Finite Group Databasebeta

← #164458

Subgroups — #164458

Order 512.

740 conjugacy classes of subgroups.

The subgroup lattice has 455 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36312C211yesmaximal, characteristic
2#22022C211yesmaximal, characteristic, complemented
3#20573C211yesmaximal, characteristic, complemented
4#7706C212yesmaximal
5#7683C211yesmaximal, characteristic, complemented
6#7680C211yesmaximal, characteristic, complemented
7#7672C212yesmaximal
8#7635C211yesmaximal, characteristic
9#7611C212yesmaximal, complemented
10#7577C212yesmaximal, complemented
11#7567C211yesmaximal, characteristic
12C2×#446—21no—
13#2860C2212yes—
14#2859C2211yescharacteristic
15#2582C411yescharacteristic
16#2538C411yescharacteristic
17#2529—22no—
18#2519—22no—
19#2508C412yes—
20#2496—21no—
21#2448C412yes—
22#2423—21no—
23#2362—21no—
24#2354C411yescharacteristic
25Q8×C4⋊C4C411yescharacteristic
26#2346—21no—
27C4×#124C2211yescharacteristic
28#2295C2212yes—
29C42×Q8C2211yescharacteristic
30#1984—21no—
31#1939—21no—
32#1917—22no—
33#1839C2212yes—
34#1831C2212yes—
35#1771—22no—
36C8×C4⋊C4—21no—
37#1767C2211yescharacteristic
38#1767C2211yescharacteristic
39C4×C4⋊C8C2212yes—
40#1745C2212yes—
41#1744C2211yescharacteristic
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2211yescharacteristic
45#1272C2212yes—
46#1271C2212yes—
47#1271C2212yes—
48#1271C2212yes—
49#1271C2212yes—
50#1270C2211yescharacteristic
51#1270C2212yes—
52#1270C2211yescharacteristic
53C2×#126—22no—
54C2×#125—21no—
55C2×#124C2×C411yescharacteristic
56C2×C22⋊Q8—21no—
57C2×C4×Q8C2×C412yes—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×Q8C2×C411yescharacteristic
60C2×C4×Q8D411yescharacteristic
61C2×C4×Q8D411yescharacteristic
62C2×C4×D4C2×C411yescharacteristic
63C2×#117—21no—
64#446—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—22no—
69C2×#97—22no—
70C2×#97—21no—
71C2×#97—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75#400—22no—
76#399—22no—
77#398—22no—
78#398—22no—
79#397—22no—
80#396—22no—
81#391—21no—
82#391C2×C412yes—
83#391C2311yescharacteristic
84#390C2×C412yes—
85#390C2×C412yes—
86#390C2311yescharacteristic
87#389—21no—
88#389—21no—
89#389—21no—
90#389—22no—
91#389C2×C412yes—
92#389C2×C412yes—
93#389C2×C412yes—
94#389C2×C412yes—
95#389C2312yes—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#388—21no—
101#388—21no—
102#388—22no—
103#388—22no—
104#387—22no—
105#387—22no—
106#387—22no—
107#384—21no—
108#384C2×C411yescharacteristic
109#384C2×C411yescharacteristic
110#382—21no—
111#382—22no—
112#382C2×C411yescharacteristic
113#380—21no—
114#380—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4C2×C412yes—
118C4×C4⋊C4C2×C411yescharacteristic
119C4×C4⋊C4C2×C412yes—
120C4×C4⋊C4D411yescharacteristic
121C4×C4⋊C4D412yes—
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4D412yes—
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126#376—22no—
127C43C2312yes—
128C43C2311yescharacteristic
129#336—22no—
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336—22no—
135#336C2311yescharacteristic
136#336C2312yes—
137#336C2311yescharacteristic
138#336C2311yescharacteristic
139#336C2311yescharacteristic
140#336C2312yes—
141C22×Q8C22⋊C411yescharacteristic
142C22×Q8C22×C411yescharacteristic
143C22×D4C22×C411yescharacteristic
144C23×C4—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151#126—82no—
152#126—82no—
153#125—81no—
154#125—81no—
155#124—22no—
156#124—41no—
157C22⋊Q8—44no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×Q8—42no—
164C4×D4—42no—
165#117—44no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22⋊C411yescharacteristic
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C411yescharacteristic
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C412yes—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C412yes—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—24no—
213C4⋊C8—28no—
214C4⋊C8—42no—
215C4⋊C8—82no—
216#97—28no—
217#97—81no—
218#97—81no—
219C4×C8—24no—
220C4×C8—42no—
221C4×C8—82no—
222#95—22no—
223#95—22no—
224#95—21no—
225#95—21no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—21no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—21no—
238C2×Q8—21no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8#13112yes—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—41no—
248C2×D4—41no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C22⋊C412yes—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C412yes—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C23×C411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C2×C8—22no—
269C2×C8—22no—
270C2×C8—24no—
271C2×C8—41no—
272C2×C8—41no—
273C2×C8—42no—
274C2×C8—41no—
275C2×C8—42no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—42no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—41no—
285C2×C8—41no—
286C4⋊C4—22no—
287C4⋊C4—24no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—81no—
304C4⋊C4—81no—
305C4⋊C4—82no—
306C4⋊C4—82no—
307C4⋊C4—82no—
308C4⋊C4#10414yes—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—44no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—82no—
319C22⋊C4—82no—
320C42—22no—
321C42—21no—
322C42—22no—
323C42—21no—
324C42—22no—
325C42—21no—
326C42—21no—
327C42—22no—
328C42—21no—
329C42—22no—
330C42—22no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—24no—
335C42—22no—
336C42—42no—
337C42—42no—
338C42—41no—
339C42—44no—
340C42—42no—
341C42—44no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23C22×C22⋊C411yescharacteristic
350Q8—42no—
351Q8—42no—
352Q8—44no—
353D4—81no—
354D4—81no—
355C2×C4—22no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
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—22no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—84no—
397C2×C4#41312yes—
398C2×C4C2×#10414yes—
399C2×C4C2×#10412yes—
400C2×C4#42112yes—
401C2×C4C2×#13111yescharacteristic
402C2×C4#56811yescharacteristic
403C2×C4#57611yescharacteristic
404C2×C4#57711yescharacteristic
405C2×C4#57711yescharacteristic
406C2×C4#57811yescharacteristic
407C8—42no—
408C8—44no—
409C8—42no—
410C8—81no—
411C8—82no—
412C8—81no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22#288011yescharacteristic
421C22#288411yescharacteristic
422C22C2×#41311yescharacteristic
423C22#289711yescharacteristic
424C22#289811yescharacteristic
425C22C22×#10411yescharacteristic
426C22C2×#42111yescharacteristic
427C4—21no—
428C4—21no—
429C4—24no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—41no—
438C4—41no—
439C4—42no—
440C4—42no—
441C4—81no—
442C4—82no—
443C4—82no—
444C4—81no—
445C4#152914yes—
446C2—161no—
447C2#1742311yescharacteristic
448C2#1751911yescharacteristic
449C2#1752611yescharacteristic
450C2#1755011yescharacteristic
451C2#1760111yescharacteristic
452C2C2×#152911yescharacteristic
453C2#1777211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22022C2semidirect
3. #20573C2semidirect
5. #7683C2semidirect
6. #7680C2semidirect
9. #7611C2semidirect
10. #7577C2semidirect

Maximal subgroups

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

subgroupindex
#363122
#220222
#205732
#77062
#76832
#76802
#76722
#76352
#76112
#75772
#75672