Finite Group Databasebeta

← #360672

Subgroups — #360672

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 539 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1847C211yesmaximal, characteristic, complemented
3C2×#1835C211yesmaximal, characteristic, complemented
4C2×#1750C211yesmaximal, characteristic
5#14340C211yesmaximal, characteristic
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10841C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C22×C22⋊C8C2212yes—
16C22×#97C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2314—24no—
19#2314—24no—
20C2×#388—22no—
21C2×#387—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380C2212yes—
27C2×#380C2211yescharacteristic
28C2×#380C2211yescharacteristic
29C2×C4×C4⋊C4C2212yes—
30#1847—24no—
31#1835—24no—
32#1750C418yescomplemented
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40#1441—22no—
41#1439—22no—
42C4⋊#95C2211yescharacteristic
43C4⋊#95C2211yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117C2×C414yes—
52C2×#117Q814yes—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4D412yes—
66C22×C42—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×#107—24no—
70C2×#107—24no—
71C2×#107—42no—
72C2×#106—24no—
73C2×#106—24no—
74C2×#106—42no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8C2×C418yes—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97C2×C414yes—
81C2×C4×C8—22no—
82C2×C4×C8—22no—
83C2×C4×C8C2×C414yes—
84#388—216no—
85#387—216no—
86#384—22no—
87#384—22no—
88#384—22no—
89#384D414yes—
90#382—22no—
91#382—22no—
92#382—22no—
93#382D414yes—
94#380—24no—
95#380—24no—
96#380—42no—
97#380—42no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—21no—
106C2×#95—22no—
107C2×#95—22no—
108#340—24no—
109#340—24no—
110#340—24no—
111#340—24no—
112#340—42no—
113#340—42no—
114#336C2×C418yes—
115C25C2×D411yescharacteristic
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C22×C412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×Q812yes—
135C23×C4C2411yescharacteristic
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8C22×C414yes—
147C22×C8C22×C414yes—
148C22×C8#4114yes—
149C22×C8#4114yes—
150#117—28no—
151#117—28no—
152#117—28no—
153#117—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—28no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C4⋊C418yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—28no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4C4⋊C418yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—28no—
206C2×C42—82no—
207C2×C42—82no—
208C2×C42C22×C412yes—
209C2×C42C22×C412yes—
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2×Q812yes—
213C2×C42C2×Q812yes—
214C2×C42#4112yes—
215C2×C42#4112yes—
216#107—48no—
217#106—48no—
218C22⋊C8—216no—
219#97—28no—
220C4×C8—28no—
221#95—28no—
222#95—28no—
223#95—28no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—82no—
228#95—82no—
229C24—21no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—21no—
234C24—21no—
235C24—22no—
236C24—22no—
237C24—22no—
238C24—22no—
239C24C2×C4⋊C412yes—
240C24C22×D411yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—24no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4C2×C4⋊C414yes—
308C22×C4C2×C4⋊C412yes—
309C22×C4C2×C4⋊C414yes—
310C22×C4#12014yes—
311C22×C4#12314yes—
312C22×C4C23×C411yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C22×Q811yescharacteristic
316C22×C4C2×#4112yes—
317C22×C4C2×#4112yes—
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+412yes—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8#11718yes—
336C4⋊C4—216no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C22⋊C4—216no—
356C22⋊C4—88no—
357C22⋊C4—84no—
358C22⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—88no—
364C42C4×D414yes—
365C42C4×D414yes—
366C42C22⋊Q814yes—
367C42C22⋊Q814yes—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—22no—
372C23—22no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—42no—
386C23—42no—
387C23—41no—
388C23—41no—
389C23—42no—
390C23—42no—
391C23C22×C4⋊C411yescharacteristic
392C23C2×#12011yescharacteristic
393C23C2×#12311yescharacteristic
394C23#53211yescharacteristic
395C23#53211yescharacteristic
396C23#53511yescharacteristic
397C23#53611yescharacteristic
398C23#57612yes—
399C23#57712yes—
400C23#57712yes—
401C23#57812yes—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#38518yes—
461C2×C4C2×#11712yes—
462C2×C4C2×C4×D411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4#52012yes—
465C2×C4C2×C22⋊Q811yescharacteristic
466C2×C4C2×C22⋊Q811yescharacteristic
467C2×C4#53312yes—
468C2×C4#54412yes—
469C2×C4#54612yes—
470C2×C4#54712yes—
471C2×C4#54812yes—
472C2×C4#55012yes—
473C2×C4#55212yes—
474C2×C4#55312yes—
475C2×C4#55512yes—
476C2×C4#55612yes—
477C8—28no—
478C8—44no—
479C22—42no—
480C22—44no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—81no—
485C22—81no—
486C22—82no—
487C22—82no—
488C22—81no—
489C22—81no—
490C22#228511yescharacteristic
491C22C2×#38511yescharacteristic
492C22#231711yescharacteristic
493C22#243111yescharacteristic
494C22#243211yescharacteristic
495C22#249911yescharacteristic
496C22#250011yescharacteristic
497C22#291212yes—
498C22#291312yes—
499C22#301212yes—
500C22#301312yes—
501C22#309612yes—
502C22#309712yes—
503C22#317812yes—
504C22#317812yes—
505C22#317912yes—
506C22#317912yes—
507C22#322011yescharacteristic
508C22#322111yescharacteristic
509C22#322211yescharacteristic
510C22#322311yescharacteristic
511C22#324312yes—
512C22#324412yes—
513C4—28no—
514C4—42no—
515C4—44no—
516C4—84no—
517C4—84no—
518C4—161no—
519C4—161no—
520C4—161no—
521C4—161no—
522C4#235212yes—
523C4#235412yes—
524C4#243912yes—
525C4#244112yes—
526C2—82no—
527C2#1387911yescharacteristic
528C2#2082212yes—
529C2#2084912yes—
530C2#2199011yescharacteristic
531C2#2199111yescharacteristic
532C2#2561111yescharacteristic
533C2#2561211yescharacteristic
534C2#2568412yes—
535C2#2788712yes—
536C2#2790211yescharacteristic
537C2#2790311yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1847C2semidirect
3. C2×#1835C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
10. #10843C2semidirect
11. #10841C2semidirect
32. #1750C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#18472
C2×#18352
C2×#17502
#143402
#110962
#110952
#109082
#109072
#108432
#108412