Finite Group Databasebeta

← #165113

Subgroups — #165113

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36416C211yesmaximal, characteristic
2#22700C211yesmaximal, characteristic
3#22700C211yesmaximal, characteristic
4#7711C212yesmaximal, complemented
5#7711C212yesmaximal, complemented
6#7663C211yesmaximal, characteristic, complemented
7#7660C211yesmaximal, characteristic, complemented
8#7652C211yesmaximal, characteristic, complemented
9#7632C212yesmaximal, complemented
10#7598C212yesmaximal, complemented
11#7570C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13C2×#436—21no—
14#2859C2211yescharacteristic
15#2851C2212yes—
16#2589—22no—
17#2587—22no—
18#2580—21no—
19#2571C411yescharacteristic
20#2566—21no—
21#2552C411yescharacteristic
22#2441C412yes—
23#2439C412yes—
24#2370—21no—
25#2355—21no—
26#2354C411yescharacteristic
27#2354C411yescharacteristic
28C4×C4⋊Q8C2211yescharacteristic
29#2296C2212yes—
30C42×D4C2211yescharacteristic
31#1939—21no—
32#1939—21no—
33#1851—22no—
34#1851—22no—
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37#1797C2212yes—
38#1797C2212yes—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41#1728C2212yes—
42#1728C2212yes—
43#1271C2211yescharacteristic
44#1271C2211yescharacteristic
45#1271C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×M16—22no—
54C22×M16—22no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#125—21no—
57C2×#125C2×C412yes—
58C2×C4×Q8—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C2×#117C2×C412yes—
63C2×#117C2×C412yes—
64C22×C42C2312yes—
65#436—44no—
66#436—44no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77#400—22no—
78#399—22no—
79#398—22no—
80#396—22no—
81#396—22no—
82#393—22no—
83#391—21no—
84#391—22no—
85#390—21no—
86#390—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2311yescharacteristic
97#389C2311yescharacteristic
98#389C2312yes—
99#388—22no—
100#388—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#384C2×C411yescharacteristic
112#384C2×C411yescharacteristic
113#382—21no—
114#382—21no—
115#382C2×C412yes—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4C2×C411yescharacteristic
125C4×C4⋊C4C2×C411yescharacteristic
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C22⋊C4C2×C412yes—
129C4×C22⋊C4D412yes—
130C4×C22⋊C4D412yes—
131C43C2311yescharacteristic
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2312yes—
140#336C2311yescharacteristic
141#336C2311yescharacteristic
142C22×Q8—21no—
143C22×D4C22⋊C411yescharacteristic
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C2×M16—44no—
148C2×M16—44no—
149C2×M16—44no—
150C2×M16—44no—
151C2×M16—82no—
152C2×M16—82no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—21no—
159C4⋊Q8—22no—
160C4⋊Q8—41no—
161#125—42no—
162#125—42no—
163#125—82no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×D4—24no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—22no—
170C4×D4—22no—
171#117—24no—
172#117—42no—
173#117—82no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22⋊C411yescharacteristic
191C2×C4⋊C4C22⋊C411yescharacteristic
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4C22⋊C412yes—
203C2×C22⋊C4C22⋊C412yes—
204C2×C22⋊C4C22⋊C412yes—
205C2×C42—22no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—22no—
209C2×C42—82no—
210C2×C42C22×C411yescharacteristic
211C2×C42C22×C411yescharacteristic
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2411yescharacteristic
220C4⋊C8—42no—
221C4⋊C8—42no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224C4⋊C8—82no—
225C4⋊C8—82no—
226C22⋊C8—44no—
227C22⋊C8—44no—
228C4×C8—42no—
229C4×C8—42no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—21no—
240#95—22no—
241#95—21no—
242#95C22⋊C412yes—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245C24C2×C22⋊C412yes—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—81no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—88no—
281M16—88no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—81no—
322C4⋊C4—81no—
323C4⋊C4—81no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—82no—
327C4⋊C4—82no—
328C4⋊C4#10114yes—
329C4⋊C4#13112yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—21no—
338C42—21no—
339C42—22no—
340C42—41no—
341C42—41no—
342C42—42no—
343C42—42no—
344C42—41no—
345C42—42no—
346C42—44no—
347C42—42no—
348C42—42no—
349C42—82no—
350C42AGL(1,ℤ/8)12yes—
351C42AGL(1,ℤ/8)12yes—
352C42#13712yes—
353C42#13712yes—
354C23—22no—
355C23—22no—
356C23—22no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23C22×C22⋊C411yescharacteristic
362Q8—84no—
363D4—44no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4C2×#10114yes—
410C2×C4C2×#10112yes—
411C2×C4#41712yes—
412C2×C4#41712yes—
413C2×C4#41712yes—
414C2×C4#41712yes—
415C2×C4C2×#13111yescharacteristic
416C2×C4#56811yescharacteristic
417C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
418C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
419C2×C4C2×#13711yescharacteristic
420C2×C4C2×#13711yescharacteristic
421C8—42no—
422C8—42no—
423C8—81no—
424C8—81no—
425C8—82no—
426C8—82no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—42no—
433C22—42no—
434C22#288011yescharacteristic
435C22#288311yescharacteristic
436C22C22×#10111yescharacteristic
437C22C2×#41711yescharacteristic
438C22C2×#41711yescharacteristic
439C22#290111yescharacteristic
440C22#290111yescharacteristic
441C4—21no—
442C4—21no—
443C4—24no—
444C4—22no—
445C4—22no—
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—41no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—81no—
455C4—82no—
456C4—81no—
457C4—82no—
458C4—81no—
459C4#156114yes—
460C2—82no—
461C2#1742611yescharacteristic
462C2#1751211yescharacteristic
463C2#1751511yescharacteristic
464C2#1763011yescharacteristic
465C2#1763011yescharacteristic
466C2#1769211yescharacteristic
467C2C2×#156111yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7711C2semidirect
5. #7711C2semidirect
6. #7663C2semidirect
7. #7660C2semidirect
8. #7652C2semidirect
9. #7632C2semidirect
10. #7598C2semidirect
11. #7570C2semidirect

Maximal subgroups

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

subgroupindex
#364162
#227002
#77112
#76632
#76602
#76522
#76322
#75982
#75702