Finite Group Databasebeta

← #147068

Subgroups — #147068

Order 512.

1090 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1800C211yesmaximal, characteristic
2C2×C4×#99C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4C2×#1390C214yesmaximal
5C22×C4×D4C2211yescharacteristic
6C2×#423—21no—
7C22×#100C2212yes—
8C22×#99C2212yes—
9C2×#386C2212yes—
10C2×#380C411yescharacteristic, complemented
11C2×C4×C22⋊C4—21no—
12C22×#95C411yescharacteristic, complemented
13#1800—22no—
14#1800C414yescomplemented
15C4×#99—22no—
16C4×#99C414yescomplemented
17C2×#343—21no—
18C2×#342—21no—
19#1441—21no—
20#1439—21no—
21#1434—21no—
22C22⋊#95—21no—
23C22⋊#95C412yescomplemented
24#1390—28no—
25C23×D4C2311yescharacteristic
26C24×C4C2×C411yescharacteristic
27C24×C4C2×C412yes—
28C24×C4D411yescharacteristic
29C22×M16—22no—
30C2×C4×D4—41no—
31C2×C4×D4—41no—
32C2×C4×D4—42no—
33C2×C4×D4C2×C412yes—
34C2×C4×D4C2×C412yes—
35C2×C4×D4D412yes—
36C2×C4×D4Q812yes—
37C22×C4⋊C4—21no—
38C22×C4⋊C4D411yescharacteristic
39C22×C22⋊C4—22no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4D411yescharacteristic
44C22×C42—41no—
45C22×C42D411yescharacteristic
46#423—24no—
47#423—42no—
48C2×C4⋊C8—41no—
49C2×C4⋊C8—41no—
50C2×#100—24no—
51C2×#100—24no—
52C2×#100—22no—
53C2×#100C2×C414yes—
54C2×#99—22no—
55C2×#99—22no—
56C2×#99—22no—
57C2×#99C2×C414yes—
58C2×#99D414yes—
59C2×#99Q814yes—
60#386—24no—
61#386—24no—
62#380—42no—
63C4×C22⋊C4—24no—
64C4×C22⋊C4—42no—
65C2×#95—22no—
66C2×#95—21no—
67C2×#95—21no—
68C2×#95—21no—
69C2×#95—22no—
70C2×#95—22no—
71C2×#95—41no—
72C2×#95—41no—
73C2×#95—41no—
74#343—24no—
75#342—24no—
76C25C22×C411yescharacteristic
77C25C2×D411yescharacteristic
78C22×D4—41no—
79C22×D4—41no—
80C22×D4—41no—
81C22×D4—41no—
82C22×D4C22×C411yescharacteristic
83C22×D4C22×C411yescharacteristic
84C22×D4C2×D411yescharacteristic
85C22×D4C2×D412yes—
86C22×D4C2×Q811yescharacteristic
87C22×D4C2×Q812yes—
88C22×D4#4112yes—
89C22×D4#4112yes—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—21no—
93C23×C4—22no—
94C23×C4—21no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—24no—
99C23×C4—44no—
100C23×C4—42no—
101C23×C4—42no—
102C23×C4—42no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4—82no—
106C23×C4C4212yes—
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C411yescharacteristic
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C4⋊C412yes—
113C23×C4C4⋊C412yes—
114C23×C4C2×D411yescharacteristic
115C2×M16—24no—
116C2×M16—42no—
117C2×M16—42no—
118C2×M16—42no—
119C2×M16—42no—
120C2×M16—42no—
121C22×C8—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C4×D4—42no—
129C4×D4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4C22⋊C412yes—
135C2×C4⋊C4C4⋊C412yes—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—41no—
143C2×C22⋊C4—41no—
144C2×C22⋊C4—42no—
145C2×C22⋊C4—42no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—44no—
148C2×C22⋊C4—44no—
149C2×C22⋊C4—42no—
150C2×C22⋊C4—42no—
151C2×C22⋊C4—82no—
152C2×C22⋊C4C4212yes—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C4⋊C412yes—
155C2×C22⋊C4C4⋊C412yes—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—44no—
160C2×C42—84no—
161C2×C42C22⋊C412yes—
162C2×C42C4⋊C412yes—
163C4⋊C8—44no—
164C4⋊C8—44no—
165#100—28no—
166#100—44no—
167#99—24no—
168#99—24no—
169#99—24no—
170#99C4⋊C418yescomplemented
171#95—44no—
172#95—42no—
173#95—44no—
174#95—41no—
175#95—41no—
176#95—42no—
177#95—42no—
178#95—42no—
179#95—82no—
180C24—21no—
181C24—21no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—42no—
189C24—42no—
190C24—42no—
191C24—42no—
192C24C2×C4211yescharacteristic
193C24C2×C22⋊C411yescharacteristic
194C24C2×C22⋊C411yescharacteristic
