1972C2214yes—
42#1970C2214yes—
43#1943C2214yes—
44#1941C2214yes—
45#1936C2218yes—
46#1934C2218yes—
47#1926C2218yes—
48#1919C2218yes—
49#1895C2218yes—
50#1879C2218yes—
51#1862C2218yes—
52#1809C2214yes—
53#1745C2214yes—
54#1744C2214yes—
55#1728C2211yescharacteristic
56#1728C2212yes—
57#1728C2211yescharacteristic
58C22×D8C2314yes—
59C22×M16C2312yes—
60C22×M16C2312yes—
61C2×C4⋊Q8C2312yes—
62C2×#127C2312yes—
63#529—416no—
64C2×#126—24no—
65C2×#125—22no—
66C2×#124—22no—
67C2×#124—22no—
68C2×#123—24no—
69C2×C22⋊Q8—22no—
70C2×#121—22no—
71C2×#121—22no—
72C2×#121C2314yes—
73C2×C4×Q8—22no—
74C2×C4×D4—24no—
75C2×C4×D4—22no—
76C2×C4×D4—22no—
77C2×C4×D4C2314yes—
78C2×#117—24no—
79C22×C42D412yes—
80C22×C42D412yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#469—216no—
85C2×#107C2314yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88C2×#102C2318yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C4×M16—28no—
92C2×#97C2314yes—
93C2×C4×C8C2314yes—
94#391—22no—
95#391—22no—
96#390—22no—
97#390—22no—
98#390—22no—
99#390—22no—
100#390—22no—
101#390—22no—
102#390C2314yes—
103#390C2314yes—
104#390C2314yes—
105#390C2314yes—
106#389—22no—
107#389—22no—
108#389—22no—
109#389—22no—
110#389C2314yes—
111#389C2314yes—
112#389C2314yes—
113#389C2314yes—
114#388—24no—
115#388—24no—
116#388—24no—
117#388—24no—
118#388—24no—
119#388—24no—
120#388—24no—
121#388—24no—
122#386—24no—
123#386—24no—
124#384C2312yes—
125#384C2312yes—
126#382—22no—
127#380—22no—
128#380C2314yes—
129C4×C4⋊C4—22no—
130C4×C4⋊C4C2314yes—
131C4×C22⋊C4—24no—
132C4×C22⋊C4—24no—
133C43D412yes—
134C43D412yes—
135C43C2312yes—
136C43C2312yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#336C2314yes—
141#336C2314yes—
142#336C2318yes—
143C22×Q8—22no—
144C22×D4—24no—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4C2414yes—
148C23×C4—24no—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C2×D8—28no—
155C2×D8—44no—
156C2×D8—44no—
157C2×M16—24no—
158C2×M16—42no—
159C2×M16—42no—
160C2×M16—42no—
161C2×M16—42no—
162C2×M16#4118yes—
163C22×C8C2414yes—
164C22×C8C2414yes—
165C4⋊Q8—44no—
166#127—44no—
167#126—416no—
168#125—48no—
169#124—48no—
170#124—48no—
171#123—416no—
172C22⋊Q8—48no—
173#121—216no—
174#121—48no—
175#121—48no—
176C4×Q8—48no—
177C4×D4—44no—
178C4×D4—44no—
179C4×D4—48no—
180C4×D4—416no—
181C4×D4—48no—
182#117—416no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—24no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—24no—
204C2×C22⋊C4—24no—
205C2×C22⋊C4—24no—
206C2×C42—24no—
207C2×C42—28no—
208C2×C42—24no—
209C2×C42—44no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—42no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2412yes—
226C2×C42C2412yes—
227C2×C42C2411yescharacteristic
228C2×C42C2411yescharacteristic
229#107—44no—
230#106—28no—
231#106—44no—
232#102—216no—
233#102—48no—
234#102—48no—
235#97—28no—
236C4×C8—28no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243C24—24no—
244C24C22×D411yescharacteristic
245C2×Q8—44no—
246C2×Q8—44no—
247C2×Q8—44no—
248C2×D4—28no—
249C2×D4—24no—
250C2×D4—28no—
251C2×D4—24no—
252C2×D4—44no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—48no—
257C2×D4—48no—
258C2×D4—48no—
259C2×D4—44no—
260C2×D4—44no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C22×D412yes—
286C22×C4C22×D412yes—
287C22×C4C2×#4114yes—
288C22×C4C2511yescharacteristic
289D8—88no—
290M16—48no—
291M16—48no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8C2×#4118yes—
298C4⋊C4—28no—
299C4⋊C4—24no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—48no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—48no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—48no—
311C22⋊C4—48no—
312C22⋊C4—48no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—48no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—24no—
333C42—41no—
334C42—42no—
335C42—41no—
336C42—42no—
337C42—42no—
338C42—48no—
339C42AGL(1,ℤ/8)18yes—
340C23—21no—
341C23—21no—
342C23—22no—
343C23—22no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—44no—
350C23—44no—
351C23#53212yes—
352C23C23×D411yescharacteristic
353Q8—88no—
354D4—48no—
355D4—48no—
356D4—88no—
357D4—88no—
358D4—816no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—48no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—816no—
391C2×C4#53212yes—
392C2×C4C2×AGL(1,ℤ/8)14yes—
393C2×C4C2×AGL(1,ℤ/8)14yes—
394C2×C4C2×AGL(1,ℤ/8)14yes—
395C2×C4#57514yes—
396C2×C4C22×#4112yes—
397C2×C4#58511yescharacteristic
398C2×C4#58512yes—
399C2×C4#58511yescharacteristic
400C8—28no—
401C8—44no—
402C22—24no—
403C22—22no—
404C22—42no—
405C22—41no—
406C22—41no—
407C22—84no—
408C22—84no—
409C22—84no—
410C22—84no—
411C22—84no—
412C22—84no—
413C22—84no—
414C22C2×#53211yescharacteristic
415C22#344911yescharacteristic
416C22#344911yescharacteristic
417C22C22×AGL(1,ℤ/8)12yes—
418C22C2×#57512yes—
419C4—24no—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—22no—
424C4—42no—
425C4—42no—
426C4—84no—
427C4—84no—
428C4—84no—
429C4#349112yes—
430C4#349312yes—
431C2—42no—
432C2—164no—
433C2#3511911yescharacteristic
434C2#6070012yes—
435C2#6094912yes—
436C2#6096711yescharacteristic
437C2#6096811yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#30247C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #36631C2semidirect
2. #33143C2semidirect
3. #30298C2semidirect
5. #30197C2semidirect
6. #29580C2semidirect
7. #29558C2semidirect
8. #29513C2semidirect
9. #29493C2semidirect
10. #22187C2semidirect
11. #20364C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#331432
#302982
#302472
#301972
#295802
#295582
#295132
#294932
#221872
#203642