Finite Group Databasebeta

← #214674

Subgroups — #214674

Order 512.

1096 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34211C211yesmaximal, characteristic
2#17369C211yesmaximal, characteristic
3#17350C211yesmaximal, characteristic
4#9103C212yesmaximal, complemented
5#9099C212yesmaximal, complemented
6#8961C212yesmaximal, complemented
7#8957C212yesmaximal, complemented
8#8731C212yesmaximal, complemented
9#8323C212yesmaximal, complemented
10C22×#124C411yescharacteristic
11C2×#519C2211yescharacteristic
12C2×#518C2211yescharacteristic
13#2975C2212yes—
14#2972C2212yes—
15#2367—22no—
16#2367—22no—
17#2342—22no—
18#2342—22no—
19#2326C411yescharacteristic
20#2326C411yescharacteristic
21#2326C2211yescharacteristic
22#2326C2211yescharacteristic
23#2310C411yescharacteristic
24#2309C411yescharacteristic
25C4×#124C411yescharacteristic
26C4×#124C411yescharacteristic
27C4×#124C2211yescharacteristic
28C4×#124C2211yescharacteristic
29#2285C411yescharacteristic
30C4×#117C2211yescharacteristic
31#1479C2214yescomplemented
32#1479C2214yescomplemented
33#1465—22no—
34#1462—22no—
35#1460C2212yes—
36#1460C2212yes—
37#1459—22no—
38#1458C2212yes—
39#1458C2212yes—
40C2×#323—22no—
41#1276C2214yescomplemented
42#1274C2214yescomplemented
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C2×#124—22no—
46C2×#124—24no—
47C2×#124—24no—
48C2×#124C2×C411yescharacteristic
49C2×#124C2×C411yescharacteristic
50C2×#124C2×C411yescharacteristic
51C2×#124C2×C411yescharacteristic
52#519—44no—
53#518—44no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8C2311yescharacteristic
59C2×C4×Q8C2311yescharacteristic
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×C4×D4D412yes—
64C2×C4×D4C2311yescharacteristic
65C2×C4×D4C2311yescharacteristic
66C2×#117—21no—
67C2×#117—21no—
68C2×#117C2×C411yescharacteristic
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C22⋊C4—22no—
73C22×C22⋊C4D412yes—
74C22×C22⋊C4D412yes—
75C22×C42C2×C411yescharacteristic
76#436—44no—
77C8×D4—44no—
78#431C2314yes—
79#431C2314yes—
80#407—24no—
81C2×C22⋊C8—24no—
82#391—22no—
83#391C2×C411yescharacteristic
84#391C2×C411yescharacteristic
85#391C2×C411yescharacteristic
86#391C2×C411yescharacteristic
87#390—22no—
88#390C2×C411yescharacteristic
89#390C2×C411yescharacteristic
90#390C2×C411yescharacteristic
91#390C2×C411yescharacteristic
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#388—22no—
100#385—22no—
101#385—22no—
102#385—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#384C2×C411yescharacteristic
106#382C2×C411yescharacteristic
107#382C2×C411yescharacteristic
108#381—22no—
109#380—22no—
110#379—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376C2×C411yescharacteristic
121#376C2311yescharacteristic
122C43C2×C411yescharacteristic
123C43C2311yescharacteristic
124Q8⋊C8—44no—
125Q8⋊C8—44no—
126#324—22no—
127#324—22no—
128#323—22no—
129#323—24no—
130#323—22no—
131#323—24no—
132C25C22⋊C412yes—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8C22×C411yescharacteristic
139C22×Q8C22×C411yescharacteristic
140C22×D4—22no—
141C22×D4—22no—
142C22×D4—24no—
143C22×D4—42no—
144C22×D4C22⋊C412yes—
145C22×D4C22×C411yescharacteristic
146C22×D4C22×C411yescharacteristic
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C22⋊C412yes—
150C23×C4C22⋊C412yes—
151C23×C4C22×C411yescharacteristic
152C23×C4C2×D412yes—
153C23×C4C2×D412yes—
154C2×M16—44no—
155C22×C8—44no—
156#124—216no—
157#124—22no—
158#124—22no—
159#124—22no—
160#124—22no—
161#124—22no—
162#124—22no—
163#124M1614yescomplemented
164#124M1614yescomplemented
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—24no—
172C4×D4—24no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—44no—
176C4×D4—44no—
177#117—24no—
178#117—24no—
179#117—44no—
180#117—44no—
181#117—42no—
182#117—42no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4C22×C411yescharacteristic
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C22⋊C4—216no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C2×D411yescharacteristic
