Finite Group Databasebeta

← #160786

Subgroups — #160786

Order 512.

1602 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#20715C211yesmaximal, characteristic
3C4×#407C211yesmaximal, characteristic
4#7627C214yesmaximal, complemented
5#7610C214yesmaximal, complemented
6#7576C214yesmaximal, complemented
7C23×C4⋊C4—21no—
8C23×C4⋊C4C411yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#432—21no—
12C2×#407—21no—
13C2×C4×M16—21no—
14C2×#389—21no—
15C2×#389—21no—
16C2×#389—21no—
17C2×#389—21no—
18C2×#389—21no—
19C2×#389—21no—
20C2×#389C412yes—
21C2×#389C412yes—
22C2×#389C412yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C22×#95—21no—
27#1817C2212yes—
28#1797—21no—
29#1797C2212yes—
30#1779—21no—
31#1749—21no—
32#1749—21no—
33#1749C2212yes—
34#1749C2212yes—
35C4×C22⋊C8—21no—
36C4×C22⋊C8C2212yes—
37#1728C2212yes—
38#1271C22116yescomplemented
39C24×C4—21no—
40C24×C4—21no—
41C24×C4C2×C411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C22×M16—22no—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—48no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C42—44no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—44no—
80C2×C4⋊C8—22no—
81#407—28no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—22no—
87C4×M16—44no—
88C2×#97—22no—
89C2×#97—22no—
90C2×C4×C8—22no—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389—22no—
100#389—22no—
101#389—22no—
102#389—22no—
103#389C2×C414yes—
104#389C2×C414yes—
105#389C2×C414yes—
106#389C2314yes—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113#336—22no—
114#336—22no—
115#336C2314yes—
116#336C2314yes—
117C25C22⋊C411yescharacteristic
118C23×C4—24no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—28no—
122C23×C4—24no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22⋊C411yescharacteristic
146C23×C4C22⋊C411yescharacteristic
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C22×C411yescharacteristic
150C23×C4C22×C411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C2×M16—28no—
158C2×M16—44no—
159C2×M16—44no—
160C22×C8—24no—
161C22×C8—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—48no—
181C2×C4⋊C4—48no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—48no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4M1618yescomplemented
191C2×C4⋊C4C22×C414yes—
192C2×C4⋊C4C22×C414yes—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22⋊C411yescharacteristic
207C2×C42C22⋊C411yescharacteristic
208C2×C42C22⋊C411yescharacteristic
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2411yescharacteristic
221C4⋊C8—44no—
222C22⋊C8—28no—
223C22⋊C8—28no—
224C22⋊C8—44no—
225C22⋊C8—44no—
226C22⋊C8—44no—
227#97—44no—
228#97—44no—
229C4×C8—44no—
230#95—24no—
231#95—24no—
232#95—24no—
233#95—24no—
234C24—22no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24—22no—
240C24#10414yes—
241C24C2×C22⋊C411yescharacteristic
242C24C2×C22⋊C411yescharacteristic
243C24C2×C22⋊C411yescharacteristic
244C22×C4—232no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—24no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—41no—
288C22×C4—41no—
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—41no—
298C22×C4—41no—
299C22×C4—416no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—416no—
303C22×C4—432no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—44no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—44no—
314C22×C4—44no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—816no—
318C22×C4#9914yes—
319C22×C4#10114yes—
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×M1614yes—
330C22×C4AGL(1,ℤ/8)14yes—
331C22×C4#13714yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335M16—416no—
336C2×C8—28no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C2×C8—44no—
342C4⋊C4—28no—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—44no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—48no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—48no—
354C4⋊C4C2×M1618yes—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C23—28no—
372C23—28no—
373C23—28no—
374C23—28no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—28no—
388C23—28no—
389C23—28no—
390C23#40712yes—
391C23C2×#9912yes—
392C23C2×#10112yes—
393C23#41812yes—
394C23#41912yes—
395C23C2×#10412yes—
396C23C2×#10412yes—
397C23C2×#10412yes—
398C23#42112yes—
399C23C22×C22⋊C411yescharacteristic
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—28no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—416no—
444C2×C4—416no—
445C2×C4—416no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—88no—
454C2×C4—88no—
455C2×C4—88no—
456C2×C4#40712yes—
457C2×C4C2×#9912yes—
458C2×C4C2×#9912yes—
459C2×C4C2×#10112yes—
460C2×C4C2×#10112yes—
461C2×C4#41812yes—
462C2×C4#41912yes—
463C2×C4C22×M1612yes—
464C2×C4C2×AGL(1,ℤ/8)12yes—
465C2×C4C2×#13712yes—
466C8—48no—
467C8—84no—
468C22—232no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22#149114yes—
488C22#150714yes—
489C22#152814yes—
490C22C2×#40711yescharacteristic
491C22C22×#9911yescharacteristic
492C22C22×#10111yescharacteristic
493C22C2×#41811yescharacteristic
494C22C2×#41911yescharacteristic
495C22C22×#10411yescharacteristic
496C22C2×#42111yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—48no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—84no—
510C4#146314yes—
511C2—24no—
512C2—24no—
513C2—24no—
514C2C2×#146311yescharacteristic
515C2C2×#149111yescharacteristic
516C2#1744811yescharacteristic
517C2C2×#150711yescharacteristic
518C2#1757811yescharacteristic
519C2C2×#152811yescharacteristic
520C2#1774211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7627C2semidirect
5. #7610C2semidirect
6. #7576C2semidirect
38. #1271C22semidirect
190. C2×C4⋊C4M16semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
#207152
C4×#4072
#76272
#76102
#75762