Finite Group Databasebeta

← #220048

Subgroups — #220048

Order 512.

1203 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
1#36127C211yesmaximal, characteristic
2#22799C211yesmaximal, characteristic, complemented
3#22795C211yesmaximal, characteristic, complemented
4C2×#1553C212yesmaximal, complemented
5#9694C211yesmaximal, characteristic, complemented
6#9692C211yesmaximal, characteristic, complemented
7#9138C212yesmaximal, complemented
8#9132C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10#8341C212yesmaximal, complemented
11C4×#331C211yesmaximal, characteristic, complemented
12C2×#547—21no—
13C2×#532C2212yes—
14C2×#432C2212yes—
15C2×#431C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2547—21no—
19#2537—21no—
20#2537C412yes—
21#2432—22no—
22#2422—22no—
23#2422C414yes—
24D4×C4⋊C4—21no—
25D4×C4⋊C4C412yes—
26C4×#127C2211yescharacteristic
27C42×D4C2211yescharacteristic
28#1954—21no—
29#1954—21no—
30#1924—22no—
31#1923—22no—
32#1842C2211yescharacteristic
33#1842C2211yescharacteristic
34#1797C2212yes—
35#1797C2212yes—
36#1770—21no—
37C8×C4⋊C4—21no—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40#1553—28no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#331C2211yescharacteristic
46C2×#331C2212yes—
47C2×#331C2211yescharacteristic
48#1296C2211yescharacteristic
49#1296C2211yescharacteristic
50#1296C2212yes—
51C22×#41—22no—
52#547—44no—
53#547—44no—
54#532—24no—
55#532—24no—
56#532—42no—
57#532—42no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8C2312yes—
61C2×#127C2311yescharacteristic
62C2×#124—22no—
63C2×#123—24no—
64C2×C22⋊Q8—22no—
65C2×#121—22no—
66C2×#121—22no—
67C2×#121C2×C414yes—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×C4×D4C2×C414yes—
72C2×C4×D4C2×C412yes—
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C22×C4⋊C4—22no—
77C22×C42C2312yes—
78#432—24no—
79#432—42no—
80#431—24no—
81#431—42no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2312yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C22⋊C8—22no—
93C2×C22⋊C8—22no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×#97—22no—
97C2×C4×C8—22no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100#397—24no—
101#394—24no—
102#394—24no—
103#391—22no—
104#390—21no—
105#390—21no—
106#390C2×C412yes—
107#389—22no—
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#380—22no—
118#380—22no—
119#380C2×C414yes—
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—24no—
133#332—42no—
134#332—42no—
135#331—24no—
136#331—22no—
137#331D414yescomplemented
138C2×#41—44no—
139C2×#41—44no—
140C2×#41—44no—
141C2×#41—44no—
142C2×#41—82no—
143C2×#41—82no—
144C22×Q8—22no—
145C22×D4—24no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4C22⋊C411yescharacteristic
149C22×D4C22×C412yes—
150C23×C4—22no—
151C23×C4—24no—
152C23×C4C22×C412yes—
153C23×C4C2×D412yes—
154C23×C4C2×D412yes—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—22no—
160C22×C8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—81no—
168C4⋊Q8—81no—
169#127—21no—
170#127—21no—
171#127C2×D412yes—
172#127#4112yes—
173#124—44no—
174#124—44no—
175#123—416no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178#121—28no—
179#121—44no—
180#121—44no—
181#121—44no—
182#121—48no—
183C4×D4—28no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—42no—
189C4×D4—44no—
190C4×D4—44no—
191C4×D4—44no—
192C4×D4—44no—
193C4×D4—42no—
194C4×D4—42no—
195C4×D4—42no—
196C4×D4—42no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—48no—
206C2×C4⋊C4C22⋊C411yescharacteristic
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22×C414yes—
210C2×C4⋊C4C22×C412yes—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—24no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C412yes—
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2411yescharacteristic
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—22no—
243C4⋊C8—22no—
244C4⋊C8—42no—
245C4⋊C8—42no—
246C4⋊C8—81no—
247C4⋊C8—81no—
248C4⋊C8C2×D412yes—
249C4⋊C8C2×D412yes—
250C22⋊C8—44no—
251C22⋊C8—44no—
252C22⋊C8—82no—
253C22⋊C8—82no—
254#97—44no—
255C4×C8—44no—
256C4×C8—42no—
257C4×C8—42no—
258#95—24no—
259#95—24no—
