Finite Group Databasebeta

← #127218

Subgroups — #127218

Order 512.

725 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#20688C211yesmaximal, characteristic
2#17357C211yesmaximal, characteristic
3#13938C211yesmaximal, characteristic
4#7213C214yesmaximal
5C22×#117C2211yescharacteristic
6#2920—21no—
7C2×#383C411yescharacteristic, complemented
8#2280—21no—
9C2×C4×C4⋊C4C411yescharacteristic, complemented
10#1820—21no—
11#1820C412yescomplemented
12#1819—21no—
13#1819C2212yes—
14#1740—21no—
15#1740C412yescomplemented
16#1464—21no—
17#1464C412yescomplemented
18#1463—21no—
19#1462C2212yes—
20#1458—21no—
21#1458C412yescomplemented
22#1441—21no—
23#1439—21no—
24#1437C2212yes—
25#1436C412yescomplemented
26#1434—21no—
27C22⋊#95—21no—
28C24×C4D411yescharacteristic
29C2×#117—22no—
30C2×#117—41no—
31C2×#117—41no—
32C2×#117—41no—
33C2×#117—41no—
34C2×#117C2×C411yescharacteristic
35C2×#117C2×C411yescharacteristic
36C2×#117C2×C411yescharacteristic
37C2×#117C2×C411yescharacteristic
38C2×#117D411yescharacteristic
39C2×#117D411yescharacteristic
40C2×#117Q811yescharacteristic
41C2×#117Q811yescharacteristic
42C22×C4⋊C4—21no—
43C22×C4⋊C4C2×C411yescharacteristic
44C22×C4⋊C4D411yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4D411yescharacteristic
47C22×C22⋊C4C2311yescharacteristic
48C22×C42C2×C411yescharacteristic
49C22×C42C2×C412yes—
50C22×C42D411yescharacteristic
51#432—22no—
52#432—41no—
53#431—22no—
54#431—41no—
55#423—41no—
56#407—22no—
57C4×M16—41no—
58#383—41no—
59#383—41no—
60#381—22no—
61#380—41no—
62#380—41no—
63#379—22no—
64C4×C4⋊C4—41no—
65C4×C4⋊C4—41no—
66C4×C22⋊C4—41no—
67C4×C22⋊C4—41no—
68C2×#95—22no—
69C2×#95—21no—
70C2×#95—21no—
71C2×#95—21no—
72C2×#95—22no—
73C2×#95—22no—
74C2×#95C2×C412yes—
75#342—22no—
76#342—22no—
77#342—22no—
78#342C2×C414yes—
79#324—22no—
80#324—22no—
81#323—22no—
82#323C2×C414yes—
83C25C2×D411yescharacteristic
84C23×C4—21no—
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—22no—
89C23×C4—22no—
90C23×C4—42no—
91C23×C4—42no—
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—42no—
103#117—41no—
104#117—41no—
105#117—41no—
106#117—41no—
107#117—41no—
108#117—41no—
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—44no—
120#117—44no—
121C2×C4⋊C4—41no—
122C2×C4⋊C4—41no—
123C2×C4⋊C4—41no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—44no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4C4211yescharacteristic
129C2×C4⋊C4C4211yescharacteristic
130C2×C4⋊C4C22⋊C411yescharacteristic
131C2×C4⋊C4C22⋊C411yescharacteristic
132C2×C4⋊C4C4⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C22⋊C4—24no—
137C2×C22⋊C4—41no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—44no—
142C2×C22⋊C4—44no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—82no—
146C2×C22⋊C4C22⋊C411yescharacteristic
147C2×C22⋊C4C22⋊C411yescharacteristic
148C2×C22⋊C4C4⋊C411yescharacteristic
149C2×C22⋊C4C4⋊C411yescharacteristic
150C2×C22⋊C4C22×C411yescharacteristic
151C2×C22⋊C4C22×C411yescharacteristic
152C2×C22⋊C4C2×D411yescharacteristic
153C2×C22⋊C4C2×Q811yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—42no—
163C2×C42—42no—
164C2×C42—82no—
165C2×C42C4211yescharacteristic
166C2×C42C4211yescharacteristic
167C2×C42C22⋊C411yescharacteristic
168C2×C42C22⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C4⋊C411yescharacteristic
171C2×C42C4⋊C411yescharacteristic
