Finite Group Databasebeta

← #147009

Subgroups — #147009

Order 512.

1070 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1800C211yesmaximal, characteristic
2C2×#1799C211yesmaximal, characteristic
3#13784C211yesmaximal, characteristic
4C2×#1390C214yesmaximal
5C22×C4×D4C2211yescharacteristic
6C2×#423—21no—
7C22×#100C2212yes—
8C22×#99C2212yes—
9C2×#386C2212yes—
10C2×#380—21no—
11C2×C4×C22⋊C4C411yescharacteristic, complemented
12C22×#95C411yescharacteristic, complemented
13#1800—22no—
14#1800C414yescomplemented
15#1799—22no—
16#1799C414yescomplemented
17C2×#343—21no—
18C2×#342—21no—
19#1437—21no—
20#1436—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—41no—
38C22×C4⋊C4D411yescharacteristic
39C22×C22⋊C4—22no—
40C22×C22⋊C4—22no—
41C22×C22⋊C4C2×C412yes—
42C22×C22⋊C4C2×C411yescharacteristic
43C22×C22⋊C4D411yescharacteristic
44C22×C42—21no—
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—24no—
55C2×#99—24no—
56C2×#99—22no—
57C2×#99C2×C414yes—
58#386—22no—
59#386—22no—
60#386D414yes—
61#386Q814yes—
62#380—24no—
63#380—42no—
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—42no—
81C22×D4C22×C411yescharacteristic
82C22×D4C22×C411yescharacteristic
83C22×D4C2×D411yescharacteristic
84C22×D4C2×D412yes—
85C22×D4C2×Q811yescharacteristic
86C22×D4C2×Q812yes—
87C22×D4#4112yes—
88C22×D4#4112yes—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—21no—
92C23×C4—22no—
93C23×C4—21no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—42no—
99C23×C4—42no—
100C23×C4—42no—
101C23×C4—82no—
102C23×C4—82no—
103C23×C4—82no—
104C23×C4—82no—
105C23×C4C4212yes—
106C23×C4C22⋊C411yescharacteristic
107C23×C4C22⋊C411yescharacteristic
108C23×C4C22⋊C411yescharacteristic
109C23×C4C22⋊C412yes—
110C23×C4C22⋊C411yescharacteristic
111C23×C4C4⋊C412yes—
112C23×C4C4⋊C412yes—
113C23×C4C4⋊C414yescomplemented
114C23×C4C2×D411yescharacteristic
115C2×M16—24no—
116C2×M16—42no—
117C2×M16—44no—
118C2×M16—44no—
119C22×C8—42no—
120C4×D4—42no—
121C4×D4—42no—
122C4×D4—42no—
123C4×D4—42no—
124C4×D4—42no—
125C4×D4—42no—
126C4×D4—42no—
127C4×D4—42no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—44no—
131C2×C4⋊C4—84no—
132C2×C4⋊C4C22⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—24no—
136C2×C22⋊C4—24no—
137C2×C22⋊C4—24no—
138C2×C22⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C22⋊C4—41no—
142C2×C22⋊C4—42no—
143C2×C22⋊C4—42no—
144C2×C22⋊C4—44no—
145C2×C22⋊C4—44no—
146C2×C22⋊C4—42no—
147C2×C22⋊C4—82no—
148C2×C22⋊C4—82no—
149C2×C22⋊C4C4212yes—
150C2×C22⋊C4C22⋊C412yes—
151C2×C22⋊C4C4⋊C412yes—
152C2×C22⋊C4C4⋊C412yes—
153C2×C22⋊C4C4⋊C414yescomplemented
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—22no—
157C2×C42—42no—
158C2×C42—82no—
159C2×C42C22⋊C412yes—
160C2×C42C4⋊C412yes—
161C4⋊C8—44no—
162C4⋊C8—44no—
163#100—28no—
164#100—42no—
165#100—42no—
166#99—28no—
167#99—42no—
168#99—42no—
169#95—24no—
170#95—24no—
171#95—42no—
172#95—44no—
173#95—41no—
174#95—41no—
175#95—42no—
176#95—82no—
177#95—82no—
178C24—21no—
179C24—21no—
180C24—21no—
181C24—22no—
182C24—21no—
183C24—22no—
184C24—42no—
185C24—41no—
186C24—41no—
187C24—41no—
188C24—42no—
189C24—41no—
190C24C2×C4211yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192C24C2×C22⋊C411yescharacteristic
193C24C2×C4⋊C411yescharacteristic
194C24C2×C4⋊C411yescharacteristic
195C24C2×C4⋊C412yes—
196C24#11712yes—
197C24#12112yes—
198C24C22⋊Q812yes—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—41no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4C4×D412yes—
216C2×D4C4×D412yes—
