Finite Group Databasebeta

← #223278

Subgroups — #223278

Order 512.

877 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36194C211yesmaximal, characteristic
2#22650C211yesmaximal, characteristic
3#22650C211yesmaximal, characteristic
4#9898C211yesmaximal, characteristic, complemented
5#9896C211yesmaximal, characteristic, complemented
6#9139C212yesmaximal, complemented
7#9137C212yesmaximal, complemented
8#8729C212yesmaximal, complemented
9#8586C211yesmaximal, characteristic, complemented
10C4×#332C211yesmaximal, characteristic, complemented
11#8041C212yesmaximal, complemented
12C2×#436—21no—
13C2×#436—21no—
14C2×#423C2212yes—
15C2×#423C2212yes—
16#2868C2211yescharacteristic
17#2857C2212yes—
18#2571C412yes—
19#2563—21no—
20#2538—21no—
21#2497—22no—
22#2492—22no—
23#2439C414yes—
24#2369—21no—
25#2355—21no—
26#2354C412yes—
27C4×C4⋊Q8C2211yescharacteristic
28#2285C2212yes—
29C42×D4C2211yescharacteristic
30#1939—21no—
31#1939—21no—
32#1851—22no—
33#1851—22no—
34#1842C2211yescharacteristic
35#1842C2211yescharacteristic
36#1797C2212yes—
37#1797C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39C4×C4⋊C8C2211yescharacteristic
40C2×#332C2211yescharacteristic
41C2×#332C2211yescharacteristic
42C2×#332C2212yes—
43#1296C2211yescharacteristic
44#1296C2212yes—
45#1296C2212yes—
46#1296C2212yes—
47#1296C2211yescharacteristic
48#1296C2212yes—
49#1296C2212yes—
50C22×M16—22no—
51C22×M16—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C22⋊Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4D411yescharacteristic
60C2×C4×D4D411yescharacteristic
61C2×#117C2×C414yes—
62C22×C42C2312yes—
63#436—44no—
64#436—44no—
65#423—24no—
66#423—24no—
67#423—42no—
68#423—42no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2311yescharacteristic
76C2×C4⋊C8C2311yescharacteristic
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2311yescharacteristic
80C2×C4⋊C8C2311yescharacteristic
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85#400—24no—
86#396—24no—
87#393—24no—
88#391—21no—
89#391—22no—
90#391—24no—
91#390—22no—
92#389—22no—
93#389—21no—
94#389—22no—
95#389C2×C412yes—
96#389C2×C414yes—
97#388—24no—
98#387—24no—
99#384—21no—
100#384—21no—
101#384C2×C412yes—
102#384C2311yescharacteristic
103#384C2312yes—
104#382—22no—
105#382C2×C414yes—
106#380—22no—
107#380—22no—
108#380—22no—
109#380C2×C414yes—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4C2×C411yescharacteristic
116C4×C4⋊C4D411yescharacteristic
117C4×C4⋊C4D411yescharacteristic
118C4×C22⋊C4C2×C412yes—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121C43C2311yescharacteristic
122#336—22no—
123#336—22no—
124#336—21no—
125#336—21no—
126#332—22no—
127#332—22no—
128#332—42no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×D4C22⋊C411yescharacteristic
132C23×C4C22×C412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D412yes—
135C2×M16—44no—
136C2×M16—44no—
137C2×M16—44no—
138C2×M16—44no—
139C2×M16—82no—
140C2×M16—82no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—81no—
154C4⋊Q8—81no—
155C4⋊Q8—82no—
156C22⋊Q8—82no—
157C22⋊Q8—82no—
158C4×Q8—42no—
159C4×Q8—42no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165#117—84no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C411yescharacteristic
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C4⋊C4C22×C414yes—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4C22⋊C412yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C412yes—
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×D412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2411yescharacteristic
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—44no—
214C4⋊C8—44no—
215C4⋊C8—42no—
216C4⋊C8—42no—
217C4⋊C8—81no—
218C4⋊C8—81no—
219C4⋊C8#4112yes—
220C4⋊C8#4112yes—
221C22⋊C8—44no—
222C22⋊C8—44no—