260#95C22⋊C412yes—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263C24—22no—
264C24C2×C22⋊C412yes—
265#41—88no—
266#41—88no—
267#41—88no—
268#41—88no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—22no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—48no—
286C2×D4—48no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—84no—
296C2×D4C4×D414yes—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—48no—
316C22×C4—48no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—44no—
320C22×C4—84no—
321C22×C4C2×C22⋊C412yes—
322C22×C4C2×C22⋊C412yes—
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C411yescharacteristic
326C22×C4C2×C22⋊C411yescharacteristic
327C22×C4C2×C22⋊C411yescharacteristic
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—41no—
346C2×C8—42no—
347C2×C8—41no—
348C2×C8—42no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—42no—
355C4⋊C4—48no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—44no—
363C4⋊C4—44no—
364C4⋊C4—82no—
365C4⋊C4—82no—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—42no—
369C22⋊C4—48no—
370C22⋊C4—48no—
371C22⋊C4—44no—
372C22⋊C4—48no—
373C22⋊C4—44no—
374C22⋊C4—44no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—22no—
386C42—41no—
387C42#13512yes—
388C42#13512yes—
389C42AGL(1,ℤ/8)12yes—
390C42AGL(1,ℤ/8)12yes—
391C42C22×D411yescharacteristic
392C42C2×#4111yescharacteristic
393C42H(5,𝔽2)11yescharacteristic
394C42H(5,𝔽2)11yescharacteristic
395C23—24no—
396C23—22no—
397C23—22no—
398C23—22no—
399C23—22no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23C22×C22⋊C411yescharacteristic
410Q8—82no—
411Q8—82no—
412Q8—82no—
413Q8—82no—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—42no—
418D4—42no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—42no—
425D4—816no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—22no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—816no—
461C2×C4—88no—
462C2×C4#41312yes—
463C2×C4#41612yes—
464C2×C4#41612yes—
465C2×C4#41812yes—
466C2×C4#41812yes—
467C2×C4C2×C4×D411yescharacteristic
468C2×C4#51811yescharacteristic
469C2×C4D4211yescharacteristic
470C2×C4D4211yescharacteristic
471C2×C4#54611yescharacteristic
472C2×C4#54611yescharacteristic
473C2×C4C2×#13511yescharacteristic
474C2×C4C2×#13511yescharacteristic
475C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
476C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
477C2×C4#57611yescharacteristic
478C2×C4#57711yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57811yescharacteristic
481C8—42no—
482C8—42no—
483C8—82no—
484C8—82no—
485C8—81no—
486C8—81no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—44no—
493C22—42no—
494C22—44no—
495C22—42no—
496C22—44no—
497C22—82no—
498C22—82no—
499C22—82no—
500C22—82no—
501C22D4×C22⋊C411yescharacteristic
502C22C2×#41311yescharacteristic
503C22#288911yescharacteristic
504C22C2×#41611yescharacteristic
505C22C2×#41811yescharacteristic
506C22#289711yescharacteristic
507C22#289811yescharacteristic
508C4—22no—
509C4—22no—
510C4—21no—
511C4—21no—
512C4—41no—
513C4—42no—
514C4—42no—
515C4—42no—
516C4—82no—
517C4—84no—
518C4#155314yescomplemented
519C4#327211yescharacteristic
520C4#327411yescharacteristic
521C4#328311yescharacteristic
522C4#328411yescharacteristic
523C4#330211yescharacteristic
524C4#330311yescharacteristic
525C4#332411yescharacteristic
526C4#332511yescharacteristic
527C2—82no—
528C2—84no—
529C2C2×#155311yescharacteristic
530C2#1772711yescharacteristic
531C2#1775911yescharacteristic
532C2#1778111yescharacteristic
533C2#2241111yescharacteristic
534C2#2241711yescharacteristic
535C2#2242811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36127C2
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
2. #22799C2semidirect
3. #22795C2semidirect
4. C2×#1553C2semidirect
5. #9694C2semidirect
6. #9692C2semidirect
7. #9138C2semidirect
8. #9132C2semidirect
9. #8585C2semidirect
10. #8341C2semidirect
11. C4×#331C2semidirect
137. #331D4semidirect
518. C4#1553semidirect

Maximal subgroups

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

subgroupindex
#361272
#227992
#227952
C2×#15532
#96942
#96922
#91382
#91322
#85852
#83412
C4×#3312