Finite Group Databasebeta

← #207445

Subgroups — #207445

Order 512.

1048 conjugacy classes of subgroups.

The subgroup lattice has 469 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#533C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic
3C4×#423C211yesmaximal, characteristic
4#9869C212yesmaximal, complemented
5#9677C212yesmaximal, complemented
6#8942C214yesmaximal
7#8171C214yesmaximal, complemented
8C2×#533C411yescharacteristic
9C2×#423C2212yes—
10C2×C4×M16—21no—
11C4×C4⋊Q8C411yescharacteristic
12C4×C4⋊Q8C411yescharacteristic
13C4×C4⋊Q8C2211yescharacteristic
14C4×C4⋊Q8C2211yescharacteristic
15C4×#125—21no—
16C4×#125—21no—
17C4×C22⋊Q8—21no—
18C4×C22⋊Q8—21no—
19C4×C22⋊Q8C412yes—
20C4×C22⋊Q8C2212yes—
21#2296—21no—
22#2295—21no—
23#2295C412yes—
24#2285C411yescharacteristic
25C42×Q8—21no—
26C42×Q8C2212yes—
27C4×#117—21no—
28C2×C43C2211yescharacteristic
29#1839—21no—
30#1831—21no—
31#1817—21no—
32#1797—21no—
33#1767—21no—
34#1767C2212yes—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yes—
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2212yes—
40#1728—21no—
41C2×Q8⋊C8C2218yes—
42#1279C2218yes—
43C22×M16—22no—
44#533—28no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2312yes—
56C2×#117—21no—
57C2×#117—21no—
58C22×C42C2×C411yescharacteristic
59C22×C42C2×C411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42C2311yescharacteristic
65#423—28no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—22no—
68C2×C4⋊C8C2314yes—
69C2×C22⋊C8—22no—
70C4×M16—44no—
71C2×#97—22no—
72C2×C4×C8—22no—
73C2×C4×C8C2314yes—
74#391—21no—
75#391—21no—
76#391C2×C412yes—
77#391C2×C412yes—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389—21no—
84#389C2×C412yes—
85#389C2×C412yes—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#385—22no—
91#385—22no—
92#384C2×C411yescharacteristic
93#384C2×C411yescharacteristic
94#382—21no—
95#380—21no—
96#380C2×C412yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115#376—22no—
116C43—22no—
117C43C2×C411yescharacteristic
118C43C2×C411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131Q8⋊C8—216no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C23×C4C22⋊C411yescharacteristic
135C23×C4C22⋊C411yescharacteristic
136C23×C4C22⋊C411yescharacteristic
137C23×C4C22⋊C411yescharacteristic
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—28no—
142C2×M16—44no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148#125—28no—
149C22⋊Q8—28no—
150C22⋊Q8—24no—
151C22⋊Q8—24no—
152C4×Q8—28no—
153C4×Q8—24no—
154C4×Q8—28no—
155C4×Q8—28no—
156C4×Q8—44no—
157C4×Q8—42no—
158C4×Q8—42no—
159C4×Q8M1618yescomplemented
160#117—28no—
161#117—44no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—42no—
193C2×C42—42no—
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×C42C22⋊C411yescharacteristic
203C2×C42C22⋊C411yescharacteristic
204C2×C42C22⋊C411yescharacteristic
205C2×C42C22⋊C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2411yescharacteristic
222C4⋊C8—28no—
223C4⋊C8—24no—
224C4⋊C8—44no—
225C4⋊C8#4118yes—
226C22⋊C8—44no—
227#97—44no—
228C4×C8—28no—
229C4×C8—44no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240C24C2×C22⋊C411yescharacteristic
241C2×Q8—24no—
242C2×Q8—22no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8C2×M1614yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—21no—
251C22×C4—21no—
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—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C411yescharacteristic
280C22×C4C2×C22⋊C411yescharacteristic
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4#13714yes—
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288M16—416no—
289C2×C8—28no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—28no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—28no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4C2×M1614yes—
312C22⋊C4—28no—
313C22⋊C4—44no—
314C42—22no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—21no—
320C42—22no—
321C42—22no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—28no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—41no—
337C42—41no—
338C42—41no—
339C42—41no—
340C42—44no—
341C42—44no—
342C42—44no—
343C42—42no—
344C42—42no—
345C42—44no—
346C42C2×M1614yes—
347C42#13114yes—
348C42AGL(1,ℤ/8)12yes—
349C42#13712yes—
350C42C2×#4114yes—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23#41912yes—
358C23C22×C22⋊C411yescharacteristic
359Q8—28no—
360Q8—48no—
361Q8—44no—
362C2×C4—28no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—21no—
386C2×C4—24no—
387C2×C4—24no—
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—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4#40712yes—
409C2×C4#40712yes—
410C2×C4#40812yes—
411C2×C4#40812yes—
412C2×C4#41712yes—
413C2×C4#41712yes—
414C2×C4#41912yes—
415C2×C4#42112yes—
416C2×C4#42112yes—
417C2×C4C4×#4112yes—
418C2×C4#52512yes—
419C2×C4#53212yes—
420C2×C4C22×M1612yes—
421C2×C4C2×#13112yes—
422C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
423C2×C4C2×#13712yes—
424C2×C4C2×#13711yescharacteristic
425C8—48no—
426C8—84no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22#229711yescharacteristic
435C22C2×#40711yescharacteristic
436C22C2×#40811yescharacteristic
437C22C2×#41911yescharacteristic
438C22C2×#41711yescharacteristic
439C22C2×#42111yescharacteristic
440C22C2×#42111yescharacteristic
441C4—22no—
442C4—22no—
443C4—22no—
444C4—22no—
445C4—22no—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—44no—
452C4—41no—
453C4—41no—
454C4—44no—
455C4—82no—
456C4—82no—
457C4#296514yes—
458C4#311012yes—
459C4#311312yes—
460C2—42no—
461C2#1744911yescharacteristic
462C2#1745111yescharacteristic
463C2#1747911yescharacteristic
464C2#1748111yescharacteristic
465C2#2137411yescharacteristic
466C2#2138311yescharacteristic
467C2#2139411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9869C2semidirect
5. #9677C2semidirect
7. #8171C2semidirect
159. C4×Q8M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5332
#207772
C4×#4232
#98692
#96772
#89422
#81712