Finite Group Databasebeta

← #238526

Subgroups — #238526

Order 512.

1414 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35011C211yesmaximal, characteristic
2C2×#1872C212yesmaximal
3C2×#1520C212yesmaximal
4#14130C211yesmaximal, characteristic
5#10407C214yesmaximal, complemented
6#10406C214yesmaximal, complemented
7#7907C211yesmaximal, characteristic
8C22×C22⋊Q8C2212yes—
9C23×C4⋊C4C2211yescharacteristic
10C22×#103—22no—
11C22×C22⋊C8C2212yes—
12#2857—21no—
13#2857—21no—
14#2659C2212yes—
15#2659C2212yes—
16#2583—24no—
17#2502—22no—
18#2434—22no—
19#2315—21no—
20#2315—21no—
21C2×#389—21no—
22C2×#385—21no—
23C2×#380C2211yescharacteristic
24C2×C4×C22⋊C4C2211yescharacteristic
25#1872—24no—
26#1872C2218yes—
27C2×#340—22no—
28#1520—28no—
29C2×#328C2218yes—
30#1444C2212yes—
31#1442C2212yes—
32C4⋊#95—21no—
33C22⋊#95—21no—
34#1280C2212yes—
35#1280C2212yes—
36C23×Q8D412yes—
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C23×C8—22no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8D414yes—
49C2×C22⋊Q8C2314yes—
50C22×C4⋊C4—22no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—44no—
57C22×C4⋊C4D412yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C2×#103—28no—
66C2×#103—48no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8C2314yes—
71#398—24no—
72#398—24no—
73#395—24no—
74#395—24no—
75#391—22no—
76#391—22no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81#389—44no—
82#385—24no—
83#385—24no—
84#385—24no—
85#385—24no—
86#385—44no—
87#384—21no—
88#384—21no—
89#380—22no—
90#380—22no—
91#380—22no—
92#380—22no—
93#380—22no—
94#380D414yes—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95C2311yescharacteristic
101C2×#95C2311yescharacteristic
102C2×#95C2312yes—
103#340—28no—
104#328—216no—
105C25C2×D411yescharacteristic
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8—24no—
109C22×Q8—44no—
110C22×Q8—44no—
111C22×Q8C2×D414yes—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4—44no—
119C23×C4—42no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—82no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C23×C4#4114yes—
132C23×C4#4112yes—
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C22×C8—44no—
136C22×C8—44no—
137C22×C8—44no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140C22⋊Q8—28no—
141C22⋊Q8—28no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—44no—
147C22⋊Q8—44no—
148C22⋊Q8—44no—
149C22⋊Q8—44no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
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—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—41no—
174C2×C4⋊C4—41no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4SD1614yes—
193C2×C22⋊C4Q1614yes—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—42no—
199C2×C42—42no—
200#103—416no—
201C22⋊C8—24no—
202C22⋊C8—24no—
203C22⋊C8—44no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—24no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—44no—
212#95—44no—
213#95—82no—
214#95C2×D414yes—
215C24—41no—
216C24—41no—
217C24—41no—
218C24—41no—
219C24#12012yes—
220C24#12112yes—
221C24#12412yes—
222C24C2×SD1612yes—
223C24C2×Q1612yes—
224C24#13712yes—
225C24#13712yes—
226C24C22×D411yescharacteristic
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—44no—
230C2×Q8—44no—
231C2×Q8—416no—
232C2×Q8—44no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—88no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—48no—
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—44no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—88no—
287C22×C4—88no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4#12011yescharacteristic
291C22×C4#12011yescharacteristic
292C22×C4#12114yes—
293C22×C4#12112yes—
294C22×C4#12114yes—
295C22×C4#12314yes—
296C22×C4#12314yes—
297C22×C4#12412yes—
298C22×C4C2×SD1612yes—
299C22×C4C2×SD1612yes—
300C22×C4C2×Q1612yes—
301C22×C4C2×Q1612yes—
302C22×C4C22×D412yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C2×#4112yes—
305C22×C4C2×#4111yescharacteristic
306C2×C8—44no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—88no—
310C4⋊C4—28no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—48no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—48no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4—82no—
329C4⋊C4—88no—
330C22⋊C4—28no—
331C22⋊C4—28no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—48no—
335C22⋊C4—48no—
336C42—42no—
337C42—42no—
338C42—81no—
339C42—81no—
340C23—21no—
341C23—22no—
342C23—21no—
343C23—22no—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—22no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23#39414yes—
361C23#44814yes—
362C23#45114yes—
363C23#46114yes—
364C23#46214yes—
365C23#48714yes—
366C23#49114yes—
367C23C2×#12011yescharacteristic
368C23C2×#12111yescharacteristic
369C23C2×#12411yescharacteristic
370C23C22×SD1611yescharacteristic
371C23C22×Q1611yescharacteristic
372C23C2×#13711yescharacteristic
373C23C2×#13711yescharacteristic
374Q8—48no—
375Q8—84no—
376Q8—88no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—28no—
383C2×C4—28no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—88no—
429C2×C4—816no—
430C2×C4—88no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#39412yes—
434C2×C4#39412yes—
435C2×C4B(4,𝔽2)12yes—
436C2×C4#45812yes—
437C2×C4C2×#12112yes—
438C2×C4C2×#12312yes—
439C8—84no—
440C22—21no—
441C22—21no—
442C22—24no—
443C22—21no—
444C22—21no—
445C22—24no—
446C22—21no—
447C22—21no—
448C22—24no—
449C22—21no—
450C22—21no—
451C22—24no—
452C22—21no—
453C22—24no—
454C22—21no—
455C22—21no—
456C22—24no—
457C22—21no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—44no—
462C22#204318yes—
463C22C2×#39411yescharacteristic
464C22C2×#44811yescharacteristic
465C22C2×#45111yescharacteristic
466C22#302111yescharacteristic
467C22#302211yescharacteristic
468C22C2×B(4,𝔽2)11yescharacteristic
469C22C2×#45811yescharacteristic
470C22C2×#46111yescharacteristic
471C22C2×#46211yescharacteristic
472C22C2×#48711yescharacteristic
473C22C2×#49111yescharacteristic
474C4—22no—
475C4—22no—
476C4—22no—
477C4—42no—
478C4—42no—
479C4—42no—
480C4—42no—
481C4—84no—
482C4—84no—
483C4—84no—
484C4—84no—
485C4#204414yes—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—28no—
491C2#1819611yescharacteristic
492C2#1819711yescharacteristic
493C2#1820411yescharacteristic
494C2#1820511yescharacteristic
495C2#2350011yescharacteristic
496C2C2×#204311yescharacteristic
497C2C2×#204411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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
5. #10407C2semidirect
6. #10406C2semidirect

Maximal subgroups

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

subgroupindex
#350112
C2×#18722
C2×#15202
#141302
#104072
#104062
#79072