Finite Group Databasebeta

← #357202

Subgroups — #357202

Order 512.

1364 conjugacy classes of subgroups.

The subgroup lattice has 449 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1958C211yesmaximal, characteristic
2C2×#1888C211yesmaximal, characteristic
3C2×#1885C211yesmaximal, characteristic
4#20417C212yesmaximal, complemented
5#20416C212yesmaximal
6#10943C212yesmaximal
7#10942C212yesmaximal, complemented
8#10892C212yesmaximal, complemented
9#10706C212yesmaximal, complemented
10C22×#124C2211yescharacteristic
11C2×#519C2211yescharacteristic
12C2×#518C2211yescharacteristic
13C2×#419C2212yes—
14C2×#418C2212yes—
15C22×#103C2212yes—
16C22×#102C2212yes—
17C22×#101C2211yescharacteristic
18C22×#100C2211yescharacteristic
19C2×#405C2212yes—
20C2×#405C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#382C2211yescharacteristic
23#2274C2212yes—
24#1958C418yescomplemented
25#1888C418yescomplemented
26#1885C418yescomplemented
27#1763—24no—
28#1762—24no—
29C4×#103—24no—
30C4×#102—24no—
31C2×#343C2212yes—
32C2×#341C2212yes—
33#1736C2212yes—
34#1736C2212yes—
35#1735C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C23×Q8C2311yescharacteristic
39C23×D4C2311yescharacteristic
40C22×M16C2312yes—
41C22×M16C2312yes—
42C23×C8C2312yes—
43C2×#124—41no—
44C2×#124—41no—
45C2×#124—41no—
46C2×#124C2×C414yes—
47C2×#124C2×C414yes—
48C2×#124C2×C414yes—
49#519—24no—
50#519—41no—
51#519C2×C414yescomplemented
52#518—24no—
53#518—41no—
54#518C2×C414yes—
55C2×C4×Q8—24no—
56C2×C4×D4—24no—
57C2×#117—41no—
58C2×#117C2312yes—
59C2×#117C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C22⋊C4—41no—
62C22×C22⋊C4—41no—
63C22×C42C2311yescharacteristic
64#419—24no—
65#419C2×C418yes—
66#418—24no—
67#418C2×C418yes—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103C2×C418yescomplemented
73C2×#102—22no—
74C2×#102—22no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102C2×C418yes—
78C2×#101—41no—
79C2×#101—41no—
80C2×#101C2×C414yescomplemented
81C2×#100—41no—
82C2×#100—41no—
83C2×#100C2×C414yes—
84#405—24no—
85#405—24no—
86#405—42no—
87#405—42no—
88C2×#97—22no—
89C2×#97—22no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8—41no—
93C2×C4×C8—41no—
94C2×C4×C8C2×C414yescomplemented
95#382—41no—
96#382C2×C414yescomplemented
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C22⋊C4—42no—
100C2×#95—41no—
101#343—42no—
102#341—24no—
103#340—24no—
104#340—24no—
105#340D414yes—
106#340D414yes—
107#339C2×C418yes—
108#339C2×C418yescomplemented
109C25—21no—
110C22×Q8—21no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8C22×C412yes—
118C22×Q8C22×C412yes—
119C22×Q8C22×C412yes—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4C22×C412yes—
128C22×D4C22×C412yes—
129C22×D4C22×C412yes—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4C2411yescharacteristic
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16C22×C414yes—
146C2×M16C22×C414yes—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8C22×C414yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8#4114yes—
157#124—44no—
158#124—44no—
159#124—44no—
160#124C42116yescomplemented
161C4×Q8—216no—
162C4×Q8—48no—
163C4×D4—216no—
164C4×D4—48no—
165#117—22no—
166#117—22no—
167#117—22no—
168#117—22no—
169#117—44no—
170#117—44no—
171#117C22×C414yes—
172#117C22×C414yes—
173#117#4114yes—
174#117#4114yes—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C22×C414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—42no—
198C2×C42—44no—
199C2×C42C22×C412yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202#103—28no—
203#103—28no—
204#103—216no—
205#102—28no—
206#102—28no—
207#102—216no—
