Finite Group Databasebeta

← #194150

Subgroups — #194150

Order 512.

1322 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34121C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4C2×#1981C211yesmaximal, characteristic, complemented
5C2×#1477C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8982C214yesmaximal, complemented
8#8045C211yesmaximal, characteristic, complemented
9#7901C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079C418yescomplemented
14C2×#477—21no—
15C2×#475C2212yes—
16C2×#476—21no—
17C2×#474C2212yes—
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C2×#417—21no—
23C22×#103—22no—
24#2357—21no—
25#2356—21no—
26#2355C2212yes—
27#2354C2212yes—
28#2353—22no—
29#2352—22no—
30#2298—22no—
31#2297—21no—
32#2285C2211yescharacteristic
33#2283—21no—
34C4×#117—21no—
35C4×#117—21no—
36C2×C4×C4⋊C4C2211yescharacteristic
37#1990C2214yes—
38#1987—22no—
39#1981—21no—
40#1981—21no—
41#1981C2214yes—
42#1824—22no—
43#1477—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C22×M16—21no—
51C23×C8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8C2312yes—
60C4×#41—22no—
61C4×#41—22no—
62C4×#41C2×C414yes—
63C4×#41D414yes—
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×D4—21no—
67C2×C4×D4—21no—
68C2×C4×D4D411yescharacteristic
69C2×C4×D4C2311yescharacteristic
70C2×#117—21no—
71C2×#117—22no—
72C2×#117—21no—
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4D412yes—
77C22×C42—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#477—28no—
81#476—28no—
82#475—24no—
83#475C2×C418yescomplemented
84#474—24no—
85#474C2×C418yescomplemented
86#431—22no—
87#431—22no—
88#428—28no—
89C2×#107—22no—
90C2×#106—22no—
91#423—22no—
92#423C2×C414yes—
93C2×C4⋊C8—21no—
94C2×C4⋊C8—22no—
95C2×C4⋊C8—22no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100#417—28no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—48no—
105C2×#102—22no—
106C2×C22⋊C8—21no—
107C2×C4×C8C2312yes—
108#391—22no—
109#390—22no—
110#388—24no—
111#388—24no—
112#387—24no—
113#387—24no—
114#386—24no—
115#384C2312yes—
116#383—22no—
117#382—22no—
118#382—21no—
119#381—22no—
120#380—21no—
121#380—22no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4D411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4C2312yes—
133C4×C22⋊C4—24no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#376—22no—
138#376—22no—
139C2×#95—21no—
140C43—22no—
141#340—28no—
142Q8⋊C8—24no—
143D4⋊C8—24no—
144C2×#41—22no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41C22⋊C414yes—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—44no—
153C22×Q8—44no—
154C22×Q8—41no—
155C22×Q8—42no—
156C22×Q8C2×D411yescharacteristic
157C22×D4—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—22no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C2×M16—24no—
168C2×M16—42no—
169C2×M16—42no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—44no—
174C22×C8—42no—
175C4⋊Q8—24no—
176C4⋊Q8—24no—
177C4⋊Q8—24no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—42no—
182C4⋊Q8—42no—
183C4⋊Q8—42no—
184C4⋊Q8C22×C414yes—
185C4×Q8—22no—
186C4×Q8—21no—
187C4×Q8—21no—
188C4×Q8C22⋊C414yes—
189C4×Q8C22×C412yes—
190C4×Q8C2×D412yes—
191C4×D4—24no—
192C4×D4—24no—
193C4×D4—24no—
194C4×D4—24no—
195C4×D4—22no—
196C4×D4—21no—
197C4×D4—21no—
198C4×D4C22⋊C414yes—
199C4×D4C22×C412yes—
200C4×D4C2×D412yes—
201#117—22no—
202#117—24no—
203#117—24no—
204#117—44no—
205#117—42no—
206#117—42no—
207#117—44no—
208#117C22⋊C414yes—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—42no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—42no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4C2×D412yes—
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D412yes—
228C2×C4⋊C4C2×D414yes—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—24no—
233C2×C22⋊C4—22no—
234C2×C22⋊C4—24no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—24no—
239C2×C42—24no—
240C2×C42—24no—
241C2×C42—22no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—22no—
245C2×C42—24no—
246C2×C42—42no—
247C2×C42—42no—
248C2×C42C22×C412yes—
249C2×C42C2×D411yescharacteristic
250C2×C42C2×D412yes—
251C2×C42C2×D411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42#4112yes—
254C2×C42#4112yes—
255C2×C42C2411yescharacteristic
256#107—28no—
257#106—28no—
258C4⋊C8—24no—
259C4⋊C8—22no—
260C4⋊C8—22no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263C4⋊C8—42no—
264C4⋊C8C22×C414yes—
265#103—28no—
266#103—48no—
267#103—48no—
268#102—28no—
269C22⋊C8—42no—
270C4×C8—22no—
271C4×C8—22no—