172C2×C42C4⋊C411yescharacteristic
173C4⋊C8—42no—
174C4⋊C8—42no—
175C22⋊C8—24no—
176C22⋊C8—42no—
177#97—42no—
178C4×C8—42no—
179#95—42no—
180#95—44no—
181#95—44no—
182#95—42no—
183#95—42no—
184#95—42no—
185#95—41no—
186#95—41no—
187#95—41no—
188#95—41no—
189#95—82no—
190C24—22no—
191C24—21no—
192C24—21no—
193C24—41no—
194C24—42no—
195C24—41no—
196C24—41no—
197C24C2×C22⋊C411yescharacteristic
198C24C2×C22⋊C411yescharacteristic
199C24C2×C4⋊C411yescharacteristic
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—24no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4#9511yescharacteristic
256C22×C4#9511yescharacteristic
257C22×C4#9511yescharacteristic
258C22×C4#9511yescharacteristic
259C22×C4#9511yescharacteristic
260C22×C4#9511yescharacteristic
261C22×C4#9511yescharacteristic
262C22×C4#9511yescharacteristic
263C22×C4#9911yescharacteristic
264C22×C4#9911yescharacteristic
265C22×C4#9911yescharacteristic
266C22×C4#9911yescharacteristic
267C22×C4C2×C4211yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C4⋊C411yescharacteristic
270C22×C4C2×C4⋊C411yescharacteristic
271M16—84no—
272C2×C8—44no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—48no—
282C4⋊C4—84no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—44no—
292C22⋊C4—44no—
293C22⋊C4—88no—
294C22⋊C4—82no—
295C22⋊C4—82no—
296C22⋊C4—84no—
297C42—21no—
298C42—21no—
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—84no—
314C42—84no—
315C23—21no—
316C23—21no—
317C23—22no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—42no—
326C23—42no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—44no—
332C23—44no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—81no—
337C23—81no—
338C23—82no—
339C23—82no—
340C23C2×#9511yescharacteristic
341C23C2×#9911yescharacteristic
342C23C2×#9911yescharacteristic
343C23#42112yes—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—81no—
395C2×C4—81no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—84no—
408C2×C4—84no—
409C2×C4—82no—
410C2×C4H(3,ℤ/4)12yes—
411C2×C4#34211yescharacteristic
412C2×C4#34211yescharacteristic
413C2×C4#34211yescharacteristic
414C2×C4#34211yescharacteristic
415C8—84no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—21no—
424C22—21no—
425C22—21no—
426C22—22no—
427C22—42no—
428C22—42no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—81no—
435C22—82no—
436C22—84no—
437C22—82no—
438C22—81no—
439C22C2×H(3,ℤ/4)11yescharacteristic
440C22#173311yescharacteristic
441C22C2×#34211yescharacteristic
442C22#212211yescharacteristic
443C22#212311yescharacteristic
444C22#212612yes—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—41no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—84no—
460C4—162no—
461C4—164no—
462C2—21no—
463C2—22no—
464C2—21no—
465C2—21no—
466C2—21no—
467C2—42no—
468C2—81no—
469C2#794311yescharacteristic
470C2#1257511yescharacteristic
471C2#1258111yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC22×C22⋊C4C23
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ C22×#117 ◁ G
LECSG ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. C2×#383C4semidirect
9. C2×C4×C4⋊C4C4semidirect
11. #1820C4semidirect
15. #1740C4semidirect
17. #1464C4semidirect
21. #1458C4semidirect
25. #1436C4semidirect

Maximal subgroups

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

subgroupindex
#206882
#173572
#139382
#72132