Finite Group Databasebeta

← #362849

Subgroups — #362849

Order 512.

2990 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#380C211yesmaximal, characteristic
2#14680C211yesmaximal, characteristic
3C4×#379C211yesmaximal, characteristic
4#11353C214yesmaximal, complemented
5#10854C214yesmaximal, complemented
6#10725C214yesmaximal
7C25×C4C2211yescharacteristic
8C23×C4⋊C4C412yescomplemented
9C23×C22⋊C4C2212yes—
10C2×#380—22no—
11C2×#380—22no—
12C2×#380—22no—
13C2×#380—22no—
14C2×#380—22no—
15C2×#380C412yescomplemented
16C2×#380C412yescomplemented
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×#380C2212yes—
20C2×#379C2212yes—
21C2×#379C2212yes—
22C2×C4×C22⋊C4—22no—
23C2×C4×C22⋊C4C2212yes—
24C22×#95C412yescomplemented
25C2×#342C22116yes—
26C22⋊#95—22no—
27C22⋊#95—22no—
28C22⋊#95C2212yes—
29C22⋊#95C2212yes—
30C26C2311yescharacteristic
31C24×C4—28no—
32C24×C4—48no—
33C24×C4C2×C411yescharacteristic
34C24×C4C2×C411yescharacteristic
35C24×C4C2×C411yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C414yes—
38C24×C4D412yes—
39C24×C4D412yes—
40C24×C4D412yes—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C24×C4C2311yescharacteristic
44C22×C4⋊C4—22no—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4—48no—
50C22×C4⋊C4C2×C414yes—
51C22×C4⋊C4C2×C414yes—
52C22×C22⋊C4—24no—
53C22×C22⋊C4—24no—
54C22×C22⋊C4—24no—
55C22×C22⋊C4—24no—
56C22×C22⋊C4—48no—
57C22×C22⋊C4C2×C414yes—
58C22×C22⋊C4C2314yes—
59C22×C22⋊C4C2314yes—
60C22×C22⋊C4C2314yes—
61C22×C42—22no—
62#380—24no—
63#380—24no—
64#380—24no—
65#380—24no—
66#380—24no—
67#380—24no—
68#380—22no—
69#380—22no—
70#380—22no—
71#380—22no—
72#380—28no—
73#380—48no—
74#379—24no—
75#379—28no—
76#379Q818yes—
77C4×C22⋊C4—28no—
78C4×C22⋊C4—48no—
79C2×#95—24no—
80C2×#95—24no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—24no—
84C2×#95—24no—
85C2×#95C2×C414yes—
86C2×#95C2×C414yes—
87#342—416no—
88C25—24no—
89C25—24no—
90C25—44no—
91C25—44no—
92C25C22×C411yescharacteristic
93C25C22×C411yescharacteristic
94C25C2×D412yes—
95C25C2×D412yes—
96C25C2×Q814yes—
97C25#4114yes—
98C25C2411yescharacteristic
99C23×C4—28no—
100C23×C4—24no—
101C23×C4—24no—
102C23×C4—28no—
103C23×C4—28no—
104C23×C4—24no—
105C23×C4—24no—
106C23×C4—24no—
107C23×C4—28no—
108C23×C4—28no—
109C23×C4—28no—
110C23×C4—28no—
111C23×C4—24no—
112C23×C4—24no—
113C23×C4—24no—
114C23×C4—24no—
115C23×C4—44no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—44no—
120C23×C4—44no—
121C23×C4—44no—
122C23×C4—44no—
123C23×C4—44no—
124C23×C4—44no—
125C23×C4—44no—
126C23×C4—48no—
127C23×C4—464no—
128C23×C4—48no—
129C23×C4—48no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C412yes—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22⋊C412yes—
135C23×C4C22⋊C412yes—
136C23×C4C22⋊C412yes—
137C23×C4C22⋊C412yes—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C22×C418yes—
141C23×C4C22×C411yescharacteristic
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4118yes—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—48no—
155C2×C4⋊C4—48no—
156C2×C4⋊C4—48no—
157C2×C4⋊C4—48no—
158C2×C4⋊C4—48no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—44no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—28no—
165C2×C22⋊C4—216no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—216no—
170C2×C22⋊C4—28no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4—48no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4—416no—
176C2×C22⋊C4—416no—
177C2×C22⋊C4—48no—
178C2×C22⋊C4—48no—
179C2×C22⋊C4—48no—
180C2×C22⋊C4—48no—
181C2×C22⋊C4C2×Q818yes—
182C2×C42—48no—
183C2×C42—44no—
184#95—28no—
185#95—24no—
186#95—24no—
187#95—48no—
188#95—48no—
189#95—44no—
190#95—44no—
191C24—22no—
192C24—22no—
193C24—24no—
194C24—24no—
195C24—22no—
196C24—22no—
197C24—24no—
198C24—24no—
199C24—24no—
200C24—24no—
201C24—216no—
202C24—216no—
203C24—24no—
204C24—24no—
205C24—24no—
206C24—24no—
207C24—42no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24—416no—
215C24—416no—
216C24—42no—
217C24—416no—
218C24—432no—
219C24—416no—
220C24C2×C22⋊C412yes—
221C24C2×C22⋊C412yes—
222C24C4×Q814yes—
223C24C4×Q814yes—
224C24C22⋊Q818yes—
225C24#12414yes—
226C24C4⋊Q814yes—
227C24C23×C411yescharacteristic
228C24C22×D411yescharacteristic
229C24C22×D411yescharacteristic
230C24C22×Q812yes—
231C24C2×#4114yes—
232C24C2×#4112yes—
233C24H(5,𝔽2)14yes—
