Finite Group Databasebeta

← #495822

Subgroups — #495822

Order 512.

784 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2206C211yesmaximal, characteristic
2C2×#2177C211yesmaximal, characteristic, complemented
3C2×C4×D16C211yesmaximal, characteristic, complemented
4#21697C211yesmaximal, characteristic, complemented
5#20508C211yesmaximal, characteristic, complemented
6#13252C211yesmaximal, characteristic, complemented
7#13178C211yesmaximal, characteristic, complemented
8#13131C211yesmaximal, characteristic, complemented
9#13130C211yesmaximal, characteristic, complemented
10#13012C211yesmaximal, characteristic, complemented
11#13012C211yesmaximal, characteristic, complemented
12#12813C211yesmaximal, characteristic, complemented
13#12654C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15C4×#357C211yesmaximal, characteristic, complemented
16C22×D16C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#493C2211yescharacteristic
19C2×#442—21no—
20C2×C4×D8C2211yescharacteristic
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×C4×D8C2211yescharacteristic
25#2328—21no—
26#2275—21no—
27#2206C418yescomplemented
28#2177—24no—
29C4×D16—24no—
30C2×#367C2211yescharacteristic
31C2×#366C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C4⋊C16C2211yescharacteristic
34C2×#357C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×#357C2211yescharacteristic
41C2×#357C2211yescharacteristic
42C2×#346C2211yescharacteristic
43C2×C4×C16C2211yescharacteristic
44#1972—21no—
45#1970C2211yescharacteristic
46#1970C2211yescharacteristic
47#1961—21no—
48#1846C2211yescharacteristic
49C4×#107C2211yescharacteristic
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1762—21no—
53#1762—21no—
54#1762—21no—
55#1726C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×D8—21no—
63C22×D8C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C22×D8C2311yescharacteristic
66C2×#127—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×D16—22no—
73C2×D16—22no—
74C2×D16—22no—
75C2×D16—22no—
76C2×D16C2×C414yescomplemented
77C2×D16C2×C414yescomplemented
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81C22×C16C2311yescharacteristic
82#493—22no—
83#493C2×C414yescomplemented
84#442—44no—
85C4×D8—22no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89C4×D8—22no—
90C4×D8—22no—
91C4×D8—22no—
92C4×D8—22no—
93C4×D8C2×C414yescomplemented
94C4×D8D414yescomplemented
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#107C2311yescharacteristic
98C2×#107C2311yescharacteristic
99C2×#107C2311yescharacteristic
100C2×#106—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×#102—21no—
108C2×#97D411yescharacteristic
109C2×#97D411yescharacteristic
110C2×#97D411yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#390—21no—
115#390—21no—
116#386—41no—
117#386—41no—
118#386—41no—
119#386—41no—
120#382—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C22⋊C4—41no—
125C4×C22⋊C4—41no—
126C4×C22⋊C4—41no—
127#376D411yescharacteristic
128C2×#95—41no—
129#367—22no—
130#366—22no—
131C4⋊C16—22no—
132C4⋊C16C2×C414yescomplemented
133#357—22no—
134#357—22no—
135#357—22no—
136#357—22no—
137#357—22no—
138#357—22no—
139#357C2×C414yescomplemented
140#357C2×C414yescomplemented
141#346—22no—
142C4×C16—22no—
143#340—21no—
144#340—21no—
145#340—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—41no—
150C22×D4—41no—
151C23×C4—41no—
152C23×C4—41no—
153C23×C4—41no—
154C23×C4—41no—
155C23×C4—41no—
156C2×D8—24no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
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—21no—
169C2×D8—21no—
170C2×D8—42no—
171C2×D8—42no—
172C2×D8C22×C412yes—
173C2×D8C22×C412yes—
174C2×D8C22×C412yes—
175C2×D8C2×D412yes—
176C2×D8#4112yes—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2411yescharacteristic
181#127—24no—
182#127—42no—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—42no—
187C4×D4—42no—
188C4×D4—42no—
189C4×D4—42no—
190C4×D4—42no—
191C4×D4—42no—
192C4×D4—42no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C22⋊C4—41no—
199C2×C22⋊C4—41no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—41no—
208C2×C42—41no—
209C2×C42—41no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213D16—28no—
214D16—28no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223C2×C16—21no—
224C2×C16—21no—
225C2×C16C22×C412yes—
226C2×C16C22×C412yes—
227#107—21no—
228#107—21no—
229#107—21no—
230#107—21no—
231#107—21no—
232#107—21no—
233#107—21no—
234#107C22×C412yes—
235#107C2×D412yes—
236#107#4112yes—
237#106—42no—
238#102—24no—
239#102—24no—
240#102—24no—
241#102—42no—
