Finite Group Databasebeta

← #194030

Subgroups — #194030

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3079C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#21804C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8983C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×#520—21no—
12C2×C4×#41C2211yescharacteristic
13#3079—24no—
14#3079C418yescomplemented
15C2×#477—21no—
16C2×#475C2212yes—
17C2×#476—21no—
18C2×#474C2212yes—
19C2×#431C2211yescharacteristic
20C2×#428—21no—
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23C22×C4×C8C2211yescharacteristic
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4C2212yes—
26D4×C4⋊C4—22no—
27D4×C4⋊C4—21no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4C2212yes—
30#2317—22no—
31#2317—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34C2×#384—21no—
35C2×#384C2211yescharacteristic
36#1990—22no—
37#1990C2214yes—
38#1981C2214yes—
39#1956—21no—
40#1956—21no—
41#1787—22no—
42C2×#340—22no—
43#1476—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×C414yescomplemented
59C2×C4⋊Q8C2312yes—
60#520—44no—
61#520—44no—
62C4×#41—22no—
63C4×#41—22no—
64C4×#41C2×C414yes—
65C4×#41D414yes—
66C2×C4×Q8—22no—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4—24no—
72C2×C4×D4—22no—
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C2×#117—22no—
76C2×#117—21no—
77C2×#117—21no—
78C2×#117D411yescharacteristic
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—22no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—22no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D412yes—
87C22×C42—21no—
88C22×C42C2311yescharacteristic
89#477—28no—
90#476—28no—
91#475—24no—
92#475C2×C418yescomplemented
93#474—24no—
94#474C2×C418yescomplemented
95#431—22no—
96#431—22no—
97#428—28no—
98C2×#107—22no—
99C2×#106—22no—
100#423—22no—
101#423C2×C414yescomplemented
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105#417—28no—
106C2×#103—22no—
107C2×#103—24no—
108C2×#103—24no—
109C2×#102—22no—
110C2×C22⋊C8—21no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2312yes—
116#389—24no—
117#389—22no—
118#389—24no—
119#389—22no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384—21no—
127#384—21no—
128#384D412yes—
129#384D411yescharacteristic
130#384D411yescharacteristic
131#384C2312yes—
132#382—22no—
133#382—21no—
134#380—22no—
135#380—21no—
136#380—22no—
137#380D412yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2312yes—
142#340—28no—
143Q8⋊C8—24no—
144D4⋊C8—24no—
145C2×#41—22no—
146C2×#41—22no—
147C2×#41—22no—
148C2×#41C22⋊C414yes—
149C2×#41SD1614yes—
150C2×#41Q1614yes—
151C22×Q8—24no—
152C22×Q8—22no—
153C22×Q8—41no—
154C22×Q8—42no—
155C22×Q8—82no—
156C22×Q8—82no—
157C22×Q8C2×D411yescharacteristic
158C22×D4—21no—
159C22×D4C2×D411yescharacteristic
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—41no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—24no—
170C2×M16—42no—
171C2×M16—42no—
172C22×C8—24no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—42no—
177C22×C8—42no—
178C4⋊Q8—24no—
179C4⋊Q8—24no—
180C4⋊Q8—24no—
181C4⋊Q8—22no—
182C4⋊Q8—22no—
183C4⋊Q8—22no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8C22×C414yes—
188C4×Q8—22no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—44no—
192C4×Q8—44no—
193C4×Q8C22⋊C414yes—
194C4×Q8C22×C412yes—
195C4×Q8C2×D412yes—
196C4×D4—24no—
197C4×D4—24no—
198C4×D4—24no—
199C4×D4—24no—
200C4×D4—22no—
201C4×D4—21no—
202C4×D4—21no—
203C4×D4—416no—
204C4×D4—44no—
205C4×D4—44no—
206C4×D4C22⋊C414yes—
207C4×D4C22×C412yes—
208C4×D4C2×D412yes—
209#117—22no—
210#117—24no—
211#117—24no—
212#117—48no—
213#117—42no—
214#117—42no—
215#117C22⋊C414yes—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—24no—
232C2×C4⋊C4—24no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—48no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—42no—
238C2×C4⋊C4—42no—
239C2×C4⋊C4—42no—
240C2×C4⋊C4—42no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—84no—
243C2×C4⋊C4—82no—
244C2×C4⋊C4—84no—
245C2×C4⋊C4C2×D412yes—
246C2×C4⋊C4C2×D412yes—
247C2×C4⋊C4C2×D411yescharacteristic
248C2×C4⋊C4C2×D411yescharacteristic
249C2×C4⋊C4C2×D414yes—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—24no—
259C2×C42—24no—
260C2×C42—24no—
261C2×C42—22no—
262C2×C42C22×C412yes—
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D412yes—
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42#4112yes—
268C2×C42#4112yes—
269C2×C42C2411yescharacteristic
270#107—28no—
271#106—28no—
272C4⋊C8—24no—
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—22no—
276C4⋊C8C22×C414yes—
277#103—28no—
278#103—84no—
279#103—84no—
280#102—28no—
281C22⋊C8—42no—
282C4×C8—22no—
283C4×C8—22no—
284C4×C8—42no—
285C4×C8—42no—
286#95—24no—
287#95—24no—
288#95—22no—
289#95—22no—
290C24—21no—
291C24#12011yescharacteristic
292#41—28no—
293#41—28no—
294#41—28no—
295#41#103116yescomplemented
