Finite Group Databasebeta

← #6435216

Subgroups — #6435216

Order 512.

1710 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35275C211yesmaximal, characteristic, complemented
2#34255C211yesmaximal, characteristic
3#29990C218yesmaximal, complemented
4#29987C214yesmaximal, complemented
5#29981C214yesmaximal, complemented
6#23745C211yesmaximal, characteristic, complemented
7#23744C211yesmaximal, characteristic, complemented
8#23731C211yesmaximal, characteristic
9C2×#2094C211yesmaximal, characteristic
10#23556C212yesmaximal, complemented
11#23456C212yesmaximal
12#21174C212yesmaximal, complemented
13#20754C211yesmaximal, characteristic, complemented
14#20650C212yesmaximal
15C2×#544C2212yes—
16C2×#518C2211yescharacteristic
17C2×#428C2212yes—
18C22×#106C2212yes—
19C2×#424C2212yes—
20C2×#418C2212yes—
21C22×#102C2212yes—
22#2868C2212yescomplemented
23#2859C2212yescomplemented
24#2660C2214yescomplemented
25#2552—22no—
26#2551—22no—
27#2440—24no—
28#2357—22no—
29#2354—22no—
30#2342—24no—
31#2319C2214yes—
32#2317C2211yescharacteristic
33#2317C2212yes—
34#2309C2211yescharacteristic
35#2296C2211yescharacteristic
36#2293C2214yes—
37C2×#390C2211yescharacteristic
38#2285C2211yescharacteristic
39C2×#384C2211yescharacteristic
40#2274C2211yescharacteristic
41C4×#117C2211yescharacteristic
42#2101C2214yescomplemented
43#2100C2212yescomplemented
44#2100C2214yescomplemented
45#2099C2212yescomplemented
46#2099C2214yescomplemented
47#2094C2212yes—
48#2094C2214yes—
49#2094C2214yes—
50#2093C2212yes—
51#2092C2214yes—
52#2059C2218yescomplemented
53#2058C2218yescomplemented
54#2025C2218yes—
55#2023C2218yes—
56#1949C2218yescomplemented
57#1948C2218yescomplemented
58#1832C2214yescomplemented
59#1831C2214yescomplemented
60#1804C2218yes—
61#1802C2218yes—
62#1736C2212yes—
63#1736C2212yes—
64#1735C2212yes—
65#1735C2212yes—
66#1729C2212yes—
67#1728C2211yescharacteristic
68C2×#336C2211yescharacteristic
69C23×D4C2311yescharacteristic
70C22×M16C2312yes—
71C23×C8C2312yes—
72#544—44no—
73#544—44no—
74C2×C4⋊Q8—22no—
75C2×C4⋊Q8—24no—
76C2×C4⋊Q8C2314yes—
77C2×#125—22no—
78C2×#125C2314yes—
79C2×C22⋊Q8—24no—
80#518—24no—
81#518Q818yescomplemented
82C2×C4×D4—22no—
83C2×C4×D4C2314yes—
84C2×#117—24no—
85C2×#117C2311yescharacteristic
86C2×#117C2312yes—
87C2×#117C2311yescharacteristic
88C2×#117C2311yescharacteristic
89C2×#117C2311yescharacteristic
90C22×C4⋊C4—22no—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2311yescharacteristic
93C22×C22⋊C4—22no—
94C22×C22⋊C4—22no—
95C22×C42C2311yescharacteristic
96#428—44no—
97C2×#107C2314yes—
98C2×#106—44no—
99C2×#106C2314yes—
100C2×#106C2318yes—
101#424—28no—
102C2×C4⋊C8C2318yes—
103#418—44no—
104C2×#102—44no—
105C2×#102C2314yes—
106C2×#102C2318yes—
107C2×#102C2314yes—
108#398—28no—
109#395—28no—
110#390—21no—
111#390—21no—
112#390—22no—
113#390C2314yes—
114#390C2312yes—
115#390C2312yes—
116#389—24no—
117#389—24no—
118#389—24no—
119#389C2314yes—
120#389C2312yes—
121#389C2312yes—
122#388—24no—
123#388—24no—
124#388—24no—
