Finite Group Databasebeta

← #512168

Subgroups — #512168

Order 512.

693 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2228C211yesmaximal, characteristic, complemented
2C2×#2210C211yesmaximal, characteristic
3#33240C211yesmaximal, characteristic, complemented
4C42×D8C211yesmaximal, characteristic, complemented
5#13277C211yesmaximal, characteristic, complemented
6#13254C211yesmaximal, characteristic, complemented
7#13133C211yesmaximal, characteristic, complemented
8#13132C211yesmaximal, characteristic, complemented
9#13131C211yesmaximal, characteristic, complemented
10#13130C211yesmaximal, characteristic, complemented
11#13014C211yesmaximal, characteristic, complemented
12#13014C211yesmaximal, characteristic, complemented
13#13012C211yesmaximal, characteristic
14#13012C211yesmaximal, characteristic
15#12879C211yesmaximal, characteristic, complemented
16C22×SD32C2211yescharacteristic
17C22×D16C2211yescharacteristic
18C2×#499C2211yescharacteristic
19C2×#495C2211yescharacteristic
20C2×#488—21no—
21C2×C4×D8C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24#2861—21no—
25C42×D4—21no—
26#2228—24no—
27#2210—24no—
28C2×#367C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×C4⋊C16C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#357C2211yescharacteristic
36C2×#357C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40#1970C2211yescharacteristic
41#1970C2211yescharacteristic
42#1969C2211yescharacteristic
43#1969C2211yescharacteristic
44#1968—21no—
45#1968—21no—
46C4×#107C2211yescharacteristic
47C4×#107C2211yescharacteristic
48C4×#107C2211yescharacteristic
49C4×#102—21no—
50C4×#102—21no—
51C4×#102—21no—
52C42×C8C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1374C2211yescharacteristic
58#1374C2211yescharacteristic
59#1374C2211yescharacteristic
60#1374C2211yescharacteristic
61C22×Q16C2311yescharacteristic
62C22×SD16—21no—
63C22×D8C2311yescharacteristic
64C22×D8C2311yescharacteristic
65C2×#125—21no—
66C2×#124—21no—
67C2×C4×D4—21no—
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C22×C42—41no—
71C2×SD32—22no—
72C2×SD32—22no—
73C2×SD32—22no—
74C2×D16—22no—
75C2×D16—22no—
76C2×D16—22no—
77C22×C16C2311yescharacteristic
78C22×C16C2311yescharacteristic
79C22×C16C2311yescharacteristic
80C22×C16C2311yescharacteristic
81#499—22no—
82#499—22no—
83#495—22no—
84#495—22no—
85#488—44no—
86C4×D8—22no—
87C4×D8—22no—
88C4×D8—22no—
89C4×D8—22no—
90C4×D8—22no—
91C4×D8D414yes—
92C2×#107C2311yescharacteristic
93C2×#107C2311yescharacteristic
94C2×#107C2311yescharacteristic
95C2×#107C2311yescharacteristic
96C2×#107C2311yescharacteristic
97C2×#106—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×C4×C8D411yescharacteristic
106C2×C4×C8D411yescharacteristic
107C2×C4×C8D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#391—21no—
113#390—21no—
114#390—21no—
115#388—41no—
116#388—41no—
117#388—41no—
118#388—41no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C22⋊C4—41no—
123C4×C22⋊C4—41no—
124C4×C22⋊C4—41no—
125C43D411yescharacteristic
126#367—22no—
127#366—22no—
128C4⋊C16—22no—
129C4⋊C16—22no—
130#358—22no—
131#358—22no—
132#358—22no—
133#357—22no—
134#357—22no—
135#357—22no—
136#357—22no—
137#357—22no—
138#340—21no—
139#340—21no—
140#340—21no—
141C22×Q8—21no—
142C22×D4—21no—
143C22×D4—21no—
144C23×C4—41no—
145C23×C4—41no—
146C23×C4—41no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×SD16—42no—
154C2×SD16—42no—
155C2×SD16—42no—
156C2×D8—21no—
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×D8C2×D412yes—
166C2×D8#4112yes—
167C2×D8#4112yes—
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2411yescharacteristic
172#125—42no—
173#125—42no—
174#124—42no—
175#124—42no—
176C4×D4—24no—
177C4×D4—24no—
178C4×D4—24no—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—41no—
192C2×C22⋊C4—41no—
193C2×C22⋊C4—41no—
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198C2×C42—41no—
199C2×C42—41no—
200C2×C42—41no—
201C2×C42—84no—
202C2×C42D811yescharacteristic
203C2×C42D811yescharacteristic
204C2×C42D811yescharacteristic
205C2×C42D811yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209SD32—44no—
