Finite Group Databasebeta

← #239821

Subgroups — #239821

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#21859C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#21171C211yesmaximal, characteristic, complemented
5C2×#1546C211yesmaximal, characteristic, complemented
6C2×#1492C211yesmaximal, characteristic
7#10421C214yesmaximal, complemented
8#10420C214yesmaximal, complemented
9#7890C211yesmaximal, characteristic, complemented
10C2×#536—21no—
11C2×#536—21no—
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×#428—21no—
16C2×#423C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2868—21no—
19#2858—21no—
20#2658—22no—
21#2542—21no—
22#2542—21no—
23#2542C2212yes—
24#2542C2212yes—
25#2501—22no—
26#2501—22no—
27C2×#384C2211yescharacteristic
28C4×#117C2211yescharacteristic
29#1990—22no—
30#1990—21no—
31#1990C2214yes—
32#1957—21no—
33#1948—21no—
34#1948C2212yes—
35#1947—21no—
36#1947C2212yes—
37#1874—22no—
38#1802C2214yes—
39#1736—21no—
40C2×#340—22no—
41#1546—24no—
42C2×#332C2214yes—
43#1492—24no—
44C2×D4⋊C8C2214yes—
45#1279C2212yes—
46#1279C2212yes—
47C22×#41—21no—
48C23×Q8—21no—
49C22×M16—21no—
50C23×C8—21no—
51#536—44no—
52#536—44no—
53#536—44no—
54#536—44no—
55#532—22no—
56#532—22no—
57#532—22no—
58#532—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—22no—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8D412yes—
69C2×C4⋊Q8C2312yes—
70C2×#127D411yescharacteristic
71C2×#124—22no—
72C2×#124—22no—
73C2×#124—21no—
74C2×#123—24no—
75C2×#123—24no—
76C2×C22⋊Q8—22no—
77C2×C22⋊Q8—22no—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×#121—21no—
81C2×#121D412yes—
82C2×C4×D4—21no—
83C2×C4×D4C2312yes—
84C2×#117D411yescharacteristic
85C2×#117D411yescharacteristic
86C22×C4⋊C4—21no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4D412yes—
91C22×C42C2311yescharacteristic
92#431—22no—
93#431—22no—
94#428—44no—
95C2×#107—22no—
96C2×#106—22no—
97#423—22no—
98#423D414yes—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×#102—24no—
103C2×C22⋊C8—21no—
104C2×C4×C8—22no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2312yes—
109#399—24no—
110#399—24no—
111#399—24no—
112#399—24no—
113#390—22no—
114#390—22no—
115#390—22no—
116#390—22no—
117#384—22no—
118#384—22no—
119#384—22no—
120#384—22no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2312yes—
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C22⋊C4—21no—
130C4×C22⋊C4—21no—
131#376—21no—
132C43—21no—
133#340—24no—
134#340—22no—
135#340—22no—
136#340—28no—
137#332—28no—
138D4⋊C8—28no—
139C2×#41—42no—
140C2×#41—42no—
141C2×#41—42no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—42no—
145C22×Q8—22no—
146C22×Q8—21no—
147C22×Q8—22no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8—42no—
151C22×Q8—42no—
152C22×Q8—42no—
153C22×Q8—42no—
154C22×D4—22no—
155C22×D4—21no—
156C22×D4—21no—
157C22×D4C2×D412yes—
158C23×C4—22no—
159C23×C4—22no—
160C23×C4—41no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—42no—
167C22×C8—24no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—42no—
172C22×C8—42no—
173C4⋊Q8—24no—
174C4⋊Q8—22no—
175C4⋊Q8—24no—
176C4⋊Q8—22no—
177C4⋊Q8—22no—
178C4⋊Q8—22no—
179C4⋊Q8—22no—
180C4⋊Q8—22no—
181C4⋊Q8—48no—
182C4⋊Q8—48no—
183C4⋊Q8—42no—
184C4⋊Q8—42no—
185C4⋊Q8—42no—
186C4⋊Q8—42no—
187#127—22no—
188#127—22no—
189#124—48no—
190#124—48no—
191#124—42no—
192#124—42no—
193#123—416no—
194#123—416no—
195C22⋊Q8—44no—
196C22⋊Q8—44no—
197C22⋊Q8—44no—
198C22⋊Q8—44no—
199C22⋊Q8—44no—
200C22⋊Q8—44no—
201C22⋊Q8—44no—
202C22⋊Q8—44no—
203#121—24no—
204#121—24no—
205#121—42no—
206#121—42no—
207C4×D4—22no—
208C4×D4—22no—
209C4×D4—22no—
210C4×D4—22no—
211C4×D4—42no—
212C4×D4—42no—
213#117—22no—
214#117—22no—
215#117D814yes—
216#117SD1614yes—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—41no—
236C2×C4⋊C4—41no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4C2×D414yes—
240C2×C4⋊C4C2×D412yes—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C4⋊C4C2×D411yescharacteristic
244C2×C4⋊C4C2×D411yescharacteristic
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—24no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42C2×D412yes—
260C2×C42C2×D411yescharacteristic
261C2×C42C2×D411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42#4112yes—
264C2×C42#4112yes—
265C2×C42#4112yes—
266C2×C42C2411yescharacteristic
267#107—44no—
268#106—44no—
269C4⋊C8—24no—
270C4⋊C8—22no—
271C4⋊C8—22no—
272C4⋊C8—22no—
273C4⋊C8C2×D414yes—
274#102—84no—
