Finite Group Databasebeta

← #243040

Subgroups — #243040

Order 512.

748 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20739C211yesmaximal, characteristic
2#20726C211yesmaximal, characteristic
3#17816C211yesmaximal, characteristic, complemented
4C2×#1478C211yesmaximal, characteristic
5#10642C212yesmaximal, complemented
6#10641C212yesmaximal, complemented
7#10638C212yesmaximal, complemented
8#10637C212yesmaximal, complemented
9#8080C211yesmaximal, characteristic, complemented
10#7958C211yesmaximal, characteristic, complemented
11#7956C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13#2920C2211yescharacteristic
14#2920C2211yescharacteristic
15C2×#432C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C2×#431C2211yescharacteristic
18#2283C2211yescharacteristic
19#1823—21no—
20#1821—21no—
21#1804C2212yes—
22#1804C2212yes—
23#1800C2212yes—
24#1800C2212yes—
25#1786—21no—
26#1783—21no—
27#1762—22no—
28C4×#100—22no—
29#1572—22no—
30#1571C2211yescharacteristic, complemented
31#1571C2211yescharacteristic, complemented
32#1570C2211yescharacteristic, complemented
33#1570C2211yescharacteristic, complemented
34#1568—22no—
35#1478—22no—
36#1478C2212yes—
37#1478C2212yes—
38C2×D4⋊C8—22no—
39#1301C2212yescomplemented
40#1301C2212yescomplemented
41#1286C2212yescomplemented
42#1286C2212yescomplemented
43#1286C2212yescomplemented
44#1286C2212yescomplemented
45C23×D4D411yescharacteristic
46C24×C4—21no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4—21no—
50C2×C4×D4—21no—
51C2×C4×D4—21no—
52C2×C4×D4—21no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—21no—
57C2×C4×D4D412yes—
58C2×C4×D4C2312yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C42C2311yescharacteristic
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432—21no—
68#432—21no—
69#432—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431D412yes—
80#431C2312yes—
81#431C2311yescharacteristic
82#431C2311yescharacteristic
83#423C2312yes—
84#423C2311yescharacteristic
85#423C2311yescharacteristic
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—22no—
91C2×#102—22no—
92C2×#100—22no—
93C2×#100—22no—
94#407—21no—
95#407—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C4×M16—21no—
100C4×M16—21no—
101C2×#97—21no—
102C2×C4×C8—21no—
103C2×C4×C8D412yes—
104#383—21no—
105#382C2312yes—
106#381—21no—
107#380—21no—
108C4×C4⋊C4—21no—
109C4×C22⋊C4—21no—
110#376—21no—
111#343—22no—
112#343—22no—
113#340—22no—
114#340—22no—
115#335—22no—
116#335—22no—
117#334—22no—
118#334—22no—
119D4⋊C8—24no—
120D4⋊C8—24no—
121C25—21no—
122C22×D4—24no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—42no—
129C22×D4C2×D412yes—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—41no—
134C23×C4—41no—
135C23×C4—41no—
136C23×C4—41no—
137C23×C4—84no—
138C23×C4C2×D411yescharacteristic
139C23×C4#4111yescharacteristic
140C23×C4#4111yescharacteristic
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—41no—
148C4×D4—22no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—22no—
154C4×D4—22no—
155C4×D4—22no—
156C4×D4—41no—
157C4×D4—41no—
158C4×D4—41no—
159C4×D4—41no—
160C4×D4—41no—
161C4×D4—41no—
162C4×D4—41no—
163C4×D4—41no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4C2×D412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—41no—
174C2×C22⋊C4—41no—
175C2×C22⋊C4—41no—
176C2×C22⋊C4—41no—
177C2×C22⋊C4—41no—
178C2×C42—41no—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×D411yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C2×C42#4111yescharacteristic
186C2×C42#4111yescharacteristic
187C2×C42#4111yescharacteristic
188C2×C42#4111yescharacteristic
189C2×C42#4111yescharacteristic
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—21no—
194C4⋊C8—21no—
195C4⋊C8—21no—
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—41no—
206#102—44no—
207#102—44no—
208#100—44no—
209#100—44no—
210C22⋊C8—22no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—22no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217C22⋊C8—41no—
