Finite Group Databasebeta

← #6434430

Subgroups — #6434430

Order 512.

1480 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35286C211yesmaximal, characteristic, complemented
2#35266C211yesmaximal, characteristic
3#30389C218yesmaximal, complemented
4#30384C214yesmaximal, complemented
5#30376C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7C2×#2093C211yesmaximal, characteristic
8#23667C212yesmaximal, complemented
9#23656C211yesmaximal, characteristic, complemented
10#23628C211yesmaximal, characteristic
11#23612C212yesmaximal
12#21158C212yesmaximal, complemented
13#21089C212yesmaximal
14#20430C211yesmaximal, characteristic, complemented
15C2×#427C2212yes—
16C22×#107C2212yes—
17C2×#424C2212yes—
18C2×#418C2212yes—
19C22×#102C2212yes—
20C2×#405C2212yes—
21#2860C2214yes—
22#2859C2214yescomplemented
23#2647C2214yescomplemented
24#2644C2214yes—
25#2571—22no—
26#2552—22no—
27#2513—22no—
28#2512—22no—
29#2480—24no—
30#2442—24no—
31#2317C2212yes—
32#2317C2211yescharacteristic
33#2316C2212yes—
34#2316C2211yescharacteristic
35#2309C2211yescharacteristic
36#2309C2211yescharacteristic
37#2296C2211yescharacteristic
38C2×#390C2211yescharacteristic
39#2285C2211yescharacteristic
40C2×#384C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2100C2214yescomplemented
43#2099C2214yescomplemented
44#2093C2212yes—
45#2093C2212yes—
46#2093C2214yes—
47#2087C2218yescomplemented
48#2086C2218yescomplemented
49#2085C2214yescomplemented
50#2084C2212yescomplemented
51#2083C2212yescomplemented
52#2073C2214yes—
53#2071C2214yes—
54#2067C2218yes—
55#2065C2218yes—
56#1943C2218yescomplemented
57#1942C2218yescomplemented
58#1936C2218yes—
59#1934C2218yes—
60#1768C2214yescomplemented
61#1767C2212yescomplemented
62C4×C4⋊C8C2212yescomplemented
63#1736C2212yes—
64#1736C2212yes—
65#1736C2212yes—
66#1735C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×D4C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72C2×#126—24no—
73C2×#125—24no—
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2312yes—
77C2×#117C2311yescharacteristic
78C2×#117C2311yescharacteristic
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2311yescharacteristic
82C22×C22⋊C4—22no—
83C22×C42C2311yescharacteristic
84#427—44no—
85C2×#107—28no—
86C2×#107C2314yes—
87C2×#107C2314yes—
88C2×#107C2314yes—
89C2×#106C2314yes—
90#424—28no—
91C2×C4⋊C8C2314yes—
92C2×C4⋊C8C2314yes—
93#418—44no—
94C2×#102—28no—
95C2×#102C2314yes—
96C2×#102C2314yes—
97C2×#102C2318yes—
98#405—44no—
99C2×#97C2314yes—
100C2×C4×C8C2314yes—
101#400—28no—
102#398—28no—
103#397—24no—
104#397—24no—
105#397—24no—
106#396—24no—
107#390—21no—
108#390—22no—
109#390—21no—
110#390—22no—
111#390C2314yes—
112#390C2314yes—
113#390C2312yes—
114#390C2312yes—
115#389—24no—
116#389—22no—
117#389—24no—
118#389—24no—
119#389C2318yes—
120#389C2314yes—
121#389C2318yes—
122#389C2314yes—
123#389C2312yes—
124#389C2312yes—
125#388—24no—
126#388—24no—
127#388—24no—
128#385—22no—
129#385—24no—
130#385—22no—
131#384—21no—
132#384—22no—
133#384—22no—
134#384C2312yes—
135#384C2312yes—
136#384C2312yes—
137#382—22no—
138#382—21no—
139#382—24no—
140#382C2312yes—
141#380—22no—
142#380—24no—
143#380—22no—
144#379—22no—
145#379—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4—22no—
148C4×C4⋊C4D412yes—
149C4×C4⋊C4D412yes—
150C4×C22⋊C4D414yes—
151#376C2312yes—
152C43C2312yes—
153#340C2318yes—
154#340C2318yes—
155#340C2314yes—
156#340C2314yes—
157#340C2318yes—
158#336C2312yes—
159#336C2312yes—
160#336C2311yescharacteristic
161#336C2311yescharacteristic
162#336C2312yes—
163C25—21no—
164C22×D4—21no—
165C22×D4—24no—
166C22×D4—22no—
167C22×D4—24no—
168C22×D4—22no—
169C22×D4C2412yes—
170C23×C4—21no—
171C23×C4—22no—
172C23×C4C2×D412yes—
173C23×C4C2411yescharacteristic
174C2×M16—24no—
175C2×M16—24no—
176C2×M16—24no—
177C22×C8—24no—
178C22×C8—24no—
179C22×C8#4118yes—
180C22×C8C2414yes—
181C22×C8C2412yes—
182C22×C8C2412yes—
183#126—88no—
184#125—88no—
185#117—24no—
186#117—22no—
187#117—22no—
188#117—24no—
189#117—22no—
190#117—24no—
191#117—22no—
