Finite Group Databasebeta

← #194106

Subgroups — #194106

Order 512.

1500 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3226C211yesmaximal, characteristic
2#34207C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#21259C211yesmaximal, characteristic, complemented
5C2×#1492C211yesmaximal, characteristic
6#8984C214yesmaximal, complemented
7#8981C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#532C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13#3226—24no—
14#3226C418yescomplemented
15C2×#500—21no—
16C2×#486—21no—
17C2×#480—22no—
18C2×#474C2214yes—
19C2×#431C2211yescharacteristic
20C2×#423C2211yescharacteristic
21C2×#419—21no—
22C2×#419—21no—
23C22×C4×C8C2211yescharacteristic
24#2361—22no—
25#2361—22no—
26#2354—21no—
27#2354C2212yes—
28D4×C4⋊C4—21no—
29D4×C4⋊C4C2212yes—
30#2326—22no—
31#2311—21no—
32C4×C4⋊Q8—21no—
33C4×#127—21no—
34C2×#384C2211yescharacteristic
35C4×#117C2211yescharacteristic
36#1990—22no—
37#1990C2214yes—
38#1981—21no—
39#1981—21no—
40#1981C2212yes—
41#1981C2212yes—
42C2×#340—22no—
43#1492—24no—
44C2×D4⋊C8C2214yes—
45#1296C2214yes—
46#1279C2212yes—
47#1279C2212yes—
48C22×#41—21no—
49C23×Q8—21no—
50C22×M16—21no—
51C23×C8—21no—
52#532—22no—
53#532—22no—
54#532C2×C414yes—
55#532D414yes—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yescomplemented
63C2×C4⋊Q8D411yescharacteristic
64C2×C4⋊Q8C2312yes—
65C2×#127D411yescharacteristic
66C2×#124—21no—
67C2×#121—21no—
68C2×#121D412yes—
69#520—44no—
70#520—44no—
71C2×C4×Q8—22no—
72C2×C4×D4—22no—
73C2×C4×D4—24no—
74C2×C4×D4—22no—
75C2×C4×D4—22no—
76C2×C4×D4—21no—
77C2×C4×D4C2312yes—
78C2×#117—22no—
79C2×#117D411yescharacteristic
80C2×#117D411yescharacteristic
81C22×C4⋊C4—21no—
82C22×C4⋊C4—21no—
83C22×C4⋊C4—21no—
84C22×C4⋊C4—21no—
85C22×C4⋊C4—22no—
86C22×C4⋊C4D412yes—
87C22×C42C2311yescharacteristic
88#500—28no—
89#486—28no—
90#480—216no—
91#474—28no—
92#474C2×C4116yescomplemented
93#431—22no—
94#431C2×C414yescomplemented
95C2×#107—24no—
96#423—22no—
97#423—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8C2312yes—
100C2×C4⋊C8C2312yes—
101#419—44no—
102#419—44no—
103C2×#103—22no—
104C2×#103—22no—
105C2×#103—22no—
106C2×#103—22no—
107C2×#102—24no—
108C2×C22⋊C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—22no—
111C2×C4×C8—22no—
112C2×C4×C8—21no—
113C2×C4×C8C2312yes—
114#391—22no—
115#390—22no—
116#390—22no—
117#390—22no—
118#389—22no—
119#389—22no—
120#388—24no—
121#388—24no—
122#386—24no—
123#385—24no—
124#385—24no—
125#384—21no—
126#384—21no—
127#384D412yes—
128#384C2312yes—
129#380—22no—
130#380—22no—
131C4×C4⋊C4—22no—
132C4×C4⋊C4—22no—
133C4×C4⋊C4D411yescharacteristic
134C4×C4⋊C4D411yescharacteristic
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C22⋊C4—21no—
138C4×C22⋊C4—21no—
139#376—21no—
140C43—21no—
141#340—28no—
142D4⋊C8—28no—
143C2×#41—24no—
144C2×#41—24no—
145C2×#41—42no—
146C2×#41—42no—
147C2×#41—42no—
148C2×#41—42no—
149C22×Q8—21no—
150C22×Q8—24no—
151C22×Q8—22no—
152C22×Q8—41no—
153C22×Q8—42no—
154C22×Q8—42no—
155C22×Q8—42no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C2×D412yes—
162C23×C4—22no—
163C23×C4—24no—
164C23×C4—22no—
165C23×C4—41no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C2×M16—42no—
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—22no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187C4⋊Q8—42no—
188C4⋊Q8C22⋊C414yes—
189C4⋊Q8C22×C414yes—
190#127—22no—
191#127C22⋊C414yes—
192#124—24no—
193#124—24no—
194#121—24no—
195#121—24no—
196#121—24no—
197#121C22⋊C418yes—
198C4×Q8—48no—
199C4×D4—24no—
200C4×D4—24no—
201C4×D4—22no—
202C4×D4—22no—
203C4×D4—48no—
204C4×D4—416no—
205C4×D4—84no—
206C4×D4—84no—
207C4×D4C22×C414yes—
208C4×D4C2×D414yes—
209#117—22no—
210#117—22no—
211#117—22no—
212#117—44no—
213#117—44no—
214#117D814yes—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—44no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—82no—
236C2×C4⋊C4C2×D411yescharacteristic
237C2×C4⋊C4C2×D411yescharacteristic
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D414yes—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C42—21no—
250C2×C42—24no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42C22×C412yes—
257C2×C42C2×D411yescharacteristic
258C2×C42C2×D412yes—
259C2×C42C2×D411yescharacteristic
260C2×C42C2×D411yescharacteristic
261C2×C42#4112yes—
262C2×C42#4112yes—
263C2×C42C2411yescharacteristic
264#107—216no—
265C4⋊C8—22no—
266C4⋊C8—22no—
267C4⋊C8—22no—
268C4⋊C8—42no—
269C4⋊C8C22×C414yes—
270#103—44no—
271#103—44no—
272#103—44no—
273#103—44no—
274#102—216no—
275C22⋊C8—24no—
276C4×C8—24no—
277C4×C8—22no—
278C4×C8—22no—
279C4×C8—42no—
280#95—24no—
281#95—24no—
282#95—21no—
283#95—21no—
284#95—21no—
285#95—21no—
286C24—22no—
287C24#12011yescharacteristic
