Finite Group Databasebeta

← #189880

Subgroups — #189880

Order 512.

1276 conjugacy classes of subgroups.

The subgroup lattice has 515 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21742C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#8714C214yesmaximal, complemented
5#8713C212yesmaximal, complemented
6#8712C212yesmaximal, complemented
7#8020C212yesmaximal, complemented
8#8016C212yesmaximal, complemented
9C22×C4×D4C2211yescharacteristic
10C2×#407C2212yes—
11#2353—22no—
12#2353—22no—
13#2352C414yescomplemented
14#2352C414yescomplemented
15#2313—21no—
16#2312—21no—
17#2308—21no—
18C2×#386—21no—
19#2283C2212yes—
20#2280C2212yes—
21C2×#380C2211yescharacteristic
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24#2273—21no—
25C22×#95C2211yescharacteristic
26#1870—22no—
27#1870—22no—
28#1817C2212yes—
29#1779C2212yes—
30#1441—21no—
31#1439—21no—
32#1434C2212yes—
33C22⋊#95—21no—
34C22⋊#95C2212yes—
35#1294C2214yes—
36#1294C2214yes—
37#1294C2214yes—
38#1294C2214yes—
39C23×D4D411yescharacteristic
40C24×C4—21no—
41C24×C4C2311yescharacteristic
42C24×C4C2311yescharacteristic
43C22×M16—22no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4C2×C412yes—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D412yes—
49C2×C4×D4D412yes—
50C2×#117C2×C414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C412yescomplemented
53C22×C4⋊C4C2×C412yescomplemented
54C22×C4⋊C4D411yescharacteristic
55C22×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D411yescharacteristic
60C22×C22⋊C4D411yescharacteristic
61C22×C42C2311yescharacteristic
62#407—24no—
63#407—24no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8C2314yes—
68#388—24no—
69#388—24no—
70#387—24no—
71#387—24no—
72#386—24no—
73#386—24no—
74#386—44no—
75#383—22no—
76#382C2×C414yescomplemented
77#381—22no—
78#381—22no—
79#380—21no—
80#380—21no—
81#380—21no—
82#380—21no—
83#380—24no—
84#380C2×C414yes—
85#380C2×C412yescomplemented
86#380C2×C412yescomplemented
87#380C2312yes—
88#380C2312yes—
89#379—22no—
90C4×C4⋊C4C2×C414yescomplemented
91C4×C22⋊C4—22no—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96#376—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—24no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95C2312yes—
105C2×#95C2312yes—
106#336—24no—
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—44no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—42no—
127C23×C4—82no—
128C23×C4—82no—
129C23×C4—82no—
130C23×C4C22×C411yescharacteristic
131C23×C4C22×C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D412yes—
140C23×C4#4114yes—
141C23×C4#4114yes—
142C23×C4C2411yescharacteristic
143C2×M16—44no—
144C2×M16—84no—
145C22×C8—24no—
146C4×D4—28no—
147C4×D4—28no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152#117—84no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—82no—
159C2×C4⋊C4—82no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22⋊C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C22×C414yes—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—42no—
179C2×C22⋊C4—44no—
180C2×C22⋊C4—42no—
181C2×C22⋊C4—42no—
182C2×C22⋊C4—42no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42C22×C412yes—
194C2×C42C2×D412yes—
195C22⋊C8—24no—
196C22⋊C8—24no—
197C22⋊C8—44no—
198C22⋊C8—82no—
199C22⋊C8—82no—
200#95—22no—
201#95—22no—
202#95—24no—
203#95—24no—
204#95—24no—
205#95—24no—
206#95—22no—
207#95—22no—
208#95—24no—
209#95—24no—
210#95—44no—
211#95—44no—
212#95—42no—
213#95—82no—
214C24—21no—
215C24—21no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—21no—
220C24—21no—
221C24—21no—
222C24—42no—
223C24—42no—
224C24—42no—
225C24—42no—
226C24C2×C22⋊C412yes—
227C24C2×C22⋊C411yescharacteristic
228C24C2×C22⋊C411yescharacteristic
229C24#12312yes—
230C24#12312yes—
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—21no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—48no—
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×C4—88no—
296C22×C4—88no—
297C22×C4—84no—
298C22×C4—84no—
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C412yes—
304C22×C4#11718yes—
305C22×C4#12314yes—
306C22×C4#12314yes—
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4C23×C411yescharacteristic
310C22×C4C22×D411yescharacteristic
311C22×C4C2×#4112yes—
312C22×C4C2×#4112yes—
313M16—88no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—84no—
324C22⋊C4—24no—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—44no—
329C22⋊C4—42no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—84no—
335C22⋊C4—84no—
336C22⋊C4—88no—
337C22⋊C4—84no—
338C42—24no—
339C42—22no—
340C42—22no—
341C42—84no—
342C42—84no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—24no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—24no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
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—44no—
374C23—44no—
375C23—81no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—81no—
380C23—81no—
381C23#38114yes—
382C23#41712yes—
383C23#41712yes—
384C23C22×C22⋊C411yescharacteristic
385C23C2×#12311yescharacteristic
386C23C2×#12311yescharacteristic
387C23C2×AGL(1,ℤ/8)11yescharacteristic
388C23C2×#13711yescharacteristic
389C23#57511yescharacteristic
390C23#57511yescharacteristic
391D4—48no—
392D4—48no—
393C2×C4—24no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—816no—
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—164no—
446C2×C4—164no—
447C2×C4#38114yes—
448C2×C4C2×#11712yes—
449C2×C4C2×#12312yes—
450C8—84no—
451C22—21no—
452C22—21no—
453C22—22no—
454C22—22no—
455C22—22no—
456C22—22no—
457C22—21no—
458C22—21no—
459C22—24no—
460C22—24no—
461C22—24no—
462C22—24no—
463C22—22no—
464C22—22no—
465C22—24no—
466C22—44no—
467C22—44no—
468C22—44no—
469C22—44no—
470C22—41no—
471C22—41no—
472C22—44no—
473C22—42no—
474C22—44no—
475C22—42no—
476C22—41no—
477C22—41no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22C2×#38111yescharacteristic
483C22#288312yes—
484C22C2×#41711yescharacteristic
485C22#289611yescharacteristic
486C22#290112yes—
487C22#309411yescharacteristic
488C22#309411yescharacteristic
489C22#309511yescharacteristic
490C22#309511yescharacteristic
491C4—22no—
492C4—22no—
493C4—24no—
494C4—41no—
495C4—41no—
496C4—42no—
497C4—42no—
498C4—84no—
499C4—84no—
500C4—84no—
501C4—162no—
502C2—21no—
503C2—21no—
504C2—21no—
505C2—21no—
506C2—44no—
507C2—42no—
508C2—42no—
509C2#1762212yes—
510C2#1762312yes—
511C2#2068911yescharacteristic
512C2#2070211yescharacteristic
513C2#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. #8714C2semidirect
5. #8713C2semidirect
6. #8712C2semidirect
7. #8020C2semidirect
8. #8016C2semidirect
13. #2352C4semidirect
14. #2352C4semidirect
52. C22×C4⋊C4C2×C4semidirect
53. C22×C4⋊C4C2×C4semidirect
76. #382C2×C4semidirect
85. #380C2×C4semidirect
86. #380C2×C4semidirect
90. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341852
#217422
#139602
#87142
#87132
#87122
#80202
#80162