Finite Group Databasebeta

← #7231136

Subgroups — #7231136

Order 512.

1481 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#46147C211yesmaximal, characteristic
2#36508C211yesmaximal, characteristic
3#33073C212yesmaximal, complemented
4#32821C212yesmaximal, complemented
5#32682C212yesmaximal, complemented
6#32039C212yesmaximal
7#31516C214yesmaximal, complemented
8#30635C214yesmaximal, complemented
9#27043C214yesmaximal, complemented
10#26353C214yesmaximal, complemented
11#25647C212yesmaximal, complemented
12#24107C212yesmaximal
13#23347C211yesmaximal, characteristic
14C2×#563—21no—
15C2×#557C2211yescharacteristic
16C2×#556C2211yescharacteristic
17C2×#487C2212yes—
18C2×#484C2212yes—
19C2×#475C2212yes—
20C2×#428C2212yes—
21#2862C2211yescharacteristic
22#2861—21no—
23#2859C2212yes—
24#2822—21no—
25#2734—21no—
26#2724—21no—
27#2698—24no—
28#2565—21no—
29#2546C2212yes—
30#2546C2211yescharacteristic
31#2545—21no—
32#2542C2211yescharacteristic
33#2519—24no—
34#2496C2214yes—
35#2486C2212yes—
36#2440C2214yes—
37#2429C2214yes—
38#2362C2214yes—
39C4×C4⋊Q8—21no—
40C4×#124C2211yescharacteristic
41C42×Q8C2211yescharacteristic
42#2095C2214yes—
43#2086C2214yes—
44#2072C2214yes—
45#2068C2214yes—
46#2068C2214yes—
47#2057C2214yes—
48#2057C2214yes—
49#2043C2214yes—
50#2043C2214yes—
51#2036C2214yes—
52#2036C2214yes—
53#2036C2214yes—
54#1984C2212yes—
55#1981C2212yes—
56#1948C2212yes—
57#1947C2212yes—
58#1942C2214yes—
59#1923C2214yes—
60#1920C2214yes—
61#1918C2214yes—
62#1918C2214yes—
63#1894C2212yes—
64#1894C2214yes—
65#1872C2214yes—
66#1870C2212yes—
67#1865C2214yes—
68#1865C2214yes—
69#1863C2214yes—
70#1854C2214yes—
71#1851C2214yes—
72#1831C2212yes—
73#1825C2212yes—
74#1803C2212yes—
75#1803C2212yes—
76C4×#103C2212yes—
77C22×M16C2312yes—
78#563—48no—
79#557—42no—
80#557—42no—
81#556—42no—
82#556—42no—
83C2×C4⋊Q8—21no—
84C2×C4⋊Q8—22no—
85C2×C4⋊Q8C2312yes—
86C2×C4⋊Q8C2311yescharacteristic
87C2×C4⋊Q8C2312yes—
88C2×C4⋊Q8C2311yescharacteristic
89C2×#126—24no—
90C2×#126—22no—
91C2×#125—21no—
92C2×#125—22no—
93C2×#125—22no—
94C2×#124—21no—
95C2×#124D411yescharacteristic
96C2×#123—22no—
97C2×C22⋊Q8—24no—
98C2×C22⋊Q8C2312yes—
99C2×C22⋊Q8C2314yes—
100C2×C4×Q8—21no—
101C2×C4×Q8—21no—
102C2×C4×Q8C2311yescharacteristic
103C2×C4×Q8C2312yes—
104C2×C4×Q8C2312yes—
105C2×C4×Q8C2311yescharacteristic
106C2×C4×D4C2311yescharacteristic
107C2×#117C2312yes—
108C22×C4⋊C4C2312yes—
109#487—44no—
110#484—44no—
111#475—44no—
112#428—44no—
113C2×#107C2314yes—
114C2×#107C2314yes—
115C2×#107C2314yes—
116C2×#106C2314yes—
117C2×C4⋊C8C2312yes—
118C2×C4⋊C8C2312yes—
119C2×C4⋊C8C2312yes—
120C2×#103C2314yes—
121C2×#103C2314yes—
122C2×#103C2314yes—
123C2×#103C2314yes—
124C2×#103C2314yes—
125C2×#103C2314yes—
126C2×C22⋊C8C2312yes—
127C2×C22⋊C8C2314yes—
128C2×C4×C8C2312yes—
129#400—24no—
130#399—22no—
131#398—22no—
132#398—22no—
133#398—22no—
134#398—22no—
135#398—24no—
136#398—24no—
137#397—22no—
138#397—24no—
139#396—24no—
140#395—24no—
141#391—21no—
142#391—22no—
143#391—21no—
144#391—21no—
145#391D411yescharacteristic
146#391C2314yes—
147#390—21no—
148#390D411yescharacteristic
