Finite Group Databasebeta

← #503638

Subgroups — #503638

Order 512.

785 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
1C2×#2226C211yesmaximal, characteristic, complemented
2C2×#2140C211yesmaximal, characteristic
3#26064C211yesmaximal, characteristic, complemented
4C22⋊C4×D8C211yesmaximal, characteristic
5#13528C211yesmaximal, characteristic, complemented
6#13476C211yesmaximal, characteristic, complemented
7#13474C211yesmaximal, characteristic, complemented
8#13363C211yesmaximal, characteristic, complemented
9#12935C211yesmaximal, characteristic, complemented
10#12935C211yesmaximal, characteristic, complemented
11#12921C211yesmaximal, characteristic, complemented
12#12913C211yesmaximal, characteristic, complemented
13#12836C211yesmaximal, characteristic, complemented
14#12813C211yesmaximal, characteristic, complemented
15#12731C211yesmaximal, characteristic
16C23×D8C2211yescharacteristic
17C22×M32C2211yescharacteristic
18C2×#500C2211yescharacteristic, complemented
19C2×C4×D8C2211yescharacteristic
20C2×C4×D8C2211yescharacteristic
21C2×#427C2211yescharacteristic
22#2439—21no—
23D4×C22⋊C4—21no—
24#2226—24no—
25C2×#367C2211yescharacteristic, complemented
26C2×#367C2211yescharacteristic, complemented
27C2×#366C2211yescharacteristic, complemented
28C2×#366C2211yescharacteristic, complemented
29C2×C4⋊C16C2211yescharacteristic, complemented
30C2×C4⋊C16C2211yescharacteristic, complemented
31#2140—24no—
32C2×#357C2211yescharacteristic
33C2×#357C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41#2085—21no—
42#2058—21no—
43#1970C2211yescharacteristic
44#1970C2211yescharacteristic
45#1942C2211yescharacteristic, complemented
46#1942C2211yescharacteristic, complemented
47#1879—21no—
48#1879—21no—
49#1854—21no—
50#1846C2211yescharacteristic, complemented
51#1808C2211yescharacteristic
52#1808C2211yescharacteristic
53#1786—21no—
54C8×C22⋊C4C2211yescharacteristic
55#1380C2211yescharacteristic, complemented
56#1380C2211yescharacteristic, complemented
57#1380C2211yescharacteristic, complemented
58#1380C2211yescharacteristic, complemented
59#1374C2211yescharacteristic, complemented
60#1374C2211yescharacteristic, complemented
61C23×D4—21no—
62C22×D8—22no—
63C22×D8—22no—
64C22×D8—21no—
65C22×D8—22no—
66C22×D8—41no—
67C22×D8—41no—
68C22×D8Q814yescomplemented
69C22×D8C2311yescharacteristic
70C22×D8C2311yescharacteristic
71C23×C8C2311yescharacteristic
72C2×C4⋊Q8—21no—
73C2×C4×D4—21no—
74C2×C4×D4—21no—
75C2×#117—21no—
76C22×C22⋊C4—41no—
77C2×M32—22no—
78C2×M32—22no—
79C2×M32—41no—
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82C22×C16C2311yescharacteristic
83C22×C16C2311yescharacteristic
84#500—22no—
85#500—41no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—41no—
89C4×D8—41no—
90#427—22no—
91#427—41no—
92C2×#107—21no—
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#106—21no—
100C2×#106—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C22⋊C8D411yescharacteristic
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#396—41no—
109#393—41no—
110#391—41no—
111#390—21no—
112#390—21no—
113#389—21no—
114#389—21no—
115#387—41no—
116#386—41no—
117#386—41no—
118#382—21no—
119#380—21no—
120#380—21no—
121#379—41no—
122C4×C22⋊C4D411yescharacteristic
123#367—22no—
124#367—22no—
125#366—41no—
126#366—41no—
127C4⋊C16—22no—
128C4⋊C16—41no—
129#357—22no—
130#357—22no—
131#357—22no—
132#357—22no—
133#357—22no—
134#357—41no—
135#357—41no—
136#357D414yes—
137C22⋊C16—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144C25—41no—
145C22×Q8—41no—
146C22×D4—24no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—42no—
150C22×D4—42no—
151C22×D4—41no—
152C22×D4—42no—
153C22×D4—81no—
154C22×D4—81no—
155C23×C4—41no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C2×D8—28no—
159C2×D8—24no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—21no—
164C2×D8—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—21no—
168C2×D8—44no—
169C2×D8—44no—
170C2×D8—44no—
171C2×D8—42no—
172C2×D8—42no—
173C2×D8C2×D412yes—
174C2×D8C2×Q812yes—
175C2×D8C2×Q812yes—
176C2×D8#4112yes—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×Q812yes—
185C22×C8#4112yes—
186C22×C8C2411yescharacteristic
187C4⋊Q8—42no—
188C4⋊Q8—81no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—81no—
192C4×D4—81no—
193#117—42no—
194#117—81no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C22⋊C4—41no—
213C2×C22⋊C4—84no—
214C2×C22⋊C4D811yescharacteristic
215C2×C22⋊C4D811yescharacteristic
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219M32—44no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16—21no—
226C2×C16—21no—
227C2×C16—21no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—21no—
231C2×C16C2×D412yes—
232#107—21no—
233#107—21no—