192#117—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—28no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2414yes—
213C2×C4⋊C4C2412yes—
214C2×C4⋊C4C2412yes—
215C2×C4⋊C4C2414yes—
216C2×C4⋊C4C2412yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—24no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—84no—
227C2×C22⋊C4C2×D412yes—
228C2×C22⋊C4C2×D412yes—
229C2×C42—24no—
230C2×C42—22no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42C2×D414yes—
234C2×C42C2×D412yes—
235C2×C42C2×D414yes—
236C2×C42C2412yes—
237C2×C42C2412yes—
238C2×C42C2411yescharacteristic
239C2×C42C2411yescharacteristic
240#107—28no—
241#107—28no—
242#107—44no—
243#106—44no—
244C4⋊C8—28no—
245C4⋊C8—28no—
246#102—28no—
247#102—28no—
248#102—48no—
249#97—44no—
250C4×C8—44no—
251#95—24no—
252#95—22no—
253#95—22no—
254#95—28no—
255#95—24no—
256#95—28no—
257#95—24no—
258#95C2×D414yes—
259#95C2×D414yes—
260C24—22no—
261C24—42no—
262C24—42no—
263C24—42no—
264C24—82no—
265C24—82no—
266C24C22×D411yescharacteristic
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—48no—
275C2×D4—48no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—42no—
286C22×C4—82no—
287C22×C4—84no—
288C22×C4C22×D412yes—
289C22×C4C22×D412yes—
290C22×C4C22×D414yes—
291C22×C4C22×D412yes—
292C22×C4C22×D412yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4114yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+414yes—
297C22×C42−1+412yes—
298C22×C4C2511yescharacteristic
299M16—48no—
300C2×C8—216no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8C2×#4114yes—
309C2×C8C2×#4114yes—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—48no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—82no—
328C4⋊C4—84no—
329C4⋊C4—88no—
330C4⋊C4—88no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—44no—
338C22⋊C4—48no—
339C22⋊C4—84no—
340C22⋊C4—84no—
341C22⋊C4—84no—
342C22⋊C4—84no—
343C22⋊C4—84no—
344C42—24no—
345C42—28no—
346C42—24no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—44no—
352C42—42no—
353C42—44no—
354C42—42no—
355C23—22no—
356C23—22no—
357C23—44no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—44no—
362C23—88no—
363C23—84no—
364C23—162no—
365C23—162no—
366C23—164no—
367C23#53212yes—
368C23#53612yes—
369C23#53712yes—
370C23C23×D411yescharacteristic
371D4—44no—
372D4—44no—
373D4—84no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—48no—
400C2×C4—162no—
401C2×C4—164no—
402C2×C4#53212yes—
403C2×C4#53612yes—
404C2×C4#53712yes—
405C2×C4#53918yes—
406C2×C4#54118yes—
407C2×C4#57614yes—
408C2×C4C22×#4112yes—
409C2×C4C2×H(5,𝔽2)11yescharacteristic
410C2×C4C2×2−1+412yes—
411C2×C4C2×2−1+411yescharacteristic
412C8—24no—
413C8—24no—
414C8—44no—
415C22—24no—
416C22—22no—
417C22—22no—
418C22—24no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—84no—
424C22—168no—
425C22#266414yes—
426C22C2×#53211yescharacteristic
427C22C2×#53611yescharacteristic
428C22C2×#53711yescharacteristic
429C22C2×#57612yes—
430C22#358612yes—
431C4—24no—
432C4—44no—
433C4—42no—
434C4—42no—
435C4—42no—
436C4—82no—
437C4—84no—
438C4—82no—
439C4—84no—
440C4—82no—
441C4#266414yes—
442C4C2×#53912yes—
443C4C2×#54112yes—
444C2—24no—
445C2—162no—
446C2C2×#266411yescharacteristic
447C2#6071411yescharacteristic
448C2#6072011yescharacteristic
449C2#6087112yes—
450C2#6092812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #35286C2semidirect
3. #30389C2semidirect
4. #30384C2semidirect
5. #30376C2semidirect
6. #23745C2semidirect
8. #23667C2semidirect
9. #23656C2semidirect
12. #21158C2semidirect
14. #20430C2semidirect
22. #2859C22semidirect
23. #2647C22semidirect
42. #2100C22semidirect
43. #2099C22semidirect
47. #2087C22semidirect
48. #2086C22semidirect
49. #2085C22semidirect
50. #2084C22semidirect
51. #2083C22semidirect
56. #1943C22semidirect
57. #1942C22semidirect
60. #1768C22semidirect
61. #1767C22semidirect
62. C4×C4⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
#352862
#352662
#303892
#303842
#303762
#237452
C2×#20932
#236672
#236562
#236282
#236122
#211582
#210892
#204302