Finite Group Databasebeta

← #6480109

Subgroups — #6480109

Order 512.

1774 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36631C212yesmaximal, complemented
2#33135C214yesmaximal, complemented
3#30293C211yesmaximal, characteristic, complemented
4#30239C211yesmaximal, characteristic
5#30185C212yesmaximal, complemented
6#29578C214yesmaximal, complemented
7#29558C214yesmaximal, complemented
8#29518C214yesmaximal, complemented
9#29499C214yesmaximal, complemented
10#22184C214yesmaximal, complemented
11C2×#1745C211yesmaximal, characteristic, complemented
12C2×#529—22no—
13C2×#466C2214yes—
14C22×#97C2212yes—
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#2099C2214yes—
34#2093C2214yes—
35#2084C2214yes—
36#2071C2214yes—
37#2058C2214yes—
38#2041C2214yes—
39#2023C2214yes—
40#2011C2214yes—
41#1972C2214yes—
42#1972C2214yes—
43#1944C2214yes—
44#1944C2214yes—
45#1936C2218yes—
46#1936C2218yes—
47#1926C2218yes—
48#1921C2218yes—
49#1893C2218yes—
50#1879C2218yes—
51#1864C2218yes—
52#1807C2214yes—
53#1745C2214yes—
54#1745C2214yes—
55C2×#336C2211yescharacteristic
56C2×#336C2212yes—
57C2×#336C2211yescharacteristic
58C22×D8C2314yes—
59C23×C8C2312yes—
60C23×C8C2312yes—
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#466—216no—
85C2×#107C2314yes—
86C2×#106C2314yes—
87C2×#106C2314yes—
88C2×#102C2318yes—
89C2×#102C2318yes—
90C2×#102C2318yes—
91C2×#97—28no—
92C2×#97C2314yes—
93C2×#97C2314yes—
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—
133C43D411yescharacteristic
134C43D411yescharacteristic
135C43D411yescharacteristic
136C43D411yescharacteristic
137C43C2312yes—
138C43C2312yes—
139#340C2318yes—
140#340C2318yes—
141#340C2318yes—
142#336C2314yes—
143#336C2314yes—
144#336C2318yes—
145C22×Q8—22no—
146C22×D4—24no—
147C22×D4—22no—
148C22×D4—22no—
149C22×D4C2414yes—
150C23×C4—24no—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2411yescharacteristic
156C2×D8—28no—
157C2×D8—44no—
158C2×D8—44no—
159C22×C8—24no—
160C22×C8—42no—
161C22×C8—42no—
162C22×C8—42no—
163C22×C8—42no—
164C22×C8#4118yes—
165C22×C8C2414yes—
166C22×C8C2414yes—
167C4⋊Q8—44no—
168#127—44no—
169#126—416no—
170#125—48no—
171#124—48no—
172#124—48no—
173#123—416no—
174C22⋊Q8—48no—
175#121—216no—
176#121—48no—
177#121—48no—
178C4×Q8—48no—
179C4×D4—44no—
180C4×D4—44no—
181C4×D4—48no—
182C4×D4—48no—
183C4×D4—416no—
184#117—416no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C4⋊C4C2414yes—
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×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C42—28no—
209C2×C42—24no—
210C2×C42—24no—
211C2×C42—44no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—42no—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2412yes—
230C2×C42C2412yes—
231C2×C42C2411yescharacteristic
232C2×C42C2411yescharacteristic
233#107—28no—
234#106—44no—
235#106—44no—
236#102—216no—
237#102—48no—
238#102—48no—
239#97—28no—
240#97—28no—
241#95—24no—
242#95—24no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247C24—24no—
248C24C22×D411yescharacteristic
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×D4—28no—
253C2×D4—24no—
254C2×D4—28no—
255C2×D4—24no—
256C2×D4—44no—
257C2×D4—44no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—48no—
261C2×D4—48no—
262C2×D4—48no—
263C2×D4—44no—
264C2×D4—44no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—48no—
281C22×C4—48no—
282C22×C4—48no—
283C22×C4AGL(1,ℤ/8)18yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D412yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C2×#4114yes—
294C22×C4C2511yescharacteristic
295D8—88no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—48no—
302C2×C8—48no—
303C2×C8C2×#4118yes—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—28no—
307C4⋊C4—24no—
308C4⋊C4—48no—
309C4⋊C4—44no—
310C4⋊C4—48no—
311C4⋊C4—48no—
312C4⋊C4—44no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C22⋊C4—48no—
318C22⋊C4—48no—
319C22⋊C4—48no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—48no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C42—22no—
329C42—22no—
330C42—24no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—24no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—42no—
346C42—42no—
347C42—48no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23#53212yes—
359C23C2×AGL(1,ℤ/8)14yes—
360C23C23×D411yescharacteristic
361Q8—88no—
362D4—48no—
363D4—48no—
364D4—88no—
365D4—88no—
366D4—816no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—816no—
402C2×C4#53212yes—
403C2×C4C2×AGL(1,ℤ/8)14yes—
404C2×C4C2×AGL(1,ℤ/8)14yes—
405C2×C4#57514yes—
406C2×C4C22×#4112yes—
407C2×C4#58511yescharacteristic
408C2×C4#58512yes—
409C2×C4#58511yescharacteristic
410C8—28no—
411C8—44no—
412C22—22no—
413C22—24no—
414C22—24no—
415C22—22no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—84no—
420C22—84no—
421C22—84no—
422C22—84no—
423C22#318414yes—
424C22C2×#53211yescharacteristic
425C22#344911yescharacteristic
426C22#344911yescharacteristic
427C22C22×AGL(1,ℤ/8)12yes—
428C22C2×#57512yes—
429C4—24no—
430C4—24no—
431C4—24no—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—42no—
438C4—84no—
439C4—84no—
440C4—84no—
441C4#349112yes—
442C4#349312yes—
443C2—24no—
444C2—164no—
445C2#3511911yescharacteristic
446C2#6067611yescharacteristic
447C2C2×#318411yescharacteristic
448C2#6095812yes—
449C2#6096212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radical#30239C2
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. #33135C2semidirect
3. #30293C2semidirect
5. #30185C2semidirect
6. #29578C2semidirect
7. #29558C2semidirect
8. #29518C2semidirect
9. #29499C2semidirect
10. #22184C2semidirect
11. C2×#1745C2semidirect

Maximal subgroups

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

subgroupindex
#366312
#331352
#302932
#302392
#301852
#295782
#295582
#295182
#294992
#221842
C2×#17452