Finite Group Databasebeta

← #189878

Subgroups — #189878

Order 512.

1210 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21790C211yesmaximal, characteristic
3#13941C211yesmaximal, characteristic
4#8715C214yesmaximal, complemented
5#8713C212yesmaximal, complemented
6#8712C212yesmaximal, complemented
7#8021C212yesmaximal, complemented
8#8017C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2353—22no—
12#2353—22no—
13#2352C414yescomplemented
14#2352C414yescomplemented
15#2313—21no—
16#2312—21no—
17#2308—21no—
18C2×#390—21no—
19#2283C2212yes—
20C2×#382C2211yescharacteristic
21#2280C2212yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2273—21no—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1955—22no—
27#1955—22no—
28#1779C2212yes—
29#1779C2212yes—
30#1439—21no—
31#1439—21no—
32C4⋊#95—21no—
33C22⋊#95C2212yes—
34C22⋊#95C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×D4D411yescharacteristic
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×C4×D4—24no—
43C2×C4×D4—24no—
44C2×C4×D4C2×C412yes—
45C2×C4×D4C2×C412yes—
46C2×C4×D4D412yes—
47C2×C4×D4D412yes—
48C2×#117C2×C414yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4D411yescharacteristic
54C22×C22⋊C4—21no—
55C22×C22⋊C4—21no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D411yescharacteristic
59C22×C22⋊C4D411yescharacteristic
60C22×C42—21no—
61C22×C42C2311yescharacteristic
62#432—24no—
63#432—24no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—24no—
68C2×C22⋊C8C2314yes—
69C2×#97—22no—
70#390—44no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—24no—
76#386—24no—
77#383—22no—
78#382—24no—
79#382C2×C414yes—
80#381—22no—
81#381—22no—
82#380—21no—
83#380—21no—
84#380—21no—
85#380—21no—
86#380C2×C412yes—
87#380C2×C412yes—
88#380C2×C414yes—
89#380C2312yes—
90#380C2312yes—
91#379—22no—
92C4×C4⋊C4—24no—
93C4×C4⋊C4C2×C414yes—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—24no—
98C4×C22⋊C4—24no—
99#376—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95C2312yes—
106C2×#95C2312yes—
107C25C2×D411yescharacteristic
108C25C2×D411yescharacteristic
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4C22⋊C412yes—
116C22×D4C22⋊C412yes—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—82no—
126C23×C4—82no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C22×C412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2411yescharacteristic
138C22×C8—24no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—24no—
142C4×D4—24no—
143C4×D4—24no—
144C4×D4—24no—
145#117—84no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—42no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—82no—
156C2×C4⋊C4—82no—
157C2×C4⋊C4C22⋊C412yes—
158C2×C4⋊C4C22⋊C412yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C22×C414yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4—82no—
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C22⋊C4C22⋊C412yes—
184C2×C22⋊C4C22⋊C412yes—
185C2×C42—24no—
186C2×C42—24no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C22×C412yes—
191C2×C42C2×D412yes—
192C2×C42#4114yes—
193C2×C42#4114yes—
194C4⋊C8—44no—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—84no—
198C22⋊C8—82no—
199C22⋊C8—82no—
200#97—44no—
201#95—22no—
202#95—22no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—24no—
207#95—22no—
208#95—22no—
209#95—24no—
210#95—24no—
211#95—42no—
212#95—82no—
213#95—82no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—21no—
223C24—21no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—42no—
228C24C2×C22⋊C412yes—
229C24C2×C22⋊C411yescharacteristic
230C24C2×C22⋊C411yescharacteristic
231C24AGL(1,ℤ/8)12yes—
232C24#13712yes—
233C24C22×D411yescharacteristic
234C2×D4—24no—
235C2×D4—24no—
236C2×D4—24no—
237C2×D4—24no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—44no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
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—44no—
279C22×C4—44no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—84no—
288C22×C4—84no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—82no—
292C22×C4—82no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C412yes—
299C22×C4C2×C22⋊C412yes—
300C22×C4#12312yes—
301C22×C4#12312yes—
302C22×C4#12312yes—
303C22×C4#12312yes—
304C22×C4C23×C411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C22×C4C2×#4112yes—
307C22×C4C2×#4112yes—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—48no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—84no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—24no—
322C22⋊C4—24no—
323C22⋊C4—44no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—44no—
329C22⋊C4—84no—
330C22⋊C4—84no—
331C22⋊C4—88no—
332C22⋊C4—88no—
333C42—84no—
334C42—84no—
335C42—84no—
336C42#11718yes—
337C42#12314yes—
338C42#12314yes—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—22no—
343C23—21no—
344C23—21no—
345C23—22no—
346C23—22no—
347C23—21no—
348C23—21no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—24no—
354C23—22no—
355C23—22no—
356C23—24no—
357C23—22no—
358C23—44no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—81no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—82no—
371C23—82no—
372C23—82no—
373C23—82no—
374C23—81no—
375C23—81no—
376C23—81no—
377C23—81no—
378C23#41712yes—
379C23#41712yes—
380C23C22×C22⋊C411yescharacteristic
381C23C2×#12311yescharacteristic
382C23C2×#12311yescharacteristic
383C23C2×AGL(1,ℤ/8)11yescharacteristic
384C23C2×#13711yescharacteristic
385C23#57511yescharacteristic
386C23#57511yescharacteristic
387D4—48no—
388D4—48no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—164no—
440C2×C4—164no—
441C2×C4#38114yes—
442C2×C4#38114yes—
443C2×C4C2×#11712yes—
444C2×C4C2×#12312yes—
445C8—84no—
446C22—21no—
447C22—21no—
448C22—22no—
449C22—22no—
450C22—22no—
451C22—22no—
452C22—21no—
453C22—21no—
454C22—24no—
455C22—24no—
456C22—22no—
457C22—22no—
458C22—41no—
459C22—41no—
460C22—42no—
461C22—42no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—44no—
466C22—44no—
467C22—42no—
468C22—42no—
469C22—82no—
470C22—82no—
471C22—82no—
472C22—82no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—82no—
477C22C2×#38111yescharacteristic
478C22#288312yes—
479C22C2×#41711yescharacteristic
480C22#289611yescharacteristic
481C22#290112yes—
482C22#309411yescharacteristic
483C22#309411yescharacteristic
484C22#309511yescharacteristic
485C22#309511yescharacteristic
486C4—22no—
487C4—22no—
488C4—24no—
489C4—24no—
490C4—42no—
491C4—41no—
492C4—41no—
493C4—84no—
494C4—84no—
495C4—84no—
496C4—162no—
497C2—21no—
498C2—21no—
499C2—21no—
500C2—21no—
501C2—42no—
502C2—42no—
503C2—82no—
504C2#1762212yes—
505C2#1762512yes—
506C2#2069011yescharacteristic
507C2#2070211yescharacteristic
508C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8715C2semidirect
5. #8713C2semidirect
6. #8712C2semidirect
7. #8021C2semidirect
8. #8017C2semidirect
13. #2352C4semidirect
14. #2352C4semidirect

Maximal subgroups

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

subgroupindex
#341852
#217902
#139412
#87152
#87132
#87122
#80212
#80172