Finite Group Databasebeta

← #218554

Subgroups — #218554

Order 512.

639 conjugacy classes of subgroups.

The subgroup lattice has 547 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22208C211yesmaximal, characteristic
2#22018C211yesmaximal, characteristic, complemented
3C4×#464C211yesmaximal, characteristic
4#9675C211yesmaximal, characteristic
5#9660C211yesmaximal, characteristic
6#9656C211yesmaximal, characteristic, complemented
7#9601C211yesmaximal, characteristic, complemented
8#9555C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic
10#8946C211yesmaximal, characteristic, complemented
11#8945C211yesmaximal, characteristic
12#8844C211yesmaximal, characteristic, complemented
13#8843C211yesmaximal, characteristic
14#8564C211yesmaximal, characteristic, complemented
15#8264C211yesmaximal, characteristic, complemented
16C2×#464C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×#124C2211yescharacteristic
22C42×Q8C2211yescharacteristic
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1984C2211yescharacteristic
26#1983C411yescharacteristic
27#1954C2211yescharacteristic
28#1949C411yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1927C411yescharacteristic
32#1925C411yescharacteristic
33#1923—21no—
34#1923—21no—
35#1916—21no—
36#1916—21no—
37#1880C411yescharacteristic
38#1878C411yescharacteristic
39#1850—21no—
40#1850—21no—
41#1842C2211yescharacteristic
42#1839—21no—
43#1831—21no—
44#1805C411yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#102C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59C2×#322C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×Q16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C23×C8—21no—
70C2×#124C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2311yescharacteristic
76#464C818yescomplemented
77C8×Q8—41no—
78C8×Q8—41no—
79C4×Q16—41no—
80C4×Q16—41no—
81C4×SD16—41no—
82C4×SD16—41no—
83C8×D4—41no—
84C8×D4—41no—
85C2×#107C2×C411yescharacteristic
86C2×#106C2×C411yescharacteristic
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2×C411yescharacteristic
94C2×C4⋊C8C2×C411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C411yescharacteristic
103C2×#103C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C411yescharacteristic
106C2×#102C2×C411yescharacteristic
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#388—21no—
121#388—21no—
122#382C2×C411yescharacteristic
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C43—21no—
132C43C2311yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336C2×C411yescharacteristic
148#336C2×C411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#336D411yescharacteristic
153#335—41no—
154#335—41no—
155Q8⋊C8—41no—
156Q8⋊C8—41no—
157Q8⋊C8—41no—
158Q8⋊C8—41no—
159D4⋊C8—41no—
160D4⋊C8—41no—
161#322—41no—
162C82—41no—
163C22×Q8C22×C411yescharacteristic
164C22×Q8C22×C411yescharacteristic
165C22×D4C22×C411yescharacteristic
166C23×C4—21no—
167C2×Q16—41no—
168C2×Q16—41no—
169C2×Q16C2×C814yescomplemented
170C2×SD16—41no—
171C2×SD16—41no—
172C2×SD16C2×C814yes—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—42no—
180C22×C8—81no—
181C22×C8—81no—
182C22×C8C22×C411yescharacteristic
183C22×C8C22×C411yescharacteristic
184C22×C8C22×C411yescharacteristic
185C22×C8C22×C411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C22×C8C2×D411yescharacteristic
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190#124—22no—
191#124C2×C814yes—
192C4×Q8—22no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—41no—
202C4×Q8—41no—
203C4×Q8C2×C814yescomplemented
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42#4111yescharacteristic
235C2×C42C2411yescharacteristic
236#107—41no—
237#106—41no—
238C4⋊C8—22no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—42no—
248C4⋊C8—42no—
249C4⋊C8—81no—
250C4⋊C8—81no—
251C4⋊C8—81no—
252C4⋊C8—81no—
253C4⋊C8C2×C814yescomplemented
254#103—41no—
255#103—41no—
256#103—41no—
257#103—41no—