296C2×Q8—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—24no—
300C2×Q8—42no—
301C2×Q8—44no—
302C2×Q8—44no—
303C2×Q8—44no—
304C2×Q8—88no—
305C2×Q8—82no—
306C2×Q8—82no—
307C2×Q8—88no—
308C2×Q8C2×C22⋊C412yes—
309C2×Q8C2×SD1612yes—
310C2×Q8C2×Q1612yes—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4C2×C22⋊C412yes—
318C2×D4C2×SD1612yes—
319C2×D4C2×Q1612yes—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—24no—
324C22×C4—24no—
325C22×C4—24no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—24no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—48no—
343C22×C4—44no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—84no—
347C22×C4—84no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4C2×C22⋊C412yes—
351C22×C4C4×D412yes—
352C22×C4C4×D412yes—
353C22×C4#12011yescharacteristic
354C22×C4#12011yescharacteristic
355C22×C4#12011yescharacteristic
356C22×C4#12112yes—
357C22×C4#12112yes—
358C22×C4#12312yes—
359C22×C4C2×SD1612yes—
360C22×C4C2×Q1612yes—
361C22×C4AGL(1,ℤ/8)12yes—
362C22×C4#13712yes—
363C22×C4C22×D411yescharacteristic
364C22×C4C22×D411yescharacteristic
365C22×C4C22×D411yescharacteristic
366M16—48no—
367C2×C8—24no—
368C2×C8—24no—
369C2×C8—42no—
370C2×C8—48no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—24no—
382C4⋊C4—24no—
383C4⋊C4—24no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—44no—
387C4⋊C4—48no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—42no—
396C4⋊C4—44no—
397C4⋊C4—42no—
398C4⋊C4—44no—
399C4⋊C4—42no—
400C4⋊C4—42no—
401C4⋊C4—82no—
402C4⋊C4—84no—
403C4⋊C4—84no—
404C4⋊C4C2×C22⋊C412yes—
405C4⋊C4C2×C22⋊C412yes—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—22no—
409C22⋊C4—24no—
410C22⋊C4—48no—
411C22⋊C4—44no—
412C42—22no—
413C42—22no—
414C42—24no—
415C42—24no—
416C42—48no—
417C42—42no—
418C42—42no—
419C42C2×C22⋊C412yes—
420C42C4×D412yes—
421C42C4×D412yes—
422C42#12112yes—
423C42#12112yes—
424C42#12312yes—
425C42C23×C411yescharacteristic
426C42C22×D411yescharacteristic
427C42C2×#4111yescharacteristic
428C42C2×#4111yescharacteristic
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23#38612yes—
441C23#45012yes—
442C23#45112yes—
443C23C2×#12011yescharacteristic
444Q8—22no—
445Q8—22no—
446Q8—22no—
447Q8—48no—
448Q8—88no—
449Q8—84no—
450Q8C2×#10314yes—
451D4—216no—
452D4—22no—
453D4—22no—
454D4—22no—
455D4C2×#10314yes—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—22no—
463C2×C4—216no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—24no—
467C2×C4—24no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—48no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—42no—
478C2×C4—44no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—44no—
482C2×C4—42no—
483C2×C4—44no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—88no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—164no—
491C2×C4—168no—
492C2×C4#38612yes—
493C2×C4#38612yes—
494C2×C4#38612yes—
495C2×C4C2×#10314yes—
496C2×C4#41714yes—
497C2×C4#45012yes—
498C2×C4#45112yes—
499C2×C4#45512yes—
500C2×C4#45612yes—
501C2×C4#48314yes—
502C2×C4#48414yes—
503C2×C4C22×C22⋊C411yescharacteristic
504C2×C4C2×C4×D411yescharacteristic
505C2×C4C2×C4×D411yescharacteristic
506C2×C4C2×#12111yescharacteristic
507C2×C4C2×#12111yescharacteristic
508C2×C4C2×#12311yescharacteristic
509C2×C4C22×SD1611yescharacteristic
510C2×C4C22×Q1611yescharacteristic
511C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
512C2×C4C2×#13711yescharacteristic
513C2×C4#57612yes—
514C2×C4#57712yes—
515C8—44no—
516C8—44no—
517C8—82no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—22no—
523C22—22no—
524C22—24no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—42no—
530C22—42no—
531C22#187214yes—
532C22C2×#38611yescharacteristic
533C22C2×#45011yescharacteristic
534C22C2×#45111yescharacteristic
535C22#300811yescharacteristic
536C22#301011yescharacteristic
537C22C2×#45511yescharacteristic
538C22C2×#45611yescharacteristic
539C4—24no—
540C4—22no—
541C4—21no—
542C4—21no—
543C4—42no—
544C4—42no—
545C4—84no—
546C4—82no—
547C4—84no—
548C4—82no—
549C4#187214yes—
550C4#188214yes—
551C4C22×#10311yescharacteristic
552C4C2×#41711yescharacteristic
553C4#294612yes—
554C4#304512yes—
555C4C2×#48311yescharacteristic
556C4C2×#48411yescharacteristic
557C2—24no—
558C2—44no—
559C2#1863111yescharacteristic
560C2#1863311yescharacteristic
561C2#1865111yescharacteristic
562C2#1865311yescharacteristic
563C2C2×#187211yescharacteristic
564C2#2093211yescharacteristic
565C2C2×#188211yescharacteristic
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. #21804C2semidirect
4. C2×#1990C2semidirect
6. #8983C2semidirect
7. #8981C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
14. #3079C4semidirect
58. C2×C4⋊Q8C2×C4semidirect
92. #475C2×C4semidirect
94. #474C2×C4semidirect
101. #423C2×C4semidirect
295. #41#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#341832
#218042
C2×#19902
C2×#14762
#89832
#89812
#80432
#78972