Finite Group Databasebeta

← #365291

Subgroups — #365291

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1935C212yesmaximal
2C2×#1805C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1767C211yesmaximal, characteristic
5#14707C211yesmaximal, characteristic
6C4×#391C211yesmaximal, characteristic
7#11316C212yesmaximal
8#11190C212yesmaximal
9#10881C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10850C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#97C2212yes—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C2212yes—
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C43C2211yescharacteristic
25#1935—28no—
26#1805—24no—
27#1805C418yescomplemented
28#1767—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36#1436—22no—
37#1436—22no—
38#1436—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42C4×#95—22no—
43C23×Q8C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8C2×C414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42—21no—
60C22×C42—21no—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8—22no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2×C414yes—
71C2×C4⋊C8D414yes—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103C2×C418yes—
78C2×#97—24no—
79C2×#97—24no—
80C2×#97—42no—
81#391—22no—
82#391—22no—
83#391—22no—
84#391D414yes—
85#389—24no—
86#389—24no—
87#382—22no—
88#382—22no—
89#382—22no—
90#382C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C43—22no—
106C43—22no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—42no—
112#340C2×C418yes—
113#336—24no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8C22×C412yes—
126C22×Q8C2×D412yes—
127C22×Q8#4112yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C22×C414yes—
152C22×C8C2×D414yes—
153C4×Q8—28no—
154C4×Q8—28no—
155C4×Q8—44no—
156C4×Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4#4112yes—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—44no—
213C2×C42—42no—
214C2×C42C22×C412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42#4112yes—
221C2×C42#4112yes—
222C4⋊C8—28no—
223C4⋊C8—28no—
224#103—216no—
225#103—48no—
226#97—48no—
227#95—28no—
228#95—24no—
229#95—24no—
230#95—28no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—44no—
237#95—42no—
238#95—44no—
239C24C22×D411yescharacteristic
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8C4×D414yes—
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—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—42no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
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×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4#12114yes—
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12412yes—
326C22×C4#12412yes—
327C22×C4#13714yes—
328C22×C4C23×C411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C42−1+411yescharacteristic
340C22×C42−1+411yescharacteristic
341C22×C42−1+411yescharacteristic
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—88no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—48no—
383C42—48no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—48no—
389C42—84no—
390C42#11714yes—
391C42C4×D414yes—
392C42#12314yes—
393C42#12614yes—
394C23C2×C4×D411yescharacteristic
395C23C2×#12111yescharacteristic
396C23C2×#12111yescharacteristic
397C23#52511yescharacteristic
398C23C2×#12411yescharacteristic
399C23#53611yescharacteristic
400C23#53711yescharacteristic
401C23C2×#13712yes—
402C23#57512yes—
403C23#57512yes—
404C23#57512yes—
405Q8—24no—
406Q8—24no—
407Q8—42no—
408Q8—42no—
409Q8—42no—
410Q8—42no—
411Q8—42no—
412Q8—42no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4#38814yes—
472C2×C4#38814yes—
473C2×C4C2×#11711yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4C2×C4×D411yescharacteristic
476C2×C4C4×#4111yescharacteristic
477C2×C4#51911yescharacteristic
478C2×C4#52011yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4C2×#12611yescharacteristic
481C2×C4#52911yescharacteristic
482C2×C4#53211yescharacteristic
483C2×C4#54111yescharacteristic
484C2×C4#54211yescharacteristic
485C2×C4D4211yescharacteristic
486C2×C4#54711yescharacteristic
487C2×C4#54711yescharacteristic
488C2×C4#54711yescharacteristic
489C2×C4#54711yescharacteristic
490C2×C4#54811yescharacteristic
491C2×C4#54811yescharacteristic
492C2×C4#55211yescharacteristic
493C2×C4#55211yescharacteristic
494C2×C4#55211yescharacteristic
495C2×C4#55311yescharacteristic
496C2×C4#55311yescharacteristic
497C2×C4#55311yescharacteristic
498C2×C4#55611yescharacteristic
499C2×C4#55611yescharacteristic
500C2×C4#55611yescharacteristic
501C8—44no—
502C8—44no—
503C22C2×#38811yescharacteristic
504C22C4×#12111yescharacteristic
505C22#232311yescharacteristic
506C22#241511yescharacteristic
507C22#241611yescharacteristic
508C22#254711yescharacteristic
509C22#256211yescharacteristic
510C22#294312yes—
511C22C4×#13712yes—
512C22#303712yes—
513C22#304012yes—
514C22#305412yes—
515C22#305612yes—
516C22#311112yes—
517C22#311712yes—
518C22#313511yescharacteristic
519C22#313511yescharacteristic
520C22#313911yescharacteristic
521C22#314011yescharacteristic
522C22#323612yes—
523C22#324012yes—
524C22#325212yes—
525C22#325612yes—
526C4—24no—
527C4—24no—
528C4—44no—
529C4—44no—
530C4—44no—
531C4—84no—
532C4—82no—
533C4—82no—
534C4—82no—
535C4—162no—
536C4#234411yescharacteristic
537C4#234711yescharacteristic
538C4#236111yescharacteristic
539C4#236211yescharacteristic
540C4#242111yescharacteristic
541C4#242211yescharacteristic
542C4#249411yescharacteristic
543C4#249611yescharacteristic
544C2#1397411yescharacteristic
545C2#2108611yescharacteristic
546C2#2109411yescharacteristic
547C2#2150812yes—
548C2#2212112yes—
549C2#2470712yes—
550C2#2475712yes—
551C2#2872311yescharacteristic
552C2#2872711yescharacteristic
553C2#2912811yescharacteristic
554C2#2912911yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
27. #1805C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19352
C2×#18052
C2×#17672
#147072
C4×#3912
#113162
#111902
#108812
#108502