Finite Group Databasebeta

← #241370

Subgroups — #241370

Order 512.

862 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35239C211yesmaximal, characteristic
2#21987C211yesmaximal, characteristic
3C2×#1948C211yesmaximal, characteristic
4#20764C211yesmaximal, characteristic
5#10582C212yesmaximal, complemented
6#10582C212yesmaximal, complemented
7#10578C212yesmaximal, complemented
8#10575C212yesmaximal, complemented
9#8044C211yesmaximal, characteristic
10#7904C211yesmaximal, characteristic
11#7895C211yesmaximal, characteristic
12C2×#432C2211yescharacteristic
13C22×#106—21no—
14C22×C4⋊C8C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2857—21no—
17#2851—21no—
18#2666—22no—
19#2571C2211yescharacteristic
20#2570—21no—
21#2552C2211yescharacteristic
22#2552C2211yescharacteristic
23#2551C2211yescharacteristic
24#2550—21no—
25#2442—21no—
26#2440—21no—
27#2438—22no—
28#2331—21no—
29#2330—21no—
30#2285C2211yescharacteristic
31C2×#384C2211yescharacteristic
32#2283C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#1948—21no—
35#1948—21no—
36#1948C2212yes—
37#1948C2212yes—
38#1948C2212yes—
39#1947C2212yes—
40#1853—21no—
41#1852—21no—
42#1847—21no—
43#1836—22no—
44#1836—21no—
45#1834C2212yes—
46#1834C2212yes—
47#1736—22no—
48C2×#340—21no—
49#1296C2212yes—
50#1296C2212yes—
51#1279C2212yes—
52#1279C2212yes—
53#1279C2212yes—
54#1279C2212yes—
55C22×M16—21no—
56C23×C8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×#125—21no—
60C2×#125—21no—
61C2×C22⋊Q8—22no—
62C2×#117D412yes—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4D411yescharacteristic
68C22×C42—21no—
69C22×C42C2311yescharacteristic
70#432—22no—
71#432Q814yescomplemented
72C2×#107—22no—
73C2×#106—22no—
74C2×#106—22no—
75C2×#106—22no—
76C2×#106—44no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8Q814yescomplemented
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—21no—
83C4×M16—22no—
84C4×M16Q814yescomplemented
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8C2312yes—
88#400—22no—
89#400—22no—
90#398—22no—
91#398—22no—
92#398—22no—
93#398—22no—
94#395—22no—
95#395—22no—
96#391—22no—
97#391—22no—
98#389—22no—
99#389—22no—
100#389—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#387—22no—
109#387—22no—
110#387—22no—
111#387—22no—
112#385—22no—
113#385—22no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2312yes—
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#383—21no—
123#382—21no—
124#382—21no—
125#382D411yescharacteristic
126#382D412yes—
127#382D411yescharacteristic
128#381—21no—
129#380—22no—
130#380—21no—
131#380—21no—
132#380D411yescharacteristic
133#380D411yescharacteristic
134C4×C4⋊C4—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4D411yescharacteristic
139C4×C4⋊C4D411yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C4⋊C4C2311yescharacteristic
143C4×C4⋊C4C2311yescharacteristic
144C4×C22⋊C4—21no—
145#376—21no—
146C2×#95—21no—
147#340—24no—
148#340—24no—
149#340—22no—
150#340—22no—
151#340—22no—
152#340—22no—
153C22×Q8—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C2×M16—24no—
161C2×M16—42no—
162C2×M16—42no—
163C22×C8—24no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—42no—
168C22×C8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—82no—
172#125—42no—
173#125—42no—
174#125—82no—
175C22⋊Q8—84no—
176#117—44no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—82no—
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—82no—
