Finite Group Databasebeta

← #194754

Subgroups — #194754

Order 512.

2340 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2409C211yesmaximal, characteristic
2#34181C211yesmaximal, characteristic
3C2×#1787C212yesmaximal, complemented
4C2×#1459C211yesmaximal, characteristic
5#9015C214yesmaximal, complemented
6#9013C214yesmaximal, complemented
7#8023C212yesmaximal, complemented
8C24×Q8C2211yescharacteristic
9C22×C22⋊Q8—21no—
10C2×#518—21no—
11C22×C4×D4C2211yescharacteristic
12C22×#103—24no—
13C22×C22⋊C8C2212yes—
14#2409—24no—
15#2409C418yescomplemented
16C2×#395C2212yes—
17C2×#393C2212yes—
18#2349—24no—
19D4×C4⋊C4—24no—
20#2316C2212yes—
21#2315—21no—
22#2315C2212yes—
23C2×#391—21no—
24C2×#386—21no—
25C2×#385—21no—
26#2285—21no—
27C2×#380—21no—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30#1787—24no—
31#1787C2218yes—
32#1459—24no—
33C2×#324C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
37C23×Q8—22no—
38C23×Q8—22no—
39C23×Q8—22no—
40C23×Q8—24no—
41C23×Q8—48no—
42C23×Q8C2×C414yescomplemented
43C23×Q8C2312yes—
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C24×C4D411yescharacteristic
47C24×C4C2311yescharacteristic
48C23×C8—22no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—48no—
51#518—44no—
52#518—44no—
53C2×C4×D4—22no—
54C2×C4×D4—22no—
55C2×C4×D4—24no—
56C2×C4×D4—28no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4D414yes—
59C2×#117—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D414yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—22no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C22×C42D411yescharacteristic
72C2×#103—216no—
73C2×#103—416no—
74C2×C22⋊C8—24no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8C2314yes—
78#395—24no—
79#395C2×C418yescomplemented
80#393—24no—
81#393C2×C418yescomplemented
82#391—28no—
83#389—28no—
84#386—28no—
85#385—28no—
86#385—28no—
87#385—28no—
88#384—22no—
89#382—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380C2×C414yescomplemented
98#380D414yes—
99#380C2314yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95C2312yes—
107#324—24no—
108#323—24no—
109C25C2×D411yescharacteristic
110C25C2×D411yescharacteristic
111C22×Q8—28no—
112C22×Q8—24no—
113C22×Q8—22no—
114C22×Q8—216no—
115C22×Q8—28no—
116C22×Q8—28no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—48no—
120C22×Q8—48no—
121C22×Q8—44no—
122C22×Q8—416no—
123C22×Q8—416no—
124C22×Q8—432no—
125C22×Q8—416no—
126C22×Q8—41no—
127C22×Q8—41no—
128C22×Q8—44no—
129C22×Q8—44no—
130C22×Q8—42no—
131C22×Q8C22×C414yes—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C22⋊C414yes—
137C22×D4SD1614yes—
138C22×D4Q1614yes—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—22no—
143C23×C4—24no—
144C23×C4—22no—
145C23×C4—24no—
146C23×C4—24no—
147C23×C4—44no—
148C23×C4—48no—
149C23×C4—44no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4C22⋊C414yes—
153C23×C4C22×C412yes—
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D412yes—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4#4114yes—
162C23×C4C2411yescharacteristic
163C22×C8—24no—
164C22×C8—48no—
165C22×C8—44no—
166C22⋊Q8—432no—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—28no—
170C4×D4—28no—
171C4×D4—432no—
172#117—44no—
173#117—44no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—28no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—48no—
185C2×C4⋊C4—84no—
186C2×C4⋊C4C22×C412yes—
187C2×C4⋊C4C2×D412yes—
188C2×C4⋊C4C2×D418yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—28no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—48no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4C22⋊C414yes—
206C2×C22⋊C4C22×C412yes—
207C2×C22⋊C4C2×D412yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—24no—
212C2×C42C22⋊C414yes—
213#103—432no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8—44no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95—28no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95C22×C414yes—
225C24—21no—
226C24—21no—
227C24—21no—
228C24—21no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C412yes—
235C24#12012yes—
236C24#12312yes—
237C24C2×SD1612yes—
238C24C2×SD1612yes—
239C24C2×Q1612yes—
240C24C2×Q1612yes—
241C24AGL(1,ℤ/8)12yes—
242C24#13712yes—
243C24C22×D411yescharacteristic
244C2×Q8—216no—
245C2×Q8—216no—
246C2×Q8—232no—
247C2×Q8—28no—
248C2×Q8—48no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—48no—
252C2×Q8—432no—
253C2×Q8—416no—
254C2×Q8—416no—
