Finite Group Databasebeta

← #203280

Subgroups — #203280

Order 512.

658 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21343C211yesmaximal, characteristic, complemented
2#21114C211yesmaximal, 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#8279C211yesmaximal, characteristic, complemented
13#8178C211yesmaximal, characteristic, complemented
14#8170C211yesmaximal, characteristic
15#8134C211yesmaximal, characteristic, complemented
16C2×C4×SD16C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C22×#97—21no—
20#2295C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1984C2211yescharacteristic
23#1984C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1928C411yescharacteristic
27#1928C411yescharacteristic
28#1919C411yescharacteristic
29#1919C411yescharacteristic
30#1917—21no—
31#1917—21no—
32#1877C411yescharacteristic
33#1877C411yescharacteristic
34#1862C411yescharacteristic
35#1862C411yescharacteristic
36C4×#107C2211yescharacteristic
37#1770C2211yescharacteristic
38#1770C2211yescharacteristic
39#1768—21no—
40#1767C2211yescharacteristic
41C4×C4⋊C8C2211yescharacteristic
42C4×#103C2211yescharacteristic
43C4×#102C2211yescharacteristic
44#1752—21no—
45#1752—21no—
46#1749—21no—
47#1746—21no—
48#1744C2211yescharacteristic
49#1744C2211yescharacteristic
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×Q8C2311yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C22×C42—21no—
75C4×SD16—22no—
76C4×SD16—22no—
77C8×D4—22no—
78C8×D4—22no—
79C8×D4—22no—
80C8×D4—22no—
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#106C2×C411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#103C2×C411yescharacteristic
91C2×#103C2×C411yescharacteristic
92C2×#103C2×C411yescharacteristic
93C2×#102C2×C411yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×#97—21no—
100C2×#97—21no—
101C2×#97—44no—
102C2×#97D411yescharacteristic
103C2×#97D411yescharacteristic
104C2×#97D411yescharacteristic
105C2×#97D411yescharacteristic
106C2×C4×C8—21no—
107C2×C4×C8C2311yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391C2×C411yescharacteristic
114#391C2×C411yescharacteristic
115#389C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#387—21no—
118#387—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127#376—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#336—21no—
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#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—
160C23×C4—21no—
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—22no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—42no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4×Q8—21no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8—21no—
189C4×D4—22no—
190C4×D4—22no—
191C4×D4—21no—
192C4×D4—21no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4#4112yes—
196C4×D4#4112yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C22×C411yescharacteristic
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—41no—
224#107—41no—
225#106—22no—
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—41no—
238#103—41no—
239#102—22no—
240#102—22no—
241#102—22no—
242C22⋊C8—24no—
243C22⋊C8—24no—
244#97—22no—
245#97—22no—
246#97—41no—
247#97—41no—
248#97—42no—
249#97—42no—
250#97—42no—
251C4×C8—21no—
252C4×C8—21no—
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—42no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274C24—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4#11712yes—
285C2×D4#13112yes—
286C2×D4#13112yes—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313SD16—44no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—22no—
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—21no—
337C2×C8—21no—
338C2×C8—48no—
339C2×C8—48no—
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×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8AGL(1,ℤ/8)12yes—
357C2×C8#13712yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—81no—
372C4⋊C4—81no—
373C4⋊C4#11712yes—
374C22⋊C4—22no—
375C22⋊C4—22no—
376C22⋊C4—42no—
377C22⋊C4—42no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—41no—
387C42—41no—
388C42—42no—
389C42—42no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—22no—
402C23—22no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C2×C4×D411yescharacteristic
408Q8—41no—
409Q8—41no—
410Q8—41no—
411Q8—41no—
412D4—22no—
413D4—22no—
414D4—22no—
415D4#43314yes—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
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—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4C2×#11711yescharacteristic
458C2×C4C4×#4111yescharacteristic
459C2×C4#52511yescharacteristic
460C2×C4#52511yescharacteristic
461C2×C4#52911yescharacteristic
462C2×C4#52911yescharacteristic
463C2×C4C2×#13111yescharacteristic
464C2×C4C2×#13111yescharacteristic
465C2×C4#56811yescharacteristic
466C2×C4#56811yescharacteristic
467C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
468C2×C4C2×#13711yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57511yescharacteristic
471C8—22no—
472C8—22no—
473C8—42no—
474C8—41no—
475C8—42no—
476C8—41no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—41no—
481C8—81no—
482C8—81no—
483C8—81no—
484C8—81no—
485C22—24no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22#229011yescharacteristic
493C22#293311yescharacteristic
494C22#293311yescharacteristic
495C22C4×AGL(1,ℤ/8)11yescharacteristic
496C22C4×#13711yescharacteristic
497C22#295711yescharacteristic
498C22#295711yescharacteristic
499C4—21no—
500C4—21no—
501C4—41no—
502C4—41no—
503C4—41no—
504C4—41no—
505C4—42no—
506C4—81no—
507C4—81no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4C2×#43311yescharacteristic
513C4#292311yescharacteristic
514C4#296811yescharacteristic
515C4#296811yescharacteristic
516C4#311011yescharacteristic
517C4#311111yescharacteristic
518C4#311411yescharacteristic
519C4#311611yescharacteristic
520C2—44no—
521C2#1800611yescharacteristic
522C2#1800811yescharacteristic
523C2#1800911yescharacteristic
524C2#1800911yescharacteristic
525C2#2110511yescharacteristic
526C2#2112111yescharacteristic
527C2#2114011yescharacteristic
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. #21343C2semidirect
4. #9548C2semidirect
5. #9547C2semidirect
6. #9540C2semidirect
7. #9537C2semidirect
12. #8279C2semidirect
13. #8178C2semidirect
15. #8134C2semidirect

Maximal subgroups

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

subgroupindex
#213432
#211142
D4×#972
#95482
#95472
#95402
#95372
#89452
#88452
#88432
#82792
#81782
#81702
#81342