Finite Group Databasebeta

← #226504

Subgroups — #226504

Order 512.

707 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#22821C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic
3C4×#440C211yesmaximal, characteristic
4#9555C211yesmaximal, characteristic, complemented
5#9553C211yesmaximal, characteristic, complemented
6#9536C211yesmaximal, characteristic, complemented
7#9533C211yesmaximal, characteristic, complemented
8#8945C211yesmaximal, characteristic
9#8945C211yesmaximal, characteristic
10#8786C211yesmaximal, characteristic, complemented
11#8566C211yesmaximal, characteristic, complemented
12#8563C211yesmaximal, characteristic
13#8262C211yesmaximal, characteristic, complemented
14#8200C211yesmaximal, characteristic
15C8×#102C211yesmaximal, characteristic
16C2×#440C411yescharacteristic
17C2×#440C411yescharacteristic
18C2×C4×SD16C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C42×Q8C2211yescharacteristic
22C42×D4C2211yescharacteristic
23#1984C2211yescharacteristic
24#1984C2211yescharacteristic
25#1939C2211yescharacteristic
26#1939C2211yescharacteristic
27#1916—21no—
28#1916—21no—
29#1850—21no—
30#1850—21no—
31#1842C2211yescharacteristic
32#1831—21no—
33#1831C2211yescharacteristic
34#1797—21no—
35#1776C411yescharacteristic
36#1776C411yescharacteristic
37C4×#107C2211yescharacteristic
38C8×C4⋊C4C2211yescharacteristic
39C8×C4⋊C4C2211yescharacteristic
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1763C411yescharacteristic
44#1763C411yescharacteristic
45#1762C411yescharacteristic
46#1762C411yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C2×#336—21no—
50C4×#97C2211yescharacteristic
51C2×Q8⋊C8C2211yescharacteristic
52C2×Q8⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16C2×C411yescharacteristic
68C23×C8—21no—
69C23×C8—21no—
70C2×C4×Q8C2×C411yescharacteristic
71C2×C4×Q8C2×C411yescharacteristic
72C2×C4×Q8C2311yescharacteristic
73C2×C4×D4C2×C411yescharacteristic
74C2×C4×D4C2×C411yescharacteristic
75C2×C4×D4C2311yescharacteristic
76C22×C42—21no—
77#440—24no—
78#440C818yescomplemented
79C4×SD16—22no—
80C4×SD16—22no—
81C8×D4—22no—
82C8×D4—22no—
83C8×D4—41no—
84C8×D4—41no—
85C2×#107C2×C411yescharacteristic
86C2×#107C2×C411yescharacteristic
87C2×#106C2×C411yescharacteristic
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8D411yescharacteristic
94C2×C4⋊C8D411yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#103C2×C411yescharacteristic
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#102C2×C411yescharacteristic
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×#97C2×C411yescharacteristic
108C2×#97C2×C411yescharacteristic
109C2×#97C2×C411yescharacteristic
110C2×#97C2×C411yescharacteristic
111C2×C4×C8—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115C2×C4×C8C2311yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118C2×C4×C8C2311yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131C43—21no—
132C43C2311yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
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#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152Q8⋊C8—41no—
153Q8⋊C8—41no—
154D4⋊C8—22no—
155D4⋊C8—22no—
156#322—41no—
157C82—41no—
158C22×Q8C22×C411yescharacteristic
159C22×D4C22×C411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C2×SD16—22no—
164C2×SD16—22no—
165C2×SD16C2×C814yes—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—42no—
173C22×C8—42no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—81no—
177C22×C8—81no—
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—41no—
193C4×Q8C2×C814yescomplemented
194C4×D4—22no—
195C4×D4—22no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4—41no—
199C4×D4C2×C814yes—
200C4×D4#4112yes—
201C4×D4#4112yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C4⋊C4C22×C411yescharacteristic
208C2×C4⋊C4C22×C411yescharacteristic
209C2×C4⋊C4C22×C411yescharacteristic
210C2×C4⋊C4C22×C411yescharacteristic
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42C22×C411yescharacteristic
226C2×C42C22×C411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42C2411yescharacteristic
232#107—22no—
233#107C2×C814yescomplemented
234#106—22no—
235C4⋊C8—22no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—41no—
245C4⋊C8—42no—
