Finite Group Databasebeta

← #164493

Subgroups — #164493

Order 512.

821 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36290C211yesmaximal, characteristic
2#22012C211yesmaximal, characteristic
3#20567C211yesmaximal, characteristic
4#7711C212yesmaximal, complemented
5#7683C211yesmaximal, characteristic, complemented
6#7680C211yesmaximal, characteristic, complemented
7#7671C212yesmaximal, complemented
8#7636C211yesmaximal, characteristic, complemented
9#7610C212yesmaximal, complemented
10#7576C212yesmaximal, complemented
11#7568C211yesmaximal, characteristic, complemented
12C2×#436—21no—
13#2859C2211yescharacteristic
14#2851C2212yes—
15#2571C411yescharacteristic
16#2552C411yescharacteristic
17#2509—21no—
18#2507—21no—
19#2505—22no—
20#2473—22no—
21#2442C412yes—
22#2442C412yes—
23#2349—21no—
24D4×C4⋊C4—21no—
25D4×C4⋊C4C411yescharacteristic
26D4×C4⋊C4C411yescharacteristic
27C4×#125C2211yescharacteristic
28C2×#389C2212yes—
29C42×D4C2211yescharacteristic
30#1954—21no—
31#1939—21no—
32#1924—22no—
33#1817C2212yes—
34#1797C2212yes—
35C8×C4⋊C4—21no—
36#1767C2211yescharacteristic
37#1767C2211yescharacteristic
38#1753—22no—
39C4×C22⋊C8C2212yes—
40#1744C2211yescharacteristic
41#1744C2211yescharacteristic
42#1728C2212yes—
43#1271C2212yes—
44#1271C2212yes—
45#1271C2212yes—
46#1271C2212yes—
47#1271C2211yescharacteristic
48#1271C2212yes—
49#1271C2211yescharacteristic
50#1271C2211yescharacteristic
51#1271C2211yescharacteristic
52#1271C2212yes—
53C22×M16—22no—
54C2×#125—21no—
55C2×#125C2×C411yescharacteristic
56C2×#125C2×C412yes—
57C2×C4×D4—21no—
58C2×C4×D4—21no—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2×C412yes—
64C22×C4⋊C4C2×C412yes—
65C22×C42C2312yes—
66#436—28no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—22no—
78#400—22no—
79#400—22no—
80#400—22no—
81#398—22no—
82#398—22no—
83#398—22no—
84#389—21no—
85#389—21no—
86#389—21no—
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389—22no—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2311yescharacteristic
96#389C2311yescharacteristic
97#389C2312yes—
98#387—21no—
99#387—21no—
100#387—22no—
101#387—22no—
102#385—22no—
103#385—22no—
104#385—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#384—21no—
109#384C2×C411yescharacteristic
110#384C2×C411yescharacteristic
111#382—21no—
112#382—21no—
113#382C2×C411yescharacteristic
114#382C2×C412yes—
115#380—22no—
116#380—22no—
117#380C2×C412yes—
118#380C2×C412yes—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2×C411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C22⋊C4C2×C412yes—
127C4×C22⋊C4D412yes—
128C4×C22⋊C4D412yes—
129C2×#95—22no—
130C43C2311yescharacteristic
131#336—22no—
132#336—21no—
133#336—22no—
134#336—21no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2312yes—
140#336C2311yescharacteristic
141C22×D4C22⋊C411yescharacteristic
142C23×C4—22no—
143C23×C4—22no—
144C23×C4C22×C412yes—
145C23×C4C2×D412yes—
146C23×C4C2×D412yes—
147C2×M16—28no—
148C2×M16—82no—
149C2×M16—82no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156#125—22no—
157#125—41no—
158#125—81no—
159#125—81no—
160#125—82no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—42no—
164C4×D4—44no—
165C4×D4—44no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22⋊C411yescharacteristic
188C2×C4⋊C4C22⋊C411yescharacteristic
189C2×C4⋊C4C22⋊C411yescharacteristic
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C412yes—
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4C22⋊C412yes—
199C2×C22⋊C4C22⋊C412yes—
200C2×C22⋊C4C22⋊C412yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—82no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2411yescharacteristic
214C4⋊C8—24no—
215C4⋊C8—42no—
216C4⋊C8—82no—
217C22⋊C8—28no—
218C22⋊C8—82no—
219C22⋊C8—82no—
220#97—81no—
221#97—81no—
222C4×C8—24no—
223C4×C8—42no—
224C4×C8—82no—
225#95—22no—
226#95—22no—
227#95—21no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—21no—
237#95—22no—
238#95—21no—
239#95C22⋊C412yes—
240#95C22⋊C412yes—
241#95C22⋊C412yes—
242C24C2×C22⋊C412yes—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4#10414yes—
246C22×C4—21no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C412yes—
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—88no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—24no—
284C2×C8—41no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—41no—
298C2×C8—41no—
299C4⋊C4—21no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—41no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—82no—
321C4⋊C4—82no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C4⋊C4#10114yes—
329C4⋊C4#13112yes—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—42no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—22no—
343C42—22no—
344C42—42no—
345C42—42no—
346C42—41no—
347C42—42no—
348C42—42no—
349C42—42no—
350C23—22no—
351C23—22no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23C2×#10414yes—
357C23C22×C22⋊C411yescharacteristic
358D4—28no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—88no—
407C2×C4—164no—
408C2×C4C2×#10114yes—
409C2×C4C2×#10112yes—
410C2×C4C2×#10412yes—
411C2×C4#42112yes—
412C2×C4C2×#13111yescharacteristic
413C2×C4#56811yescharacteristic
414C2×C4#57611yescharacteristic
415C2×C4#57711yescharacteristic
416C2×C4#57711yescharacteristic
417C2×C4#57811yescharacteristic
418C8—42no—
419C8—44no—
420C8—42no—
421C8—81no—
422C8—82no—
423C8—81no—
424C22—24no—
425C22—24no—
426C22—24no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22#288011yescharacteristic
432C22#288311yescharacteristic
433C22C22×#10111yescharacteristic
434C22#289711yescharacteristic
435C22#289811yescharacteristic
436C22C22×#10411yescharacteristic
437C22C2×#42111yescharacteristic
438C4—21no—
439C4—21no—
440C4—24no—
441C4—21no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—42no—
449C4—42no—
450C4—42no—
451C4—81no—
452C4—82no—
453C4—82no—
454C4—81no—
455C4—81no—
456C4#152814yes—
457C2—44no—
458C2#1742611yescharacteristic
459C2#1751911yescharacteristic
460C2#1752211yescharacteristic
461C2#1754711yescharacteristic
462C2#1760011yescharacteristic
463C2C2×#152811yescharacteristic
464C2#1777711yescharacteristic
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. #7683C2semidirect
6. #7680C2semidirect
7. #7671C2semidirect
8. #7636C2semidirect
9. #7610C2semidirect
10. #7576C2semidirect
11. #7568C2semidirect

Maximal subgroups

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

subgroupindex
#362902
#220122
#205672
#77112
#76832
#76802
#76712
#76362
#76102
#75762
#75682