Finite Group Databasebeta

← #368261

Subgroups — #368261

Order 512.

1304 conjugacy classes of subgroups.

The subgroup lattice has 537 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1776C211yesmaximal, characteristic
5#14150C211yesmaximal, characteristic
6#11328C212yesmaximal
7#11292C212yesmaximal
8#10908C212yesmaximal
9#10885C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2369—24no—
18#2369—24no—
19C2×#391—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2211yescharacteristic
29#1993C418yescomplemented
30#1988—24no—
31#1776—24no—
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1444—21no—
43#1442—21no—
44#1436—22no—
45C4⋊#95C2212yes—
46C23×Q8—21no—
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2311yescharacteristic
67C22×C42—21no—
68C22×C42—22no—
69C22×C42—21no—
70C22×C42C2312yes—
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×#106Q814yescomplemented
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×#97C2×C414yes—
84#391—216no—
85#389—28no—
86#389—24no—
87#389—24no—
88#389—24no—
89#389—42no—
90#382—22no—
91#382C2×C414yes—
92C4×C4⋊C4—28no—
93C4×C4⋊C4—24no—
94C4×C4⋊C4—24no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4D414yes—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95D412yes—
111#340—24no—
112#340—24no—
113#340—22no—
114#340—22no—
115#340—24no—
116#340—42no—
117#340C2×C418yes—
118#340C2×C418yes—
119#336—24no—
120#336—42no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8—24no—
124C22×Q8—82no—
125C22×Q8—82no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C22×C412yes—
155C22×C8C22×C412yes—
156C22×C8C2×Q812yes—
157C22×C8#4112yes—
158C4⋊Q8—28no—
159C4⋊Q8—28no—
160C4⋊Q8—28no—
161C4⋊Q8C4⋊C4116yescomplemented
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—82no—
199C2×C4⋊C4—82no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4C22×C414yes—
202C2×C4⋊C4C22×C414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q812yes—
209C2×C4⋊C4#4112yes—
210C2×C4⋊C4#4112yes—
211C2×C42—24no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—28no—
219C2×C42—44no—
220C2×C42—42no—
221C2×C42—82no—
222C2×C42—82no—
223C2×C42—82no—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×Q812yes—
228C2×C42#4112yes—
229C2×C42#4112yes—
230#107—28no—
231#106—28no—
232C4⋊C8—44no—
233#97—28no—
234#95—28no—
235#95—28no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—44no—
241#95—42no—
242#95—44no—
243#95—82no—
244C24C22×D411yescharacteristic
245C2×Q8—216no—
246C2×Q8—88no—
247C2×Q8—88no—
248C2×Q8—84no—
249C2×Q8—84no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4C4×D412yes—
302C22×C4C4×D412yes—
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4C22⋊Q812yes—
306C22×C4C22⋊Q812yes—
307C22×C4#13714yes—
308C22×C4C23×C411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C22×Q811yescharacteristic
312C22×C4C22×Q811yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C42−1+411yescharacteristic
320C22×C42−1+411yescharacteristic
321C22×C42−1+411yescharacteristic
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8C4×Q814yes—
343C4⋊C4—28no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—216no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—44no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C4⋊C4C2×C4⋊C418yes—
371C4⋊C4C22⋊Q814yes—
372C42—48no—
373C42—42no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—84no—
378C42—88no—
379C42C2×C4⋊C414yes—
380C42C4×D414yes—
381C42C22⋊Q814yes—
382C42#12314yes—
383C23C2×C4×D411yescharacteristic
384C23C2×#12011yescharacteristic
385C23C2×C22⋊Q811yescharacteristic
386C23#54611yescharacteristic
387C23#54711yescharacteristic
388C23#54811yescharacteristic
389C23D4×Q811yescharacteristic
390C23C2×#13712yes—
391C23#57512yes—
392C23#57612yes—
393C23#57712yes—
394Q8—816no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—28no—
407C2×C4—24no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#38514yes—
452C2×C4#38514yes—
453C2×C4C22×C4⋊C411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C2×C4×Q811yescharacteristic
456C2×C4C4×#4111yescharacteristic
457C2×C4#51911yescharacteristic
458C2×C4#52011yescharacteristic
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4C2×#12311yescharacteristic
462C2×C4#52511yescharacteristic
463C2×C4#52911yescharacteristic
464C2×C4#53311yescharacteristic
465C2×C4#53611yescharacteristic
466C2×C4#53711yescharacteristic
467C2×C4#54111yescharacteristic
468C2×C4#54211yescharacteristic
469C2×C4#54811yescharacteristic
470C2×C4#54811yescharacteristic
471C2×C4D4×Q811yescharacteristic
472C2×C4#55011yescharacteristic
473C2×C4#55611yescharacteristic
474C2×C4#55611yescharacteristic
475C2×C4#55711yescharacteristic
476C2×C4#55711yescharacteristic
477C2×C4#55711yescharacteristic
478C2×C4#55711yescharacteristic
479C2×C4Q8211yescharacteristic
480C2×C4#55911yescharacteristic
481C8—24no—
482C8—42no—
483C8—82no—
484C22C2×#38511yescharacteristic
485C22#236511yescharacteristic
486C22#236711yescharacteristic
487C22#240711yescharacteristic
488C22#240911yescharacteristic
489C22#245111yescharacteristic
490C22#245311yescharacteristic
491C22C4×#13712yes—
492C22#294612yes—
493C22#300411yescharacteristic
494C22#300511yescharacteristic
495C22#300811yescharacteristic
496C22#300911yescharacteristic
497C22#307812yes—
498C22#308112yes—
499C22#331612yes—
500C22#331912yes—
501C22#334412yes—
502C22#334712yes—
503C22#337512yes—
504C22#337612yes—
505C22#338512yes—
506C22#338812yes—
507C4—24no—
508C4—24no—
509C4—44no—
510C4—44no—
511C4—48no—
512C4—82no—
513C4—82no—
514C4—82no—
515C4—82no—
516C4—84no—
517C4C4×C22⋊Q811yescharacteristic
518C4#232311yescharacteristic
519C4Q8×C4⋊C411yescharacteristic
520C4#235111yescharacteristic
521C4#244611yescharacteristic
522C4#244811yescharacteristic
523C4#258111yescharacteristic
524C4#258211yescharacteristic
525C2#1401711yescharacteristic
526C2#2087411yescharacteristic
527C2#2087611yescharacteristic
528C2#2316912yes—
529C2#2323012yes—
530C2#2412911yescharacteristic
531C2#2413411yescharacteristic
532C2#2434111yescharacteristic
533C2#2434311yescharacteristic
534C2#2645312yes—
535C2#2652012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
29. #1993C4semidirect
77. C2×#106Q8semidirect
161. C4⋊Q8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#19932
C2×#19882
C2×#17762
#141502
#113282
#112922
#109082
#108852
#108502
#108352