Finite Group Databasebeta

← #410545

Subgroups — #410545

Order 512.

1136 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1936C212yesmaximal
2C2×#1934C211yesmaximal, characteristic
3C2×#1770C211yesmaximal, characteristic, complemented
4#15194C211yesmaximal, characteristic, complemented
5#15147C211yesmaximal, characteristic
6#11228C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11123C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10862C211yesmaximal, characteristic, complemented
11#10821C212yesmaximal, complemented
12C22×#107C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×#102C2212yes—
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#97C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#397—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1936—28no—
26#1934—24no—
27#1770—24no—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1444—21no—
37#1441—22no—
38#1440—22no—
39#1439—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2212yes—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—22no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4D411yescharacteristic
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#107—24no—
64C2×#107—24no—
65C2×#107—42no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—22no—
68C2×#102—24no—
69C2×#102—22no—
70C2×#102—24no—
71C2×#102—22no—
72C2×#102—24no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102D414yes—
76C2×#97—24no—
77C2×#97—24no—
78C2×#97—24no—
79C2×C4×C8—24no—
80C2×C4×C8D414yes—
81#397—44no—
82#390—24no—
83#390—22no—
84#390—22no—
85#390—42no—
86#389—24no—
87#389—22no—
88#389—22no—
89#389—42no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—21no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95D412yes—
103#340—22no—
104#340—22no—
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#336—24no—
110C25—21no—
111C22×D4—21no—
112C22×D4—21no—
113C22×D4—21no—
114C22×D4—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4C2×D412yes—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C22×C8#4114yes—
146C22×C8#4112yes—
147C22×C8#4114yes—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4#4114yes—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C2×D412yes—
204C2×C42#4112yes—
205#107—48no—
206C4⋊C8—28no—
207#102—216no—
208#102—28no—
209#102—44no—
210#97—48no—
211C4×C8—28no—
212#95—24no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—42no—
217#95—42no—
218#95—81no—
219#95—81no—
220#95—82no—
221#95—82no—
222#95—82no—
223C24—22no—
224C24—41no—
225C24—42no—
226C24—41no—
227C24—42no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—42no—
232C24—41no—
233C24—41no—
234C24C22×D411yescharacteristic
235C2×D4—22no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—24no—
239C2×D4—41no—
240C2×D4—41no—
241C2×D4—42no—
242C2×D4—42no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—42no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4#12112yes—
303C22×C4#12112yes—
304C22×C4AGL(1,ℤ/8)12yes—
305C22×C4#13712yes—
306C22×C4C22×D411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4112yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C42−1+412yes—
315C22×C42−1+411yescharacteristic
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8#12114yes—
332C2×C8#12314yes—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—22no—
336C4⋊C4—24no—
337C4⋊C4—44no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—41no—
342C4⋊C4—44no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—84no—
354C4⋊C4—88no—
355C22⋊C4—84no—
356C22⋊C4—88no—
357C22⋊C4—88no—
358C22⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C23—44no—
367C23—44no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—81no—
372C23—82no—
373C23—81no—
374C23—81no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—82no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23—81no—
389C23C2×#12111yescharacteristic
390C23#53212yes—
391C23D4211yescharacteristic
392C23#54711yescharacteristic
393C23#54812yes—
394C23C2×AGL(1,ℤ/8)11yescharacteristic
395C23C2×#13711yescharacteristic
396C23#57512yes—
397C23#57612yes—
398C23#57611yescharacteristic
399C23#57811yescharacteristic
400D4—42no—
401D4—44no—
402D4—42no—
403D4—81no—
404D4—81no—
405D4—82no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#46814yes—
459C2×C4C2×#12111yescharacteristic
460C2×C4C2×#12311yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#52512yes—
463C2×C4#52912yes—
464C2×C4#52912yes—
465C2×C4#53211yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#53711yescharacteristic
468C2×C4#54112yes—
469C2×C4#54212yes—
470C2×C4#54711yescharacteristic
471C2×C4#54712yes—
472C2×C4#55112yes—
473C2×C4#55212yes—
474C2×C4#55212yes—
475C2×C4#55511yescharacteristic
476C2×C4#55612yes—
477C8—24no—
478C8—24no—
479C8—44no—
480C22—82no—
481C22—82no—
482C22—84no—
483C22—84no—
484C22—82no—
485C22—161no—
486C22—161no—
487C22—162no—
488C22—162no—
489C22—161no—
490C22—161no—
491C22#246112yes—
492C22#251912yes—
493C22#253711yescharacteristic
494C22#254511yescharacteristic
495C22#285411yescharacteristic
496C22C2×#46811yescharacteristic
497C22#305411yescharacteristic
498C22#305811yescharacteristic
499C22#306111yescharacteristic
500C22#317512yes—
501C22#317612yes—
502C22#318012yes—
503C22#318112yes—
504C22#327312yes—
505C22#327411yescharacteristic
506C22#327611yescharacteristic
507C22#335311yescharacteristic
508C22#335512yes—
509C22#335711yescharacteristic
510C22#336612yes—
511C22#337012yes—
512C22#337512yes—
513C22#337712yes—
514C4—24no—
515C4—42no—
516C4—82no—
517C4—82no—
518C4—82no—
519C4—84no—
520C4—161no—
521C4—162no—
522C4—161no—
523C4#239211yescharacteristic
524C4#247812yes—
525C4#248611yescharacteristic
526C4#249811yescharacteristic
527C4#253212yes—
528C4#263711yescharacteristic
529C2—162no—
530C2#1508111yescharacteristic
531C2#2684012yes—
532C2#2684312yes—
533C2#2805812yes—
534C2#2806112yes—
535C2#2823011yescharacteristic
536C2#2823311yescharacteristic
537C2#2864311yescharacteristic
538C2#2864511yescharacteristic
539C2#3313411yescharacteristic
540C2#3313811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1770C2semidirect
4. #15194C2semidirect
6. #11228C2semidirect
7. #11219C2semidirect
10. #10862C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19362
C2×#19342
C2×#17702
#151942
#151472
#112282
#112192
#111232
#110762
#108622
#108212