Finite Group Databasebeta

← #160789

Subgroups — #160789

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2295C211yesmaximal, characteristic
2#20778C211yesmaximal, characteristic
3C4×#432C211yesmaximal, characteristic
4#7629C212yesmaximal
5#7627C212yesmaximal
6#7611C214yesmaximal, complemented
7#7577C214yesmaximal, complemented
8C22×C4×Q8C411yescharacteristic
9C2×#432—21no—
10#2295—21no—
11#2295—21no—
12#2295—21no—
13#2295—21no—
14#2295C412yes—
15#2295C412yes—
16#2295C2212yes—
17#2295C2212yes—
18C2×#391C411yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#389C411yescharacteristic
21C2×#389C2211yescharacteristic
22C2×#387—21no—
23C2×#387—21no—
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C411yescharacteristic
26C2×#376—21no—
27C2×C43C2211yescharacteristic
28#1839C2212yes—
29#1831C2212yes—
30#1817—21no—
31#1797—21no—
32#1767—21no—
33#1767—21no—
34C4×C4⋊C8C2212yes—
35#1749C2212yes—
36#1749C2212yes—
37C4×C22⋊C8—21no—
38#1745—21no—
39#1745C2212yes—
40C4×#97—21no—
41#1272C2218yes—
42#1271C2218yes—
43C23×Q8C2×C411yescharacteristic
44C2×C4×Q8—28no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8C2×C412yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2×C411yescharacteristic
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C42—21no—
56C22×C42—21no—
57C22×C42C2×C411yescharacteristic
58C22×C42C2×C411yescharacteristic
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64#432—28no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×#97—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×C4×C8—22no—
73#391—21no—
74#391—21no—
75#391—21no—
76#391—21no—
77#391C2×C412yes—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2312yes—
81#389—21no—
82#389—21no—
83#389—21no—
84#389—21no—
85#389C2×C412yes—
86#389C2×C412yes—
87#389C2×C412yes—
88#389C2312yes—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105#376—22no—
106#376—22no—
107#376—22no—
108#376—22no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C43C2×C411yescharacteristic
117C43C2×C411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43C2311yescharacteristic
123C43C2311yescharacteristic
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336C2314yes—
129#336C2314yes—
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8C22×C412yes—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C22⋊C411yescharacteristic
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—24no—
147C22×C8—24no—
148C4×Q8—28no—
149C4×Q8—28no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—24no—
153C4×Q8M1618yes—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C42—28no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42—44no—
185C2×C42C22⋊C411yescharacteristic
186C2×C42C22⋊C411yescharacteristic
187C2×C42C22⋊C411yescharacteristic
188C2×C42C22⋊C411yescharacteristic
189C2×C42C22⋊C411yescharacteristic
190C2×C42C22⋊C411yescharacteristic
191C2×C42C22⋊C411yescharacteristic
192C2×C42C22⋊C411yescharacteristic
193C2×C42C22⋊C411yescharacteristic
194C2×C42C22⋊C411yescharacteristic
195C2×C42C22⋊C411yescharacteristic
196C2×C42C22⋊C411yescharacteristic
197C2×C42C22×C411yescharacteristic
198C2×C42C22×C411yescharacteristic
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2411yescharacteristic
213C4⋊C8—28no—
214C4⋊C8—44no—
215C22⋊C8—28no—
216C22⋊C8—44no—
217#97—28no—
218#97—44no—
219#97—44no—
220C4×C8—44no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235C24C2×C22⋊C411yescharacteristic
236C2×Q8—24no—
237C2×Q8—22no—
238C2×Q8—416no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8C2×M1614yes—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4#10414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C23×C411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C2×C8—28no—
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—44no—
299C2×C8—44no—
300C2×C8—44no—
301C4⋊C4—24no—
302C4⋊C4—28no—
303C4⋊C4—22no—
304C4⋊C4—28no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4C2×M1614yes—
314C42—28no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—28no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—41no—
331C42—41no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—44no—
343C42—44no—
344C42C2×M1614yes—
345C42#13714yes—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23#41312yes—
351C23C2×#10412yes—
352C23#42112yes—
353C23C22×C22⋊C411yescharacteristic
354Q8—44no—
355Q8—44no—
356Q8—48no—
357C2×C4—28no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—28no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—48no—
398C2×C4—48no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4#40712yes—
413C2×C4#40712yes—
414C2×C4#41012yes—
415C2×C4#41912yes—
416C2×C4#41912yes—
417C2×C4C2×#10412yes—
418C2×C4C2×#10412yes—
419C2×C4C22×M1612yes—
420C2×C4C2×#13712yes—
421C2×C4#57512yes—
422C8—48no—
423C8—84no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22#152914yes—
431C22C2×#40711yescharacteristic
432C22C2×#41011yescharacteristic
433C22C2×#41311yescharacteristic
434C22C2×#41911yescharacteristic
435C22#289611yescharacteristic
436C22C22×#10411yescharacteristic
437C22C2×#42111yescharacteristic
438C4—22no—
439C4—22no—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—41no—
446C4—44no—
447C4—41no—
448C4—41no—
449C4—41no—
450C4—44no—
451C4—41no—
452C4—41no—
453C4—82no—
454C4—82no—
455C4#149114yes—
456C2—24no—
457C2#1735711yescharacteristic
458C2C2×#149111yescharacteristic
459C2#1745211yescharacteristic
460C2#1753711yescharacteristic
461C2#1758411yescharacteristic
462C2C2×#152911yescharacteristic
463C2#1774811yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #7611C2semidirect
7. #7577C2semidirect

Maximal subgroups

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

subgroupindex
C2×#22952
#207782
C4×#4322
#76292
#76272
#76112
#75772