125#388—24no—
126#385—22no—
127#385—22no—
128#385—24no—
129#384—21no—
130#384—24no—
131#384—22no—
132#384C2314yes—
133#384C2311yescharacteristic
134#384C2312yes—
135#384C2312yes—
136#384C2311yescharacteristic
137#382—21no—
138#382C2314yes—
139#382C2312yes—
140#381—24no—
141#380—22no—
142#380—24no—
143#379—22no—
144C4×C4⋊C4—22no—
145C4×C4⋊C4—22no—
146C4×C4⋊C4—22no—
147C4×C4⋊C4D412yes—
148C4×C4⋊C4D412yes—
149C4×C4⋊C4C2314yes—
150C4×C22⋊C4—22no—
151C4×C22⋊C4—24no—
152C4×C22⋊C4D414yes—
153#376C2312yes—
154C43C2312yes—
155#340C2314yes—
156#340C2318yes—
157#340C2314yes—
158#340C2318yes—
159#340C2314yes—
160#340C2314yes—
161#336C2318yes—
162#336C2312yes—
163#336C2312yes—
164#336C2312yes—
165#336C2312yes—
166C25—21no—
167C22×Q8—24no—
168C22×D4—21no—
169C22×D4—22no—
170C22×D4—24no—
171C22×D4—24no—
172C22×D4C2×Q814yes—
173C22×D4C2412yes—
174C23×C4—24no—
175C23×C4—21no—
176C23×C4—22no—
177C23×C4C2×D412yes—
178C23×C4C2411yescharacteristic
179C2×M16—24no—
180C2×M16—24no—
181C2×M16—24no—
182C22×C8—24no—
183C22×C8—24no—
184C22×C8—24no—
185C22×C8C2414yes—
186C22×C8C2414yes—
187C4⋊Q8—44no—
188C4⋊Q8—44no—
189C4⋊Q8—48no—
190C4⋊Q8—88no—
191#125—44no—
192#125—44no—
193#125—48no—
194C22⋊Q8—416no—
195C4×D4—216no—
196C4×D4—28no—
197C4×D4C2×Q8116yes—
198#117—22no—
199#117—24no—
200#117—24no—
201#117—24no—
202#117—24no—
203#117—88no—
204#117C2×Q814yes—
205#117C2×Q814yes—
206#117#4114yes—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—28no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—84no—
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D412yes—
227C2×C4⋊C4C2414yes—
228C2×C4⋊C4C2412yes—
229C2×C4⋊C4C2412yes—
230C2×C4⋊C4C2412yes—
231C2×C4⋊C4C2412yes—
232C2×C4⋊C4C2412yes—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—24no—
235C2×C22⋊C4—216no—
236C2×C22⋊C4—22no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—24no—
239C2×C22⋊C4—24no—
240C2×C22⋊C4—24no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—84no—
243C2×C22⋊C4C2×D412yes—
244C2×C22⋊C4C2×D412yes—
245C2×C42—21no—
246C2×C42—24no—
247C2×C42—22no—
248C2×C42—21no—
249C2×C42—24no—
250C2×C42C2×D414yes—
251C2×C42C2×D412yes—
252C2×C42C2×D414yes—
253C2×C42C2412yes—
254C2×C42C2411yescharacteristic
255C2×C42C2411yescharacteristic
256C2×C42C2412yes—
257#107—44no—
258#106—48no—
259#106—44no—
260C4⋊C8—216no—
261#102—48no—
262#102—44no—
263#102—44no—
264#95—22no—
265#95—24no—
266#95—24no—
267#95—24no—
268#95—22no—
269#95—28no—
270#95C2×D414yes—
271#95C2×D414yes—
272C24—22no—
273C24—24no—
274C24—42no—
275C24—42no—
276C24—82no—
277C24—82no—
278C24C22×D411yescharacteristic
279C2×Q8—48no—
280C2×Q8—88no—
281C2×D4—24no—
282C2×D4—216no—
283C2×D4—22no—
284C2×D4—24no—
285C2×D4—24no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—48no—
289C2×D4C22×Q814yes—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—216no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—84no—
303C22×C4—84no—
304C22×C4—82no—
305C22×C4—84no—
306C22×C4C22×D412yes—
307C22×C4C22×D412yes—
308C22×C4C22×D412yes—
