Finite Group Databasebeta

← #409274

Subgroups — #409274

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1809C211yesmaximal, characteristic
2C2×#1728C211yesmaximal, characteristic
3#14697C212yesmaximal
4#11379C211yesmaximal, characteristic, complemented
5#11377C211yesmaximal, characteristic, complemented
6#11327C211yesmaximal, characteristic, complemented
7#11327C211yesmaximal, characteristic, complemented
8#11293C212yesmaximal, complemented
9#11292C212yesmaximal, complemented
10#10722C212yesmaximal
11#10722C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#385—22no—
20C2×#380C2212yes—
21C2×#380C2212yes—
22#1809—24no—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31#1728—24no—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35C2×#336C2211yescharacteristic
36#1436—22no—
37#1436—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C22⋊#95—22no—
41C24×C4D412yes—
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C22×M16—24no—
45C22×M16—24no—
46C22×M16—22no—
47C22×M16—22no—
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16D414yes—
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42—22no—
66C22×C42—22no—
67C22×C42—42no—
68C22×C42—42no—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—42no—
77#389—42no—
78#389—42no—
79#389—42no—
80#389—42no—
81#385—48no—
82#380—24no—
83#380—24no—
84#380—24no—
85#380Q818yes—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94#340—24no—
95#340—22no—
96#340—22no—
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—24no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—24no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4—42no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D412yes—
128C23×C4C2×Q814yes—
129C23×C4#4112yes—
130C23×C4#4114yes—
131C23×C4C2411yescharacteristic
132C2×M16—28no—
133C2×M16—28no—
134C2×M16—28no—
135C2×M16—44no—
136C2×M16—48no—
137C2×M16—48no—
138C2×M16—48no—
139C2×M16—48no—
140C2×M16—44no—
141C2×M16—44no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8#4112yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4C2×Q814yes—
191C2×C4⋊C4C2×Q814yes—
192C2×C22⋊C4—28no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C42—24no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—44no—
210C2×C42—44no—
211C2×C42—48no—
212#107—44no—
213#106—44no—
214#95—28no—
215#95—44no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24—22no—
229C24#12112yes—
230C24C22⋊Q814yes—
231C24C22×D411yescharacteristic
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—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—48no—
268C22×C4—48no—
269C22×C4—48no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4#12112yes—
298C22×C4C22⋊Q814yes—
299C22×C4#12314yes—
300C22×C4#12518yes—
301C22×C4C4⋊Q814yes—
302C22×C4AGL(1,ℤ/8)14yes—
303C22×C4#13714yes—
304C22×C4C22×D411yescharacteristic
305C22×C4C22×Q812yes—
306C22×C4C2×#4112yes—
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4111yescharacteristic
311C22×C4C2×#4111yescharacteristic
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)12yes—
314M16—416no—
315M16—416no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8#12314yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C22⋊C4—416no—
355C22⋊C4—88no—
356C42—48no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—42no—
362C42—42no—
363C23—24no—
364C23—24no—
365C23—24no—
366C23—24no—
367C23—24no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—24no—
373C23—42no—
374C23—42no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23#39814yes—
384C23C2×#12111yescharacteristic
385C23C2×C22⋊Q812yes—
386C23#53211yescharacteristic
387C23#53211yescharacteristic
388C23#53412yes—
389C23C2×AGL(1,ℤ/8)12yes—
390C23C2×#13712yes—
391C23#57512yes—
392C23#57512yes—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—432no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4#39814yes—
447C2×C4C2×#12311yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4#52511yescharacteristic
450C2×C4C2×#12512yes—
451C2×C4#52911yescharacteristic
452C2×C4#52911yescharacteristic
453C2×C4C2×C4⋊Q811yescharacteristic
454C2×C4#53411yescharacteristic
455C2×C4#53511yescharacteristic
456C2×C4#53812yes—
457C2×C4#55112yes—
458C2×C4#55112yes—
459C2×C4#55112yes—
460C2×C4#55112yes—
461C2×C4#55412yes—
462C2×C4#55412yes—
463C2×C4#55412yes—
464C2×C4#55412yes—
465C8—24no—
466C8—42no—
467C8—42no—
468C8—42no—
469C22—28no—
470C22—24no—
471C22—44no—
472C22—42no—
473C22—42no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—44no—
478C22C2×#39811yescharacteristic
479C22#256412yes—
480C22#256412yes—
481C22#261711yescharacteristic
482C22#285411yescharacteristic
483C22#305512yes—
484C22#305612yes—
485C22#307712yes—
486C22#307812yes—
487C22#307912yes—
488C22#308012yes—
489C22#317112yes—
490C22#317212yes—
491C22#318311yescharacteristic
492C22#318311yescharacteristic
493C22#318411yescharacteristic
494C22#318511yescharacteristic
495C22#319312yes—
496C22#319412yes—
497C22#319612yes—
498C22#319712yes—
499C4—24no—
500C4—44no—
501C4—44no—
502C4—84no—
503C4—84no—
504C4—162no—
505C4—162no—
506C4#248211yescharacteristic
507C4#248311yescharacteristic
508C4#250211yescharacteristic
509C4#250512yes—
510C4#250512yes—
511C4#258311yescharacteristic
512C2—44no—
513C2#1501411yescharacteristic
514C2#2367012yes—
515C2#2916312yes—
516C2#2916412yes—
517C2#2916612yes—
518C2#2916712yes—
519C2#2974412yes—
520C2#3314311yescharacteristic
521C2#3314411yescharacteristic
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
4. #11379C2semidirect
5. #11377C2semidirect
6. #11327C2semidirect
7. #11327C2semidirect
8. #11293C2semidirect
9. #11292C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18092
C2×#17282
#146972
#113792
#113772
#113272
#112932
#112922
#107222