208#101—44no—
209#100—44no—
210#97—44no—
211#97—44no—
212C4×C8—44no—
213C4×C8—44no—
214C4×C8—44no—
215#95—44no—
216#95—44no—
217C24—22no—
218C24—22no—
219C24—22no—
220C24—41no—
221C24—41no—
222C24—41no—
223C24—41no—
224C24—84no—
225C24H(5,𝔽2)11yescharacteristic
226C2×Q8—22no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8C2×C4214yes—
242C2×D4—24no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4C2×C4214yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—84no—
276C22×C4—164no—
277C22×C4C23×C411yescharacteristic
278C22×C4C23×C411yescharacteristic
279C22×C4C23×C411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C2×#4112yes—
282C22×C4C2×#4112yes—
283C22×C4C2×#4112yes—
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286C22×C4H(5,𝔽2)11yescharacteristic
287M16—24no—
288M16—24no—
289M16—24no—
290M16—24no—
291M16—42no—
292M16—42no—
293M16—42no—
294M16—42no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8C4×D414yes—
310C2×C8C4×D414yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—48no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4C4×D418yes—
319C4⋊C4C4×D418yes—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—416no—
325C22⋊C4—48no—
326C42—24no—
327C42—24no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—24no—
332C42—24no—
333C42—48no—
334C42—164no—
335C42C2×C4214yes—
336C23—24no—
337C23—24no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—84no—
345C23—84no—
346C23—84no—
347C23—84no—
348C23—162no—
349C23—162no—
350C23—162no—
351C23#51811yescharacteristic
352C23#51811yescharacteristic
353C23#51811yescharacteristic
354C23D4211yescharacteristic
355C23#55111yescharacteristic
356C23#55111yescharacteristic
357C23#55911yescharacteristic
358Q8—28no—
359Q8—44no—
360Q8—44no—
361Q8—44no—
362D4—28no—
363D4—44no—
364D4—44no—
365D4—44no—
366C2×C4—28no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—168no—
387C2×C4—162no—
388C2×C4C22×C4211yescharacteristic
389C2×C4C2×C4×D412yes—
390C2×C4C2×C4×D412yes—
391C2×C4C2×C4×D412yes—
392C2×C4C4×#4112yes—
393C2×C4C4×#4112yes—
394C2×C4C4×#4112yes—
395C2×C4#51811yescharacteristic
396C2×C4#51811yescharacteristic
397C2×C4#51811yescharacteristic
398C2×C4#54612yes—
399C2×C4#54612yes—
400C2×C4#55012yes—
401C2×C4#55312yes—
402C2×C4#55312yes—
403C2×C4#55512yes—
404C8—24no—
405C8—24no—
406C8—44no—
407C8—44no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—22no—
412C22—22no—
413C22—22no—
414C22—44no—
415C22—84no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—162no—
420C22—162no—
421C22—162no—
422C22#227311yescharacteristic
423C22#235811yescharacteristic
424C22#235811yescharacteristic
425C22#236611yescharacteristic
426C22#237211yescharacteristic
427C22#237411yescharacteristic
428C22#237411yescharacteristic
429C22#329014yes—
430C4—44no—
431C4—44no—
432C4—48no—
433C4—48no—
434C4—84no—
435C4C42×D412yes—
436C4#236312yes—
437C4#236312yes—
438C4#236412yes—
439C2—21no—
440C2—21no—
441C2—21no—
442C2—21no—
443C2—84no—
444C2#1382411yescharacteristic
445C2#2289512yes—
446C2#2289612yes—
447C2#2319912yes—
subgroupG/H
center, socleC23D42
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #20417C2semidirect
7. #10942C2semidirect
8. #10892C2semidirect
9. #10706C2semidirect
24. #1958C4semidirect
25. #1888C4semidirect
26. #1885C4semidirect
51. #519C2×C4semidirect
72. C2×#103C2×C4semidirect
80. C2×#101C2×C4semidirect
94. C2×C4×C8C2×C4semidirect
96. #382C2×C4semidirect
108. #339C2×C4semidirect
160. #124C42semidirect

Maximal subgroups

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

subgroupindex
C2×#19582
C2×#18882
C2×#18852
#204172
#204162
#109432
#109422
#108922
#107062