Finite Group Databasebeta

← #374538

Subgroups — #374538

Order 512.

1468 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#433C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3C2×#1989C212yesmaximal, complemented
4#13777C211yesmaximal, characteristic
5#11200C212yesmaximal
6#11199C212yesmaximal
7#10881C212yesmaximal, complemented
8#10880C212yesmaximal, complemented
9#10849C212yesmaximal, complemented
10C22×#125C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#433—28no—
13C2×#433—24no—
14C22×C4⋊C8C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2629—24no—
19#2629C418yescomplemented
20C2×#398—22no—
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26C2×#380C2212yes—
27C2×C4×C4⋊C4C2212yes—
28#1989—28no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37#1437—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C24×C4C2311yescharacteristic
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×#125—22no—
44C2×#125—24no—
45C2×#125—24no—
46C2×#125C2×C414yes—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117C2×C414yes—
50C2×#117D414yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D412yes—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#433—232no—
67C2×C4⋊C8—24no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8D418yes—
70C2×C22⋊C8—24no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—24no—
73C2×#97—24no—
74C2×#97—22no—
75C2×C4×C8—24no—
76C2×C4×C8—22no—
77#398—216no—
78#396—216no—
79#389—24no—
80#389C2×C418yes—
81#384—22no—
82#384C2×C414yes—
83#380—24no—
84#380—24no—
85#380—24no—
86#380C2×C418yes—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104C25C2×D411yescharacteristic
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C22×C412yes—
116C23×C4C2×D412yes—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D412yes—
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×C8C2×D414yes—
134C22×C8C2×D414yes—
135#125—216no—
136#125—44no—
137#125—44no—
138#117—216no—
139#117—28no—
140#117—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
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—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22⋊C418yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4#4114yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4—82no—
186C2×C22⋊C4C22⋊C418yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—44no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42C22×C412yes—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D414yes—
196C2×C42C2×D412yes—
197C2×C42#4114yes—
198C4⋊C8—216no—
199C22⋊C8—216no—
200#97—28no—
201C4×C8—28no—
202#95—24no—
203#95—24no—
204#95—28no—
205#95—28no—
206#95—44no—
207#95—82no—
208#95—82no—
209C24—22no—
210C24—21no—
211C24—21no—
212C24—22no—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—22no—
217C24C2×C22⋊C412yes—
218C24C22×D411yescharacteristic
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
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×C4C2×C22⋊C412yes—
278C22×C4C2×C22⋊C414yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4#12114yes—
281C22×C4#12114yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4C2×#4112yes—
287C22×C4C2×#4112yes—
288C22×C42−1+412yes—
289C22×C42−1+412yes—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—216no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—416no—
322C4⋊C4—88no—
323C4⋊C4—88no—
324C22⋊C4—216no—
325C22⋊C4—88no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C42—24no—
329C42—22no—
330C42—22no—
331C42—88no—
332C42C4×D418yes—
333C42#12014yes—
334C42#12314yes—
335C23—22no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—22no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23C22×C22⋊C411yescharacteristic
357C23C2×#12112yes—
358C23#52511yescharacteristic
359C23#52511yescharacteristic
360C23#53612yes—
361C23#57612yes—
362C23#57712yes—
363C23#57712yes—
364C23#57812yes—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4#38618yes—
416C2×C4C2×C4×D412yes—
417C2×C4C4×#4112yes—
418C2×C4#51911yescharacteristic
419C2×C4#51911yescharacteristic
420C2×C4C2×#12011yescharacteristic
421C2×C4C2×#12311yescharacteristic
422C2×C4#53212yes—
423C2×C4#53612yes—
424C2×C4#54712yes—
425C2×C4#54712yes—
426C2×C4#54812yes—
427C2×C4#54812yes—
428C2×C4#55212yes—
429C2×C4#55212yes—
430C2×C4#55612yes—
431C2×C4#55612yes—
432C8—44no—
433C8—44no—
434C22—42no—
435C22—44no—
436C22—42no—
437C22—42no—
438C22—44no—
439C22—82no—
440C22—82no—
441C22—82no—
442C22—82no—
443C22C2×#38611yescharacteristic
444C22#229711yescharacteristic
445C22#231011yescharacteristic
446C22#241512yes—
447C22#241512yes—
448C22#289712yes—
449C22#289812yes—
450C22#306012yes—
451C22#306112yes—
452C22#306212yes—
453C22#306312yes—
454C22#311912yes—
455C22#312012yes—
456C22#312211yescharacteristic
457C22#312311yescharacteristic
458C22#312611yescharacteristic
459C22#312711yescharacteristic
460C22#324112yes—
461C22#324212yes—
462C22#324312yes—
463C22#324412yes—
464C4—28no—
465C4—42no—
466C4—44no—
467C4—84no—
468C4—84no—
469C4—84no—
470C4—84no—
471C4#236212yes—
472C4#236212yes—
473C4#243212yes—
474C4#249912yes—
475C2—82no—
476C2#1413811yescharacteristic
477C2#2097612yes—
478C2#2139811yescharacteristic
479C2#2139911yescharacteristic
480C2#2184312yes—
481C2#2471312yes—
482C2#2471512yes—
483C2#2471712yes—
484C2#2471912yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1989C2semidirect
7. #10881C2semidirect
8. #10880C2semidirect
9. #10849C2semidirect
19. #2629C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4332
C2×#26292
C2×#19892
#137772
#112002
#111992
#108812
#108802
#108492