288#41—48no—
289#41—48no—
290#41—48no—
291#41—48no—
292C2×Q8—24no—
293C2×Q8—24no—
294C2×Q8—42no—
295C2×Q8—44no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—48no—
303C2×Q8—48no—
304C2×Q8—82no—
305C2×Q8—82no—
306C2×Q8—84no—
307C2×Q8—84no—
308C2×D4—28no—
309C2×D4—22no—
310C2×D4—24no—
311C2×D4—24no—
312C2×D4—24no—
313C2×D4—24no—
314C2×D4—44no—
315C2×D4—44no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C2×D4—42no—
320C2×D4C2×C22⋊C414yes—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—24no—
329C22×C4—24no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—48no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—84no—
344C22×C4—84no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4C2×C22⋊C412yes—
348C22×C4C4×D412yes—
349C22×C4C4×D412yes—
350C22×C4#12011yescharacteristic
351C22×C4#12011yescharacteristic
352C22×C4#12011yescharacteristic
353C22×C4#12112yes—
354C22×C4#12112yes—
355C22×C4#12312yes—
356C22×C4C2×D812yes—
357C22×C4#13512yes—
358C22×C4AGL(1,ℤ/8)12yes—
359C22×C4AGL(1,ℤ/8)12yes—
360C22×C4C22×D411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×D411yescharacteristic
363M16—84no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—42no—
367C2×C8—48no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C4⋊C4—22no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—44no—
385C4⋊C4—48no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—42no—
390C4⋊C4—44no—
391C4⋊C4—44no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—82no—
396C4⋊C4C2×C22⋊C414yes—
397C4⋊C4C2×D812yes—
398C4⋊C4C2×D812yes—
399C22⋊C4—24no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4—48no—
404C22⋊C4—42no—
405C22⋊C4—84no—
406C22⋊C4—84no—
407C42—24no—
408C42—48no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C42—42no—
414C42C2×C22⋊C412yes—
415C42C4×D412yes—
416C42C4×D412yes—
417C42#12112yes—
418C42#12112yes—
419C42#12312yes—
420C42C23×C411yescharacteristic
421C42C22×D411yescharacteristic
422C42C2×#4111yescharacteristic
423C42C2×#4111yescharacteristic
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—24no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23—42no—
434C23—42no—
435C23#38612yes—
436C23#44712yes—
437C23#44912yes—
438C23C2×#12011yescharacteristic
439Q8—48no—
440Q8—44no—
441Q8—44no—
442Q8—88no—
443Q8—82no—
444Q8—82no—
445D4—28no—
446D4—416no—
447D4—44no—
448D4—48no—
449D4—48no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—24no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—44no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—44no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—88no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—168no—
485C2×C4—164no—
486C2×C4#38612yes—
487C2×C4#38612yes—
488C2×C4#38612yes—
489C2×C4C4×D814yes—
490C2×C4#44214yes—
491C2×C4#44712yes—
492C2×C4#44912yes—
493C2×C4#45512yes—
494C2×C4#45612yes—
495C2×C4#45914yes—
496C2×C4#46314yes—
497C2×C4C22×C22⋊C411yescharacteristic
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×D411yescharacteristic
500C2×C4C2×#12111yescharacteristic
501C2×C4C2×#12111yescharacteristic
502C2×C4C2×#12311yescharacteristic
503C2×C4C22×D811yescharacteristic
504C2×C4C2×#13511yescharacteristic
505C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
506C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
507C2×C4#57712yes—
508C2×C4#57812yes—
509C8—44no—
510C8—44no—
511C8—82no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—82no—
522C22—82no—
523C22—82no—
524C22—82no—
525C22#187914yes—
526C22C2×#38611yescharacteristic
527C22C2×#44711yescharacteristic
528C22C2×#44911yescharacteristic
529C22#300911yescharacteristic
530C22#301111yescharacteristic
531C22C2×#45511yescharacteristic
532C22C2×#45611yescharacteristic
533C4—22no—
534C4—21no—
535C4—21no—
536C4—44no—
537C4—42no—
538C4—42no—
539C4—84no—
540C4—84no—
541C4—81no—
542C4—81no—
543C4#187914yes—
544C4#188214yes—
545C4#289812yes—
546C4C2×C4×D811yescharacteristic
547C4C2×#44211yescharacteristic
548C4C2×#45911yescharacteristic
549C4C2×#46311yescharacteristic
550C4#309712yes—
551C2—24no—
552C2—84no—
553C2#1863011yescharacteristic
554C2#1863611yescharacteristic
555C2#1865011yescharacteristic
556C2#1865611yescharacteristic
557C2#2091311yescharacteristic
558C2C2×#187911yescharacteristic
559C2C2×#188211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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. C2×#1990C2semidirect
4. #21259C2semidirect
6. #8984C2semidirect
7. #8981C2semidirect
8. #8043C2semidirect
9. #7890C2semidirect
14. #3226C4semidirect
62. C2×C4⋊Q8C2×C4semidirect
92. #474C2×C4semidirect
94. #431C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32262
#342072
C2×#19902
#212592
C2×#14922
#89842
#89812
#80432
#78902