Finite Group Databasebeta

← #275730

Subgroups — #275730

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29941C211yesmaximal, characteristic
2#27908C211yesmaximal, characteristic
3#22633C211yesmaximal, characteristic
4#21963C211yesmaximal, characteristic
5#10654C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10596C211yesmaximal, characteristic, complemented
8#10596C211yesmaximal, characteristic, complemented
9#9620C211yesmaximal, characteristic, complemented
10#9611C211yesmaximal, characteristic, complemented
11#9569C211yesmaximal, characteristic, complemented
12#9561C211yesmaximal, characteristic, complemented
13#9137C211yesmaximal, characteristic
14#8833C211yesmaximal, characteristic
15#8727C211yesmaximal, characteristic
16C2×#485—21no—
17C2×#485—21no—
18C2×#423C2211yescharacteristic
19#2857C2211yescharacteristic
20#2285C2211yescharacteristic
21#2283C2211yescharacteristic
22#2101—21no—
23#2097—21no—
24#2097—21no—
25#2085—21no—
26#2077—21no—
27#2076—21no—
28#1988C2211yescharacteristic
29#1988C2211yescharacteristic
30#1956C2211yescharacteristic
31#1956C2211yescharacteristic
32#1949—21no—
33#1949—21no—
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1939C2211yescharacteristic
41#1939C2211yescharacteristic
42#1924—21no—
43#1923—21no—
44#1845C2211yescharacteristic
45#1844C2211yescharacteristic
46#1832—21no—
47#1809—21no—
48#1809—21no—
49#1797C2211yescharacteristic
50#1797C2211yescharacteristic
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C22⋊C8C2211yescharacteristic
54#1746—21no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×#117—21no—
71C22×C42C2311yescharacteristic
72#485—44no—
73#485—44no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8D411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#391—21no—
108#389—21no—
109#389—21no—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#383—21no—
114#382—21no—
115#382D411yescharacteristic
116#382D411yescharacteristic
117#381—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122#380D411yescharacteristic
123#380D411yescharacteristic
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129#376—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336D411yescharacteristic
147#336D411yescharacteristic
148#334—22no—
149#334—22no—
150#332—22no—
151#332—22no—
152C22×Q8—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4#4111yescharacteristic
156C2×M16—24no—
157C2×M16—81no—
158C2×M16—81no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#117—42no—
176#117—42no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4C2×D411yescharacteristic
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—41no—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—42no—
209#107—42no—
210#107—81no—
211#107—81no—
212#107—81no—
213#106—42no—
214#106—42no—
215#106—81no—
216#106—81no—
217#106—81no—
218C4⋊C8—24no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—81no—
230C4⋊C8—81no—
231C4⋊C8#4112yes—
232C4⋊C8#4112yes—
233#103—42no—
234#103—42no—
235#103—42no—
236#103—42no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—81no—
240C22⋊C8—81no—
241#97—42no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8—42no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24#12111yescharacteristic
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296M16—84no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8AGL(1,ℤ/8)12yes—
326C2×C8AGL(1,ℤ/8)12yes—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—82no—
355C22⋊C4—42no—
356C22⋊C4—81no—
357C22⋊C4—82no—
358C22⋊C4—82no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—82no—
370C42C2×#4111yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C422−1+411yescharacteristic
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23C2×#12111yescharacteristic
382Q8—81no—
383Q8—81no—
384Q8—81no—
385Q8—81no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
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—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4#52511yescharacteristic
432C2×C4#53611yescharacteristic
433C2×C4#54711yescharacteristic
434C2×C4#54711yescharacteristic
435C2×C4#55211yescharacteristic
436C2×C4#55211yescharacteristic
437C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
438C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57811yescharacteristic
444C2×C4#57811yescharacteristic
445C8—22no—
446C8—22no—
447C8—41no—
448C8—41no—
449C8—42no—
450C8—42no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—41no—
461C22—41no—
462C22#241311yescharacteristic
463C22#303811yescharacteristic
464C22#304611yescharacteristic
465C22#306011yescharacteristic
466C22#306311yescharacteristic
467C22#306711yescharacteristic
468C22#306911yescharacteristic
469C4—21no—
470C4—21no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—161no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#312111yescharacteristic
483C4#312311yescharacteristic
484C4#324211yescharacteristic
485C4#324411yescharacteristic
486C4#334311yescharacteristic
487C4#334311yescharacteristic
488C4#335411yescharacteristic
489C4#335411yescharacteristic
490C2—81no—
491C2#1935511yescharacteristic
492C2#1937711yescharacteristic
493C2#1954811yescharacteristic
494C2#1958711yescharacteristic
495C2#2459411yescharacteristic
496C2#2461911yescharacteristic
497C2#2462411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10654C2semidirect
6. #10652C2semidirect
7. #10596C2semidirect
8. #10596C2semidirect
9. #9620C2semidirect
10. #9611C2semidirect
11. #9569C2semidirect
12. #9561C2semidirect

Maximal subgroups

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

subgroupindex
#299412
#279082
#226332
#219632
#106542
#106522
#105962
#96202
#96112
#95692
#95612
#91372
#88332
#87272