272#95—24no—
273#95—24no—
274#95—24no—
275#95—24no—
276#95—21no—
277#95—21no—
278#95—22no—
279#95—22no—
280#95—22no—
281C24—21no—
282C24#12011yescharacteristic
283#41—28no—
284#41—28no—
285#41—44no—
286#41—44no—
287C2×Q8—21no—
288C2×Q8—24no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8—21no—
293C2×Q8—416no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—44no—
297C2×Q8—44no—
298C2×Q8—84no—
299C2×Q8—84no—
300C2×Q8—84no—
301C2×Q8C2×C22⋊C412yes—
302C2×D4—24no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—42no—
309C2×D4—42no—
310C2×D4C2×C22⋊C412yes—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—24no—
320C22×C4—24no—
321C22×C4—22no—
322C22×C4—24no—
323C22×C4—42no—
324C22×C4—44no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—42no—
329C22×C4—42no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—82no—
333C22×C4—82no—
334C22×C4—84no—
335C22×C4C2×C22⋊C412yes—
336C22×C4C4×D412yes—
337C22×C4C4×D412yes—
338C22×C4#12011yescharacteristic
339C22×C4#12011yescharacteristic
340C22×C4#12011yescharacteristic
341C22×C4#12112yes—
342C22×C4#12112yes—
343C22×C4#12312yes—
344C22×C4#13512yes—
345C22×C4#13512yes—
346C22×C4#13712yes—
347C22×C4#13712yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D411yescharacteristic
351M16—48no—
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—44no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—44no—
359C2×C8—84no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—22no—
365C4⋊C4—24no—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—44no—
371C4⋊C4—44no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—44no—
380C4⋊C4C2×C22⋊C412yes—
381C4⋊C4C2×C22⋊C412yes—
382C4⋊C4AGL(1,ℤ/8)14yes—
383C22⋊C4—24no—
384C22⋊C4—22no—
385C22⋊C4—22no—
386C22⋊C4—24no—
387C22⋊C4—48no—
388C22⋊C4—44no—
389C22⋊C4—84no—
390C42—22no—
391C42—22no—
392C42—24no—
393C42—24no—
394C42—48no—
395C42—42no—
396C42—42no—
397C42—42no—
398C42—42no—
399C42—44no—
400C42—84no—
401C42C2×C22⋊C412yes—
402C42C4×D412yes—
403C42C4×D412yes—
404C42#12112yes—
405C42#12112yes—
406C42#12312yes—
407C42C23×C411yescharacteristic
408C42C22×D411yescharacteristic
409C42C2×#4111yescharacteristic
410C42C2×#4111yescharacteristic
411C23—22no—
412C23—22no—
413C23—22no—
414C23—41no—
415C23—41no—
416C23—42no—
417C23—42no—
418C23#38612yes—
419C23#45212yes—
420C23#45212yes—
421C23C2×#12011yescharacteristic
422Q8—22no—
423Q8—22no—
424Q8—41no—
425Q8—41no—
426Q8—48no—
427Q8—88no—
428Q8—82no—
429Q8—82no—
430D4—22no—
431D4—22no—
432D4—48no—
433D4—41no—
434D4—41no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—48no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—84no—
465C2×C4—88no—
466C2×C4—88no—
467C2×C4—84no—
468C2×C4#38612yes—
469C2×C4#38612yes—
470C2×C4#38612yes—
471C2×C4#41614yes—
472C2×C4#41914yes—
473C2×C4#45212yes—
474C2×C4#45212yes—
475C2×C4#48514yes—
476C2×C4#48514yes—
477C2×C4C22×C22⋊C411yescharacteristic
478C2×C4C2×C4×D411yescharacteristic
479C2×C4C2×C4×D411yescharacteristic
480C2×C4C2×#12111yescharacteristic
481C2×C4C2×#12111yescharacteristic
482C2×C4C2×#12311yescharacteristic
483C2×C4C2×#13511yescharacteristic
484C2×C4C2×#13511yescharacteristic
485C2×C4C2×AGL(1,ℤ/8)12yes—
486C2×C4C2×#13711yescharacteristic
487C2×C4C2×#13711yescharacteristic
488C2×C4#57512yes—
489C8—44no—
490C8—82no—
491C8—82no—
492C22—24no—
493C22—24no—
494C22—22no—
495C22—22no—
496C22—42no—
497C22—42no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22#187314yes—
502C22C2×#38611yescharacteristic
503C22C2×#45211yescharacteristic
504C22C2×#45211yescharacteristic
505C22#300612yes—
506C22#302212yes—
507C4—22no—
508C4—21no—
509C4—22no—
510C4—21no—
511C4—44no—
512C4—42no—
513C4—42no—
514C4—42no—
515C4—84no—
516C4—82no—
517C4—84no—
518C4#187314yes—
519C4C2×#41611yescharacteristic
520C4C2×#41911yescharacteristic
521C4C4×AGL(1,ℤ/8)12yes—
522C4#303811yescharacteristic
523C4#304011yescharacteristic
524C4C2×#48511yescharacteristic
525C4C2×#48511yescharacteristic
526C2—24no—
527C2—82no—
528C2#1880312yes—
529C2#1880512yes—
530C2C2×#187311yescharacteristic
531C2#2092811yescharacteristic
532C2#2094411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ 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
3. #21834C2semidirect
4. C2×#1981C2semidirect
6. #8984C2semidirect
7. #8982C2semidirect
8. #8045C2semidirect
9. #7901C2semidirect
13. #3079C4semidirect
83. #475C2×C4semidirect
85. #474C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341212
#218342
C2×#19812
C2×#14772
#89842
#89822
#80452
#79012