Finite Group Databasebeta

← #218561

Subgroups — #218561

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23677C211yesmaximal, characteristic
2#22238C211yesmaximal, characteristic, complemented
3#22013C211yesmaximal, characteristic
4#9677C211yesmaximal, characteristic, complemented
5#9660C211yesmaximal, characteristic
6#9655C211yesmaximal, characteristic
7#9606C211yesmaximal, characteristic, complemented
8#9555C211yesmaximal, characteristic, complemented
9#9553C211yesmaximal, characteristic, complemented
10#8948C211yesmaximal, characteristic
11#8948C211yesmaximal, characteristic
12#8846C211yesmaximal, characteristic, complemented
13#8844C211yesmaximal, characteristic, complemented
14#8569C211yesmaximal, characteristic
15#8264C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C4×SD16C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21#2293C2211yescharacteristic
22#2084C411yescharacteristic
23#2071C411yescharacteristic
24#2066C411yescharacteristic
25#2066C411yescharacteristic
26#2055C411yescharacteristic
27#2055C411yescharacteristic
28#2043C411yescharacteristic
29#2013C411yescharacteristic
30#1984C2211yescharacteristic
31#1984C2211yescharacteristic
32#1954C2211yescharacteristic
33#1954C2211yescharacteristic
34#1939C2211yescharacteristic
35#1939C2211yescharacteristic
36#1924—21no—
37#1924—21no—
38#1916—21no—
39#1916—21no—
40#1916—21no—
41#1916—21no—
42#1831C2211yescharacteristic
43#1818—21no—
44#1798—21no—
45#1767C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×C4⋊C8C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1744C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#334C2211yescharacteristic
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×SD16C2×C411yescharacteristic
68C22×SD16C2×C411yescharacteristic
69C2×C4⋊Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C8×Q8—41no—
75C8×Q8—41no—
76C8×Q8—41no—
77C8×Q8—41no—
78C4×SD16—41no—
79C4×SD16—41no—
80C4×SD16—41no—
81C4×SD16—41no—
82C2×#106C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2×C411yescharacteristic
90C2×C4⋊C8C2×C411yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×#103C2×C411yescharacteristic
98C2×#103C2×C411yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#102C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116#391C2×C411yescharacteristic
117#391C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#390C2×C411yescharacteristic
120#389—21no—
121#389—21no—
122#389C2×C411yescharacteristic
123#389C2×C411yescharacteristic
124#388—21no—
125#388—21no—
126#384C2×C411yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C43C2311yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#336—21no—
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#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—41no—
153#334—41no—
154Q8⋊C8—41no—
155Q8⋊C8—41no—
156Q8⋊C8—41no—
157Q8⋊C8—41no—
158D4⋊C8—41no—
159D4⋊C8—41no—
160#322—41no—
161C82—41no—
162C22×Q8C22×C411yescharacteristic
163C22×Q8C22×C411yescharacteristic
164C22×D4C22×C411yescharacteristic
165C23×C4—21no—
166C2×SD16—22no—
167C2×SD16—22no—
168C2×SD16—22no—
169C2×SD16—22no—
170C2×SD16—41no—
171C2×SD16—41no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C22×C411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C4⋊Q8—22no—
187C4⋊Q8—22no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×Q8—21no—
193C4×Q8—21no—
194C4×Q8—21no—
195C4×Q8—21no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—21no—
199C4×D4—21no—
200#117—81no—
201#117—81no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C4⋊C4C22×C411yescharacteristic
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C411yescharacteristic
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42C2411yescharacteristic
230#106—41no—
231#106—41no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—21no—
235C4⋊C8—21no—
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—42no—
245C4⋊C8—81no—
246C4⋊C8—81no—
247#103—22no—
248#103—22no—
249#103—22no—
250#103—22no—
251#103—41no—
252#103—41no—
253#102—41no—
254#102—41no—
255C22⋊C8—81no—
256C22⋊C8—81no—
257C22⋊C8—81no—
258C22⋊C8—81no—
259#97—81no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—21no—
270C4×C8—21no—
271C4×C8—21no—
272C4×C8—42no—
273C4×C8—42no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281C24—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8#13112yes—
293C2×Q8#13112yes—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D411yescharacteristic
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320SD16—44no—
321SD16—44no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
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—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
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#13112yes—
363C2×C8#13112yes—
364C4⋊C4—24no—
365C4⋊C4—24no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—81no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—81no—
384C22⋊C4—41no—
385C22⋊C4—41no—
386C22⋊C4—81no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42H(5,𝔽2)11yescharacteristic
407C42H(5,𝔽2)11yescharacteristic
408C42H(5,𝔽2)11yescharacteristic
409C422−1+411yescharacteristic
410C23—41no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23C2×C4×D411yescharacteristic
418Q8—22no—
419Q8—22no—
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424Q8—41no—
425Q8—41no—
426D4—41no—
427D4—41no—
428D4—41no—
429D4—41no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
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—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4#51811yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#53511yescharacteristic
472C2×C4#53711yescharacteristic
473C2×C4#54011yescharacteristic
474C2×C4#54211yescharacteristic
475C2×C4C2×#13111yescharacteristic
476C2×C4C2×#13111yescharacteristic
477C2×C4C2×#13111yescharacteristic
478C2×C4C2×#13111yescharacteristic
479C2×C4#57711yescharacteristic
480C2×C4#57711yescharacteristic
481C2×C4#57711yescharacteristic
482C2×C4#57811yescharacteristic
483C8—22no—
484C8—22no—
485C8—42no—
486C8—41no—
487C8—41no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—81no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—81no—
502C22—81no—
503C22—81no—
504C22#232411yescharacteristic
505C22#292911yescharacteristic
506C22#293111yescharacteristic
507C22#294711yescharacteristic
508C22#294911yescharacteristic
509C22#295611yescharacteristic
510C22#295711yescharacteristic
511C4—21no—
512C4—21no—
513C4—41no—
514C4—41no—
515C4—41no—
516C4—42no—
517C4—42no—
518C4—41no—
519C4—81no—
520C4—81no—
521C4—81no—
522C4—81no—
523C4—81no—
524C4#297711yescharacteristic
525C4#297711yescharacteristic
526C4#299411yescharacteristic
527C4#299411yescharacteristic
528C4#322111yescharacteristic
529C4#322511yescharacteristic
530C4#325811yescharacteristic
531C4#326011yescharacteristic
532C2—161no—
533C2#1791911yescharacteristic
534C2#1792311yescharacteristic
535C2#1796911yescharacteristic
536C2#1797111yescharacteristic
537C2#2213411yescharacteristic
538C2#2216211yescharacteristic
539C2#2216711yescharacteristic
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 ◁ #1744 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22238C2semidirect
4. #9677C2semidirect
7. #9606C2semidirect
8. #9555C2semidirect
9. #9553C2semidirect
12. #8846C2semidirect
13. #8844C2semidirect
15. #8264C2semidirect

Maximal subgroups

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

subgroupindex
#236772
#222382
#220132
#96772
#96602
#96552
#96062
#95552
#95532
#89482
#88462
#88442
#85692
#82642