149#389—22no—
150#389—22no—
151#389—24no—
152#389C2314yes—
153#389C2314yes—
154#389C2314yes—
155#389C2314yes—
156#388—24no—
157#388—24no—
158#388—24no—
159#388D414yes—
160#387—24no—
161#387—24no—
162#385—24no—
163#384—21no—
164#384C2312yes—
165#384C2312yes—
166#382—22no—
167#382—22no—
168#382D411yescharacteristic
169#381—22no—
170#380C2312yes—
171#380C2314yes—
172C4×C4⋊C4—21no—
173C4×C4⋊C4—22no—
174C4×C4⋊C4—24no—
175C4×C4⋊C4C2312yes—
176C4×C4⋊C4C2311yescharacteristic
177C4×C4⋊C4C2312yes—
178C4×C22⋊C4C2314yes—
179C43—21no—
180C43C2311yescharacteristic
181#340C2314yes—
182#340C2314yes—
183#340C2314yes—
184#340C2314yes—
185#340C2314yes—
186#340C2314yes—
187#336C2314yes—
188#336C2314yes—
189#336C2314yes—
190C22×Q8—21no—
191C22×Q8—22no—
192C22×Q8C2×D411yescharacteristic
193C22×Q8C2412yes—
194C22×Q8C2411yescharacteristic
195C22×D4C2×D411yescharacteristic
196C23×C4—22no—
197C23×C4C2412yes—
198C2×M16—44no—
199C2×M16—42no—
200C22×C8C2414yes—
201C22×C8C2414yes—
202C4⋊Q8—24no—
203C4⋊Q8—41no—
204C4⋊Q8—41no—
205C4⋊Q8—42no—
206C4⋊Q8—42no—
207C4⋊Q8—42no—
208C4⋊Q8—41no—
209C4⋊Q8—41no—
210C4⋊Q8—44no—
211C4⋊Q8—48no—
212#126—416no—
213#126—48no—
214#125—44no—
215#125—48no—
216#125—48no—
217#124—24no—
218#124—44no—
219#123—48no—
220C22⋊Q8—24no—
221C22⋊Q8—28no—
222C22⋊Q8—416no—
223C22⋊Q8—44no—
224C22⋊Q8—42no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—41no—
230C4×Q8—41no—
231C4×Q8—42no—
232C4×Q8—42no—
233C4×Q8—44no—
234C4×Q8—44no—
235C4×D4—41no—
236C4×D4—41no—
237#117—24no—
238#117—42no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—22no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—24no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—22no—
252C2×C4⋊C4—22no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—24no—
255C2×C4⋊C4—22no—
256C2×C4⋊C4—22no—
257C2×C4⋊C4—22no—
258C2×C4⋊C4—22no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C2×D411yescharacteristic
262C2×C4⋊C4C2×D411yescharacteristic
263C2×C4⋊C4C2×D412yes—
264C2×C4⋊C4C2414yes—
265C2×C4⋊C4C2412yes—
266C2×C4⋊C4C2414yes—
267C2×C4⋊C4C2411yescharacteristic
268C2×C4⋊C4C2412yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—24no—
271C2×C22⋊C4—22no—
272C2×C22⋊C4—24no—
273C2×C22⋊C4—22no—
274C2×C22⋊C4C2×D414yes—
275C2×C22⋊C4C2×D414yes—
276C2×C22⋊C4C2×D412yes—
277C2×C42—21no—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—22no—
281C2×C42—22no—
282C2×C42—21no—
283C2×C42—22no—
284C2×C42C2×D411yescharacteristic
285C2×C42C2×D412yes—
286C2×C42C2×D411yescharacteristic
287C2×C42C2411yescharacteristic
288C2×C42C2412yes—
289C2×C42C2412yes—
290#107—28no—
291#107—44no—
292#107—44no—
293#106—28no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8—42no—
297#103—28no—
298#103—44no—
299#103—44no—
300#103—44no—
301#103—44no—
302#103—44no—
303C22⋊C8—44no—
304C22⋊C8—42no—
305C4×C8—42no—
306#95—22no—
307#95—21no—
308#95—24no—
309#95—22no—
310#95—24no—
311#95—22no—
312#95—24no—
313#95—24no—
314#95—24no—
315#95—24no—
316#95—24no—
317#95C2×D414yes—
318#95C2×D414yes—
319C24C22×D412yes—
320C2×Q8—21no—
321C2×Q8—22no—
322C2×Q8—22no—
323C2×Q8—22no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—41no—
