Finite Group Databasebeta

← #6436178

Subgroups — #6436178

Order 512.

2050 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3160C211yesmaximal, characteristic, complemented
2C22×#431C211yesmaximal, characteristic
3#36625C212yesmaximal
4#29879C218yesmaximal, complemented
5#29878C212yesmaximal, complemented
6#29877C214yesmaximal, complemented
7#29876C212yesmaximal, complemented
8#23666C214yesmaximal, complemented
9#23638C214yesmaximal
10#21318C212yesmaximal, complemented
11#20760C211yesmaximal, characteristic, complemented
12C2×#533C2212yes—
13C23×C42C2211yescharacteristic
14#3160—28no—
15C2×#500C2211yescharacteristic, complemented
16C2×#500C2211yescharacteristic, complemented
17C2×#499C2214yescomplemented
18C2×#498C2212yescomplemented
19C2×#431—28no—
20C2×#431C2212yes—
21C2×#431C2212yes—
22C2×#431C2214yes—
23C2×#427C2214yes—
24C22×C4⋊C8C2212yes—
25C22×C22⋊C8C2212yes—
26C22×C4×C8C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2857C2214yes—
29#2857C2214yes—
30#2855—22no—
31#2855—22no—
32#2851C2218yes—
33#2848—22no—
34#2848—22no—
35C4×C22⋊Q8—24no—
36C4×C22⋊Q8—24no—
37#2296C2214yes—
38#2285C2212yes—
39#2280C2214yes—
40#2088C22116yescomplemented
41#2087C2218yescomplemented
42#2086C2218yescomplemented
43#2077C22116yes—
44#2076C2218yes—
45#2075C2218yes—
46#1991C2218yescomplemented
47#1990C2214yescomplemented
48#1990C2214yescomplemented
49#1835C2214yescomplemented
50#1834C2212yescomplemented
51#1834C2212yescomplemented
52#1735C2214yes—
53#1735C2218yes—
54C24×C4D411yescharacteristic
55C24×C4D411yescharacteristic
56C24×C4C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2312yes—
59#533—28no—
60#533—28no—
61C2×C4⋊Q8C2312yes—
62C2×C4⋊Q8C2312yes—
63C2×#125C2314yes—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—24no—
67C2×C22⋊Q8—24no—
68C2×C4×Q8—22no—
69C2×C4×Q8—22no—
70C2×#117C2314yes—
71C2×#117C2314yes—
72C22×C42—24no—
73C22×C42—24no—
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D412yes—
77C22×C42D412yes—
78C22×C42C2312yes—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82#500—24no—
83#500—24no—
84#499—216no—
85#498—28no—
86#431—28no—
87#431—216no—
88#431—28no—
89#427—216no—
90C2×#107C2314yes—
91C2×#107C2314yes—
92C2×#106C2318yes—
93C2×C4⋊C8—28no—
94C2×C4⋊C8C2314yes—
95C2×C4⋊C8C2314yes—
96C2×C22⋊C8—28no—
97C2×C22⋊C8C2314yes—
98C2×C22⋊C8C2314yes—
99C2×C4×C8—24no—
100C2×C4×C8D418yes—
101C2×C4×C8C2312yes—
102C2×C4×C8C2312yes—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2312yes—
106#391—24no—
107#391—24no—
108#389—24no—
109#389—24no—
110#389C2318yes—
111#389C2318yes—
112#387—28no—
113#387—28no—
114#385—28no—
115#385—28no—
116#385—24no—
117#385—24no—
118#384C2314yes—
119#382C2314yes—
120#381—24no—
121#380—22no—
122#380—22no—
123#380—24no—
124#380—24no—
125#380C2318yes—
126#380C2318yes—
127#379—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4—22no—
130C4×C4⋊C4—28no—
131C4×C22⋊C4—28no—
132#340C2318yes—
133#340C23116yes—
134#340C23116yes—
135#340C2318yes—
136C25C2×D411yescharacteristic
137C22×Q8—22no—
138C22×Q8—22no—
139C23×C4—24no—
140C23×C4—28no—
141C23×C4—24no—
142C23×C4—22no—
143C23×C4—22no—
144C23×C4—28no—
145C23×C4—44no—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D412yes—
156C23×C4C2412yes—
157C23×C4C2411yescharacteristic
158C22×C8—28no—
159C22×C8—24no—
160C22×C8—28no—
161C22×C8C2×D418yes—
162C22×C8C2414yes—
163C22×C8C2412yes—
164C22×C8C2412yes—
165C4⋊Q8—24no—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169#125—28no—
170#125—28no—
171C22⋊Q8—416no—
172C22⋊Q8—48no—
173C22⋊Q8—48no—
174C22⋊Q8—416no—
175C4×Q8—48no—
176C4×Q8—48no—
177#117—28no—
178#117—28no—
179#117—216no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—28no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4C2414yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2418yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—28no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—28no—
