Finite Group Databasebeta

← #303435

Subgroups — #303435

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#127C211yesmaximal, characteristic
2#33289C211yesmaximal, characteristic, complemented
3#33196C211yesmaximal, characteristic, complemented
4#28716C211yesmaximal, characteristic
5#10420C214yesmaximal, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9638C211yesmaximal, characteristic
8#8447C214yesmaximal, complemented
9C4×Q8⋊C8C211yesmaximal, characteristic, complemented
10C2×D4×Q8—21no—
11C2×D4×Q8—21no—
12C2×D4×Q8C2212yes—
13C2×#486—21no—
14C2×#475—21no—
15#2868—21no—
16#2868C2211yescharacteristic
17#2868C2211yescharacteristic
18#2542—22no—
19#2542—22no—
20#2542—22no—
21#2542—21no—
22#2542—21no—
23#2542C2212yes—
24C4×#127—21no—
25C4×#127C2211yescharacteristic
26C42×Q8C2211yescharacteristic
27#2071—22no—
28#1990—22no—
29#1990—21no—
30#1990C2214yes—
31#1988—21no—
32#1948—21no—
33#1948C2212yes—
34#1842C2211yescharacteristic
35#1834—22no—
36#1803—21no—
37#1803C2212yes—
38#1802C2214yes—
39C4×C4⋊C8C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×#332C2214yes—
42C2×Q8⋊C8C2212yes—
43C2×D4⋊C8C2214yes—
44#1279C2212yes—
45C23×Q8—22no—
46D4×Q8—28no—
47D4×Q8—44no—
48D4×Q8—44no—
49D4×Q8—44no—
50D4×Q8—44no—
51D4×Q8—44no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2312yes—
61C2×#127D411yescharacteristic
62C2×C22⋊Q8—24no—
63C2×C22⋊Q8—24no—
64C2×C22⋊Q8—22no—
65C2×C22⋊Q8—22no—
66C2×C22⋊Q8—22no—
67C2×C22⋊Q8—24no—
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8D411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4—22no—
72C2×C4×D4—22no—
73C2×C4×D4—22no—
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4C2312yes—
77#486—44no—
78#475—44no—
79C2×#106—22no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×#103—22no—
85C2×#102—24no—
86C2×C4×C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8—21no—
89C2×C4×C8C2312yes—
90#399—24no—
91#399—24no—
92#399—24no—
93#390—22no—
94#390—22no—
95#390—22no—
96#390—21no—
97#390—21no—
98#390D412yes—
99#384—22no—
100#384—22no—
101#384—22no—
102#384—21no—
103#384—21no—
104#384—21no—
105#384—21no—
106#384—21no—
107#384D411yescharacteristic
108#384D412yes—
109#384D412yes—
110#384C2312yes—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C43—21no—
118C43C2311yescharacteristic
119#340—24no—
120#340—22no—
121#340—22no—
122#340—28no—
123#336—21no—
124#332—44no—
125Q8⋊C8—24no—
126D4⋊C8—44no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—44no—
130C22×Q8—44no—
131C22×Q8—44no—
132C22×Q8—44no—
133C22×Q8—44no—
134C22×Q8—44no—
135C22×Q8—41no—
136C22×Q8C2×D411yescharacteristic
137C22×D4—21no—
138C22×D4—21no—
139C22×D4C2×D412yes—
140C23×C4—24no—
141C23×C4—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—48no—
148C4⋊Q8—48no—
149C4⋊Q8—48no—
150C4⋊Q8—44no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—42no—
156C4⋊Q8—42no—
157C4⋊Q8—42no—
158C4⋊Q8—42no—
159#127—22no—
160#127SD1614yes—
161C22⋊Q8—416no—
162C22⋊Q8—44no—
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C22⋊Q8—416no—
166C22⋊Q8—416no—
167C22⋊Q8—44no—
168C22⋊Q8—44no—
169C22⋊Q8—44no—
170C4×Q8—22no—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8D814yes—
174C4×Q8SD1614yes—
175C4×Q8#4114yes—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—48no—
179C4×D4—48no—
180C4×D4—48no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D414yes—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—24no—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2411yescharacteristic
228#106—44no—
229C4⋊C8—22no—
230C4⋊C8—24no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234#103—44no—
