Finite Group Databasebeta

← #410220

Subgroups — #410220

Order 512.

1814 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
1C22×#389C211yesmaximal, characteristic
2C2×#1852C211yesmaximal, characteristic
3C2×#1823C211yesmaximal, characteristic
4C2×#1823C211yesmaximal, characteristic
5C2×#1817C211yesmaximal, characteristic
6#14685C211yesmaximal, characteristic
7#11292C214yesmaximal, complemented
8#11219C214yesmaximal, complemented
9#10730C211yesmaximal, characteristic
10C23×C4⋊C4—21no—
11C23×C4⋊C4C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C42C2211yescharacteristic
14C22×#107C2212yes—
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C2×#389—22no—
18C2×#389—24no—
19C2×#389—24no—
20C2×#389C2214yes—
21C2×#381—21no—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C22×#95—21no—
25#1852—24no—
26#1823—24no—
27#1823—24no—
28#1817—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2214yes—
32C2×#340C2214yes—
33C2×#340C2212yes—
34C2×#336C2214yes—
35C4⋊#95C2212yes—
36C4⋊#95C2212yes—
37#1434—22no—
38C22⋊#95—21no—
39C24×C4—21no—
40C24×C4—21no—
41C24×C4D411yescharacteristic
42C24×C4D411yescharacteristic
43C24×C4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C22×C4⋊C4—21no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—28no—
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—28no—
54C22×C4⋊C4—24no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—44no—
58C22×C4⋊C4—44no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C42—22no—
69C22×C42—22no—
70C22×C42—22no—
71C22×C42—22no—
72C22×C42—22no—
73C22×C42—22no—
74C22×C42D412yes—
75C22×C42C2312yes—
76C2×#107—24no—
77C2×#107—28no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C22⋊C8D414yes—
84#389—216no—
85#389—28no—
86#389—28no—
87#381—44no—
88#380—22no—
89#380—22no—
90#380—22no—
91#380—22no—
92C2×#95—28no—
93C2×#95—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100#340—28no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—28no—
105#336—28no—
106C25C2×D411yescharacteristic
107C23×C4—24no—
108C23×C4—24no—
109C23×C4—28no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—44no—
129C23×C4—42no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4D814yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4#4112yes—
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—24no—
155C22×C8—22no—
156C22×C8—24no—
157C22×C8C2×D414yes—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—216no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—432no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—48no—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4#4118yes—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—42no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—44no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—44no—
212#107—48no—
213C22⋊C8—28no—
214C22⋊C8—44no—
215#95—216no—
216#95—44no—
217#95—42no—
218#95—42no—
219#95—44no—
220#95—44no—
221#95—44no—
222C24—22no—
223C24—22no—
224C24—22no—
225C24—22no—
226C24#12012yes—
227C24#12112yes—
228C24#12312yes—
229C24C2×D812yes—
230C24#13512yes—
231C24AGL(1,ℤ/8)12yes—
232C24#13712yes—
233C24C22×D411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—28no—
244C22×C4—216no—
245C22×C4—28no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—416no—
267C22×C4—416no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—84no—
308C22×C4—84no—
309C22×C4#12012yes—
310C22×C4#12114yes—
311C22×C4#12114yes—
312C22×C4#12112yes—
313C22×C4#12314yes—
314C22×C4#12312yes—
315C22×C4#12414yes—
316C22×C4C2×D814yes—
317C22×C4#13714yes—
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4114yes—
324C22×C4C2×#4111yescharacteristic
325C22×C42−1+414yes—
326C2×C8—24no—
327C2×C8—28no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—44no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—216no—
337C4⋊C4—24no—
338C4⋊C4—28no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—432no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—48no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4#12318yes—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—48no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—24no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—24no—
370C23—24no—
371C23—24no—
372C23—22no—
373C23—22no—
374C23—24no—
375C23#39414yes—
376C23#44714yes—
377C23#45214yes—
378C23#46614yes—
379C23#46814yes—
380C23#48014yes—
381C23#48214yes—
382C23C2×#12011yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×#12311yescharacteristic
385C23#52512yes—
386C23#53612yes—
387C23C22×D811yescharacteristic
388C23C2×#13511yescharacteristic
389C23C2×AGL(1,ℤ/8)11yescharacteristic
390C23C2×#13711yescharacteristic
391C23C2×#13712yes—
392C23#57512yes—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—28no—
398C2×C4—28no—
399C2×C4—28no—
400C2×C4—28no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—416no—
409C2×C4—432no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—416no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—816no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4#39414yes—
449C2×C4C2×#12111yescharacteristic
450C2×C4C2×#12111yescharacteristic
451C2×C4C2×#12311yescharacteristic
452C2×C4C2×#12312yes—
453C2×C4#52512yes—
454C2×C4C2×#12411yescharacteristic
455C2×C4#53612yes—
456C2×C4#53612yes—
457C2×C4#54714yes—
458C2×C4#55214yes—
459C2×C4#55214yes—
460C2×C4#55214yes—
461C8—44no—
462C8—82no—
463C22—28no—
464C22—28no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—28no—
469C22—28no—
470C22—24no—
471C22—24no—
472C22—24no—
473C22#203318yes—
474C22C2×#39411yescharacteristic
475C22#241312yes—
476C22#248412yes—
477C22#261411yescharacteristic
478C22#284911yescharacteristic
479C22C2×#44711yescharacteristic
480C22C2×#45211yescharacteristic
481C22#300411yescharacteristic
482C22#300511yescharacteristic
483C22C2×#46611yescharacteristic
484C22C2×#46811yescharacteristic
485C22#305612yes—
486C22C2×#48011yescharacteristic
487C22C2×#48211yescharacteristic
488C22#309512yes—
489C22#310012yes—
490C22#310812yes—
491C22#311312yes—
492C22#311412yes—
493C22#322612yes—
494C22#323212yes—
495C22#323612yes—
496C22#323812yes—
497C4—24no—
498C4—44no—
499C4—44no—
500C4—48no—
501C4—84no—
502C4—162no—
503C4—161no—
504C4—161no—
505C4#241312yes—
506C4#248412yes—
507C4#248612yes—
508C4#250312yes—
509C2—28no—
510C2#1507311yescharacteristic
511C2C2×#203311yescharacteristic
512C2#2348111yescharacteristic
513C2#2460012yes—
514C2#2460912yes—
515C2#2761012yes—
516C2#2761612yes—
517C2#2963811yescharacteristic
518C2#2964311yescharacteristic
519C2#3300211yescharacteristic
520C2#3300811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11292C2semidirect
8. #11219C2semidirect

Maximal subgroups

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

subgroupindex
C22×#3892
C2×#18522
C2×#18232
C2×#18172
#146852
#112922
#112192
#107302