Finite Group Databasebeta

← #127609

Subgroups — #127609

Order 512.

826 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20717C211yesmaximal, characteristic
2C2×#1740C211yesmaximal, characteristic
3#17357C211yesmaximal, characteristic
4#7252C214yesmaximal, complemented
5C22×#117—21no—
6C22×#117C2211yescharacteristic
7#2920—21no—
8C2×#432C411yescharacteristic, complemented
9C2×#431C411yescharacteristic, complemented
10#1817—21no—
11#1798C412yescomplemented
12#1798C2212yes—
13#1779—21no—
14#1750—21no—
15#1750—21no—
16#1740—21no—
17#1740—21no—
18#1740C412yes—
19#1740C2212yes—
20C2×#342—21no—
21C2×#342C412yes—
22#1464—21no—
23#1464C412yescomplemented
24#1463C412yescomplemented
25#1462—21no—
26#1458—21no—
27#1458C2212yes—
28C24×C4D411yescharacteristic
29C2×#117—28no—
30C2×#117—22no—
31C2×#117—22no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117—41no—
35C2×#117—41no—
36C2×#117—41no—
37C2×#117—41no—
38C2×#117—44no—
39C2×#117C2×C411yescharacteristic
40C2×#117C2×C411yescharacteristic
41C2×#117C2×C411yescharacteristic
42C2×#117D411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117D411yescharacteristic
45C2×#117Q811yescharacteristic
46C2×#117C2311yescharacteristic
47C22×C4⋊C4—41no—
48C22×C4⋊C4C2×C411yescharacteristic
49C22×C4⋊C4Q811yescharacteristic
50C22×C22⋊C4—22no—
51C22×C22⋊C4C2×C411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—41no—
54C22×C42C2×C411yescharacteristic
55C22×C42D411yescharacteristic
56#432—22no—
57#432—41no—
58#432—41no—
59#432—41no—
60#431—22no—
61#431—41no—
62#431—41no—
63#431—41no—
64#423—41no—
65C2×C4⋊C8C2×C412yescomplemented
66C2×C4⋊C8C2×C412yescomplemented
67#407—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8—22no—
72C4×M16—41no—
73C2×#97—21no—
74C2×C4×C8—21no—
75#342—22no—
76#342—22no—
77#342—22no—
78#342C2×C414yes—
79#336—22no—
80#336—22no—
81#324—22no—
82#324C2×C414yescomplemented
83#323—22no—
84#323—22no—
85C25C22⋊C411yescharacteristic
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—24no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—42no—
94C23×C4C4211yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C4⋊C411yescharacteristic
102C23×C4C4⋊C411yescharacteristic
103C23×C4C2×D411yescharacteristic
104C2×M16—42no—
105C22×C8—22no—
106C22×C8—22no—
107#117—28no—
108#117—28no—
109#117—41no—
110#117—41no—
111#117—41no—
112#117—41no—
113#117—41no—
114#117—41no—
115#117—41no—
116#117—41no—
117#117—41no—
118#117—41no—
119#117—41no—
120#117—41no—
121#117—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—48no—
126#117—48no—
127#117—44no—
128#117—44no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—48no—
134C2×C4⋊C4—42no—
135C2×C4⋊C4—42no—
136C2×C4⋊C4C4211yescharacteristic
137C2×C4⋊C4C22⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C4⋊C411yescharacteristic
140C2×C4⋊C4C4⋊C411yescharacteristic
141C2×C4⋊C4C4⋊C411yescharacteristic
142C2×C4⋊C4C22×C411yescharacteristic
143C2×C4⋊C4C2×Q811yescharacteristic
144C2×C22⋊C4—28no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C22⋊C4—41no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—44no—
152C2×C22⋊C4C4211yescharacteristic
153C2×C22⋊C4C22⋊C411yescharacteristic
154C2×C22⋊C4C22⋊C411yescharacteristic
155C2×C22⋊C4C22⋊C411yescharacteristic
156C2×C22⋊C4C4⋊C411yescharacteristic
157C2×C22⋊C4C4⋊C411yescharacteristic
158C2×C22⋊C4C22×C411yescharacteristic
159C2×C22⋊C4C2×D411yescharacteristic
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42—48no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42C4211yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C22⋊C411yescharacteristic