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q812yes—
231C2×C42C2×Q812yes—
232C2×C42C2×Q812yes—
233C2×C42#4112yes—
234C2×C42C2411yescharacteristic
235#107—82no—
236#106—44no—
237#106—44no—
238#106—82no—
239C4⋊C8—24no—
240C4⋊C8—22no—
241C4⋊C8—22no—
242C4⋊C8C2×Q814yes—
243C4⋊C8C2×Q814yes—
244C22⋊C8—24no—
245#97—24no—
246#97—24no—
247C4×C8—22no—
248C4×C8C2×Q814yes—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—22no—
254#95—22no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—22no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269C24#12011yescharacteristic
270C2×Q8—44no—
271C2×Q8—82no—
272C2×Q8—82no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4#12011yescharacteristic
302C22×C4#12011yescharacteristic
303C22×C4#12011yescharacteristic
304C22×C4C22⋊Q812yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4C4⋊Q812yes—
308C22×C4C4⋊Q812yes—
309C22×C4C4⋊Q812yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313M16—48no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—48no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4AGL(1,ℤ/8)12yes—
362C4⋊C4#13712yes—
363C22⋊C4—42no—
364C22⋊C4—44no—
365C22⋊C4—42no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—42no—
371C42—42no—
372C42—81no—
373C42—81no—
374C42—82no—
375C42C22⋊Q812yes—
376C42C22⋊Q812yes—
377C42C22⋊Q812yes—
378C42C4⋊Q812yes—
379C42C4⋊Q812yes—
380C42C4⋊Q812yes—
381C42C22×Q811yescharacteristic
382C42C22×Q811yescharacteristic
383C42C22×Q811yescharacteristic
384C42C2×#4111yescharacteristic
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23#39512yes—
390C23#45412yes—
391C23#45512yes—
392C23C2×#12011yescharacteristic
393Q8—164no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—164no—
446C2×C4#39512yes—
447C2×C4#39512yes—
448C2×C4#39512yes—
449C2×C4C2×C22⋊Q811yescharacteristic
450C2×C4C2×C22⋊Q811yescharacteristic
451C2×C4C2×C22⋊Q811yescharacteristic
452C2×C4C2×C4⋊Q811yescharacteristic
453C2×C4C2×C4⋊Q811yescharacteristic
454C2×C4C2×C4⋊Q811yescharacteristic
455C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
456C2×C4C2×#13711yescharacteristic
457C2×C4#57511yescharacteristic
458C2×C4#57511yescharacteristic
459C2×C4#57712yes—
460C2×C4#57712yes—
461C8—44no—
462C8—44no—
463C8—44no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22#206114yes—
471C22C2×#39511yescharacteristic
472C22#300611yescharacteristic
473C22#300711yescharacteristic
474C22#300911yescharacteristic
475C22#301011yescharacteristic
476C22C2×#45511yescharacteristic
477C22C2×#45411yescharacteristic
478C4—22no—
479C4—22no—
480C4—21no—
481C4—21no—
482C4—22no—
483C4—42no—
484C4—42no—
485C4—81no—
486C4—82no—
487C4—81no—
488C4—81no—
489C4—82no—
490C4—81no—
491C4—81no—
492C4—81no—
493C4—162no—
494C4#307711yescharacteristic
495C4#307811yescharacteristic
496C4#308412yes—
497C4#316312yes—
498C4#316511yescharacteristic
499C4#316611yescharacteristic
500C2—24no—
501C2#1857011yescharacteristic
502C2#1857311yescharacteristic
503C2#1857911yescharacteristic
504C2#1858011yescharacteristic
505C2#2355411yescharacteristic
506C2#2355611yescharacteristic
507C2C2×#206111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10582C2semidirect
6. #10582C2semidirect
7. #10578C2semidirect
8. #10575C2semidirect
71. #432Q8semidirect
79. C2×C4⋊C8Q8semidirect
84. C4×M16Q8semidirect

Maximal subgroups

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

subgroupindex
#352392
#219872
C2×#19482
#207642
#105822
#105782
#105752
#80442
#79042
#78952