234C22×C4—232no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—28no—
263C22×C4—28no—
264C22×C4—216no—
265C22×C4—416no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
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—432no—
279C22×C4—432no—
280C22×C4—416no—
281C22×C4—432no—
282C22×C4—432no—
283C22×C4—416no—
284C22×C4—416no—
285C22×C4—416no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—48no—
289C22×C4—48no—
290C22×C4—48no—
291C22×C4—48no—
292C22×C4—44no—
293C22×C4—44no—
294C22×C4—48no—
295C22×C4—48no—
296C22×C4—48no—
297C22×C4—48no—
298C22×C4—48no—
299C22×C4—48no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—44no—
303C22×C4—832no—
304C22×C4#9918yes—
305C22×C4C2×C22⋊C412yes—
306C22×C4C2×C22⋊C411yescharacteristic
307C22×C4C2×C22⋊C411yescharacteristic
308C22×C4C2×C22⋊C411yescharacteristic
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C411yescharacteristic
311C22×C4#11714yes—
312C22×C4#11714yes—
313C22×C4#12314yes—
314C22×C4#12314yes—
315C22×C4#12314yes—
316C22×C4#12314yes—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—48no—
320C4⋊C4—48no—
321C4⋊C4—88no—
322C22⋊C4—216no—
323C22⋊C4—432no—
324C22⋊C4—416no—
325C22⋊C4—48no—
326C22⋊C4—48no—
327C22⋊C4—48no—
328C22⋊C4—416no—
329C22⋊C4—816no—
330C42—88no—
331C23—28no—
332C23—216no—
333C23—28no—
334C23—22no—
335C23—28no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—22no—
344C23—216no—
345C23—22no—
346C23—22no—
347C23—28no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—216no—
353C23—28no—
354C23—28no—
355C23—216no—
356C23—28no—
357C23—28no—
358C23—22no—
359C23—44no—
360C23—48no—
361C23—48no—
362C23—44no—
363C23—44no—
364C23—44no—
365C23—44no—
366C23—44no—
367C23—48no—
368C23—48no—
369C23—48no—
370C23—48no—
371C23—48no—
372C23—48no—
373C23—48no—
374C23—48no—
375C23—48no—
376C23—44no—
377C23—464no—
378C23—464no—
379C23—48no—
380C23—44no—
381C23#39114yes—
382C23#39114yes—
383C23C2×#9914yes—
384C23#41014yes—
385C23C22×C22⋊C411yescharacteristic
386C23C2×#11712yes—
387C23C2×C4×Q812yes—
388C23#51812yes—
389C23C2×C22⋊Q812yes—
390C23C2×#12312yes—
391C23C2×#12312yes—
392C23C2×#12411yescharacteristic
393C23C2×C4⋊Q811yescharacteristic
394C23#53512yes—
395C23#53512yes—
396C23#55318yes—
397C23#55418yes—
398C2×C4—216no—
399C2×C4—28no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—28no—
406C2×C4—28no—
407C2×C4—216no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—48no—
416C2×C4—48no—
417C2×C4—48no—
418C2×C4—48no—
419C2×C4—48no—
420C2×C4—48no—
421C2×C4—48no—
422C2×C4—44no—
423C2×C4—48no—
424C2×C4—416no—
425C2×C4—48no—
426C2×C4—48no—
427C2×C4—48no—
428C2×C4—48no—
429C2×C4—84no—
430C2×C4—88no—
431C2×C4—84no—
432C2×C4—816no—
433C2×C4—816no—
434C2×C4—816no—
435C2×C4—88no—
436C2×C4—84no—
437C2×C4—88no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4#38112yes—
441C2×C4#38112yes—
442C2×C4#38112yes—
443C2×C4#38112yes—
444C2×C4C2×#9914yes—
445C2×C4C2×#9914yes—
446C22—232no—
447C22—24no—
448C22—24no—
449C22—24no—
450C22—24no—
451C22—24no—
452C22—24no—
453C22—22no—
454C22—24no—
455C22—22no—
456C22—24no—
457C22—24no—
458C22—22no—
459C22—24no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—24no—
466C22—24no—
467C22—24no—
468C22—24no—
469C22—48no—
470C22—48no—
471C22—432no—
472C22—416no—
473C22—416no—
474C22—416no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22#199618yes—
480C22C2×#38111yescharacteristic
481C22C2×#39111yescharacteristic
482C22#230911yescharacteristic
483C22#234514yes—
484C22#248714yes—
485C22#250212yes—
486C22#250412yes—
487C22C22×#9912yes—
488C22C2×#41012yes—
489C4—22no—
490C4—24no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—88no—
495C4—88no—
496C4—164no—
497C4#181914yes—
498C2—24no—
499C2—22no—
500C2—24no—
501C2—28no—
502C2—22no—
503C2—44no—
504C2#1392411yescharacteristic
505C2C2×#181911yescharacteristic
506C2#2068811yescharacteristic
507C2C2×#199612yes—
508C2#2179312yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC26C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C25×C4 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11353C2semidirect
5. #10854C2semidirect
8. C23×C4⋊C4C4semidirect
15. C2×#380C4semidirect
16. C2×#380C4semidirect
24. C22×#95C4semidirect

Maximal subgroups

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

subgroupindex
C22×#3802
#146802
C4×#3792
#113532
#108542
#107252