Finite Group Databasebeta

← #162260

Subgroups — #162260

Order 512.

1060 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2#20808C211yesmaximal, characteristic, complemented
3C4×#432C211yesmaximal, characteristic, complemented
4#7706C214yesmaximal, complemented
5#7686C214yesmaximal, complemented
6#7576C212yesmaximal, complemented
7#7572C212yesmaximal
8C22×#126—21no—
9C2×#432—21no—
10C2×#431—21no—
11#2860—21no—
12#2860—21no—
13#2860—21no—
14#2860—21no—
15#2860C412yes—
16#2860C412yes—
17#2860C2212yes—
18#2860C2212yes—
19C2×#390C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C411yescharacteristic
23C2×#389C411yescharacteristic
24C2×#389C2211yescharacteristic
25C2×#388—21no—
26C2×#388—21no—
27C2×#388—21no—
28C2×C43C2211yescharacteristic
29#1842—21no—
30#1831C2212yescomplemented
31#1797—21no—
32#1797C2212yes—
33#1767C2212yescomplemented
34C4×C4⋊C8—21no—
35C4×C4⋊C8—21no—
36C4×C4⋊C8C2212yescomplemented
37#1749—21no—
38C4×C22⋊C8C2212yes—
39#1745C2212yescomplemented
40C4×#97—21no—
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C2×#126—22no—
45C2×#126—22no—
46C2×#126—22no—
47C2×#126—22no—
48C2×#126—48no—
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×C22⋊C4—21no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C42C2×C411yescharacteristic
62C22×C42C2×C411yescharacteristic
63C22×C42D411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42C2311yescharacteristic
68#432—28no—
69#431—44no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8—22no—
76C2×#97—22no—
77C2×#97—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80#390—21no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390C2×C412yes—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2312yes—
92#389—21no—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389C2×C412yes—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2312yes—
104#388—22no—
105#388—22no—
106#388—22no—
107#388—22no—
108#388—22no—
109#388—22no—
110#388—22no—
111#388—22no—
112#388—22no—
113#388—22no—
114#388—22no—
115#388—22no—
116C2×#95—21no—
117C2×#95—21no—
118C2×#95—21no—
119C43C2×C411yescharacteristic
120C43C2×C411yescharacteristic
121C43D411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43C2311yescharacteristic
126C43C2311yescharacteristic
127#336—22no—
128#336—22no—
129#336C2314yes—
130#336C2314yes—
131C25—21no—
132C22×D4—21no—
133C22×D4—24no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4C22×C412yes—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4C22⋊C411yescharacteristic
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—24no—
148C22×C8—24no—
149#126—48no—
150#126—48no—
151#126—48no—
152#126—48no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—44no—
185C2×C22⋊C4—44no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42—42no—
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×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2411yescharacteristic
226C4⋊C8—28no—
227C4⋊C8—44no—
228C4⋊C8—44no—
229C4⋊C8—44no—
230C22⋊C8—28no—
231C22⋊C8—44no—
232#97—28no—
233#97—44no—
234C4×C8—44no—
235C4×C8—44no—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24—22no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24—42no—
250C24C2×C22⋊C411yescharacteristic
251C2×D4—24no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—88no—
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—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4#10414yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C411yescharacteristic
292C22×C4C2×C22⋊C411yescharacteristic
293C22×C4C2×C22⋊C411yescharacteristic
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4#13114yes—
301C22×C4AGL(1,ℤ/8)12yes—
302C22×C4#13712yes—
303C22×C4C23×C411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D411yescharacteristic
306C2×C8—28no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C2×C8—44no—
312C4⋊C4—24no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C22⋊C4—44no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—44no—
328C22⋊C4—42no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—44no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C42—22no—
338C42—22no—
339C42—21no—
340C42—21no—
341C42—22no—
342C42—21no—
343C42—21no—
344C42—22no—
345C42—22no—
346C42—22no—
347C42—22no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42AGL(1,ℤ/8)14yes—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—84no—
379C23—84no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—84no—
385C23#40812yes—
386C23#41712yes—
387C23C2×#10412yes—
388C23C22×C22⋊C411yescharacteristic
389D4—84no—
390D4—84no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—28no—
410C2×C4—28no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—84no—
439C2×C4—84no—
440C2×C4—84no—
441C2×C4#40812yes—
442C2×C4#41012yes—
443C2×C4#41312yes—
444C2×C4#41712yes—
445C2×C4#41812yes—
446C2×C4#41812yes—
447C2×C4C2×#10412yes—
448C2×C4C2×#10412yes—
449C2×C4#42112yes—
450C2×C4C2×#13112yes—
451C2×C4C2×AGL(1,ℤ/8)12yes—
452C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
453C2×C4C2×#13711yescharacteristic
454C8—48no—
455C8—84no—
456C22—22no—
457C22—22no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—82no—
463C22—82no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—82no—
468C22—82no—
469C22—168no—
470C22#148314yes—
471C22C2×#40811yescharacteristic
472C22C2×#41011yescharacteristic
473C22C2×#41311yescharacteristic
474C22C2×#41811yescharacteristic
475C22C2×#41711yescharacteristic
476C22C22×#10411yescharacteristic
477C22C2×#42111yescharacteristic
478C4—22no—
479C4—22no—
480C4—22no—
481C4—22no—
482C4—22no—
483C4—21no—
484C4—21no—
485C4—22no—
486C4—22no—
487C4—22no—
488C4—41no—
489C4—41no—
490C4—41no—
491C4—41no—
492C4—82no—
493C4—82no—
494C4—82no—
495C4#147314yes—
496C2—24no—
497C2—162no—
498C2C2×#147311yescharacteristic
499C2C2×#148311yescharacteristic
500C2#1747811yescharacteristic
501C2#1754411yescharacteristic
502C2#1758111yescharacteristic
503C2#1764211yescharacteristic
504C2#1774511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×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
2. #20808C2semidirect
3. C4×#432C2semidirect
4. #7706C2semidirect
5. #7686C2semidirect
6. #7576C2semidirect
30. #1831C22semidirect
33. #1767C22semidirect
36. C4×C4⋊C8C22semidirect
39. #1745C22semidirect
41. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
#208082
C4×#4322
#77062
#76862
#75762
#75722