Finite Group Databasebeta

← #203382

Subgroups — #203382

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#21264C211yesmaximal, characteristic
2#21114C211yesmaximal, characteristic
3#20516C211yesmaximal, characteristic, complemented
4#9545C211yesmaximal, characteristic
5#9542C211yesmaximal, characteristic
6#9537C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic
9#8948C211yesmaximal, characteristic
10#8850C211yesmaximal, characteristic, complemented
11#8844C211yesmaximal, characteristic, complemented
12#8279C211yesmaximal, characteristic, complemented
13#8180C211yesmaximal, characteristic, complemented
14#8172C211yesmaximal, characteristic
15#8132C211yesmaximal, characteristic, complemented
16C2×#446C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×#431—21no—
20#2295C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#1954C2211yescharacteristic
23#1954C2211yescharacteristic
24#1939C2211yescharacteristic
25#1939C2211yescharacteristic
26#1928C411yescharacteristic
27#1928C411yescharacteristic
28#1923—21no—
29#1923—21no—
30#1919C411yescharacteristic
31#1919C411yescharacteristic
32#1877C411yescharacteristic
33#1877C411yescharacteristic
34#1862C411yescharacteristic
35#1862C411yescharacteristic
36C4×#107C2211yescharacteristic
37#1771—21no—
38#1770C2211yescharacteristic
39C8×C4⋊C4—21no—
40C8×C4⋊C4C2211yescharacteristic
41#1768—21no—
42#1767C2211yescharacteristic
43C4×C4⋊C8C2211yescharacteristic
44C4×#103C2211yescharacteristic
45C4×#102C2211yescharacteristic
46C4×C22⋊C8—21no—
47#1744C2211yescharacteristic
48#1726—21no—
49C4×#97C2211yescharacteristic
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×Q8C2311yescharacteristic
69C2×C4×D4C2×C411yescharacteristic
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2311yescharacteristic
72C22×C42—21no—
73#446—22no—
74#446—22no—
75C8×Q8—22no—
76C8×Q8—22no—
77C4×SD16—22no—
78C4×SD16—22no—
79#431—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×#97—21no—
105C2×#97—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#391C2×C411yescharacteristic
120#391C2×C411yescharacteristic
121#389C2×C411yescharacteristic
122#389C2×C411yescharacteristic
123#387—21no—
124#387—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C4×C22⋊C4—21no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—21no—
133#376—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—
160C23×C4—21no—
161C23×C4—21no—
162C2×SD16—22no—
163C2×SD16—22no—
164C2×SD16—41no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C22×C411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8#4112yes—
182C4×Q8#4112yes—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C4⋊C4C22×C411yescharacteristic
199C2×C4⋊C4C22×C411yescharacteristic
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#107—22no—
218#107—41no—
219#106—22no—
220C4⋊C8—24no—
221C4⋊C8—24no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—42no—
229C4⋊C8—42no—
230C4⋊C8#4112yes—
231C4⋊C8#4112yes—
232#103—22no—
233#103—22no—
234#103—41no—
235#102—22no—
236#102—22no—
237#102—41no—
238C22⋊C8—42no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241C22⋊C8—42no—
242#97—24no—
243#97—24no—
244#97—42no—
245#97—42no—
246#97—42no—
247C4×C8—24no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—21no—
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—41no—
263C4×C8—41no—
264C4×C8—42no—
265C4×C8—42no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
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—21no—
281C2×Q8—21no—
282C2×Q8#11712yes—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4#13112yes—
289C22×C4—24no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4C4×D411yescharacteristic
309C22×C4C4×D411yescharacteristic
310C22×C4C4×D411yescharacteristic
311C22×C4C4×D411yescharacteristic
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C2×#4111yescharacteristic
315SD16—44no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—22no—
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—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—22no—
360C4⋊C4—22no—
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—21no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4#11712yes—
373C4⋊C4#13112yes—
374C22⋊C4—24no—
375C22⋊C4—42no—
376C22⋊C4—41no—
377C22⋊C4—41no—
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—42no—
391C42—42no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—41no—
396C42C2×#4111yescharacteristic
397C42C2×#4111yescharacteristic
398C42C2×#4111yescharacteristic
399C42C2×#4111yescharacteristic
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23C2×C4×D411yescharacteristic
408Q8—22no—
409Q8—22no—
410Q8—41no—
411Q8—41no—
412D4—22no—
413D4—22no—
414D4—41no—
415D4—41no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—48no—
432C2×C4—41no—
433C2×C4—41no—
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—41no—
450C2×C4—84no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4#44312yes—
456C2×C4#44412yes—
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—42no—
475C8—41no—
476C8—42no—
477C8—41no—
478C8—42no—
479C8—41no—
480C8—41no—
481C8—41no—
482C8—41no—
483C8—81no—
484C8—81no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—81no—
489C22—81no—
490C22—81no—
491C22—81no—
492C22#229011yescharacteristic
493C22#292911yescharacteristic
494C22#293011yescharacteristic
495C22C4×AGL(1,ℤ/8)11yescharacteristic
496C22C4×#13711yescharacteristic
497C22C2×#44311yescharacteristic
498C22C2×#44411yescharacteristic
499C4—22no—
500C4—21no—
501C4—21no—
502C4—22no—
503C4—41no—
504C4—42no—
505C4—42no—
506C4—41no—
507C4—42no—
508C4—81no—
509C4—81no—
510C4—81no—
511C4—81no—
512C4#292411yescharacteristic
513C4#292511yescharacteristic
514C4#296811yescharacteristic
515C4#296911yescharacteristic
516C4#311111yescharacteristic
517C4#311211yescharacteristic
518C4#311511yescharacteristic
519C4#311711yescharacteristic
520C2—82no—
521C2#1791311yescharacteristic
522C2#1791411yescharacteristic
523C2#1794111yescharacteristic
524C2#1794211yescharacteristic
525C2#2110911yescharacteristic
526C2#2112411yescharacteristic
527C2#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. #20516C2semidirect
6. #9537C2semidirect
7. #9536C2semidirect
10. #8850C2semidirect
11. #8844C2semidirect
12. #8279C2semidirect
13. #8180C2semidirect
15. #8132C2semidirect

Maximal subgroups

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

subgroupindex
#212642
#211142
#205162
#95452
#95422
#95372
#95362
#89522
#89482
#88502
#88442
#82792
#81802
#81722
#81322