309C22×C4C22×D414yes—
310C22×C4C22×D412yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×Q812yes—
313C22×C4C2×#4112yes—
314C22×C4H(5,𝔽2)14yes—
315C22×C42−1+414yes—
316C22×C4C2511yescharacteristic
317M16—48no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—48no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—48no—
339C4⋊C4—48no—
340C4⋊C4—44no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—82no—
346C4⋊C4—84no—
347C4⋊C4—84no—
348C4⋊C4—88no—
349C4⋊C4C22⋊Q818yes—
350C4⋊C4C22×Q814yes—
351C22⋊C4—24no—
352C22⋊C4—24no—
353C22⋊C4—216no—
354C22⋊C4—44no—
355C22⋊C4—42no—
356C22⋊C4—48no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C22⋊C4—84no—
360C22⋊C4C22⋊Q818yes—
361C42—24no—
362C42—24no—
363C42—28no—
364C42—24no—
365C42—22no—
366C42—22no—
367C42—44no—
368C42—44no—
369C42—42no—
370C42—84no—
371C42C22×Q814yes—
372C42C2×#4114yes—
373C23—22no—
374C23—22no—
375C23—24no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—44no—
380C23—48no—
381C23—88no—
382C23—162no—
383C23—162no—
384C23—164no—
385C23C2×C22⋊Q812yes—
386C23#53512yes—
387C23#53612yes—
388C23C23×D411yescharacteristic
389Q8—168no—
390D4—28no—
391D4—48no—
392D4—44no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—816no—
420C2×C4—162no—
421C2×C4—164no—
422C2×C4C2×C22⋊Q818yes—
423C2×C4C2×C22⋊Q812yes—
424C2×C4#53512yes—
425C2×C4#53612yes—
426C2×C4#54218yes—
427C2×C4#54418yes—
428C2×C4#57714yes—
429C2×C4C23×Q811yescharacteristic
430C2×C4C22×#4111yescharacteristic
431C2×C4C2×H(5,𝔽2)12yes—
432C2×C4C2×2−1+412yes—
433C8—44no—
434C8—44no—
435C22—24no—
436C22—22no—
437C22—22no—
438C22—24no—
439C22—44no—
440C22—48no—
441C22—82no—
442C22—82no—
443C22—84no—
444C22—168no—
445C22#262814yes—
446C22C22×C22⋊Q811yescharacteristic
447C22C2×#53511yescharacteristic
448C22C2×#53611yescharacteristic
449C22C2×#57712yes—
450C22#358512yes—
451C4—28no—
452C4—24no—
453C4—44no—
454C4—42no—
455C4—42no—
456C4—42no—
457C4—82no—
458C4—82no—
459C4—84no—
460C4—82no—
461C4—82no—
462C4#262814yes—
463C4C2×#54212yes—
464C4C2×#54412yes—
465C2—24no—
466C2—84no—
467C2C2×#262811yescharacteristic
468C2#6049512yes—
469C2#6082011yescharacteristic
470C2#6082711yescharacteristic
471C2#6087312yes—
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. #35275C2semidirect
3. #29990C2semidirect
4. #29987C2semidirect
5. #29981C2semidirect
6. #23745C2semidirect
7. #23744C2semidirect
10. #23556C2semidirect
12. #21174C2semidirect
13. #20754C2semidirect
22. #2868C22semidirect
23. #2859C22semidirect
24. #2660C22semidirect
42. #2101C22semidirect
43. #2100C22semidirect
44. #2100C22semidirect
45. #2099C22semidirect
46. #2099C22semidirect
52. #2059C22semidirect
53. #2058C22semidirect
56. #1949C22semidirect
57. #1948C22semidirect
58. #1832C22semidirect
59. #1831C22semidirect
81. #518Q8semidirect

Maximal subgroups

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

subgroupindex
#352752
#342552
#299902
#299872
#299812
#237452
#237442
#237312
C2×#20942
#235562
#234562
#211742
#207542
#206502