Finite Group Databasebeta

← #220094

Subgroups — #220094

Order 512.

1145 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic
3#22799C211yesmaximal, characteristic
4C2×#1556C212yesmaximal, complemented
5#9694C211yesmaximal, characteristic, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9140C212yesmaximal, complemented
8#9138C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10#8340C212yesmaximal, complemented
11C4×#332C211yesmaximal, characteristic, complemented
12C2×D4×Q8—21no—
13C22×C4⋊Q8C2212yes—
14C2×#432C2212yes—
15C2×#432C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2551—21no—
19#2551C412yes—
20#2546—21no—
21#2440—22no—
22#2440C414yes—
23#2434—22no—
24D4×C4⋊C4—21no—
25D4×C4⋊C4C412yes—
26C4×C4⋊Q8C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1924—22no—
31#1924—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36#1770—21no—
37#1770—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1556—28no—
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C23×Q8—22no—
52D4×Q8—44no—
53D4×Q8—44no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—42no—
62C2×C4⋊Q8—42no—
63C2×C4⋊Q8C2×C414yes—
64C2×C4⋊Q8C2312yes—
65C2×C4⋊Q8C2311yescharacteristic
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C4×Q8—21no—
69C2×C4×D4—22no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4D411yescharacteristic
73C22×C4⋊C4—22no—
74C22×C4⋊C4C2×C414yes—
75C22×C42C2312yes—
76#432—24no—
77#432—24no—
78#432—42no—
79#432—42no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C22⋊C8—22no—
91C2×C22⋊C8—22no—
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×#97—22no—
95C2×#97—22no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98#398—24no—
99#395—24no—
100#395—24no—
101#391—21no—
102#391—22no—
103#389—22no—
104#389—22no—
105#389—21no—
106#389—22no—
107#389C2×C412yes—
108#389C2×C414yes—
109#385—24no—
110#385—24no—
111#384—21no—
112#384—21no—
113#384C2×C412yes—
114#384C2×C412yes—
115#384C2311yescharacteristic
116#380—22no—
117#380C2×C414yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C22⋊C4C2×C412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4D412yes—
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336—21no—
130#336—21no—
131#332—24no—
132#332—22no—
133#332—24no—
134#332—24no—
135#332—42no—
136#332—42no—
137#332D414yescomplemented
138C22×Q8—24no—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—44no—
142C22×Q8—44no—
143C22×Q8—44no—
144C22×Q8—44no—
145C22×Q8—82no—
146C22×Q8—82no—
147C22×D4C22⋊C411yescharacteristic
148C23×C4—24no—
149C23×C4C22×C412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C4⋊Q8—28no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—44no—
166C4⋊Q8—44no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169C4⋊Q8—44no—
170C4⋊Q8—48no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—81no—
174C4⋊Q8—81no—
175C4⋊Q8C2×D412yes—
176C4⋊Q8#4112yes—
177C22⋊Q8—416no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C4×Q8—42no—
181C4×Q8—42no—
182C4×D4—24no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—48no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—28no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22×C414yes—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C22⋊C4C22⋊C412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C412yes—
221C2×C42C22×C411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2411yescharacteristic
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—42no—
240C4⋊C8—42no—
241C4⋊C8—81no—
242C4⋊C8—81no—
243C4⋊C8C2×D412yes—
244C4⋊C8C2×D412yes—
245C22⋊C8—44no—
246C22⋊C8—44no—
247C22⋊C8—82no—
248C22⋊C8—82no—
249#97—44no—
250#97—44no—
251C4×C8—42no—
252C4×C8—42no—
253#95—22no—
254#95—22no—
255#95—24no—
256#95—24no—
257#95C22⋊C412yes—
258#95C22⋊C412yes—
259#95C22⋊C412yes—
260C24C2×C22⋊C412yes—
261C2×Q8—22no—
262C2×Q8—22no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—41no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—42no—
275C2×Q8—84no—
276C2×Q8—88no—
277C2×Q8—88no—
278C2×Q8—88no—
279C2×Q8—88no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4#10114yes—
283C22×C4—22no—
284C22×C4—24no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—42no—
302C22×C4—48no—
303C22×C4—48no—
304C22×C4—84no—
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×C22⋊C412yes—
307C22×C4C2×C22⋊C412yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C411yescharacteristic
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4C2×C22⋊C411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—41no—
330C2×C8—42no—
331C2×C8—41no—
332C2×C8—42no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—48no—
342C4⋊C4—48no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—44no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4C4×D414yes—
357C22⋊C4—24no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—48no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—22no—
372C42—41no—
373C42—44no—
374C42AGL(1,ℤ/8)12yes—
375C42AGL(1,ℤ/8)12yes—
376C42#13712yes—
377C42#13712yes—
378C42C22×D411yescharacteristic
379C42C2×#4111yescharacteristic
380C42H(5,𝔽2)11yescharacteristic
381C42H(5,𝔽2)11yescharacteristic
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23C2×#10114yes—
389C23C22×C22⋊C411yescharacteristic
390Q8—42no—
391Q8—42no—
392Q8—42no—
393Q8—42no—
394Q8—816no—
395Q8—82no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399D4—28no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—816no—
438C2×C4C2×#10112yes—
439C2×C4#41812yes—
440C2×C4#41812yes—
441C2×C4#41912yes—
442C2×C4#41912yes—
443C2×C4C2×C4×D411yescharacteristic
444C2×C4#51811yescharacteristic
445C2×C4D4211yescharacteristic
446C2×C4D4211yescharacteristic
447C2×C4#54611yescharacteristic
448C2×C4#54611yescharacteristic
449C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
450C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
451C2×C4C2×#13711yescharacteristic
452C2×C4C2×#13711yescharacteristic
453C2×C4#57611yescharacteristic
454C2×C4#57711yescharacteristic
455C2×C4#57711yescharacteristic
456C2×C4#57811yescharacteristic
457C8—42no—
458C8—42no—
459C8—81no—
460C8—82no—
461C8—82no—
462C8—81no—
463C22—24no—
464C22—24no—
465C22—24no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22D4×C22⋊C411yescharacteristic
471C22C22×#10111yescharacteristic
472C22#288911yescharacteristic
473C22C2×#41811yescharacteristic
474C22C2×#41911yescharacteristic
475C22#289711yescharacteristic
476C22#289811yescharacteristic
477C4—22no—
478C4—22no—
479C4—21no—
480C4—21no—
481C4—41no—
482C4—42no—
483C4—42no—
484C4—44no—
485C4—42no—
486C4—82no—
487C4—84no—
488C4#155614yescomplemented
489C4#327211yescharacteristic
490C4#327411yescharacteristic
491C4#327611yescharacteristic
492C4#327811yescharacteristic
493C4#332411yescharacteristic
494C4#332511yescharacteristic
495C4#332811yescharacteristic
496C4#332911yescharacteristic
497C2—44no—
498C2C2×#155611yescharacteristic
499C2#1775911yescharacteristic
500C2#1776111yescharacteristic
501C2#1777411yescharacteristic
502C2D4×#10111yescharacteristic
503C2#2242811yescharacteristic
504C2#2243011yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1556C2semidirect
5. #9694C2semidirect
6. #9692C2semidirect
7. #9140C2semidirect
8. #9138C2semidirect
9. #8586C2semidirect
10. #8340C2semidirect
11. C4×#332C2semidirect
137. #332D4semidirect
488. C4#1556semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#227992
C2×#15562
#96942
#96922
#91402
#91382
#85862
#83402
C4×#3322