223C4×C8—42no—
224C4×C8—42no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—22no—
229#95—22no—
230#95—24no—
231#95C22⋊C412yes—
232#95C22⋊C412yes—
233#95C22⋊C412yes—
234C24C2×C22⋊C412yes—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—41no—
241C2×Q8—44no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
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—24no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—42no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C411yescharacteristic
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C412yes—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C23×C411yescharacteristic
274C22×C4C22×D411yescharacteristic
275C22×C4C22×D411yescharacteristic
276M16—88no—
277M16—88no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8—41no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—41no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—41no—
310C4⋊C4—42no—
311C4⋊C4—41no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—84no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4#10114yes—
318C22⋊C4—24no—
319C22⋊C4—24no—
320C22⋊C4—24no—
321C22⋊C4—84no—
322C22⋊C4—84no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—22no—
334C42—41no—
335C42—44no—
336C42—84no—
337C42AGL(1,ℤ/8)12yes—
338C42AGL(1,ℤ/8)12yes—
339C42#13712yes—
340C42#13712yes—
341C42C2×#4111yescharacteristic
342C42C2×#4111yescharacteristic
343C42H(5,𝔽2)11yescharacteristic
344C422−1+411yescharacteristic
345C23—22no—
346C23—22no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—22no—
352C23C22×C22⋊C411yescharacteristic
353Q8—81no—
354Q8—81no—
355Q8—81no—
356Q8—81no—
357Q8—164no—
358D4—44no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—22no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4C2×#10114yes—
396C2×C4C2×#10112yes—
397C2×C4#41712yes—
398C2×C4#41712yes—
399C2×C4#41712yes—
400C2×C4#41712yes—
401C2×C4C4×#4111yescharacteristic
402C2×C4#52011yescharacteristic
403C2×C4#54611yescharacteristic
404C2×C4#54611yescharacteristic
405C2×C4#54711yescharacteristic
406C2×C4#54711yescharacteristic
407C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
408C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
409C2×C4C2×#13711yescharacteristic
410C2×C4C2×#13711yescharacteristic
411C2×C4#57611yescharacteristic
412C2×C4#57711yescharacteristic
413C2×C4#57711yescharacteristic
414C2×C4#57811yescharacteristic
415C8—42no—
416C8—42no—
417C8—81no—
418C8—81no—
419C8—82no—
420C8—82no—
421C22—42no—
422C22—42no—
423C22—42no—
424C22—42no—
425C22—42no—
426C22—42no—
427C22—42no—
428C22#233911yescharacteristic
429C22C22×#10111yescharacteristic
430C22#288811yescharacteristic
431C22C2×#41711yescharacteristic
432C22C2×#41711yescharacteristic
433C22#289711yescharacteristic
434C22#289811yescharacteristic
435C4—22no—
436C4—22no—
437C4—21no—
438C4—21no—
439C4—24no—
440C4—41no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—82no—
445C4—82no—
446C4#156114yes—
447C4#332411yescharacteristic
448C4#332511yescharacteristic
449C4#332811yescharacteristic
450C4#332911yescharacteristic
451C4#335311yescharacteristic
452C4#335411yescharacteristic
453C4#335611yescharacteristic
454C4#335711yescharacteristic
455C2—82no—
456C2C2×#156111yescharacteristic
457C2#1776211yescharacteristic
458C2#1776411yescharacteristic
459C2#1777711yescharacteristic
460C2#2248111yescharacteristic
461C2#2250311yescharacteristic
462C2#2250411yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9898C2semidirect
5. #9896C2semidirect
6. #9139C2semidirect
7. #9137C2semidirect
8. #8729C2semidirect
9. #8586C2semidirect
10. C4×#332C2semidirect
11. #8041C2semidirect

Maximal subgroups

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

subgroupindex
#361942
#226502
#98982
#98962
#91392
#91372
#87292
#85862
C4×#3322
#80412