275C22⋊C8—42no—
276C4×C8—22no—
277C4×C8—22no—
278C4×C8—42no—
279C4×C8—42no—
280#95—21no—
281#95—21no—
282#95—21no—
283#95—21no—
284C24—22no—
285C24#12011yescharacteristic
286#41—84no—
287#41—84no—
288#41—84no—
289#41—84no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—44no—
298C2×Q8—44no—
299C2×Q8—42no—
300C2×Q8—44no—
301C2×Q8—42no—
302C2×Q8—84no—
303C2×Q8—84no—
304C2×Q8—84no—
305C2×Q8—84no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—24no—
309C2×D4—24no—
310C2×D4—44no—
311C2×D4—44no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—42no—
315C2×D4—42no—
316C2×D4—42no—
317C2×D4—42no—
318C2×D4—42no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—24no—
328C22×C4—22no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—44no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—82no—
340C22×C4—82no—
341C22×C4—82no—
342C22×C4#12011yescharacteristic
343C22×C4#12011yescharacteristic
344C22×C4#12011yescharacteristic
345C22×C4#12112yes—
346C22×C4#12112yes—
347C22×C4#12112yes—
348C22×C4#12312yes—
349C22×C4#12312yes—
350C22×C4#12412yes—
351C22×C4C2×D812yes—
352C22×C4C2×SD1612yes—
353C22×C4AGL(1,ℤ/8)12yes—
354C22×C4AGL(1,ℤ/8)12yes—
355C22×C4C22×D411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C22×D411yescharacteristic
358M16—48no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—42no—
362C2×C8—48no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—48no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—48no—
381C4⋊C4—44no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—82no—
391C4⋊C4—82no—
392C4⋊C4—82no—
393C4⋊C4C2×D812yes—
394C4⋊C4C2×D812yes—
395C4⋊C4C2×SD1612yes—
396C4⋊C4C2×SD1612yes—
397C22⋊C4—24no—
398C22⋊C4—24no—
399C22⋊C4—48no—
400C22⋊C4—48no—
401C22⋊C4—48no—
402C22⋊C4—48no—
403C22⋊C4—42no—
404C22⋊C4—42no—
405C22⋊C4—44no—
406C42—24no—
407C42—24no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42#12112yes—
413C42#12112yes—
414C42#12112yes—
415C42#12312yes—
416C42#12312yes—
417C42#12412yes—
418C42C22×D411yescharacteristic
419C42C2×#4111yescharacteristic
420C42C2×#4111yescharacteristic
421C42C2×#4111yescharacteristic
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—42no—
433C23#39412yes—
434C23#44712yes—
435C23#45012yes—
436C23C2×#12011yescharacteristic
437Q8—82no—
438Q8—82no—
439Q8—82no—
440Q8—82no—
441Q8—88no—
442Q8—84no—
443D4—44no—
444D4—44no—
445D4—88no—
446D4—84no—
447D4—84no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—42no—
474C2×C4—88no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—164no—
483C2×C4#39412yes—
484C2×C4#39412yes—
485C2×C4#39412yes—
486C2×C4#44712yes—
487C2×C4#45012yes—
488C2×C4#45412yes—
489C2×C4#45612yes—
490C2×C4#45914yes—
491C2×C4#46014yes—
492C2×C4#48614yes—
493C2×C4#49014yes—
494C2×C4C2×#12111yescharacteristic
495C2×C4C2×#12111yescharacteristic
496C2×C4C2×#12111yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C2×C4C2×#12311yescharacteristic
499C2×C4C2×#12411yescharacteristic
500C2×C4C22×D811yescharacteristic
501C2×C4C22×SD1611yescharacteristic
502C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
503C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
504C2×C4#57712yes—
505C2×C4#57812yes—
506C8—44no—
507C8—44no—
508C8—82no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—82no—
520C22—82no—
521C22—82no—
522C22#204114yes—
523C22C2×#39411yescharacteristic
524C22C2×#44711yescharacteristic
525C22C2×#45011yescharacteristic
526C22#301011yescharacteristic
527C22#301111yescharacteristic
528C22C2×#45411yescharacteristic
529C22C2×#45611yescharacteristic
530C4—22no—
531C4—21no—
532C4—21no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—42no—
537C4—82no—
538C4—84no—
539C4—162no—
540C4#203814yes—
541C4#204114yes—
542C4C2×#45911yescharacteristic
543C4C2×#46011yescharacteristic
544C4#306312yes—
545C4#309712yes—
546C4C2×#48611yescharacteristic
547C4C2×#49011yescharacteristic
548C2—24no—
549C2—162no—
550C2#1864011yescharacteristic
551C2#1864111yescharacteristic
552C2#1865011yescharacteristic
553C2#1865111yescharacteristic
554C2#2348411yescharacteristic
555C2C2×#203811yescharacteristic
556C2C2×#204111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
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
3. C2×#1990C2semidirect
4. #21171C2semidirect
5. C2×#1546C2semidirect
7. #10421C2semidirect
8. #10420C2semidirect
9. #7890C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#218592
C2×#19902
#211712
C2×#15462
C2×#14922
#104212
#104202
#78902