246C4⋊C8—42no—
247C4⋊C8—81no—
248C4⋊C8—81no—
249C4⋊C8—81no—
250#103—22no—
251#103—22no—
252#103C2×C814yescomplemented
253#102—22no—
254#102C2×C814yes—
255#102M1614yes—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—22no—
259#97—22no—
260#97—41no—
261#97C2×C814yescomplemented
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—21no—
273C4×C8—21no—
274C4×C8—81no—
275C4×C8—81no—
276#95—21no—
277#95—21no—
278#95—21no—
279#95—21no—
280#95—21no—
281#95—21no—
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8C22×C812yes—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4#11712yes—
293C2×D4C22×C812yes—
294C2×D4C2×M1612yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321SD16—28no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—22no—
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—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×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—84no—
361C2×C8—84no—
362C2×C8C22×C812yes—
363C2×C8C22×C812yes—
364C2×C8C2×M1612yes—
365C2×C8#13712yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—24no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—42no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4#11712yes—
379C4⋊C4C22×C812yes—
380C4⋊C4C22×C812yes—
381C4⋊C4C2×M1612yes—
382C22⋊C4—22no—
383C22⋊C4—22no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—21no—
392C42—24no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—81no—
404C42—81no—
405C42—81no—
406C42C22×C812yes—
407C42#13112yes—
408C42C2×#4111yescharacteristic
409C42C2×#4111yescharacteristic
410C42H(5,𝔽2)11yescharacteristic
411C422−1+411yescharacteristic
412C23—22no—
413C23—22no—
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23C2×C4×D411yescharacteristic
420Q8—22no—
421Q8—22no—
422Q8—41no—
423Q8—41no—
424D4—22no—
425D4—22no—
426D4—22no—
427D4#43114yes—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
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—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—84no—
466C2×C4—84no—
467C2×C4C8×D412yes—
468C2×C4C8×D412yes—
469C2×C4C2×#11711yescharacteristic
470C2×C4#52011yescharacteristic
471C2×C4#54611yescharacteristic
472C2×C4#54811yescharacteristic
473C2×C4#55211yescharacteristic
474C2×C4#55311yescharacteristic
475C2×C4C23×C811yescharacteristic
476C2×C4C22×M1611yescharacteristic
477C2×C4C2×#13111yescharacteristic
478C2×C4#56811yescharacteristic
479C2×C4C2×#13711yescharacteristic
480C2×C4#57511yescharacteristic
481C2×C4#57611yescharacteristic
482C2×C4#57711yescharacteristic
483C8—22no—
484C8—22no—
485C8—22no—
486C8—22no—
487C8—42no—
488C8—41no—
489C8—41no—
490C8—41no—
491C8—41no—
492C8—81no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C22—24no—
498C22—42no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—42no—
503C22—42no—
504C22#235211yescharacteristic
505C22C2×C8×D411yescharacteristic
506C22D4×M1611yescharacteristic
507C22C4×#13711yescharacteristic
508C22#294611yescharacteristic
509C22#295511yescharacteristic
510C22#295611yescharacteristic
511C4—21no—
512C4—21no—
513C4—22no—
514C4—22no—
515C4—41no—
516C4—41no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—81no—
523C4—81no—
524C4C2×#43111yescharacteristic
525C4#292511yescharacteristic
526C4#298811yescharacteristic
527C4#299011yescharacteristic
528C4#331111yescharacteristic
529C4#331411yescharacteristic
530C4#337011yescharacteristic
531C4#337211yescharacteristic
532C2—44no—
533C2C8×#13711yescharacteristic
534C2#1785211yescharacteristic
535C2#1804711yescharacteristic
536C2#1805011yescharacteristic
537C2#2274811yescharacteristic
538C2#2277611yescharacteristic
539C2#2278411yescharacteristic
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 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #22821C2semidirect
4. #9555C2semidirect
5. #9553C2semidirect
6. #9536C2semidirect
7. #9533C2semidirect
10. #8786C2semidirect
11. #8566C2semidirect
13. #8262C2semidirect
78. #440C8semidirect
193. C4×Q8C2×C8semidirect
233. #107C2×C8semidirect
252. #103C2×C8semidirect
261. #97C2×C8semidirect

Maximal subgroups

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

subgroupindex
#228212
#226962
C4×#4402
#95552
#95532
#95362
#95332
#89452
#87862
#85662
#85632
#82622
#82002
C8×#1022