Finite Group Databasebeta

← #160686

Subgroups — #160686

Order 512.

1314 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2293C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3C4×#423C211yesmaximal, characteristic, complemented
4#7614C214yesmaximal, complemented
5#7610C212yesmaximal, complemented
6#7606C212yesmaximal
7#7560C214yesmaximal, complemented
8C22×C4×D4C411yescharacteristic
9C22×#117—21no—
10C2×#423—21no—
11C2×C4×M16—21no—
12#2293—21no—
13#2293—21no—
14#2293—21no—
15#2293—21no—
16#2293C412yes—
17#2293C412yes—
18#2293C2212yes—
19#2293C2212yes—
20C2×#390C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C2211yescharacteristic
24C2×#388—21no—
25C2×#388—21no—
26C2×C4×C4⋊C4C411yescharacteristic
27C2×C4×C22⋊C4—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1831C2212yescomplemented
31#1817C2212yes—
32#1797—21no—
33#1767—21no—
34#1767C2212yescomplemented
35C4×C4⋊C8—21no—
36C4×C4⋊C8—21no—
37C4×C4⋊C8C2212yescomplemented
38C4×C22⋊C8—21no—
39#1744C2212yescomplemented
40#1728C2212yes—
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C24×C4—21no—
45C22×M16—22no—
46C2×C4×D4—28no—
47C2×C4×D4—21no—
48C2×C4×D4—21no—
49C2×C4×D4C2×C412yes—
50C2×C4×D4C2×C412yes—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—44no—
56C2×#117—44no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42C2311yescharacteristic
75#423—28no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C22⋊C8—22no—
80C4×M16—44no—
81C2×#97—22no—
82C2×C4×C8—22no—
83C2×C4×C8—22no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390—21no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2312yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389C2×C412yes—
97#389C2×C412yes—
98#389C2×C412yes—
99#389C2312yes—
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131#336C2314yes—
132#336C2314yes—
133C25—21no—
134C22×D4—24no—
135C22×D4—21no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4C22×C412yes—
139C23×C4—28no—
140C23×C4—21no—
141C23×C4—22no—
142C23×C4—22no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—44no—
146C23×C4—44no—
147C23×C4C22⋊C411yescharacteristic
148C23×C4C22⋊C411yescharacteristic
149C23×C4C22⋊C411yescharacteristic
150C23×C4C22⋊C411yescharacteristic
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C2×M16—28no—
155C2×M16—44no—
156C2×M16—44no—
157C22×C8—24no—
158C22×C8—24no—
159C4×D4—28no—
160C4×D4—28no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4M1618yescomplemented
165#117—48no—
166#117—48no—
167#117—44no—
168#117—44no—
169#117—44no—
170#117—44no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C22×C412yes—
185C2×C4⋊C4C22×C412yes—
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—44no—
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42D814yes—
225C2×C42SD1614yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42C2×D411yescharacteristic
241C2×C42C2411yescharacteristic
242C4⋊C8—28no—
243C4⋊C8—28no—
244C4⋊C8—44no—
245C22⋊C8—44no—
246#97—44no—
247C4×C8—44no—
248C4×C8—44no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255C24—24no—
256C24—22no—
257C24—42no—
258C24—42no—
259C24—42no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24C2×C22⋊C411yescharacteristic
264C2×D4—24no—
265C2×D4—22no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—416no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4C2×M1614yes—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—28no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—44no—
296C22×C4—432no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4#10214yes—
309C22×C4#10214yes—
310C22×C4#10414yes—
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C411yescharacteristic
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×D812yes—
323C22×C4C2×SD1612yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327M16—416no—
328C2×C8—28no—
329C2×C8—44no—
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—44no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4C2×M1614yes—
346C22⋊C4—28no—
347C22⋊C4—28no—
348C22⋊C4—44no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—44no—
352C22⋊C4—44no—
353C22⋊C4—44no—
354C22⋊C4—44no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—44no—
381C42—44no—
382C42#10214yes—
383C42#10214yes—
384C42C2×M1614yes—
385C42C2×D812yes—
386C42C2×D812yes—
387C42C2×SD1612yes—
388C42C2×SD1612yes—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—24no—
394C23—48no—
395C23—48no—
396C23—48no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—84no—
404C23—84no—
405C23—84no—
406C23—84no—
407C23#41012yes—
408C23C2×#10212yes—
409C23C2×#10412yes—
410C23C22×C22⋊C411yescharacteristic
411D4—44no—
412D4—44no—
413D4—48no—
414C2×C4—28no—
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—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—28no—
440C2×C4—21no—
441C2×C4—24no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—416no—
452C2×C4—48no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—416no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4#40712yes—
465C2×C4#40712yes—
466C2×C4#41312yes—
467C2×C4C2×#10212yes—
468C2×C4C2×#10212yes—
469C2×C4C2×#10212yes—
470C2×C4C2×#10212yes—
471C2×C4C2×#10212yes—
472C2×C4C2×#10412yes—
473C2×C4C2×#10412yes—
474C2×C4#42112yes—
475C2×C4C22×M1612yes—
476C2×C4C22×D811yescharacteristic
477C2×C4C22×SD1611yescharacteristic
478C2×C4#57512yes—
479C8—48no—
480C8—84no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—44no—
488C22—44no—
489C22—44no—
490C22—816no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22#150214yes—
496C22C2×#40711yescharacteristic
497C22C2×#41011yescharacteristic
498C22C2×#41311yescharacteristic
499C22C22×#10211yescharacteristic
500C22#289611yescharacteristic
501C22C22×#10411yescharacteristic
502C22C2×#42111yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—21no—
508C4—21no—
509C4—22no—
510C4—22no—
511C4—22no—
512C4—22no—
513C4—22no—
514C4—22no—
515C4—44no—
516C4—41no—
517C4—41no—
518C4—82no—
519C4—82no—
520C4#148614yes—
521C2—24no—
522C2—84no—
523C2#1735711yescharacteristic
524C2C2×#148611yescharacteristic
525C2#1744711yescharacteristic
526C2C2×#150211yescharacteristic
527C2#1755411yescharacteristic
528C2#1765011yescharacteristic
529C2#1770411yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. C4×#423C2semidirect
4. #7614C2semidirect
5. #7610C2semidirect
7. #7560C2semidirect
30. #1831C22semidirect
34. #1767C22semidirect
37. C4×C4⋊C8C22semidirect
39. #1744C22semidirect
41. #1271C22semidirect
164. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
C2×#22932
#207772
C4×#4232
#76142
#76102
#76062
#75602