235#102—48no—
236C4×C8—24no—
237C4×C8—22no—
238C4×C8—22no—
239C4×C8—42no—
240C4×C8—42no—
241#95—21no—
242#95—21no—
243#95—21no—
244C24—22no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—88no—
258C2×Q8—88no—
259C2×Q8—88no—
260C2×Q8—88no—
261C2×Q8—82no—
262C2×Q8—82no—
263C2×Q8—82no—
264C2×Q8#12412yes—
265C2×Q8C2×D812yes—
266C2×Q8C2×SD1612yes—
267C2×D4—24no—
268C2×D4—22no—
269C2×D4—24no—
270C2×D4—24no—
271C2×D4—24no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4C2×SD1614yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—48no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4#12011yescharacteristic
294C22×C4#12011yescharacteristic
295C22×C4#12011yescharacteristic
296C22×C4#12011yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C4⋊C4—21no—
310C4⋊C4—24no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—28no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—42no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—82no—
333C4⋊C4—82no—
334C4⋊C4—82no—
335C4⋊C4#12412yes—
336C4⋊C4C2×D812yes—
337C4⋊C4C2×SD1612yes—
338C22⋊C4—48no—
339C22⋊C4—48no—
340C22⋊C4—48no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C42—24no—
345C42—22no—
346C42—24no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42C2×D812yes—
354C42C2×SD1612yes—
355C42C2×SD1612yes—
356C42AGL(1,ℤ/8)12yes—
357C42AGL(1,ℤ/8)12yes—
358C42#13712yes—
359C42C2×#4112yes—
360C42H(5,𝔽2)12yes—
361C23—24no—
362C23—42no—
363C23—42no—
364C23—42no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23C2×#12011yescharacteristic
369Q8—22no—
370Q8—22no—
371Q8—22no—
372Q8—816no—
373Q8—82no—
374Q8—82no—
375Q8—82no—
376Q8—82no—
377Q8—164no—
378Q8#48614yescomplemented
379D4—28no—
380D4—44no—
381D4—48no—
382D4—48no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—816no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#44712yes—
416C2×C4#44712yes—
417C2×C4#44812yes—
418C2×C4#44812yes—
419C2×C4#45012yes—
420C2×C4#45012yes—
421C2×C4#45412yes—
422C2×C4#45612yes—
423C2×C4C2×#12411yescharacteristic
424C2×C4#53511yescharacteristic
425C2×C4#54612yes—
426C2×C4#54612yes—
427C2×C4C22×D811yescharacteristic
428C2×C4C22×SD1611yescharacteristic
429C2×C4C22×SD1611yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
431C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
432C2×C4C2×#13711yescharacteristic
433C2×C4#57812yes—
434C8—44no—
435C8—82no—
436C8—82no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22—82no—
444C22#243511yescharacteristic
445C22C2×#44711yescharacteristic
446C22C2×#45011yescharacteristic
447C22C2×#44811yescharacteristic
448C22#301111yescharacteristic
449C22C2×#45411yescharacteristic
450C22C2×#45611yescharacteristic
451C4—22no—
452C4—22no—
453C4—22no—
454C4—24no—
455C4—42no—
456C4—42no—
457C4—42no—
458C4—84no—
459C4—82no—
460C4—162no—
461C4#162714yes—
462C4#165114yes—
463C4C2×#48611yescharacteristic
464C4#313811yescharacteristic
465C4#320511yescharacteristic
466C4#320811yescharacteristic
467C4#329812yes—
468C4#332812yes—
469C2—84no—
470C2C2×#162711yescharacteristic
471C2C2×#165111yescharacteristic
472C2#1864111yescharacteristic
473C2#1865011yescharacteristic
474C2#2585511yescharacteristic
475C2#2587611yescharacteristic
476C2#2588411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33289C2semidirect
3. #33196C2semidirect
5. #10420C2semidirect
6. #9692C2semidirect
8. #8447C2semidirect
9. C4×Q8⋊C8C2semidirect
378. Q8#486semidirect

Maximal subgroups

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

subgroupindex
Q8×#1272
#332892
#331962
#287162
#104202
#96922
#96382
#84472
C4×Q8⋊C82