Finite Group Databasebeta

← #163454

Subgroups — #163454

Order 512.

1032 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2860C211yesmaximal, characteristic
2#20810C211yesmaximal, characteristic, complemented
3#20808C211yesmaximal, characteristic, complemented
4#7732C214yesmaximal, complemented
5#7686C214yesmaximal, complemented
6#7641C212yesmaximal, complemented
7#7640C212yesmaximal
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#1842—21no—
31#1831C2212yescomplemented
32#1831C2212yescomplemented
33#1797—21no—
34#1797—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1767—21no—
38#1767C2212yescomplemented
39C4×C4⋊C8—21no—
40C4×C4⋊C8C2212yescomplemented
41#1271C2218yescomplemented
42#1270C2218yes—
43C23×D4C2×C411yescharacteristic
44C2×#126—22no—
45C2×#126—22no—
46C2×#126—22no—
47C2×#126—22no—
48C2×#126—44no—
49C2×#126—44no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4C2×C411yescharacteristic
55C22×C4⋊C4C2×C411yescharacteristic
56C22×C22⋊C4—21no—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C42C2×C411yescharacteristic
63C22×C42C2×C411yescharacteristic
64C22×C42D411yescharacteristic
65C22×C42D411yescharacteristic
66C22×C42D411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42C2311yescharacteristic
69#432—44no—
70#431—44no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×C4×C8—22no—
81#390—21no—
82#390—21no—
83#390—21no—
84#390—21no—
85#390—21no—
86#390—21no—
87#390C2×C412yes—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2312yes—
93#389—21no—
94#389—21no—
95#389—21no—
96#389—21no—
97#389—21no—
98#389—21no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C412yes—
103#389C2×C412yes—
104#389C2312yes—
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—
116#388—22no—
117C2×#95—21no—
118C2×#95—21no—
119C2×#95—21no—
120C43C2×C411yescharacteristic
121C43C2×C411yescharacteristic
122C43D411yescharacteristic
123C43D411yescharacteristic
124C43D411yescharacteristic
125C43D411yescharacteristic
126C43C2311yescharacteristic
127C43C2311yescharacteristic
128#336—22no—
129#336—22no—
130#336C2314yes—
131#336C2314yes—
132C25—21no—
133C22×D4—21no—
134C22×D4—42no—
135C22×D4—42no—
136C22×D4—42no—
137C22×D4C22×C412yes—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4C22⋊C411yescharacteristic
142C23×C4C22⋊C411yescharacteristic
143C23×C4C22⋊C411yescharacteristic
144C23×C4C22⋊C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C22×C8—24no—
149C22×C8—24no—
150#126—48no—
151#126—48no—
152#126—44no—
153#126—44no—
154#126—44no—
155#126—44no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
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—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—42no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—42no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—42no—
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×C42D814yes—
215C2×C42SD1614yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C22×C411yescharacteristic
218C2×C42C22×C411yescharacteristic
219C2×C42C22×C411yescharacteristic
220C2×C42C22×C411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2411yescharacteristic
232C4⋊C8—44no—
233C4⋊C8—44no—
234C4⋊C8—44no—
235C4⋊C8—44no—
236C4⋊C8—44no—
237C4⋊C8—44no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240#97—44no—
241C4×C8—44no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248C24—22no—
249C24—42no—
250C24—42no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24C2×C22⋊C411yescharacteristic
257C2×D4—24no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—88no—
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—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4#10214yes—
294C22×C4#10214yes—
295C22×C4C2×C22⋊C411yescharacteristic
296C22×C4C2×C22⋊C411yescharacteristic
297C22×C4C2×C22⋊C411yescharacteristic
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C411yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4C2×C22⋊C411yescharacteristic
304C22×C4C2×C22⋊C411yescharacteristic
305C22×C4C2×C22⋊C411yescharacteristic
306C22×C4C2×D812yes—
307C22×C4C2×SD1612yes—
308C22×C4AGL(1,ℤ/8)12yes—
309C22×C4#13712yes—
310C22×C4C23×C411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C2×C8—44no—
314C2×C8—44no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—44no—
318C2×C8—44no—
319C4⋊C4—24no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—44no—
331C4⋊C4—44no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—44no—
342C22⋊C4—44no—
343C22⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—44no—
346C42—22no—
347C42—22no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
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—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42#10214yes—
373C42#10214yes—
374C42C2×D812yes—
375C42C2×D812yes—
376C42C2×SD1612yes—
377C42C2×SD1612yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—42no—
389C23—84no—
390C23—84no—
391C23—84no—
392C23—84no—
393C23—84no—
394C23—84no—
395C23—84no—
396C23#41312yes—
397C23C2×#10212yes—
398C23#41712yes—
399C23C22×C22⋊C411yescharacteristic
400D4—84no—
401D4—84no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
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—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4#41012yes—
454C2×C4C2×#10212yes—
455C2×C4C2×#10212yes—
456C2×C4C2×#10212yes—
457C2×C4C2×#10212yes—
458C2×C4C2×#10212yes—
459C2×C4#41712yes—
460C2×C4#42112yes—
461C2×C4#42112yes—
462C2×C4#56812yes—
463C2×C4C22×D811yescharacteristic
464C2×C4C22×SD1611yescharacteristic
465C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
466C2×C4C2×#13711yescharacteristic
467C8—84no—
468C8—84no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—82no—
479C22—82no—
480C22—82no—
481C22—82no—
482C22—168no—
483C22#154714yes—
484C22#287911yescharacteristic
485C22C2×#41011yescharacteristic
486C22C2×#41311yescharacteristic
487C22C22×#10211yescharacteristic
488C22C2×#41711yescharacteristic
489C22C2×#42111yescharacteristic
490C22C2×#42111yescharacteristic
491C4—22no—
492C4—22no—
493C4—22no—
494C4—22no—
495C4—22no—
496C4—21no—
497C4—21no—
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—22no—
503C4—22no—
504C4—22no—
505C4—41no—
506C4—41no—
507C4—82no—
508C4—82no—
509C4—82no—
510C2—24no—
511C2—162no—
512C2#1739811yescharacteristic
513C2#1748511yescharacteristic
514C2#1749211yescharacteristic
515C2#1757411yescharacteristic
516C2#1758111yescharacteristic
517C2#1765911yescharacteristic
518C2C2×#154711yescharacteristic
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. #20810C2semidirect
3. #20808C2semidirect
4. #7732C2semidirect
5. #7686C2semidirect
6. #7641C2semidirect
31. #1831C22semidirect
32. #1831C22semidirect
38. #1767C22semidirect
40. C4×C4⋊C8C22semidirect
41. #1271C22semidirect

Maximal subgroups

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

subgroupindex
C2×#28602
#208102
#208082
#77322
#76862
#76412
#76402