217C2×D4C4×Q812yes—
218C2×D4C4×Q812yes—
219C2×D4#12411yescharacteristic
220C2×D4#12411yescharacteristic
221C2×D4#12512yes—
222C2×D4#12711yescharacteristic
223C2×D4C4⋊Q812yes—
224C2×D4C4⋊Q811yescharacteristic
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—41no—
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—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—82no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—84no—
272C22×C4—84no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—88no—
277C22×C4—84no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4—84no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4#9512yes—
284C22×C4#9512yes—
285C22×C4#9512yes—
286C22×C4#9512yes—
287C22×C4#9911yescharacteristic
288C22×C4#9911yescharacteristic
289C22×C4#9912yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C4⋊C411yescharacteristic
292C22×C4#12112yes—
293C22×C4C22⋊Q812yes—
294M16—44no—
295M16—44no—
296M16—84no—
297C2×C8—44no—
298C2×C8—84no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—88no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—42no—
305C22⋊C4—42no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C42—22no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—84no—
318C23—22no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—21no—
336C23—21no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—44no—
347C23—42no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—81no—
364C23—81no—
365C23—82no—
366C23—82no—
367C23C2×#9511yescharacteristic
368C23C4×C4⋊C412yes—
369C23#38012yes—
370C23#38211yescharacteristic
371C23#38411yescharacteristic
372C23#38912yes—
373C23#39011yescharacteristic
374C23#39111yescharacteristic
375C23C2×#9911yescharacteristic
376C23C2×#9911yescharacteristic
377C23#41012yes—
378D4—42no—
379D4—42no—
380D4—42no—
381D4—42no—
382D4—41no—
383D4—41no—
384D4—42no—
385D4—41no—
386D4—42no—
387D4—41no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—88no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—88no—
437C2×C4—84no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—164no—
441C2×C4—164no—
442C2×C4#34212yes—
443C2×C4#34212yes—
444C2×C4#38912yes—
445C2×C4#39011yescharacteristic
446C2×C4#39111yescharacteristic
447C8—84no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—22no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—21no—
457C22—22no—
458C22—21no—
459C22—21no—
460C22—22no—
461C22—21no—
462C22—42no—
463C22—41no—
464C22—41no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—44no—
469C22—44no—
470C22—42no—
471C22—42no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—81no—
491C22—81no—
492C22C4⋊#9511yescharacteristic
493C22C2×#34211yescharacteristic
494C22#174011yescharacteristic
495C22#179912yes—
496C22#179912yes—
497C22#212112yes—
498C22#212211yescharacteristic
499C22#212311yescharacteristic
500C4—22no—
501C4—22no—
502C4—21no—
503C4—22no—
504C4—21no—
505C4—42no—
506C4—42no—
507C4—42no—
508C4—84no—
509C4—84no—
510C4—164no—
511C2—22no—
512C2—21no—
513C2—21no—
514C2—41no—
515C2—42no—
516C2—41no—
517C2—41no—
518C2—42no—
519C2—41no—
520C2#1084611yescharacteristic
521C2#1257411yescharacteristic
522C2#1257511yescharacteristic
523C2#1279512yes—
524C2#1279612yes—
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
11. C2×C4×C22⋊C4C4semidirect
12. C22×#95C4semidirect
14. #1800C4semidirect
16. #1799C4semidirect
23. C22⋊#95C4semidirect
113. C23×C4C4⋊C4semidirect
153. C2×C22⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18002
C2×#17992
#137842
C2×#13902