213C2×C22⋊C4C2×D411yescharacteristic
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×D412yes—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42C22⋊C412yes—
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2411yescharacteristic
235C4⋊C8—24no—
236C4⋊C8—24no—
237C22⋊C8—24no—
238C22⋊C8—24no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C4×C8—24no—
242C4×C8—24no—
243#95—21no—
244#95—21no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253C24—24no—
254C24—24no—
255C24—42no—
256C24—82no—
257C24—82no—
258C24C2×C22⋊C412yes—
259C24C2×C22⋊C412yes—
260C24C2×C22⋊C411yescharacteristic
261C2×Q8—24no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—22no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—48no—
269C2×Q8—48no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8C2×M1612yes—
273C2×Q8C2×M1612yes—
274C2×D4—24no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—22no—
280C2×D4—22no—
281C2×D4—24no—
282C2×D4—48no—
283C2×D4—48no—
284C2×D4—42no—
285C2×D4C2×M1612yes—
286C2×D4C2×M1612yes—
287C22×C4—22no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—24no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—81no—
307C22×C4—81no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—82no—
311C22×C4—82no—
312C22×C4C2×C22⋊C411yescharacteristic
313C22×C4C2×C22⋊C412yes—
314C22×C4C2×C22⋊C412yes—
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4#13114yes—
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321M16—164no—
322C2×C8—44no—
323C2×C8—44no—
324C2×C8—164no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C22⋊C4—28no—
339C22⋊C4—28no—
340C22⋊C4—28no—
341C22⋊C4—28no—
342C22⋊C4—22no—
343C22⋊C4—22no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—42no—
351C22⋊C4—82no—
352C22⋊C4—82no—
353C22⋊C4—82no—
354C22⋊C4—82no—
355C42—24no—
356C42—22no—
357C42—21no—
358C42—21no—
359C42—22no—
360C42—22no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—81no—
368C42—81no—
369C42—81no—
370C42—81no—
371C42C2×M1612yes—
372C42C2×M1612yes—
373C42H(5,𝔽2)14yes—
374C23—21no—
375C23—22no—
376C23—21no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—22no—
381C23—48no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23—88no—
389C23—162no—
390C23—162no—
391C23—164no—
392C23#40714yes—
393C23#40812yes—
394C23C22×C22⋊C411yescharacteristic
395Q8—24no—
396Q8—24no—
397Q8—48no—
398Q8—42no—
399Q8—42no—
400D4—24no—
401D4—24no—
402D4—48no—
403D4—42no—
404D4—42no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—48no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—84no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4—162no—
446C2×C4—162no—
447C2×C4#40812yes—
448C2×C4#51812yes—
449C2×C4#53412yes—
450C2×C4#53412yes—
451C2×C4C22×M1611yescharacteristic
452C2×C4C22×M1611yescharacteristic
453C2×C4C2×#13112yes—
454C8—84no—
455C8—84no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—24no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22—42no—
465C22—42no—
466C22—41no—
467C22—41no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—82no—
472C22—82no—
473C22—168no—
474C22#230811yescharacteristic
475C22C2×#40711yescharacteristic
476C22C2×#40811yescharacteristic
477C4—22no—
478C4—22no—
479C4—22no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—42no—
487C4—42no—
488C4—81no—
489C4—81no—
490C4—81no—
491C4—81no—
492C4#297412yes—
493C4#297412yes—
494C2—21no—
495C2—21no—
496C2—42no—
497C2—44no—
498C2—82no—
499C2#2174011yescharacteristic
500C2#2178411yescharacteristic
501C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9103C2semidirect
5. #9099C2semidirect
6. #8961C2semidirect
7. #8957C2semidirect
8. #8731C2semidirect
9. #8323C2semidirect
31. #1479C22semidirect
32. #1479C22semidirect
41. #1276C22semidirect
42. #1274C22semidirect
163. #124M16semidirect
164. #124M16semidirect

Maximal subgroups

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

subgroupindex
#342112
#173692
#173502
#91032
#90992
#89612
#89572
#87312
#83232