Finite Group Databasebeta

← #203235

Subgroups — #203235

Order 512.

661 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21266C211yesmaximal, characteristic
2#21120C211yesmaximal, characteristic
3C4×C8×Q8C211yesmaximal, characteristic, complemented
4#9542C211yesmaximal, characteristic
5#9541C211yesmaximal, characteristic
6#9536C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8948C211yesmaximal, characteristic
9#8948C211yesmaximal, characteristic
10#8846C211yesmaximal, characteristic, complemented
11#8844C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8181C211yesmaximal, characteristic
14#8173C211yesmaximal, characteristic, complemented
15#8132C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#433—21no—
20#2293C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1925C411yescharacteristic
27#1925C411yescharacteristic
28#1923—21no—
29#1923—21no—
30#1920C411yescharacteristic
31#1920C411yescharacteristic
32#1880C411yescharacteristic
33#1880C411yescharacteristic
34#1863C411yescharacteristic
35#1863C411yescharacteristic
36C4×#107C2211yescharacteristic
37C8×C4⋊C4—21no—
38C8×C4⋊C4—21no—
39C8×C4⋊C4C2211yescharacteristic
40C8×C4⋊C4C2211yescharacteristic
41#1767C2211yescharacteristic
42C4×C4⋊C8—21no—
43C4×C4⋊C8C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46#1750—21no—
47#1744C2211yescharacteristic
48C4×#97C2211yescharacteristic
49C42×C8—21no—
50C42×C8C2211yescharacteristic
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
68C2×C4×Q8C2×C411yescharacteristic
69C2×C4×Q8C2×C411yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C8×Q8—22no—
74C8×Q8—22no—
75C8×Q8—22no—
76C8×Q8—22no—
77C4×SD16—22no—
78C4×SD16—22no—
79#433—44no—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#106C2×C411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×#97—21no—
102C2×#97—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8D411yescharacteristic
110C2×C4×C8D411yescharacteristic
111C2×C4×C8D411yescharacteristic
112C2×C4×C8D411yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119#390C2×C411yescharacteristic
120#390C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#388—21no—
124#388—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4C2×C411yescharacteristic
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4C2311yescharacteristic
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C43—21no—
134C43C2311yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#340C2×C411yescharacteristic
140#340C2×C411yescharacteristic
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#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#335—22no—
152#334—22no—
153Q8⋊C8—22no—
154Q8⋊C8—22no—
155D4⋊C8—22no—
156D4⋊C8—22no—
157C22×Q8C22×C411yescharacteristic
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C2×SD16—22no—
161C2×SD16—22no—
162C2×SD16—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8C22×C411yescharacteristic
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C4×Q8—22no—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8#4112yes—
184C4×Q8#4112yes—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C22×C411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—22no—
222#107—22no—
223#106—22no—
224C4⋊C8—24no—
225C4⋊C8—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—42no—
233C4⋊C8—42no—
234C4⋊C8#4112yes—
235C4⋊C8#4112yes—
236#103—22no—
237#103—22no—
238#103—22no—
239#102—22no—
240#102—22no—
241#102—22no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—42no—
245C22⋊C8—42no—
246#97—24no—
247#97—42no—
248#97—42no—
249C4×C8—24no—
250C4×C8—24no—
251C4×C8—22no—
252C4×C8—22no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—41no—
266C4×C8—41no—
267C4×C8—42no—
268C4×C8—42no—
269C4×C8—42no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8#11712yes—
281C2×Q8#13112yes—
282C2×Q8#13112yes—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C4×D411yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309SD16—44no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
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—21no—
335C2×C8—21no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8AGL(1,ℤ/8)12yes—
351C2×C8#13712yes—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—21no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4#11712yes—
368C4⋊C4#13112yes—
369C4⋊C4#13112yes—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—41no—
373C22⋊C4—41no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C42—24no—
377C42—24no—
378C42—22no—
379C42—22no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—22no—
389C42—22no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—42no—
395C42—42no—
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C2×C4×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—22no—
411Q8#43314yescomplemented
412D4—41no—
413D4—41no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—22no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4#44312yes—
454C2×C4#44412yes—
455C2×C4C2×#11711yescharacteristic
456C2×C4C4×#4111yescharacteristic
457C2×C4#52511yescharacteristic
458C2×C4#52511yescharacteristic
459C2×C4#52911yescharacteristic
460C2×C4#52911yescharacteristic
461C2×C4C2×#13111yescharacteristic
462C2×C4C2×#13111yescharacteristic
463C2×C4C2×#13111yescharacteristic
464C2×C4C2×#13111yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C2×C4#57511yescharacteristic
468C2×C4#57511yescharacteristic
469C8—22no—
470C8—22no—
471C8—42no—
472C8—42no—
473C8—41no—
474C8—42no—
475C8—41no—
476C8—42no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—81no—
482C8—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—81no—
487C22—81no—
488C22—81no—
489C22—81no—
490C22#229011yescharacteristic
491C22#292911yescharacteristic
492C22#292911yescharacteristic
493C22C4×AGL(1,ℤ/8)11yescharacteristic
494C22C4×#13711yescharacteristic
495C22C2×#44311yescharacteristic
496C22C2×#44411yescharacteristic
497C4—24no—
498C4—21no—
499C4—21no—
500C4—22no—
501C4—22no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—41no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4C2×#43311yescharacteristic
511C4#292211yescharacteristic
512C4#296611yescharacteristic
513C4#296611yescharacteristic
514C4#311111yescharacteristic
515C4#311211yescharacteristic
516C4#311511yescharacteristic
517C4#311711yescharacteristic
518C2—161no—
519C2#1791111yescharacteristic
520C2#1791211yescharacteristic
521C2#1791511yescharacteristic
522C2#1791711yescharacteristic
523C2#2110311yescharacteristic
524C2#2112411yescharacteristic
525C2#2113711yescharacteristic
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 ◁ C42×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C4×C8×Q8C2semidirect
6. #9536C2semidirect
7. #9533C2semidirect
10. #8846C2semidirect
11. #8844C2semidirect
12. #8279C2semidirect
14. #8173C2semidirect
15. #8132C2semidirect
411. Q8#433semidirect

Maximal subgroups

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

subgroupindex
#212662
#211202
C4×C8×Q82
#95422
#95412
#95362
#95332
#89482
#88462
#88442
#82792
#81812
#81732
#81322