Finite Group Databasebeta

← #207830

Subgroups — #207830

Order 512.

1046 conjugacy classes of subgroups.

The subgroup lattice has 475 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2C4×#431C211yesmaximal, characteristic
3C4×#382C211yesmaximal, characteristic
4#9884C212yesmaximal
5#9679C212yesmaximal
6#9112C214yesmaximal
7C4×#328C214yesmaximal
8C2×#431C2212yes—
9#2864—21no—
10#2863—21no—
11#2863C412yescomplemented
12#2862C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2857C2211yescharacteristic
16#2855—21no—
17#2855C2212yes—
18#2851—21no—
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#125—21no—
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8C2212yes—
24#2295—21no—
25#2295C412yescomplemented
26#2283—21no—
27C2×#382—21no—
28C2×C4×C4⋊C4—21no—
29C2×C43C2211yescharacteristic
30C4×C4⋊C8—21no—
31C4×C4⋊C8—21no—
32C4×C22⋊C8C2212yes—
33C4×C22⋊C8C2212yes—
34#1744—21no—
35C42×C8—21no—
36C2×#328C2218yes—
37C4⋊#95C2212yes—
38#1434—21no—
39#1434C2212yes—
40C4×#95—21no—
41C4×#95C2212yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125—21no—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8C2312yes—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C22×C4⋊C4—22no—
51C22×C42—22no—
52C22×C42D411yescharacteristic
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59#431—28no—
60C2×C4⋊C8—22no—
61C2×C22⋊C8C2314yes—
62C2×C4×C8—22no—
63C2×C4×C8—22no—
64C2×C4×C8—22no—
65#391—21no—
66#391—21no—
67#391—21no—
68#391—21no—
69#391C2×C412yes—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#389—21no—
74#389—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389C2×C412yes—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#387—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#385—22no—
94#385—22no—
95#384C2×C411yescharacteristic
96#384C2×C411yescharacteristic
97#383—22no—
98#382—28no—
99#382—21no—
100#382—21no—
101#382—21no—
102#381—22no—
103#380—21no—
104#380—21no—
105#380—21no—
106#380C2312yes—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—21no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—44no—
113C4×C4⋊C4C2×C412yes—
114C4×C22⋊C4—22no—
115#376—22no—
116C2×#95—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95C2314yes—
121C43C2×C411yescharacteristic
122C43C2×C411yescharacteristic
123C43C2×C411yescharacteristic
124C43C2×C411yescharacteristic
125C43D411yescharacteristic
126C43D411yescharacteristic
127C43D411yescharacteristic
128C43D411yescharacteristic
129#336—22no—
130#336—22no—
131#328—216no—
132C22×Q8—21no—
133C22×Q8C22×C412yes—
134C23×C4—24no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—24no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145#125—44no—
146C22⋊Q8—28no—
147C22⋊Q8—44no—
148C4×Q8—44no—
149C4×Q8—44no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4C22×C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C42—24no—
172C2×C42—24no—
173C2×C42—24no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—42no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—44no—
182C2×C42C22⋊C411yescharacteristic
183C2×C42C22⋊C411yescharacteristic
184C2×C42C22⋊C411yescharacteristic
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×C42SD1614yes—
199C2×C42Q1614yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C4⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8#4118yes—
215C4×C8—28no—
216C4×C8—44no—
217C4×C8—44no—
218#95—28no—
219#95—28no—
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—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—28no—
235#95—44no—
236#95—44no—
237C24C22×D411yescharacteristic
238C2×Q8—24no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—28no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
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×SD1612yes—
284C22×C4C2×SD1612yes—
285C22×C4C2×SD1612yes—
286C22×C4C2×Q1612yes—
287C22×C4C2×Q1612yes—
288C22×C4C2×Q1612yes—
289C22×C4#13514yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C2×#4114yes—
293C2×C8—28no—
294C2×C8—44no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—416no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C42—22no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—41no—
333C42—41no—
334C42—41no—
335C42—41no—
336C42—41no—
337C42—41no—
338C42—44no—
339C42—44no—
340C42—88no—
341C42#10314yes—
342C42#10314yes—
343C42#10314yes—
344C42#10314yes—
345C42#10414yes—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23C22×C22⋊C411yescharacteristic
351C23#53212yes—
352C23C22×SD1611yescharacteristic
353C23C22×Q1611yescharacteristic
354C23C2×#13512yes—
355Q8—84no—
356Q8—84no—
357C2×C4—21no—
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—21no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—84no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4#41012yes—
413C2×C4#41012yes—
414C2×C4C2×#10312yes—
415C2×C4C2×#10312yes—
416C2×C4C2×#10312yes—
417C2×C4C2×#10312yes—
418C2×C4C2×#10312yes—
419C2×C4C2×#10312yes—
420C2×C4#41612yes—
421C2×C4#41612yes—
422C2×C4C2×#10412yes—
423C2×C4C2×#10412yes—
424C2×C4C2×#10412yes—
425C2×C4#42112yes—
426C2×C4C4×#4112yes—
427C2×C4#52512yes—
428C8—48no—
429C22—21no—
430C22—21no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—21no—
435C22—21no—
436C22—22no—
437C22#229711yescharacteristic
438C22C2×#41011yescharacteristic
439C22C2×#41011yescharacteristic
440C22C22×#10311yescharacteristic
441C22C2×#41611yescharacteristic
442C22C22×#10411yescharacteristic
443C22C2×#42111yescharacteristic
444C22#316712yes—
445C22#317012yes—
446C4—22no—
447C4—22no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—22no—
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—88no—
459C4—82no—
460C4—82no—
461C4—82no—
462C4—82no—
463C4#150314yes—
464C4#151814yes—
465C2—22no—
466C2—22no—
467C2C2×#150311yescharacteristic
468C2#1753311yescharacteristic
469C2#1756711yescharacteristic
470C2C2×#151811yescharacteristic
471C2#2138211yescharacteristic
472C2#2140011yescharacteristic
473C2#2142111yescharacteristic
subgroupG/H
centerC22×C4C2×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C22×C22⋊C4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
11. #2863C4semidirect
12. #2862C4semidirect
13. #2859C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect
25. #2295C4semidirect

Maximal subgroups

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

subgroupindex
#366352
C4×#4312
C4×#3822
#98842
#96792
#91122
C4×#3282