Finite Group Databasebeta

← #407795

Subgroups — #407795

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1936C212yesmaximal
2#14707C211yesmaximal, characteristic, complemented
3#14700C211yesmaximal, characteristic
4#11315C212yesmaximal, complemented
5#11208C211yesmaximal, characteristic, complemented
6#11208C211yesmaximal, characteristic, complemented
7#11189C212yesmaximal
8#11051C211yesmaximal, characteristic
9#11051C211yesmaximal, characteristic
10#10881C212yesmaximal, complemented
11#10693C211yesmaximal, characteristic, complemented
12C22×#102C2212yes—
13C22×#102C2212yes—
14C22×#97C2212yes—
15C2×#390C2211yescharacteristic
16C2×#390C2211yescharacteristic
17C2×#390C2211yescharacteristic
18C2×#389C2212yes—
19C2×#389C2211yescharacteristic
20C2×#388—22no—
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C43C2211yescharacteristic
24#1936—28no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34#1437—22no—
35#1436—22no—
36#1436—22no—
37C4⋊#95C2211yescharacteristic
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×D4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C22×C4⋊C4—22no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2311yescharacteristic
47C22×C4⋊C4C2311yescharacteristic
48C22×C4⋊C4C2311yescharacteristic
49C22×C4⋊C4C2312yes—
50C22×C22⋊C4—22no—
51C22×C22⋊C4—21no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—21no—
54C22×C42D412yes—
55C22×C42D412yes—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C2×#102—24no—
60C2×#102—24no—
61C2×#102—24no—
62C2×#102—24no—
63C2×#102—24no—
64C2×#102—42no—
65C2×#97—24no—
66C2×#97—24no—
67C2×#97—42no—
68#390—22no—
69#390—22no—
70#390—22no—
71#390—42no—
72#390D414yes—
73#389—24no—
74#389—24no—
75#389—22no—
76#389—22no—
77#388—48no—
78#382—24no—
79#382—42no—
80C2×#95—21no—
81C2×#95—21no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—21no—
87C2×#95—21no—
88C2×#95—22no—
89C2×#95—21no—
90C43—42no—
91#340—24no—
92#340—24no—
93#340—24no—
94#340—24no—
95#340—24no—
96#340—42no—
97#336—24no—
98#336—22no—
99#336—22no—
100C25—21no—
101C22×D4—21no—
102C22×D4—21no—
103C22×D4—22no—
104C22×D4—22no—
105C22×D4—22no—
106C22×D4—21no—
107C22×D4—22no—
108C22×D4—21no—
109C22×D4—21no—
110C22×D4C2×D412yes—
111C23×C4—21no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2411yescharacteristic
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8#4114yes—
134C22×C8#4114yes—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4C2×D412yes—
165C2×C4⋊C4#4114yes—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—44no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C2×D412yes—
191C2×C42#4112yes—
192C2×C42#4112yes—
193C2×C42#4112yes—
194#102—216no—
195#102—48no—
196#97—48no—
197#95—24no—
198#95—24no—
199#95—44no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—44no—
206#95—44no—
207C24—22no—
208C24—41no—
209C24—41no—
210C24—41no—
211C24—41no—
212C24—41no—
213C24—41no—
214C24—42no—
215C24—42no—
216C24—42no—
217C24—42no—
218C24C22×D411yescharacteristic
219C2×D4—24no—
220C2×D4—24no—
221C2×D4—22no—
222C2×D4—22no—
223C2×D4—42no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4#12114yes—
285C22×C4#12412yes—
286C22×C4#12412yes—
287C22×C4AGL(1,ℤ/8)14yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4C2×#4112yes—
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4H(5,𝔽2)12yes—
296C22×C42−1+411yescharacteristic
297C22×C42−1+411yescharacteristic
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—84no—
329C4⋊C4—88no—
330C22⋊C4—416no—
331C22⋊C4—84no—
332C22⋊C4—84no—
333C22⋊C4—88no—
334C42—24no—
335C42—24no—
336C42—24no—
337C42—24no—
338C42—48no—
339C42—48no—
340C42—42no—
341C42—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—42no—
346C23—42no—
347C23—81no—
348C23—82no—
349C23—82no—
350C23—81no—
351C23—81no—
352C23—81no—
353C23—81no—
354C23—81no—
355C23—81no—
356C23—82no—
357C23—81no—
358C23—82no—
359C23—82no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23C2×#12111yescharacteristic
366C23#52511yescharacteristic
367C23C2×#12411yescharacteristic
368C23#53211yescharacteristic
369C23#53211yescharacteristic
370C23#53411yescharacteristic
371C23#53611yescharacteristic
372C23C2×AGL(1,ℤ/8)12yes—
373C23#57512yes—
374C23#57512yes—
375C23#57512yes—
376D4—44no—
377D4—44no—
378D4—82no—
379D4—82no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—81no—
417C2×C4—82no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#52512yes—
434C2×C4#52912yes—
435C2×C4#52912yes—
436C2×C4#52912yes—
437C2×C4#53912yes—
438C2×C4#54212yes—
439C2×C4#54612yes—
440C2×C4#54711yescharacteristic
441C2×C4#54711yescharacteristic
442C2×C4#55212yes—
443C2×C4#55312yes—
444C2×C4#55312yes—
445C2×C4#55312yes—
446C2×C4#55611yescharacteristic
447C2×C4#55612yes—
448C2×C4#55611yescharacteristic
449C8—44no—
450C8—44no—
451C22—82no—
452C22—82no—
453C22—82no—
454C22—84no—
455C22—84no—
456C22—161no—
457C22—161no—
458C22—162no—
459C22—161no—
460C22—162no—
461C22—161no—
462C22#239211yescharacteristic
463C22#254311yescharacteristic
464C22#254711yescharacteristic
465C22#259111yescharacteristic
466C22#260411yescharacteristic
467C22#266311yescharacteristic
468C22#286111yescharacteristic
469C22#303612yes—
470C22#304012yes—
471C22#311112yes—
472C22#311512yes—
473C22#313511yescharacteristic
474C22#313511yescharacteristic
475C22#313811yescharacteristic
476C22#314011yescharacteristic
477C22#317112yes—
478C22#318012yes—
479C22#318312yes—
480C22#318312yes—
481C22#319412yes—
482C22#319712yes—
483C22#323412yes—
484C22#324012yes—
485C4—42no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4—82no—
490C4—84no—
491C4—161no—
492C4—161no—
493C4—162no—
494C4#246612yes—
495C4#247012yes—
496C4#252512yes—
497C4#259412yes—
498C2—162no—
499C2#1497211yescharacteristic
500C2#2368612yes—
501C2#2857912yes—
502C2#2872211yescharacteristic
503C2#2872711yescharacteristic
504C2#2933711yescharacteristic
505C2#2933911yescharacteristic
506C2#2952312yes—
507C2#3036212yes—
508C2#3328011yescharacteristic
509C2#3328811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #14707C2semidirect
4. #11315C2semidirect
5. #11208C2semidirect
6. #11208C2semidirect
10. #10881C2semidirect
11. #10693C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19362
#147072
#147002
#113152
#112082
#111892
#110512
#108812
#106932