Finite Group Databasebeta

← #410689

Subgroups — #410689

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1972C211yesmaximal, characteristic
2C2×#1936C212yesmaximal, complemented
3C2×#1934C212yesmaximal, complemented
4C2×#1771C211yesmaximal, characteristic, complemented
5#15148C212yesmaximal, complemented
6#11327C211yesmaximal, characteristic, complemented
7#11327C211yesmaximal, characteristic, complemented
8#11076C212yesmaximal, complemented
9#11022C211yesmaximal, characteristic
10#11022C211yesmaximal, characteristic
11#10821C211yesmaximal, characteristic, complemented
12C23×D8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#102C2212yes—
18C22×#97C2211yescharacteristic
19C22×#97C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#399—22no—
22C2×#390C2212yes—
23C2×#390C2212yes—
24C2×#390C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28#1972—24no—
29#1936—28no—
30#1934—28no—
31#1771—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1439—22no—
40#1439—22no—
41#1439—22no—
42C4⋊#95C2212yes—
43C23×D4C2312yes—
44C22×D8—24no—
45C22×D8—22no—
46C22×D8—22no—
47C22×D8—22no—
48C22×D8—42no—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—22no—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68C2×#106—22no—
69C2×#106—42no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—24no—
80C2×#102—24no—
81C2×#102—24no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—24no—
85C2×#97—22no—
86C2×#97D414yes—
87C2×C4×C8—24no—
88C2×C4×C8—22no—
89#399—48no—
90#390—24no—
91#390—24no—
92#390—24no—
93#390—24no—
94#390—42no—
95#390—42no—
96#389—24no—
97#389—24no—
98#389—42no—
99#389—42no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—22no—
108#340—22no—
109#340—24no—
110#340—24no—
111#340D418yes—
112#336—22no—
113#336—22no—
114C25—22no—
115C22×D4—22no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4—22no—
128C22×D4#4114yes—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
137C2×D8—28no—
138C2×D8—48no—
139C2×D8—48no—
140C2×D8—44no—
141C2×D8—44no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8C2×D412yes—
156C22×C8C2×D414yes—
157C22×C8C2×D412yes—
158C22×C8#4112yes—
159C22×C8#4112yes—
160C22×C8#4112yes—
161C22×C8#4112yes—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—82no—
202C2×C22⋊C4—82no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42C2×D412yes—
220C2×C42#4112yes—
221#106—44no—
222C4⋊C8—28no—
223#102—216no—
224#102—216no—
225#102—48no—
226#97—28no—
227#97—44no—
228C4×C8—44no—
229#95—24no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—82no—
234C24—24no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24—42no—
240C24—42no—
241C24—42no—
242C24—42no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C22×D411yescharacteristic
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—24no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12112yes—
321C22×C4#12112yes—
322C22×C4AGL(1,ℤ/8)14yes—
323C22×C4C22×D411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333D8—416no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8#12314yes—
353C2×C8#12714yes—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—48no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—88no—
368C4⋊C4—88no—
369C22⋊C4—88no—
370C22⋊C4—88no—
371C22⋊C4—88no—
372C22⋊C4—88no—
373C22⋊C4—88no—
374C22⋊C4—88no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—41no—
383C42—42no—
384C42—42no—
385C23—44no—
386C23—44no—
387C23—44no—
388C23—44no—
389C23—44no—
390C23—44no—
391C23—44no—
392C23—82no—
393C23—82no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—82no—
410C23—82no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23—82no—
418C23—82no—
419C23—82no—
420C23C2×#12111yescharacteristic
421C23#53211yescharacteristic
422C23#53211yescharacteristic
423C23D4212yes—
424C23#54812yes—
425C23C2×AGL(1,ℤ/8)12yes—
426C23#57512yes—
427C23#57612yes—
428C23#57712yes—
429D4—44no—
430D4—44no—
431D4—44no—
432D4—44no—
433D4—82no—
434D4—82no—
435D4—82no—
436D4—82no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—42no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4C2×#12311yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4#52512yes—
485C2×C4#52512yes—
486C2×C4#52511yescharacteristic
487C2×C4#52911yescharacteristic
488C2×C4#52911yescharacteristic
489C2×C4C2×#12711yescharacteristic
490C2×C4#53511yescharacteristic
491C2×C4#53611yescharacteristic
492C2×C4#54212yes—
493C2×C4#54212yes—
494C2×C4#54712yes—
495C2×C4#54712yes—
496C2×C4#55012yes—
497C2×C4#55112yes—
498C2×C4#55112yes—
499C2×C4#55212yes—
500C8—24no—
501C8—24no—
502C8—42no—
503C8—42no—
504C22—84no—
505C22—84no—
506C22—84no—
507C22—84no—
508C22—84no—
509C22—84no—
510C22—84no—
511C22—162no—
512C22—162no—
513C22—162no—
514C22—162no—
515C22—162no—
516C22—162no—
517C22—162no—
518C22—162no—
519C22#246112yes—
520C22#246112yes—
521C22#253611yescharacteristic
522C22#254111yescharacteristic
523C22#285411yescharacteristic
524C22#305512yes—
525C22#306212yes—
526C22#317112yes—
527C22#317311yescharacteristic
528C22#317611yescharacteristic
529C22#317912yes—
530C22#318311yescharacteristic
531C22#318411yescharacteristic
532C22#326412yes—
533C22#326512yes—
534C22#326812yes—
535C22#327112yes—
536C22#336612yes—
537C22#337012yes—
538C22#337312yes—
539C22#337612yes—
540C4—44no—
541C4—42no—
542C4—42no—
543C4—82no—
544C4—84no—
545C4—84no—
546C4—162no—
547C4#239711yescharacteristic
548C4#247912yes—
549C4#247912yes—
550C4#248211yescharacteristic
551C4#250111yescharacteristic
552C4#263211yescharacteristic
553C2—164no—
554C2#1508411yescharacteristic
555C2#2682712yes—
556C2#2682812yes—
557C2#2683412yes—
558C2#2683712yes—
559C2#2819712yes—
560C2#2849312yes—
561C2#3314011yescharacteristic
562C2#3314311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D8C22
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1936C2semidirect
3. C2×#1934C2semidirect
4. C2×#1771C2semidirect
5. #15148C2semidirect
6. #11327C2semidirect
7. #11327C2semidirect
8. #11076C2semidirect
11. #10821C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19722
C2×#19362
C2×#19342
C2×#17712
#151482
#113272
#110762
#110222
#108212