218C22⋊C8—41no—
219C22⋊C8—41no—
220C22⋊C8—41no—
221#97—41no—
222#97—41no—
223#97—41no—
224#97—41no—
225C4×C8—22no—
226C4×C8—22no—
227C4×C8—41no—
228C4×C8—41no—
229C4×C8—41no—
230C4×C8C2×D412yes—
231C4×C8C2×D412yes—
232#95—22no—
233#95—22no—
234#95—41no—
235#95—41no—
236C24—22no—
237C24—44no—
238C24—41no—
239C24—41no—
240C24—82no—
241C24—82no—
242C24#12311yescharacteristic
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—44no—
251C2×D4—44no—
252C2×D4—44no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4#13514yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—84no—
270C22×C4—84no—
271C22×C4—81no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4—82no—
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4#12311yescharacteristic
282C22×C4#12311yescharacteristic
283C22×C4#12311yescharacteristic
284C22×C4#12311yescharacteristic
285C22×C4#12311yescharacteristic
286C22×C4#12411yescharacteristic
287C22×C4#12411yescharacteristic
288C22×C4#12611yescharacteristic
289C22×C4#12611yescharacteristic
290C22×C4#12611yescharacteristic
291C22×C4#12611yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295M16—42no—
296M16—42no—
297M16—42no—
298M16—42no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—82no—
310C2×C8—82no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—82no—
322C42—82no—
323C42#12111yescharacteristic
324C42#12111yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C42#12411yescharacteristic
330C42#12411yescharacteristic
331C42#12611yescharacteristic
332C42#12611yescharacteristic
333C42#12611yescharacteristic
334C42#12611yescharacteristic
335C42C22×D411yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C42C2×#4111yescharacteristic
339C23—22no—
340C23—22no—
341C23—24no—
342C23—41no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—82no—
348C23—82no—
349C23—82no—
350C23—82no—
351C23—82no—
352C23—816no—
353C23—82no—
354C23#39711yescharacteristic
355C23#39711yescharacteristic
356C23C2×#12311yescharacteristic
357D4—24no—
358D4—24no—
359D4—44no—
360D4—44no—
361D4—42no—
362D4—42no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4#39711yescharacteristic
394C2×C4#39711yescharacteristic
395C2×C4#39711yescharacteristic
396C2×C4#39711yescharacteristic
397C2×C4#39711yescharacteristic
398C2×C4#39711yescharacteristic
399C2×C4C2×#12111yescharacteristic
400C2×C4C2×#12311yescharacteristic
401C2×C4C2×#12311yescharacteristic
402C2×C4C2×#12411yescharacteristic
403C2×C4C2×#12611yescharacteristic
404C2×C4C2×#12611yescharacteristic
405C2×C4C2×#13512yes—
406C2×C4#57512yes—
407C8—42no—
408C8—42no—
409C8—82no—
410C8—82no—
411C8—81no—
412C8—81no—
413C8—81no—
414C8—81no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—44no—
423C22—44no—
424C22—84no—
425C22—84no—
426C22—88no—
427C22—88no—
428C22—82no—
429C22—82no—
430C22#208112yes—
431C22C2×#39711yescharacteristic
432C22#309112yes—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—21no—
442C4—21no—
443C4—82no—
444C4—162no—
445C4#305011yescharacteristic
446C4#305411yescharacteristic
447C4#309112yes—
448C2—22no—
449C2—22no—
450C2—41no—
451C2—84no—
452C2#2363811yescharacteristic
453C2#2364911yescharacteristic
454C2C2×#208111yescharacteristic
subgroupG/H
center, socleC22C2×#397
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #17816C2semidirect
5. #10642C2semidirect
6. #10641C2semidirect
7. #10638C2semidirect
8. #10637C2semidirect
9. #8080C2semidirect
10. #7958C2semidirect
11. #7956C2semidirect
30. #1571C22semidirect
31. #1571C22semidirect
32. #1570C22semidirect
33. #1570C22semidirect
39. #1301C22semidirect
40. #1301C22semidirect
41. #1286C22semidirect
42. #1286C22semidirect
43. #1286C22semidirect
44. #1286C22semidirect

Maximal subgroups

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

subgroupindex
#207392
#207262
#178162
C2×#14782
#106422
#106412
#106382
#106372
#80802
#79582
#79562