242#102—42no—
243#102—42no—
244#102—42no—
245#97—22no—
246#97—22no—
247#97—41no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8C22×C412yes—
251C4×C8C2×D412yes—
252C4×C8#4112yes—
253C4×C8#4112yes—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95D811yescharacteristic
260#95D811yescharacteristic
261#95D811yescharacteristic
262#95D811yescharacteristic
263C24—41no—
264C24—41no—
265C24—41no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—44no—
285C2×D4—44no—
286C2×D4—161no—
287C2×D4—161no—
288C2×D4—161no—
289C2×D4—161no—
290C22×C4—44no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—161no—
301C22×C4—161no—
302C22×C4—161no—
303C22×C4—161no—
304C22×C4—161no—
305C22×C4—161no—
306C22×C4—161no—
307C22×C4—161no—
308C22×C4—161no—
309C22×C4—161no—
310C22×C4C2×D811yescharacteristic
311C22×C4C2×D811yescharacteristic
312C22×C4C2×D811yescharacteristic
313C22×C4C2×D811yescharacteristic
314C22×C4C2×D811yescharacteristic
315C22×C4C2×D811yescharacteristic
316C22×C4C22×D411yescharacteristic
317D8—22no—
318D8—22no—
319D8—22no—
320D8—22no—
321D8—22no—
322D8—22no—
323D8—22no—
324D8—48no—
325D8—41no—
326D8—41no—
327D8—41no—
328D8—41no—
329D8C4×D414yes—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8C4×D412yes—
338C2×C8#12112yes—
339C2×C8C23×C411yescharacteristic
340C2×C8C22×D411yescharacteristic
341C2×C8C2×#4111yescharacteristic
342C2×C8C2×#4111yescharacteristic
343C2×C8C2×#4111yescharacteristic
344C2×C8C2×#4111yescharacteristic
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—82no—
359C22⋊C4—82no—
360C22⋊C4—161no—
361C22⋊C4—161no—
362C22⋊C4—161no—
363C22⋊C4—161no—
364C42—21no—
365C42—21no—
366C42—161no—
367C42—161no—
368C42—161no—
369C42C4×D412yes—
370C42#12112yes—
371C42#13512yes—
372C42AGL(1,ℤ/8)12yes—
373C16—22no—
374C16—22no—
375C16—41no—
376C16—41no—
377C23—42no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23—81no—
390C23—81no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23C22×D811yescharacteristic
399D4—24no—
400D4—42no—
401D4—42no—
402D4—42no—
403D4—42no—
404D4—42no—
405D4—42no—
406D4—42no—
407D4—81no—
408D4—81no—
409D4—81no—
410D4—81no—
411D4—164no—
412D4—164no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
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—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—164no—
443C2×C4—164no—
444C2×C4—164no—
445C2×C4—164no—
446C2×C4—164no—
447C2×C4C4×D812yes—
448C2×C4C4×D812yes—
449C2×C4#45912yes—
450C2×C4#45912yes—
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×#12111yescharacteristic
453C2×C4#52511yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
456C2×C4#57511yescharacteristic
457C8—22no—
458C8—22no—
459C8—41no—
460C8C2×C4×D411yescharacteristic
461C8C4×#4111yescharacteristic
462C8#52911yescharacteristic
463C8#53211yescharacteristic
464C22—44no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22—161no—
478C22—161no—
479C22—161no—
480C22—161no—
481C22—161no—
482C22—161no—
483C22—161no—
484C22—161no—
485C22C2×C4×D811yescharacteristic
486C22C2×#45911yescharacteristic
487C22#310011yescharacteristic
488C22#341511yescharacteristic
489C22#341511yescharacteristic
490C22#341511yescharacteristic
491C22#341611yescharacteristic
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—41no—
497C4—41no—
498C4—82no—
499C4—82no—
500C4—82no—
501C4—161no—
502C4—161no—
503C4C4×#12111yescharacteristic
504C4C4×AGL(1,ℤ/8)11yescharacteristic
505C4#303511yescharacteristic
506C4#310611yescharacteristic
507C2—84no—
508C2—162no—
509C2—162no—
510C2C4×#45911yescharacteristic
511C2#3345411yescharacteristic
512C2#3345611yescharacteristic
513C2#3358511yescharacteristic
514C2#3358911yescharacteristic
515C2#3367911yescharacteristic
516C2#3368211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC2×#2206C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ #376 ◁ #1726 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2177C2semidirect
3. C2×C4×D16C2semidirect
4. #21697C2semidirect
5. #20508C2semidirect
6. #13252C2semidirect
7. #13178C2semidirect
8. #13131C2semidirect
9. #13130C2semidirect
10. #13012C2semidirect
11. #13012C2semidirect
12. #12813C2semidirect
13. #12654C2semidirect
14. #12637C2semidirect
15. C4×#357C2semidirect
27. #2206C4semidirect
76. C2×D16C2×C4semidirect
77. C2×D16C2×C4semidirect
83. #493C2×C4semidirect
93. C4×D8C2×C4semidirect
94. C4×D8D4semidirect
132. C4⋊C16C2×C4semidirect
139. #357C2×C4semidirect
140. #357C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#22062
C2×#21772
C2×C4×D162
#216972
#205082
#132522
#131782
#131312
#131302
#130122
#128132
#126542
#126372
C4×#3572