328C2×Q8—41no—
329C2×Q8—42no—
330C2×Q8—42no—
331C2×Q8—42no—
332C2×Q8—44no—
333C2×Q8—82no—
334C2×Q8—82no—
335C2×Q8AGL(1,ℤ/8)14yes—
336C2×Q8H(5,𝔽2)12yes—
337C2×D4—22no—
338C2×D4—41no—
339C2×D4—41no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—24no—
346C22×C4—24no—
347C22×C4—24no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—24no—
351C22×C4—44no—
352C22×C4—82no—
353C22×C4—82no—
354C22×C4C22×D412yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D412yes—
357C22×C4C22×D412yes—
358C22×C4C22×D412yes—
359C22×C4C22×D411yescharacteristic
360C22×C4C22×D412yes—
361C22×C4C2511yescharacteristic
362M16—48no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—24no—
369C4⋊C4—24no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—22no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—42no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—44no—
389C4⋊C4—48no—
390C4⋊C4—42no—
391C4⋊C4—44no—
392C4⋊C4—42no—
393C4⋊C4—48no—
394C4⋊C4—48no—
395C4⋊C4—48no—
396C4⋊C4—44no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4H(5,𝔽2)12yes—
401C4⋊C42−1+414yes—
402C22⋊C4—28no—
403C22⋊C4—28no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C22⋊C4—48no—
407C22⋊C4—44no—
408C22⋊C4—48no—
409C22⋊C4—44no—
410C42—24no—
411C42—22no—
412C42—24no—
413C42—22no—
414C42—22no—
415C42—21no—
416C42—21no—
417C42—22no—
418C42—22no—
419C42—41no—
420C42—44no—
421C42—44no—
422C42—44no—
423C42—44no—
424C42—42no—
425C42—44no—
426C23—22no—
427C23—22no—
428C23—24no—
429C23—22no—
430C23—24no—
431C23C23×D411yescharacteristic
432Q8—24no—
433Q8—44no—
434Q8—41no—
435Q8—41no—
436Q8—42no—
437Q8—42no—
438Q8—44no—
439Q8—84no—
440Q8—164no—
441D4—82no—
442C2×C4—22no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—21no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—44no—
478C2×C4—164no—
479C2×C4C2×AGL(1,ℤ/8)12yes—
480C2×C4C2×AGL(1,ℤ/8)14yes—
481C2×C4#57814yes—
482C2×C4C2×H(5,𝔽2)11yescharacteristic
483C2×C4C2×H(5,𝔽2)11yescharacteristic
484C2×C4C2×2−1+412yes—
485C2×C4#58512yes—
486C8—44no—
487C8—44no—
488C22—42no—
489C22—42no—
490C22—44no—
491C22—44no—
492C22—42no—
493C22#348411yescharacteristic
494C22#348912yes—
495C22C22×AGL(1,ℤ/8)11yescharacteristic
496C22C2×#57812yes—
497C22#358611yescharacteristic
498C4—22no—
499C4—24no—
500C4—24no—
501C4—42no—
502C4—42no—
503C4—42no—
504C4—44no—
505C4—41no—
506C4—42no—
507C4—42no—
508C4—84no—
509C4—82no—
510C4—82no—
511C4—84no—
512C4#352911yescharacteristic
513C4#355011yescharacteristic
514C4#355912yes—
515C2—82no—
516C2#4622811yescharacteristic
517C2#6157211yescharacteristic
518C2#6166511yescharacteristic
519C2#6214812yes—
520C2#6216812yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#124 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #33073C2semidirect
4. #32821C2semidirect
5. #32682C2semidirect
7. #31516C2semidirect
8. #30635C2semidirect
9. #27043C2semidirect
10. #26353C2semidirect
11. #25647C2semidirect

Maximal subgroups

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

subgroupindex
#461472
#365082
#330732
#328212
#326822
#320392
#315162
#306352
#270432
#263532
#256472
#241072
#233472