255C2×Q8—48no—
256C2×Q8—432no—
257C2×Q8—416no—
258C2×Q8—432no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—48no—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—28no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4#103116yescomplemented
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—24no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—28no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—416no—
292C22×C4—48no—
293C22×C4—44no—
294C22×C4—416no—
295C22×C4—416no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—44no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—44no—
303C22×C4—44no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—48no—
307C22×C4—48no—
308C22×C4—84no—
309C22×C4—82no—
310C22×C4—84no—
311C22×C4—84no—
312C22×C4—84no—
313C22×C4—84no—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4C2×C22⋊C412yes—
318C22×C4C4×D414yes—
319C22×C4C4×D414yes—
320C22×C4#12011yescharacteristic
321C22×C4#12011yescharacteristic
322C22×C4#12114yes—
323C22×C4#12114yes—
324C22×C4#12312yes—
325C22×C4C2×SD1612yes—
326C22×C4C2×Q1612yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D412yes—
329C22×C4C22×D411yescharacteristic
330C22×C4C2×#4112yes—
331C2×C8—48no—
332C2×C8—44no—
333C2×C8—88no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—48no—
337C4⋊C4—48no—
338C4⋊C4—416no—
339C4⋊C4—48no—
340C4⋊C4—48no—
341C4⋊C4—84no—
342C4⋊C4—84no—
343C22⋊C4—28no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—416no—
347C22⋊C4—48no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—48no—
351C22⋊C4—48no—
352C42—24no—
353C42—24no—
354C42—48no—
355C23—22no—
356C23—24no—
357C23—22no—
358C23—24no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—22no—
364C23—22no—
365C23—42no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23—44no—
373C23—42no—
374C23—42no—
375C23—44no—
376C23—44no—
377C23—42no—
378C23—42no—
379C23—81no—
380C23—82no—
381C23—82no—
382C23—81no—
383C23#38614yes—
384C23C2×#10314yes—
385C23C2×#10314yescomplemented
386C23#41714yes—
387C23#45014yes—
388C23#45114yes—
389C23#48314yes—
390C23#48414yes—
391C23C22×C22⋊C411yescharacteristic
392C23C2×#12011yescharacteristic
393C23C2×#12311yescharacteristic
394C23C22×SD1611yescharacteristic
395C23C22×Q1611yescharacteristic
396C23C2×AGL(1,ℤ/8)11yescharacteristic
397C23C2×#13711yescharacteristic
398Q8—216no—
399Q8—416no—
400Q8—416no—
401Q8—48no—
402Q8—416no—
403D4—232no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—28no—
414C2×C4—24no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—48no—
426C2×C4—432no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—48no—
430C2×C4—44no—
431C2×C4—48no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—48no—
438C2×C4—48no—
439C2×C4—48no—
440C2×C4—84no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—168no—
446C2×C4—168no—
447C2×C4#38612yes—
448C2×C4#38612yes—
449C2×C4C2×#10314yescomplemented
450C2×C4B(4,𝔽2)14yes—
451C2×C4C2×C4×D412yes—
452C2×C4C2×#12112yes—
453C8—84no—
454C22—22no—
455C22—28no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—28no—
460C22—21no—
461C22—21no—
462C22—24no—
463C22—21no—
464C22—21no—
465C22—24no—
466C22—24no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—44no—
471C22—44no—
472C22—44no—
473C22—44no—
474C22—42no—
475C22—42no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22#187218yes—
480C22C2×#38611yescharacteristic
481C22C22×#10311yescharacteristic
482C22C2×#41711yescharacteristic
483C22C2×#45011yescharacteristic
484C22C2×#45111yescharacteristic
485C22#302212yes—
486C22C2×B(4,𝔽2)12yes—
487C22C2×#48311yescharacteristic
488C22C2×#48411yescharacteristic
489C4—24no—
490C4—22no—
491C4—22no—
492C4—22no—
493C4—42no—
494C4—48no—
495C4—42no—
496C4—84no—
497C4—88no—
498C4#188914yes—
499C2—21no—
500C2—21no—
501C2—21no—
502C2—21no—
503C2—28no—
504C2—44no—
505C2#1819512yes—
506C2#1819712yes—
507C2C2×#187211yescharacteristic
508C2#2095511yescharacteristic
509C2C2×#188911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1787C2semidirect
5. #9015C2semidirect
6. #9013C2semidirect
7. #8023C2semidirect
15. #2409C4semidirect
42. C23×Q8C2×C4semidirect
79. #395C2×C4semidirect
81. #393C2×C4semidirect
97. #380C2×C4semidirect
269. C2×D4#103semidirect
385. C23C2×#103semidirect
449. C2×C4C2×#103semidirect

Maximal subgroups

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

subgroupindex
C2×#24092
#341812
C2×#17872
C2×#14592
#90152
#90132
#80232