Finite Group Databasebeta

← #203328

Subgroups — #203328

Order 512.

718 conjugacy classes of subgroups.

The subgroup lattice has 545 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21345C211yesmaximal, characteristic, complemented
2C4×#440C211yesmaximal, characteristic
3D4×#97C211yesmaximal, characteristic
4#9548C211yesmaximal, characteristic, complemented
5#9547C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8945C211yesmaximal, characteristic
9#8945C211yesmaximal, characteristic
10#8845C211yesmaximal, characteristic
11#8843C211yesmaximal, characteristic
12#8276C211yesmaximal, characteristic, complemented
13#8173C211yesmaximal, characteristic, complemented
14#8172C211yesmaximal, characteristic
15#8136C211yesmaximal, characteristic, complemented
16C2×#440C411yescharacteristic
17C2×#440C411yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C22×#97—21no—
22C42×Q8C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1939C2211yescharacteristic
27#1939C2211yescharacteristic
28#1917—21no—
29#1917—21no—
30#1776C411yescharacteristic
31#1776C411yescharacteristic
32C4×#107C2211yescharacteristic
33#1770C2211yescharacteristic
34#1770C2211yescharacteristic
35#1767—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1763C411yescharacteristic
39#1763C411yescharacteristic
40#1762C411yescharacteristic
41#1762C411yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1752—21no—
45#1752—21no—
46#1749—21no—
47#1745—21no—
48C4×#97C2211yescharacteristic
49C4×#97C2211yescharacteristic
50C4×#97C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×D4⋊C8C2211yescharacteristic
56C2×D4⋊C8C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1273C2211yescharacteristic
64#1273C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#440—24no—
78#440—24no—
79C4×SD16—22no—
80C4×SD16—22no—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4—22no—
84C8×D4—22no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—44no—
109C2×#97C2×C411yescharacteristic
110C2×#97C2×C411yescharacteristic
111C2×#97C2×C411yescharacteristic
112C2×#97C2×C411yescharacteristic
113C2×#97D411yescharacteristic
114C2×#97D411yescharacteristic
115C2×#97D411yescharacteristic
116C2×#97D411yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C43—21no—
136C43C2311yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
141#340C2×C411yescharacteristic
142#340C2×C411yescharacteristic
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#335—22no—
154#334—22no—
155Q8⋊C8—22no—
156Q8⋊C8—22no—
157D4⋊C8—22no—
158D4⋊C8—22no—
159C22×Q8C22×C411yescharacteristic
160C22×D4C22×C411yescharacteristic
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—21no—
164C2×SD16—22no—
165C2×SD16—22no—
166C2×SD16M1614yes—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—42no—
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C22×C411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—41no—
194C4×Q8—41no—
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4#4112yes—
202C4×D4#4112yes—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42C22×C411yescharacteristic
227C2×C42C22×C411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—22no—
234#107—22no—
235#106—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—42no—
243C4⋊C8—42no—
244C4⋊C8#4112yes—
245C4⋊C8#4112yes—
246#103—22no—
247#103—22no—
248#103—22no—
249#102—22no—
250#102—22no—
251#102M1614yes—
252C22⋊C8—24no—
253C22⋊C8—24no—
254#97—22no—
255#97—22no—
256#97—22no—
257#97—22no—
258#97—41no—
259#97—41no—
260#97—41no—
261#97—41no—
262#97—42no—
263#97—42no—
264#97—42no—
265#97—42no—
266#97—42no—
267#97—42no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—21no—
273C4×C8—21no—
274C4×C8—21no—
275C4×C8—21no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286C24—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8C2×M1612yes—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4#11712yes—
297C2×D4C2×M1612yes—
298C2×D4C2×M1612yes—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—81no—
317C22×C4—81no—
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325SD16—28no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—22no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—48no—
351C2×C8—48no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8C2×M1612yes—
367C2×C8C2×M1612yes—
368C2×C8AGL(1,ℤ/8)12yes—
369C2×C8#13712yes—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4#11712yes—
385C4⋊C4C2×M1612yes—
386C22⋊C4—22no—
387C22⋊C4—22no—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C42—81no—
411C42—81no—
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42C2×#4111yescharacteristic
415C42C2×#4111yescharacteristic
416C23—22no—
417C23—22no—
418C23—22no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23C2×C4×D411yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—41no—
427Q8—41no—
428D4—22no—
429D4—22no—
430D4—22no—
431D4#43214yes—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—22no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4C2×#11711yescharacteristic
474C2×C4C4×#4111yescharacteristic
475C2×C4#52511yescharacteristic
476C2×C4#52511yescharacteristic
477C2×C4#52911yescharacteristic
478C2×C4#52911yescharacteristic
479C2×C4C22×M1611yescharacteristic
480C2×C4C22×M1611yescharacteristic
481C2×C4#56811yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
484C2×C4C2×#13711yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57511yescharacteristic
487C8—22no—
488C8—22no—
489C8—22no—
490C8—22no—
491C8—42no—
492C8—42no—
493C8—42no—
494C8—41no—
495C8—42no—
496C8—41no—
497C8—41no—
498C8—41no—
499C8—81no—
500C8—81no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—42no—
507C22—42no—
508C22#229011yescharacteristic
509C22D4×M1611yescharacteristic
510C22D4×M1611yescharacteristic
511C22C4×AGL(1,ℤ/8)11yescharacteristic
512C22C4×#13711yescharacteristic
513C22#295611yescharacteristic
514C22#295611yescharacteristic
515C4—21no—
516C4—21no—
517C4—42no—
518C4—41no—
519C4—42no—
520C4—41no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—81no—
525C4—81no—
526C4—81no—
527C4—81no—
528C4C2×#43211yescharacteristic
529C4#292311yescharacteristic
530C4#297011yescharacteristic
531C4#297011yescharacteristic
532C4#311011yescharacteristic
533C4#311111yescharacteristic
534C4#311411yescharacteristic
535C4#311611yescharacteristic
536C2—44no—
537C2#1804611yescharacteristic
538C2#1804711yescharacteristic
539C2#1804711yescharacteristic
540C2#1804811yescharacteristic
541C2#2110711yescharacteristic
542C2#2112111yescharacteristic
543C2#2113911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#97 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #21345C2semidirect
4. #9548C2semidirect
5. #9547C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
12. #8276C2semidirect
13. #8173C2semidirect
15. #8136C2semidirect

Maximal subgroups

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

subgroupindex
#213452
C4×#4402
D4×#972
#95482
#95472
#95402
#95372
#89452
#88452
#88432
#82762
#81732
#81722
#81362