174C2×C42C22⋊C411yescharacteristic
175C2×C42C4⋊C411yescharacteristic
176C2×C42C4⋊C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—42no—
183C22⋊C8—24no—
184C22⋊C8—42no—
185C22⋊C8—41no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—44no—
190C22⋊C8—42no—
191C22⋊C8—42no—
192#97—42no—
193#97—42no—
194C4×C8—42no—
195C4×C8—42no—
196C24—21no—
197C24—22no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—42no—
202C24—81no—
203C24#9511yescharacteristic
204C24#9511yescharacteristic
205C24C2×C22⋊C411yescharacteristic
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—44no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4—84no—
250C22×C4—84no—
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9511yescharacteristic
255C22×C4#9511yescharacteristic
256C22×C4#9511yescharacteristic
257C22×C4#9911yescharacteristic
258C22×C4#9911yescharacteristic
259C22×C4#9911yescharacteristic
260C22×C4#9911yescharacteristic
261C22×C4#10414yes—
262C22×C4C2×C4211yescharacteristic
263C22×C4C2×C22⋊C411yescharacteristic
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C4⋊C411yescharacteristic
266C22×C4C2×C4⋊C411yescharacteristic
267C22×C4C2×C4⋊C411yescharacteristic
268C22×C4AGL(1,ℤ/8)12yes—
269C22×C4#13712yes—
270M16—84no—
271C2×C8—44no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C22⋊C4—28no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—48no—
298C22⋊C4—44no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—21no—
310C42—21no—
311C42—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—48no—
316C42—48no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—41no—
329C23—42no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23—84no—
338C23—82no—
339C23—82no—
340C23#33912yes—
341C23#34211yescharacteristic
342C23#34211yescharacteristic
343C23C2×#9511yescharacteristic
344C23#41811yescharacteristic
345C23#41911yescharacteristic
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—48no—
391C2×C4—81no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—816no—
396C2×C4—84no—
397C2×C4—84no—
398C2×C4#33912yes—
399C2×C4#34211yescharacteristic
400C2×C4#34211yescharacteristic
401C2×C4C2×#9911yescharacteristic
402C2×C4C2×#9911yescharacteristic
403C2×C4#41712yes—
404C2×C4C2×#10412yes—
405C2×C4#42812yes—
406C8—84no—
407C8—82no—
408C8—82no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—22no—
417C22—21no—
418C22—21no—
419C22—42no—
420C22—42no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—44no—
425C22—44no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22—81no—
432C22—82no—
433C22—82no—
434C22#139512yes—
435C22C2×#33911yescharacteristic
436C22#173611yescharacteristic
437C22C2×#34211yescharacteristic
438C4—22no—
439C4—21no—
440C4—21no—
441C4—22no—
442C4—41no—
443C4—41no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—88no—
451C4#150412yes—
452C4#150412yes—
453C2—22no—
454C2—21no—
455C2—21no—
456C2—41no—
457C2—41no—
458C2—42no—
459C2—41no—
460C2#802011yescharacteristic
461C2#1257711yescharacteristic
462C2C2×#139511yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7252C2semidirect
8. C2×#432C4semidirect
9. C2×#431C4semidirect
11. #1798C4semidirect
23. #1464C4semidirect
24. #1463C4semidirect
65. C2×C4⋊C8C2×C4semidirect
66. C2×C4⋊C8C2×C4semidirect
82. #324C2×C4semidirect

Maximal subgroups

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

subgroupindex
#207172
C2×#17402
#173572
#72522