Finite Group Databasebeta

← #160977

Subgroups — #160977

Order 512.

1586 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#389C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#20626C211yesmaximal, characteristic
4#7641C214yesmaximal, complemented
5#7610C214yesmaximal, complemented
6#7573C214yesmaximal, complemented
7C23×C4⋊C4—21no—
8C23×C4⋊C4C411yescharacteristic
9C23×C4⋊C4C411yescharacteristic
10C23×C42C2211yescharacteristic
11C2×#431—21no—
12C2×#423—21no—
13C2×#407—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#1797—21no—
30#1797—21no—
31#1797C2212yes—
32#1797C2212yes—
33#1797C2212yes—
34#1779—21no—
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—44no—
59C22×C4⋊C4—44no—
60C22×C4⋊C4—44no—
61C22×C4⋊C4C2×C412yes—
62C22×C4⋊C4C2×C412yes—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C4⋊C4C2×C412yes—
66C22×C4⋊C4C2×C412yes—
67C22×C42—44no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#431—44no—
81#423—44no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—44no—
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—22no—
91C2×C4×C8—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#389—22no—
104#389C2×C414yes—
105#389C2×C414yes—
106#389C2×C414yes—
107#389C2314yes—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95—22no—
114#336—22no—
115#336—22no—
116#336C2314yes—
117#336C2314yes—
118C25C22⋊C411yescharacteristic
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—24no—
122C23×C4—28no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—42no—
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×C4D814yes—
149C23×C4SD1614yes—
150C23×C4C22×C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C22×C411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C2×M16—44no—
160C2×M16—44no—
161C2×M16—44no—
162C22×C8—24no—
163C22×C8—24no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—48no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4M1618yescomplemented
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C22×C414yes—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—42no—
201C2×C42—42no—
202C2×C42—42no—
203C2×C42—42no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—44no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C22⋊C8—44no—
230C22⋊C8—44no—
231C22⋊C8—44no—
232C22⋊C8—44no—
233C22⋊C8—44no—
234C4×C8—44no—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239C24—22no—
240C24—22no—
241C24—22no—
242C24—22no—
243C24—22no—
244C24—22no—
245C24#10214yes—
246C24C2×C22⋊C411yescharacteristic
247C24C2×C22⋊C411yescharacteristic
248C24C2×C22⋊C411yescharacteristic
249C22×C4—232no—
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—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—28no—
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—24no—
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—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—416no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—432no—
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—44no—
318C22×C4—816no—
319C22×C4#9914yes—
320C22×C4#10114yes—
321C22×C4#10214yes—
322C22×C4#10214yes—
323C22×C4#10214yes—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C2×C22⋊C411yescharacteristic
329C22×C4C2×C22⋊C411yescharacteristic
330C22×C4C2×C22⋊C411yescharacteristic
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4C2×M1614yes—
334C22×C4C2×D812yes—
335C22×C4C2×D812yes—
336C22×C4C2×D812yes—
337C22×C4C2×SD1612yes—
338C22×C4C2×SD1612yes—
339C22×C4C2×SD1612yes—
340C22×C4#13714yes—
341C22×C4C23×C411yescharacteristic
342C22×C4C22×D411yescharacteristic
343C22×C4C22×D411yescharacteristic
344M16—88no—
345C2×C8—44no—
346C2×C8—44no—
347C2×C8—44no—
348C2×C8—44no—
349C2×C8—44no—
350C2×C8—44no—
351C4⋊C4—28no—
352C4⋊C4—28no—
353C4⋊C4—24no—
354C4⋊C4—28no—
355C4⋊C4—44no—
356C4⋊C4—44no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—44no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4C2×M1618yes—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—22no—
371C42—22no—
372C42—22no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C23—28no—
382C23—28no—
383C23—28no—
384C23—28no—
385C23—28no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—28no—
399C23—28no—
400C23#40712yes—
401C23C2×#9912yes—
402C23C2×#10112yes—
403C23C2×#10212yes—
404C23C2×#10212yes—
405C23C2×#10212yes—
406C23#41912yes—
407C23#42112yes—
408C23#42112yes—
409C23C22×C22⋊C411yescharacteristic
410C2×C4—28no—
411C2×C4—28no—
412C2×C4—28no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
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—24no—
432C2×C4—28no—
433C2×C4—24no—
434C2×C4—28no—
435C2×C4—28no—
436C2×C4—28no—
437C2×C4—28no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—416no—
456C2×C4—416no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—416no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—88no—
465C2×C4—88no—
466C2×C4—88no—
467C2×C4#40712yes—
468C2×C4C2×#9912yes—
469C2×C4C2×#9912yes—
470C2×C4C2×#10112yes—
471C2×C4C2×#10112yes—
472C2×C4C2×#10212yes—
473C2×C4C2×#10212yes—
474C2×C4C2×#10212yes—
475C2×C4#41912yes—
476C2×C4C22×M1612yes—
477C2×C4C22×D811yescharacteristic
478C2×C4C22×SD1611yescharacteristic
479C2×C4C2×#13712yes—
480C8—84no—
481C8—84no—
482C22—232no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22—22no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22#148614yes—
502C22#151914yes—
503C22#154614yes—
504C22C2×#40711yescharacteristic
505C22C22×#9911yescharacteristic
506C22C22×#10111yescharacteristic
507C22C22×#10211yescharacteristic
508C22C2×#41911yescharacteristic
509C22C2×#42111yescharacteristic
510C22C2×#42111yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—22no—
516C4—22no—
517C4—22no—
518C4—22no—
519C4—22no—
520C4—22no—
521C4—48no—
522C4—41no—
523C4—41no—
524C4—84no—
525C4#146314yes—
526C2—24no—
527C2—24no—
528C2—24no—
529C2C2×#146311yescharacteristic
530C2C2×#148611yescharacteristic
531C2#1744911yescharacteristic
532C2C2×#151911yescharacteristic
533C2#1757811yescharacteristic
534C2#1766711yescharacteristic
535C2C2×#154611yescharacteristic
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. #7641C2semidirect
5. #7610C2semidirect
6. #7573C2semidirect
38. #1271C22semidirect
195. 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
#207142
#206262
#76412
#76102
#75732