Finite Group Databasebeta

← #470357

Subgroups — #470357

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2647C211yesmaximal, characteristic, complemented
2C2×#2094C211yesmaximal, characteristic
3C2×#2073C211yesmaximal, characteristic
4#14260C211yesmaximal, characteristic
5#11373C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11327C212yesmaximal, complemented
8#11220C212yesmaximal, complemented
9#11066C212yesmaximal
10#11066C212yesmaximal
11#10796C211yesmaximal, characteristic, complemented
12C22×#106C2212yes—
13C22×#102C2212yes—
14C22×#102C2212yes—
15#2647—24no—
16#2647—24no—
17C2×#400—22no—
18C2×#398—22no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2211yescharacteristic
25C2×#384C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×#376C2211yescharacteristic
28#2094—24no—
29#2073—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1440—22no—
40#1439—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—21no—
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#102—24no—
66C2×#102—24no—
67C2×#102—24no—
68C2×#102—24no—
69C2×#102—42no—
70C2×#102—42no—
71#400—216no—
72#398—216no—
73#390—22no—
74#390—22no—
75#390—22no—
76#390—22no—
77#390—42no—
78#389—24no—
79#389—24no—
80#389—24no—
81#389—24no—
82#389—42no—
83#384—22no—
84#384D414yes—
85#382—22no—
86#382D414yes—
87#376—22no—
88#376—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D412yes—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—24no—
103#336—22no—
104#336—22no—
105#336—42no—
106C25—21no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4—22no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—21no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—21no—
116C22×D4#4112yes—
117C23×C4—21no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2411yescharacteristic
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8#4114yes—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
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—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—82no—
167C2×C4⋊C4—82no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4#4112yes—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200#106—48no—
201#102—48no—
202#102—48no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95—24no—
208#95—42no—
209#95—82no—
210#95—82no—
211C24—22no—
212C24—41no—
213C24—41no—
214C24—41no—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—42no—
219C24—42no—
220C24—42no—
221C24—42no—
222C24C22×D411yescharacteristic
223C2×D4—24no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—24no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—41no—
234C2×D4—42no—
235C2×D4—41no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4#12014yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)12yes—
299C22×C42−1+412yes—
300C22×C42−1+412yes—
301C22×C42−1+411yescharacteristic
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—88no—
331C4⋊C4—88no—
332C4⋊C4—88no—
333C4⋊C4—84no—
334C22⋊C4—88no—
335C22⋊C4—88no—
336C22⋊C4—84no—
337C22⋊C4—84no—
338C42—24no—
339C42—24no—
340C42—41no—
341C42—42no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—42no—
346C23—44no—
347C23—42no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—81no—
352C23—81no—
353C23—82no—
354C23—81no—
355C23—82no—
356C23—81no—
357C23—81no—
358C23—81no—
359C23—81no—
360C23—81no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—82no—
365C23—82no—
366C23—82no—
367C23—82no—
368C23—82no—
369C23C2×#12011yescharacteristic
370C23#52511yescharacteristic
371C23#53211yescharacteristic
372C23#53511yescharacteristic
373C23#53611yescharacteristic
374C23#53611yescharacteristic
375C23#53711yescharacteristic
376C23#57612yes—
377C23#57712yes—
378C23#57712yes—
379C23#57712yes—
380D4—44no—
381D4—44no—
382D4—82no—
383D4—82no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4#53612yes—
438C2×C4#53612yes—
439C2×C4#53912yes—
440C2×C4#53912yes—
441C2×C4#53912yes—
442C2×C4#54112yes—
443C2×C4#54612yes—
444C2×C4#54611yescharacteristic
445C2×C4#54711yescharacteristic
446C2×C4#54811yescharacteristic
447C2×C4#55011yescharacteristic
448C2×C4#55012yes—
449C2×C4#55212yes—
450C2×C4#55212yes—
451C2×C4#56212yes—
452C2×C4#56312yes—
453C8—44no—
454C8—82no—
455C22—82no—
456C22—84no—
457C22—82no—
458C22—82no—
459C22—84no—
460C22—161no—
461C22—161no—
462C22—161no—
463C22—161no—
464C22—162no—
465C22—162no—
466C22#240711yescharacteristic
467C22#243111yescharacteristic
468C22#261111yescharacteristic
469C22#261411yescharacteristic
470C22#263311yescharacteristic
471C22#266411yescharacteristic
472C22#282011yescharacteristic
473C22#301012yes—
474C22#301212yes—
475C22#311812yes—
476C22#312512yes—
477C22#317412yes—
478C22#317912yes—
479C22#322011yescharacteristic
480C22#322111yescharacteristic
481C22#322211yescharacteristic
482C22#322211yescharacteristic
483C22#324112yes—
484C22#324112yes—
485C22#324412yes—
486C22#324412yes—
487C22#325712yes—
488C22#326312yes—
489C4—44no—
490C4—81no—
491C4—81no—
492C4—84no—
493C4—82no—
494C4—84no—
495C4—82no—
496C4—161no—
497C4—161no—
498C4#269012yes—
499C4#269412yes—
500C4#269412yes—
501C4#272112yes—
502C2—162no—
503C2#1668211yescharacteristic
504C2#2416612yes—
505C2#2560711yescharacteristic
506C2#2561111yescharacteristic
507C2#2958812yes—
508C2#2964712yes—
509C2#3008411yescharacteristic
510C2#3009011yescharacteristic
511C2#3039512yes—
512C2#3283011yescharacteristic
513C2#3283111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2647C2semidirect
5. #11373C2semidirect
6. #11358C2semidirect
7. #11327C2semidirect
8. #11220C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#26472
C2×#20942
C2×#20732
#142602
#113732
#113582
#113272
#112202
#110662
#107962