Finite Group Databasebeta

← #207253

Subgroups — #207253

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic
3C42×C4⋊C4C211yesmaximal, characteristic
4#9910C214yesmaximal
5#9909C212yesmaximal
6#9908C212yesmaximal
7#8021C214yesmaximal
8C2×#432C2212yes—
9#2864C412yescomplemented
10#2863—21no—
11#2863—21no—
12#2862C411yescharacteristic, complemented
13#2859C411yescharacteristic, complemented
14#2857C2211yescharacteristic
15#2857C2211yescharacteristic
16#2855—21no—
17#2855—21no—
18#2851C2212yes—
19C4×C4⋊Q8C411yescharacteristic, complemented
20C4×C4⋊Q8C411yescharacteristic, complemented
21C4×#125C412yescomplemented
22C4×C22⋊Q8—21no—
23C4×C22⋊Q8—21no—
24#2295—21no—
25#2295—21no—
26#2283C2212yes—
27C2×C4×C4⋊C4—21no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C43—21no—
31C2×C43C2211yescharacteristic
32#1842—21no—
33#1831—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767—21no—
37#1767—21no—
38C4×#95—21no—
39C4×#95C2212yes—
40C4×#95C2212yes—
41#1294C2218yes—
42#1294C2218yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125C2×C412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8—21no—
50C22×C4⋊C4—22no—
51C22×C42—22no—
52C22×C42—22no—
53C22×C42—22no—
54C22×C42D411yescharacteristic
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42D411yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C22×C42C2311yescharacteristic
61#432—28no—
62C2×C4⋊C8—22no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C22⋊C8C2314yes—
66C2×#97—22no—
67#391—21no—
68#391—21no—
69#391—21no—
70#391—21no—
71#391—21no—
72#391—21no—
73#391—21no—
74#391—21no—
75#389—21no—
76#389—21no—
77#389—21no—
78#389—21no—
79#389—21no—
80#389—21no—
81#389—21no—
82#389—21no—
83#389C2×C412yes—
84#389C2×C412yes—
85#389C2×C412yes—
86#389C2×C412yes—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#387—22no—
94#387—22no—
95#385—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#384C2×C411yescharacteristic
99#383—22no—
100#382C2×C412yes—
101#382C2×C412yes—
102#382C2×C412yes—
103#381—22no—
104#380—21no—
105#380—21no—
106#380C2312yes—
107#380C2312yes—
108C4×C4⋊C4—28no—
109C4×C4⋊C4—28no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—24no—
115C4×C4⋊C4C2×C412yes—
116C4×C4⋊C4C2×C412yes—
117C4×C4⋊C4Q818yescomplemented
118C4×C22⋊C4—22no—
119#376—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95C2314yes—
123C43—44no—
124C43C2×C411yescharacteristic
125C43C2×C411yescharacteristic
126C43C2×C411yescharacteristic
127C43C2×C411yescharacteristic
128C43D411yescharacteristic
129C43D411yescharacteristic
130C43D411yescharacteristic
131C43D411yescharacteristic
132#336—22no—
133#336—22no—
134C22×Q8—21no—
135C22×Q8—21no—
136C23×C4—24no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—24no—
145C4⋊Q8—42no—
146C4⋊Q8—42no—
147#125—44no—
148C22⋊Q8—44no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C4×Q8—44no—
152C4×Q8—44no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
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—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—28no—
179C2×C42—28no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—28no—
183C2×C42—42no—
184C2×C42—42no—
185C2×C42—42no—
186C2×C42—44no—
187C2×C42—44no—
188C2×C42—44no—
189C2×C42—44no—
190C2×C42—44no—
191C2×C42—44no—
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×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×C42C22×C411yescharacteristic
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×Q814yes—
221C2×C42#4114yes—
222C4⋊C8—44no—
223C4⋊C8—44no—
224C4⋊C8—44no—
225C22⋊C8—28no—
226#97—44no—
227#95—28no—
228#95—28no—
229#95—22no—
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—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—24no—
244#95C2×Q818yes—
245C24C22×D411yescharacteristic
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—28no—
264C22×C4—28no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—44no—
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×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C22⋊Q814yes—
295C22×C4C22⋊Q814yes—
296C22×C4AGL(1,ℤ/8)12yes—
297C22×C4#13712yes—
298C22×C4C23×C411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q812yes—
301C22×C4C2×#4112yes—
302C2×C8—44no—
303C2×C8—44no—
304C2×C8—44no—
305C4⋊C4—232no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—44no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C22⋊C4—42no—
323C22⋊C4—42no—
324C22⋊C4—44no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—22no—
331C42—22no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—41no—
340C42—41no—
341C42—41no—
342C42—41no—
343C42—41no—
344C42—41no—
345C42—41no—
346C42—41no—
347C42—416no—
348C42—416no—
349C42—416no—
350C42—44no—
351C42—44no—
352C42#10414yes—
353C42#10414yes—
354C42C4×Q814yes—
355C42C4×Q814yes—
356C42#12412yes—
357C42#12412yes—
358C42C4⋊Q812yes—
359C42C4⋊Q812yes—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23C22×C22⋊C411yescharacteristic
365C23C2×C22⋊Q812yes—
366C23C2×AGL(1,ℤ/8)11yescharacteristic
367C23C2×#13711yescharacteristic
368C23#57512yes—
369Q8—84no—
370Q8—84no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
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—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—21no—
394C2×C4—21no—
395C2×C4—28no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—216no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—48no—
410C2×C4—48no—
411C2×C4—48no—
412C2×C4—48no—
413C2×C4—48no—
414C2×C4—48no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4#39112yes—
428C2×C4#39112yes—
429C2×C4#39112yes—
430C2×C4#39112yes—
431C2×C4#41012yes—
432C2×C4#41012yes—
433C2×C4#41712yes—
434C2×C4#41712yes—
435C2×C4C2×#10412yes—
436C2×C4C2×#10412yes—
437C2×C4C2×#10412yes—
438C2×C4C2×#10412yes—
439C2×C4C2×#10412yes—
440C2×C4C2×#10412yes—
441C2×C4C2×C4×Q812yes—
442C2×C4C2×#12411yescharacteristic
443C2×C4C2×C4⋊Q811yescharacteristic
444C8—84no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—22no—
450C22—22no—
451C22—21no—
452C22—21no—
453C22C2×#39111yescharacteristic
454C22C2×#41011yescharacteristic
455C22C2×#41011yescharacteristic
456C22C2×#41711yescharacteristic
457C22#289611yescharacteristic
458C22C22×#10411yescharacteristic
459C22C22×#10411yescharacteristic
460C22#307712yes—
461C22#307812yes—
462C4—22no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—22no—
467C4—21no—
468C4—22no—
469C4—22no—
470C4—22no—
471C4—22no—
472C4—22no—
473C4—416no—
474C4—41no—
475C4—41no—
476C4—82no—
477C4—82no—
478C4—82no—
479C4—82no—
480C4#150914yes—
481C4#199414yes—
482C2—22no—
483C2—22no—
484C2C2×#150911yescharacteristic
485C2#1753711yescharacteristic
486C2#1754311yescharacteristic
487C2#1754611yescharacteristic
488C2#2131211yescharacteristic
489C2C2×#199411yescharacteristic
490C2#2133211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
9. #2864C4semidirect
12. #2862C4semidirect
13. #2859C4semidirect
19. C4×C4⋊Q8C4semidirect
20. C4×C4⋊Q8C4semidirect
21. C4×#125C4semidirect
117. C4×C4⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#366352
#208102
C42×C4⋊C42
#99102
#99092
#99082
#80212