258#103C2×C814yescomplemented
259#102—41no—
260#102—41no—
261#102C2×C814yes—
262C22⋊C8—42no—
263C22⋊C8—81no—
264C22⋊C8—81no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
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—42no—
278C4×C8—81no—
279#95—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×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8C22×C812yes—
298C2×Q8C22×C812yes—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C22×C812yes—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4C4×D411yescharacteristic
319C22×C4C4×D411yescharacteristic
320C22×C4C4×D411yescharacteristic
321C22×C4C4×D411yescharacteristic
322C22×C4C23×C411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C2×#4111yescharacteristic
325Q16—44no—
326SD16—44no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
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—41no—
351C2×C8—41no—
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×C8—41no—
367C2×C8—84no—
368C2×C8C22×C812yes—
369C2×C8C22×C812yes—
370C4⋊C4—21no—
371C4⋊C4—24no—
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—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4C22×C812yes—
386C22⋊C4—24no—
387C22⋊C4—24no—
388C22⋊C4—41no—
389C22⋊C4—41no—
390C22⋊C4—81no—
391C22⋊C4—81no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—24no—
399C42—41no—
400C42—41no—
401C42—42no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—81no—
409C42—81no—
410C42C22×C812yes—
411C42#13112yes—
412C42H(5,𝔽2)11yescharacteristic
413C42H(5,𝔽2)11yescharacteristic
414C42H(5,𝔽2)11yescharacteristic
415C422−1+411yescharacteristic
416C23—22no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23C2×C4×D411yescharacteristic
424Q8—22no—
425Q8—22no—
426Q8—22no—
427Q8—22no—
428Q8—41no—
429Q8—41no—
430Q8—41no—
431Q8—41no—
432D4—22no—
433D4—22no—
434D4—41no—
435D4—41no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
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—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4C8×D412yes—
474C2×C4C8×D412yes—
475C2×C4#51811yescharacteristic
476C2×C4#52011yescharacteristic
477C2×C4#53511yescharacteristic
478C2×C4#53711yescharacteristic
479C2×C4#54011yescharacteristic
480C2×C4#54211yescharacteristic
481C2×C4C23×C811yescharacteristic
482C2×C4C2×#13111yescharacteristic
483C2×C4#56811yescharacteristic
484C2×C4#56811yescharacteristic
485C2×C4#57611yescharacteristic
486C2×C4#57711yescharacteristic
487C2×C4#57811yescharacteristic
488C2×C4#57811yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22#232411yescharacteristic
511C22C2×C8×D411yescharacteristic
512C22#293011yescharacteristic
513C22#294711yescharacteristic
514C22#295111yescharacteristic
515C22#295611yescharacteristic
516C22#295711yescharacteristic
517C4—21no—
518C4—21no—
519C4—22no—
520C4—22no—
521C4—41no—
522C4—41no—
523C4—42no—
524C4—42no—
525C4—42no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4#297211yescharacteristic
531C4#298111yescharacteristic
532C4#298811yescharacteristic
533C4#299311yescharacteristic
534C4#322311yescharacteristic
535C4#322411yescharacteristic
536C4#325811yescharacteristic
537C4#326211yescharacteristic
538C2—82no—
539C2#1785311yescharacteristic
540C2#1785711yescharacteristic
541C2#1794711yescharacteristic
542C2#1795411yescharacteristic
543C2#2213311yescharacteristic
544C2#2216311yescharacteristic
545C2#2216711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22018C2semidirect
6. #9656C2semidirect
7. #9601C2semidirect
8. #9555C2semidirect
10. #8946C2semidirect
12. #8844C2semidirect
14. #8564C2semidirect
15. #8264C2semidirect
76. #464C8semidirect
169. C2×Q16C2×C8semidirect
203. C4×Q8C2×C8semidirect
253. C4⋊C8C2×C8semidirect
258. #103C2×C8semidirect

Maximal subgroups

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

subgroupindex
#222082
#220182
C4×#4642
#96752
#96602
#96562
#96012
#95552
#95542
#89462
#89452
#88442
#88432
#85642
#82642