Finite Group Databasebeta

← #162442

Subgroups — #162442

Order 512.

724 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35206C211yesmaximal, characteristic
2#22835C211yesmaximal, characteristic
3#22008C211yesmaximal, characteristic
4#7731C212yesmaximal, complemented
5#7721C212yesmaximal, complemented
6#7671C212yesmaximal, complemented
7#7661C212yesmaximal, complemented
8#7588C211yesmaximal, characteristic, complemented
9#7577C211yesmaximal, characteristic, complemented
10#7570C211yesmaximal, characteristic, complemented
11#7562C211yesmaximal, characteristic, complemented
12#2855C2212yes—
13#2851C2212yes—
14#2652—22no—
15#2568C411yescharacteristic
16#2552C411yescharacteristic
17#2547C411yescharacteristic
18#2538C411yescharacteristic
19#2515—22no—
20#2511—21no—
21#2495C412yes—
22#2489C412yes—
23#2447—21no—
24#2326—21no—
25#2320—21no—
26C4×C4⋊Q8C2211yescharacteristic
27#2295C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1954—21no—
30#1954—21no—
31#1939—21no—
32#1923—22no—
33#1923—22no—
34#1831C2211yescharacteristic
35#1817C2212yes—
36#1817C2212yes—
37#1797C2212yes—
38C8×C4⋊C4—21no—
39C4×C4⋊C8C2211yescharacteristic
40C4×C4⋊C8C2211yescharacteristic
41C4×C22⋊C8C2212yes—
42#1744C2211yescharacteristic
43#1272C2211yescharacteristic
44#1272C2212yes—
45#1272C2211yescharacteristic
46#1271C2212yes—
47#1271C2212yes—
48#1271C2211yescharacteristic
49#1271C2212yes—
50#1271C2212yes—
51#1271C2211yescharacteristic
52#1271C2212yes—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×#125C2×C412yes—
55C2×#124—21no—
56C2×C22⋊Q8C2×C412yes—
57C2×C4×Q8—21no—
58C2×C4×Q8C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×C4×D4D411yescharacteristic
62C22×C42C2312yes—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8—21no—
74C2×C4×C8—22no—
75#400—22no—
76#400—22no—
77#400—22no—
78#398—22no—
79#397—22no—
80#397—22no—
81#397—22no—
82#396—22no—
83#391—21no—
84#391—21no—
85#391—21no—
86#391—21no—
87#391C2×C412yes—
88#391C2×C411yescharacteristic
89#391C2311yescharacteristic
90#390—21no—
91#389—21no—
92#389—21no—
93#389—21no—
94#389C2×C412yes—
95#389C2×C411yescharacteristic
96#389C2312yes—
97#389C2311yescharacteristic
98#388—22no—
99#388—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#387—22no—
104#387—21no—
105#387—21no—
106#385—22no—
107#385—22no—
108#385—22no—
109#384—21no—
110#384C2×C411yescharacteristic
111#383—22no—
112#382—22no—
113#382C2×C412yes—
114#381—22no—
115#380C2×C412yes—
116#380C2×C412yes—
117#380C2×C412yes—
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4C2×C412yes—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4D411yescharacteristic
124C4×C4⋊C4D411yescharacteristic
125C4×C22⋊C4C2×C412yes—
126C4×C22⋊C4D412yes—
127C4×C22⋊C4D412yes—
128#376—21no—
129C43C2311yescharacteristic
130#336—22no—
131#336—22no—
132#336—21no—
133#336—21no—
134#336C2311yescharacteristic
135#336C2312yes—
136#336C2311yescharacteristic
137#336C2311yescharacteristic
138#336C2312yes—
139#336C2311yescharacteristic
140C22×Q8—21no—
141C22×Q8C22×C411yescharacteristic
142C22×D4C22⋊C411yescharacteristic
143C23×C4C22×C412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D412yes—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C4⋊Q8—24no—
153#125—82no—
154#124—44no—
155C22⋊Q8—82no—
156C4×Q8—24no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×D4—24no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—42no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22×C411yescharacteristic
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C22×C411yescharacteristic
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—82no—
193C2×C42C22×C412yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—42no—
204C4⋊C8—42no—
205C4⋊C8—42no—
206C4⋊C8—82no—
207C22⋊C8—44no—
208C22⋊C8—82no—
209C22⋊C8—82no—
210C22⋊C8—82no—
211C4×C8—42no—
212C4×C8—42no—
213C4×C8—81no—
214C4×C8—82no—
215#95—21no—
216#95—21no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235C24C2×C22⋊C412yes—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C2×C22⋊C411yescharacteristic
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C23×C411yescharacteristic
270C22×C4C22×D411yescharacteristic
271C22×C4C22×D411yescharacteristic
272C2×C8—22no—
273C2×C8—41no—
274C2×C8—41no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—41no—
284C2×C8—41no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—41no—
289C2×C8—41no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—24no—
295C4⋊C4—41no—
296C4⋊C4—42no—
297C4⋊C4—41no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C4⋊C4—82no—
308C4⋊C4—82no—
309C4⋊C4—82no—
310C4⋊C4—82no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C22⋊C4—82no—
318C42—21no—
319C42—21no—
320C42—22no—
321C42—42no—
322C42—41no—
323C42—42no—
324C42—41no—
325C42—42no—
326C42—41no—
327C42—42no—
328C42—44no—
329C42—42no—
330C42#13114yes—
331C42#13512yes—
332C42#13512yes—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23C22×C22⋊C411yescharacteristic
341Q8—82no—
342Q8—84no—
343D4—44no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—41no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—84no—
389C2×C4#40812yes—
390C2×C4#40812yes—
391C2×C4#41612yes—
392C2×C4#41612yes—
393C2×C4#42112yes—
394C2×C4C2×#13112yes—
395C2×C4C2×#13511yescharacteristic
396C2×C4C2×#13511yescharacteristic
397C2×C4#57711yescharacteristic
398C2×C4#57811yescharacteristic
399C8—42no—
400C8—81no—
401C8—82no—
402C8—81no—
403C8—81no—
404C8—82no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—42no—
412C22C2×#40811yescharacteristic
413C22#288311yescharacteristic
414C22#288911yescharacteristic
415C22C2×#41611yescharacteristic
416C22#289811yescharacteristic
417C22C2×#42111yescharacteristic
418C22#290111yescharacteristic
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—22no—
426C4—41no—
427C4—42no—
428C4—42no—
429C4—42no—
430C4—81no—
431C4—82no—
432C4—81no—
433C4—81no—
434C4—82no—
435C4—81no—
436C4#148214yes—
437C2—82no—
438C2#1738811yescharacteristic
439C2C2×#148211yescharacteristic
440C2#1748411yescharacteristic
441C2#1760011yescharacteristic
442C2#1761711yescharacteristic
443C2#1768211yescharacteristic
444C2#1772511yescharacteristic
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. #7731C2semidirect
5. #7721C2semidirect
6. #7671C2semidirect
7. #7661C2semidirect
8. #7588C2semidirect
9. #7577C2semidirect
10. #7570C2semidirect
11. #7562C2semidirect

Maximal subgroups

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

subgroupindex
#352062
#228352
#220082
#77312
#77212
#76712
#76612
#75882
#75772
#75702
#75622