195C24C2×C4⋊C411yescharacteristic
196C24C2×C4⋊C411yescharacteristic
197C24C4×D412yes—
198C24C4×Q812yes—
199C24#12112yes—
200C24C22⋊Q812yes—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4C2×C4⋊C412yes—
218C2×D4#11712yes—
219C2×D4C4×D412yes—
220C2×D4C4×Q812yes—
221C2×D4#12112yes—
222C2×D4C22⋊Q812yes—
223C2×D4#12512yes—
224C2×D4C4⋊Q812yes—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—44no—
250C22×C4—42no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—88no—
277C22×C4—88no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—84no—
282C22×C4#9512yes—
283C22×C4#9512yes—
284C22×C4#9512yes—
285C22×C4#9512yes—
286C22×C4#9911yescharacteristic
287C22×C4#9911yescharacteristic
288C22×C4#9912yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C4⋊C411yescharacteristic
291C22×C4#12411yescharacteristic
292C22×C4#12411yescharacteristic
293C22×C4#12711yescharacteristic
294C22×C4C4⋊Q811yescharacteristic
295M16—44no—
296M16—44no—
297M16—82no—
298M16—82no—
299C2×C8—44no—
300C2×C8—82no—
301C2×C8—82no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—84no—
305C22⋊C4—28no—
306C22⋊C4—42no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C22⋊C4—42no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—88no—
314C22⋊C4—84no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—88no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—41no—
339C23—41no—
340C23—44no—
341C23—41no—
342C23—41no—
343C23—44no—
344C23—42no—
345C23—44no—
346C23—42no—
347C23—44no—
348C23—44no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—81no—
359C23—81no—
360C23—82no—
361C23—82no—
362C23C2×#9511yescharacteristic
363C23C4×C4⋊C412yes—
364C23#38012yes—
365C23#38912yes—
366C23#38912yes—
367C23#39011yescharacteristic
368C23#39111yescharacteristic
369C23C2×#9911yescharacteristic
370C23C2×#9911yescharacteristic
371C23#41011yescharacteristic
372C23#41011yescharacteristic
373D4—42no—
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—42no—
379D4—42no—
380D4—42no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4—84no—
425C2×C4—88no—
426C2×C4—88no—
427C2×C4—88no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—88no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#34212yes—
439C2×C4#34212yes—
440C2×C4#38211yescharacteristic
441C2×C4#38411yescharacteristic
442C2×C4#39011yescharacteristic
443C2×C4#39111yescharacteristic
444C8—84no—
445C22—21no—
446C22—21no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—44no—
462C22—42no—
463C22—42no—
464C22—44no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—41no—
470C22—41no—
471C22—41no—
472C22—41no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—82no—
478C22—81no—
479C22—81no—
480C22—82no—
481C22C4⋊#9511yescharacteristic
482C22C2×#34211yescharacteristic
483C22#174011yescharacteristic
484C22#196411yescharacteristic
485C22#196411yescharacteristic
486C22#199611yescharacteristic
487C22#199611yescharacteristic
488C22#212012yes—
489C22#212112yes—
490C4—22no—
491C4—22no—
492C4—21no—
493C4—21no—
494C4—21no—
495C4—21no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—88no—
500C4—84no—
501C4—162no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2—42no—
507C2—42no—
508C2—42no—
509C2—42no—
510C2#1085411yescharacteristic
511C2#1257311yescharacteristic
512C2#1257411yescharacteristic
513C2#1322212yes—
514C2#1329712yes—
subgroupG/H
center, socleC23C2×#99
commutatorC24C2×C42
Frattini, prime radical, cosocleC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
10. C2×#380C4semidirect
12. C22×#95C4semidirect
14. #1800C4semidirect
16. C4×#99C4semidirect
23. C22⋊#95C4semidirect
170. #99C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×C4×#992
#139602
C2×#13902