200C2×C42—28no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D414yes—
215C2×C42#4114yes—
216C2×C42C2412yes—
217C2×C42C2411yescharacteristic
218C2×C42C2411yescharacteristic
219#107—28no—
220#107—28no—
221#106—216no—
222C4⋊C8—28no—
223C4⋊C8—28no—
224C22⋊C8—28no—
225C22⋊C8—28no—
226C4×C8—24no—
227C4×C8—24no—
228C4×C8C2×D414yes—
229C4×C8C2×D418yes—
230C4×C8C2×D414yes—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—24no—
235#95—28no—
236#95—28no—
237C24—22no—
238C24—24no—
239C24—24no—
240C24—42no—
241C24C22×D412yes—
242C24C22×D411yescharacteristic
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—48no—
246C2×Q8—48no—
247C22×C4—232no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—28no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—44no—
272C22×C4—42no—
273C22×C4—416no—
274C22×C4—48no—
275C22×C4—48no—
276C22×C4#12114yes—
277C22×C4#12114yes—
278C22×C4#13518yes—
279C22×C4C22×D412yes—
280C22×C4C22×D412yes—
281C22×C4C22×D411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D412yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4112yes—
289C22×C42−1+418yes—
290C22×C4C2511yescharacteristic
291C2×C8—216no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—28no—
296C2×C8—48no—
297C2×C8C22×D414yes—
298C2×C8C22×D414yes—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—216no—
304C4⋊C4—416no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—48no—
310C4⋊C4—416no—
311C4⋊C4—48no—
312C22⋊C4—416no—
313C22⋊C4—48no—
314C22⋊C4—48no—
315C22⋊C4—416no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—22no—
322C42—22no—
323C42—48no—
324C42—48no—
325C42#12114yes—
326C42#12114yes—
327C42C22×D412yes—
328C42C22×D412yes—
329C42C2×#4112yes—
330C42C2×#4112yes—
331C23—22no—
332C23—22no—
333C23—28no—
334C23—22no—
335C23—24no—
336C23—22no—
337C23—24no—
338C23—416no—
339C23—44no—
340C23—44no—
341C23C2×#12112yes—
342C23#53614yes—
343C23C2×#13514yes—
344C23C23×D411yescharacteristic
345Q8—88no—
346Q8—88no—
347C2×C4—216no—
348C2×C4—28no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—216no—
355C2×C4—28no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—48no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—48no—
370C2×C4—48no—
371C2×C4—48no—
372C2×C4—48no—
373C2×C4—48no—
374C2×C4C2×#12112yes—
375C2×C4C2×#12112yes—
376C2×C4C2×#12112yes—
377C2×C4C2×#12112yes—
378C2×C4C2×#12112yes—
379C2×C4#53614yes—
380C2×C4#53618yes—
381C2×C4#54118yes—
382C2×C4C2×#13514yes—
383C2×C4C2×#13514yes—
384C2×C4#57514yes—
385C2×C4C23×D411yescharacteristic
386C2×C4C22×#4111yescharacteristic
387C2×C4C2×2−1+414yes—
388C8—24no—
389C8—24no—
390C8—44no—
391C22—22no—
392C22—22no—
393C22—28no—
394C22—24no—
395C22—28no—
396C22—416no—
397C22—42no—
398C22—42no—
399C22—44no—
400C22#262114yes—
401C22C22×#12111yescharacteristic
402C22C2×#53612yes—
403C22C22×#13512yes—
404C22C2×#57512yes—
405C4—22no—
406C4—24no—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—84no—
411C4—84no—
412C4—88no—
413C4#262114yes—
414C4#305014yes—
415C4#305414yes—
416C4C2×#53612yes—
417C4C2×#54112yes—
418C2—24no—
419C2—44no—
420C2C2×#262111yescharacteristic
421C2C2×#305011yescharacteristic
422C2C2×#305411yescharacteristic
423C2#6084614yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C23×C42 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3160C2semidirect
4. #29879C2semidirect
5. #29878C2semidirect
6. #29877C2semidirect
7. #29876C2semidirect
8. #23666C2semidirect
10. #21318C2semidirect
11. #20760C2semidirect
15. C2×#500C22semidirect
16. C2×#500C22semidirect
17. C2×#499C22semidirect
18. C2×#498C22semidirect
40. #2088C22semidirect
41. #2087C22semidirect
42. #2086C22semidirect
46. #1991C22semidirect
47. #1990C22semidirect
48. #1990C22semidirect
49. #1835C22semidirect
50. #1834C22semidirect
51. #1834C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31602
C22×#4312
#366252
#298792
#298782
#298772
#298762
#236662
#236382
#213182
#207602