234#107—21no—
235#107—21no—
236#107—21no—
237#107—21no—
238#107—21no—
239#107—21no—
240#107—21no—
241#107—21no—
242#107—21no—
243#107—42no—
244#107C2×D412yes—
245#106—42no—
246#106—81no—
247#102—42no—
248#102—42no—
249#102—81no—
250#102—81no—
251C22⋊C8—22no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260#95—41no—
261#95—41no—
262#95D811yescharacteristic
263#95D811yescharacteristic
264C24—41no—
265C24—41no—
266C24—44no—
267C24—82no—
268C24—82no—
269C24—82no—
270C24—82no—
271C24—161no—
272C24—161no—
273C24C2×D811yescharacteristic
274C2×Q8—82no—
275C2×Q8—82no—
276C2×Q8—161no—
277C2×D4—22no—
278C2×D4—22no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—48no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—44no—
291C2×D4—84no—
292C2×D4—84no—
293C2×D4—84no—
294C2×D4—82no—
295C2×D4—82no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—161no—
311C22×C4—161no—
312C22×C4C22⋊Q812yes—
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C2×D811yescharacteristic
317C22×C4C2×D811yescharacteristic
318C22×C4#13712yes—
319C22×C4C22×D411yescharacteristic
320D8—22no—
321D8—22no—
322D8—22no—
323D8—22no—
324D8—22no—
325D8—48no—
326D8—41no—
327D8—41no—
328D8C22⋊Q814yes—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—22no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8C22⋊Q812yes—
342C2×C8C22×D411yescharacteristic
343C2×C8C22×Q811yescharacteristic
344C2×C8C2×#4111yescharacteristic
345C2×C8C2×#4111yescharacteristic
346C2×C8H(5,𝔽2)11yescharacteristic
347C2×C8H(5,𝔽2)11yescharacteristic
348C4⋊C4—22no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C4⋊C4—161no—
363C22⋊C4—22no—
364C22⋊C4—41no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C22⋊C4—82no—
368C22⋊C4—82no—
369C22⋊C4—82no—
370C22⋊C4—82no—
371C22⋊C4—161no—
372C22⋊C4—161no—
373C22⋊C4—161no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—82no—
379C16—22no—
380C16—41no—
381C16—41no—
382C16—41no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—88no—
394C23—88no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—81no—
400C23—81no—
401C23—81no—
402C23—81no—
403C23—81no—
404C23—81no—
405C23—164no—
406C23—164no—
407C23—164no—
408C23—164no—
409C23—164no—
410C23#47412yes—
411C23C22×D811yescharacteristic
412Q8—164no—
413D4—24no—
414D4—42no—
415D4—42no—
416D4—42no—
417D4—42no—
418D4—42no—
419D4—88no—
420D4—81no—
421D4—81no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—164no—
458C2×C4—164no—
459C2×C4#47412yes—
460C2×C4#50912yes—
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4D4211yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4C2×#13711yescharacteristic
465C2×C4#57611yescharacteristic
466C2×C4#57711yescharacteristic
467C8—22no—
468C8—41no—
469C8—41no—
470C8C2×C22⋊Q811yescharacteristic
471C8#54311yescharacteristic
472C8#54611yescharacteristic
473C8#55111yescharacteristic
474C22—22no—
475C22—22no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—44no—
482C22—82no—
483C22—82no—
484C22—82no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22—1616no—
492C22—161no—
493C22—161no—
494C22—161no—
495C22—161no—
496C22C2×#47411yescharacteristic
497C22D4×D811yescharacteristic
498C22#329411yescharacteristic
499C22C2×#50911yescharacteristic
500C22#341411yescharacteristic
501C22#341511yescharacteristic
502C22#341611yescharacteristic
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—82no—
509C4—161no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#243711yescharacteristic
515C4#308111yescharacteristic
516C4#326611yescharacteristic
517C4#331411yescharacteristic
518C2—42no—
519C2—84no—
520C2—162no—
521C2#2595011yescharacteristic
522C2#3362711yescharacteristic
523C2#3363111yescharacteristic
524C2#3383111yescharacteristic
525C2#3383911yescharacteristic
526C2#3387811yescharacteristic
527C2#3389211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC23×D8C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2226C2semidirect
3. #26064C2semidirect
5. #13528C2semidirect
6. #13476C2semidirect
7. #13474C2semidirect
8. #13363C2semidirect
9. #12935C2semidirect
10. #12935C2semidirect
11. #12921C2semidirect
12. #12913C2semidirect
13. #12836C2semidirect
14. #12813C2semidirect
18. C2×#500C22semidirect
25. C2×#367C22semidirect
26. C2×#367C22semidirect
27. C2×#366C22semidirect
28. C2×#366C22semidirect
29. C2×C4⋊C16C22semidirect
30. C2×C4⋊C16C22semidirect
45. #1942C22semidirect
46. #1942C22semidirect
50. #1846C22semidirect
55. #1380C22semidirect
56. #1380C22semidirect
57. #1380C22semidirect
58. #1380C22semidirect
59. #1374C22semidirect
60. #1374C22semidirect
68. C22×D8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#22262
C2×#21402
#260642
C22⋊C4×D82
#135282
#134762
#134742
#133632
#129352
#129212
#129132
#128362
#128132
#127312