210D16—44no—
211C2×C16—21no—
212C2×C16—21no—
213C2×C16—21no—
214C2×C16—21no—
215C2×C16—21no—
216C2×C16—21no—
217C2×C16—21no—
218C2×C16—21no—
219C2×C16—21no—
220C2×C16—21no—
221C2×C16—21no—
222C2×C16—21no—
223#107—21no—
224#107—21no—
225#107—21no—
226#107—21no—
227#107—21no—
228#107—21no—
229#107—21no—
230#107C2×D412yes—
231#107#4112yes—
232#107#4112yes—
233#106—42no—
234#103—42no—
235#103—42no—
236#102—24no—
237#102—24no—
238#102—24no—
239#102—42no—
240#102—42no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—22no—
246C4×C8—41no—
247C4×C8—41no—
248C4×C8C2×D412yes—
249C4×C8#4112yes—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257C24—41no—
258C24—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4—82no—
292C22×C4C2×D811yescharacteristic
293C22×C4C2×D811yescharacteristic
294C22×C4C2×D811yescharacteristic
295C22×C4C2×D811yescharacteristic
296C22×C4C2×D811yescharacteristic
297C22×C4C2×D811yescharacteristic
298C22×C4C22×D411yescharacteristic
299Q16—41no—
300Q16—41no—
301Q16—41no—
302Q16—41no—
303SD16—84no—
304D8—22no—
305D8—22no—
306D8—22no—
307D8—41no—
308D8—41no—
309D8—41no—
310D8—41no—
311D8#12414yes—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8#12112yes—
321C2×C8C22×D411yescharacteristic
322C2×C8C2×#4111yescharacteristic
323C2×C8C2×#4111yescharacteristic
324C2×C8C2×#4111yescharacteristic
325C2×C8C2×#4111yescharacteristic
326C2×C8C2×#4111yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—82no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—22no—
353C42—22no—
354C42—41no—
355C42—41no—
356C42—82no—
357C42—82no—
358C42—82no—
359C42#12112yes—
360C42AGL(1,ℤ/8)12yes—
361C16—41no—
362C16—41no—
363C16—41no—
364C16—41no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23—81no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23—81no—
379C23C22×D811yescharacteristic
380Q8—81no—
381Q8—81no—
382Q8—81no—
383Q8—81no—
384D4—24no—
385D4—42no—
386D4—42no—
387D4—42no—
388D4—81no—
389D4—81no—
390D4—81no—
391D4—81no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—88no—
410C2×C4—88no—
411C2×C4—88no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4#45912yes—
428C2×C4#45912yes—
429C2×C4#50812yes—
430C2×C4#50912yes—
431C2×C4C2×#12111yescharacteristic
432C2×C4#52511yescharacteristic
433C2×C4#52511yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57511yescharacteristic
437C8—22no—
438C8—41no—
439C8—41no—
440C8C2×#12411yescharacteristic
441C8#52911yescharacteristic
442C8#52911yescharacteristic
443C8#53211yescharacteristic
444C22—44no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—82no—
450C22—82no—
451C22—161no—
452C22—161no—
453C22—161no—
454C22—161no—
455C22—161no—
456C22—161no—
457C22—161no—
458C22C2×#45911yescharacteristic
459C22#310011yescharacteristic
460C22#310011yescharacteristic
461C22C2×#50811yescharacteristic
462C22C2×#50911yescharacteristic
463C22#341411yescharacteristic
464C22#341411yescharacteristic
465C4—22no—
466C4—22no—
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4—82no—
473C4—82no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#285011yescharacteristic
478C4#304011yescharacteristic
479C4#311511yescharacteristic
480C4#311511yescharacteristic
481C2—84no—
482C2—321no—
483C2#3302411yescharacteristic
484C2#3357911yescharacteristic
485C2#3358011yescharacteristic
486C2#3366211yescharacteristic
487C2#3366511yescharacteristic
488C2#3367111yescharacteristic
489C2#3367411yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutator, Frattini, cosocleC22×C8C24
prime radicalC22×D16C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ C42×C8 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2228C2semidirect
3. #33240C2semidirect
4. C42×D8C2semidirect
5. #13277C2semidirect
6. #13254C2semidirect
7. #13133C2semidirect
8. #13132C2semidirect
9. #13131C2semidirect
10. #13130C2semidirect
11. #13014C2semidirect
12. #13014C2semidirect
15. #12879C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22282
C2×#22102
#332402
C42×D82
#132772
#132542
#131332
#131322
#131312
#131302
#130142
#130122
#128792