Finite Group Databasebeta

← #190973

Subgroups — #190973

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 481 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34151C211yesmaximal, characteristic
2#21431C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#8708C212yesmaximal, complemented
5#8568C212yesmaximal, complemented
6#8563C212yesmaximal, complemented
7#8315C212yesmaximal
8#8181C211yesmaximal, characteristic, complemented
9#8179C211yesmaximal, characteristic, complemented
10#8173C211yesmaximal, characteristic, complemented
11#8171C211yesmaximal, characteristic, complemented
12C2×#529C2211yescharacteristic
13C2×#433—21no—
14#2374—22no—
15#2373—22no—
16#2370C412yes—
17#2367C412yes—
18#2363C412yes—
19#2362C412yes—
20#2298—21no—
21#2295C2211yescharacteristic
22#2293—21no—
23#2293C2211yescharacteristic
24#2290—21no—
25C42×Q8C2211yescharacteristic
26C42×D4C2211yescharacteristic
27C4×#117—21no—
28C4×#117C2211yescharacteristic
29C4×#117C2211yescharacteristic
30#1924—22no—
31#1917—22no—
32#1842C2212yes—
33#1832—21no—
34#1798—21no—
35#1797C2212yes—
36#1768—21no—
37#1767C2211yescharacteristic
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40C4×C22⋊C8C2212yes—
41#1744C2211yescharacteristic
42C4×#97C2211yescharacteristic
43C2×Q8⋊C8C2212yes—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46C2×D4⋊C8C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49#1279C2212yes—
50#1279C2212yes—
51#529—24no—
52#529—24no—
53C2×#126—21no—
54C2×#125C2×C411yescharacteristic
55C2×#124C2×C411yescharacteristic
56C2×#123—21no—
57C2×C22⋊Q8C2×C411yescharacteristic
58C2×#121C2×C411yescharacteristic
59C2×C4×Q8C2×C412yes—
60C2×C4×Q8C2311yescharacteristic
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×C4×D4C2311yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117C2×C412yes—
67C2×#117D411yescharacteristic
68C2×#117D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71#433—44no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—21no—
80C2×#97—22no—
81C2×C4×C8—21no—
82C2×C4×C8C2312yes—
83C2×C4×C8C2312yes—
84#391C2×C412yes—
85#390—22no—
86#390C2×C412yes—
87#389—22no—
88#389C2×C412yes—
89#389C2×C412yes—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#387—22no—
100#386—22no—
101#385—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C412yes—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2311yescharacteristic
113C4×C4⋊C4C2311yescharacteristic
114C4×C22⋊C4—21no—
115C4×C22⋊C4—21no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4D412yes—
123#376—21no—
124#376—21no—
125#376D411yescharacteristic
126#376D411yescharacteristic
127C43—21no—
128C43—21no—
129C43C2311yescharacteristic
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336—22no—
136#336—22no—
137Q8⋊C8—24no—
138Q8⋊C8—24no—
139D4⋊C8—24no—
140D4⋊C8—24no—
141C22×Q8C22×C411yescharacteristic
142C22×D4—21no—
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C23×C4—22no—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153#126—28no—
154#125—22no—
155#125—22no—
156#124—22no—
157#124—22no—
158#123—28no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161#121—22no—
162#121—22no—
163C4×Q8—22no—
164C4×Q8—24no—
165C4×Q8—42no—
166C4×Q8#4114yes—
167C4×D4—28no—
168C4×D4—24no—
169C4×D4—22no—
170C4×D4—42no—
171C4×D4#4114yes—
172#117—28no—
173#117—22no—
174#117—22no—
175#117—24no—
176#117—42no—
177#117—44no—
178#117—42no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C412yes—
188C2×C4⋊C4C22×C412yes—
189C2×C4⋊C4C22×C411yescharacteristic
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C412yes—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C22×C412yes—
216C2×C42C22×C412yes—
217C2×C42C22×C411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42C2411yescharacteristic
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—44no—
232C4⋊C8—44no—
233C4⋊C8—44no—
234C22⋊C8—44no—
235C22⋊C8—44no—
236#97—42no—
237#97—44no—
238C4×C8—22no—
239C4×C8—22no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—42no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—21no—
251#95—21no—
252#95C22⋊C412yes—
253#95C22⋊C412yes—
254#95C22⋊C412yes—
255#95C22⋊C412yes—
256C24—21no—
257C24C2×C22⋊C411yescharacteristic
258C2×Q8—21no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8#11712yes—
262C2×D4—24no—
263C2×D4—21no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—82no—
267C2×D4#11712yes—
268C22×C4—22no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—44no—
289C22×C4—82no—
290C22×C4C2×C22⋊C412yes—
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C2×C22⋊C412yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C2×C8—24no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—24no—
318C4⋊C4—22no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—42no—
323C4⋊C4—44no—
324C4⋊C4—44no—
325C4⋊C4—82no—
326C4⋊C4#11712yes—
327C4⋊C4#11712yes—
328C4⋊C4#13114yes—
329C22⋊C4—24no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—42no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—82no—
340C42—21no—
341C42—21no—
342C42—22no—
343C42—22no—
344C42—24no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—21no—
350C42—21no—
351C42—22no—
352C42—22no—
353C42—42no—
354C42—42no—
355C42—44no—
356C42—44no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—44no—
361C42—82no—
362C42—82no—
363C42AGL(1,ℤ/8)12yes—
364C42#13712yes—
365C42C2×#4112yes—
366C42C2×#4112yes—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—22no—
372C23—21no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23C22×C22⋊C411yescharacteristic
378Q8—22no—
379Q8—22no—
380Q8—42no—
381D4—24no—
382D4—41no—
383D4—41no—
384D4—84no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—42no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—88no—
426C2×C4#41714yes—
427C2×C4C2×#11711yescharacteristic
428C2×C4C2×#11711yescharacteristic
429C2×C4#52512yes—
430C2×C4#52512yes—
431C2×C4C2×#13112yes—
432C2×C4#56812yes—
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57512yes—
436C8—44no—
437C8—82no—
438C8—82no—
439C8—82no—
440C22—41no—
441C22—42no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—41no—
447C22—42no—
448C22—82no—
449C22—82no—
450C22#228011yescharacteristic
451C22#288012yes—
452C22C2×#41711yescharacteristic
453C22#289611yescharacteristic
454C22#290112yes—
455C4—22no—
456C4—24no—
457C4—41no—
458C4—42no—
459C4—41no—
460C4—42no—
461C4—42no—
462C4—44no—
463C4—42no—
464C4—42no—
465C4—41no—
466C4—42no—
467C4—82no—
468C4#292211yescharacteristic
469C4#292311yescharacteristic
470C4#292512yes—
471C4#311512yes—
472C4#311612yes—
473C2—81no—
474C2—82no—
475C2#1751212yes—
476C2#1751612yes—
477C2#2072011yescharacteristic
478C2#2074311yescharacteristic
479C2#2074911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#121C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21431C2semidirect
3. #20812C2semidirect
4. #8708C2semidirect
5. #8568C2semidirect
6. #8563C2semidirect
8. #8181C2semidirect
9. #8179C2semidirect
10. #8173C2semidirect
11. #8171C2semidirect

Maximal subgroups

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

subgroupindex
#341512
#214312
#208122
#87082
#85682
#85632
#83152
#81812
#81792
#81732
#81712