Finite Group Databasebeta

← #215289

Subgroups — #215289

Order 512.

1290 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#124C211yesmaximal, characteristic
2#20777C211yesmaximal, characteristic, complemented
3#20777C211yesmaximal, characteristic, complemented
4#9675C214yesmaximal, complemented
5#9645C214yesmaximal, complemented
6#8311C212yesmaximal, complemented
7#8310C212yesmaximal
8C22×#124C411yescharacteristic
9C22×C4×Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C4×#124—21no—
12C4×#124—21no—
13C4×#124—21no—
14C4×#124—21no—
15C4×#124C412yes—
16C4×#124C412yes—
17C4×#124C2212yes—
18C4×#124C2212yes—
19C2×#391C411yescharacteristic
20C2×#390C411yescharacteristic
21C2×#388—21no—
22C2×#388—21no—
23C2×#382C411yescharacteristic
24C2×C4×C22⋊C4—21no—
25C2×C4×C22⋊C4—21no—
26C2×C43C2211yescharacteristic
27#1839—21no—
28#1839—21no—
29#1831—21no—
30#1831C2212yescomplemented
31#1817—21no—
32#1817—21no—
33#1797—21no—
34#1797C2212yes—
35C4×C4⋊C8C2212yescomplemented
36C4×C4⋊C8C2212yescomplemented
37C4×C22⋊C8—21no—
38C4×C22⋊C8C2212yes—
39#1744—21no—
40#1744C2212yescomplemented
41C2×Q8⋊C8C2218yescomplemented
42C2×D4⋊C8C2218yes—
43C23×Q8C2×C411yescharacteristic
44C23×D4C2×C411yescharacteristic
45C24×C4—21no—
46C2×#124—28no—
47C2×#124—21no—
48C2×#124—21no—
49C2×#124C2×C412yes—
50C2×#124C2×C412yes—
51C2×C4×Q8—21no—
52C2×C4×Q8—21no—
53C2×C4×Q8—44no—
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2312yes—
56C2×C4×D4—21no—
57C2×C4×D4—21no—
58C2×C4×D4—44no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2312yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2×C411yescharacteristic
63C22×C4⋊C4C2×C411yescharacteristic
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C42—21no—
70C22×C42C2×C411yescharacteristic
71C22×C42C2×C411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42C2311yescharacteristic
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2314yes—
79C2×C22⋊C8—22no—
80C2×C22⋊C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8C2314yes—
83#391—21no—
84#391—21no—
85#391C2×C412yes—
86#391C2×C412yes—
87#390—21no—
88#390—21no—
89#390C2×C412yes—
90#390C2×C412yes—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#388—22no—
97#388—22no—
98#388—22no—
99#382—21no—
100#382—21no—
101#382C2×C412yes—
102#382C2×C412yes—
103C4×C22⋊C4—22no—
104C4×C22⋊C4—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C43C2×C411yescharacteristic
116C43C2×C411yescharacteristic
117C43D411yescharacteristic
118C43D411yescharacteristic
119C43D411yescharacteristic
120C43D411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130#336—22no—
131Q8⋊C8—48no—
132D4⋊C8—48no—
133C25—21no—
134C22×Q8—24no—
135C22×Q8—21no—
136C22×Q8—24no—
137C22×Q8—42no—
138C22×Q8C22×C412yes—
139C22×D4—24no—
140C22×D4—21no—
141C22×D4—24no—
142C22×D4—42no—
143C22×D4C22×C412yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—22no—
148C23×C4—22no—
149C23×C4—44no—
150C23×C4—44no—
151C23×C4—44no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22×C411yescharacteristic
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C22×C8—24no—
160C22×C8—24no—
161#124—28no—
162#124—28no—
163#124—24no—
164#124—24no—
165#124—24no—
166#124M1618yes—
167C4×Q8—28no—
168C4×Q8—44no—
169C4×Q8—44no—
170C4×Q8—44no—
171C4×D4—28no—
172C4×D4—44no—
173C4×D4—44no—
174C4×D4—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—44no—
209C2×C42C22⋊C411yescharacteristic
210C2×C42C22⋊C411yescharacteristic
211C2×C42C22⋊C411yescharacteristic
212C2×C42C22⋊C411yescharacteristic
213C2×C42C22⋊C411yescharacteristic
214C2×C42C22⋊C411yescharacteristic
215C2×C42C22⋊C411yescharacteristic
216C2×C42C22⋊C411yescharacteristic
217C2×C42C22⋊C411yescharacteristic
218C2×C42C22⋊C411yescharacteristic
219C2×C42C22⋊C411yescharacteristic
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C22×C411yescharacteristic
224C2×C42C22×C411yescharacteristic
225C2×C42C22×C411yescharacteristic
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2411yescharacteristic
237C4⋊C8—28no—
238C4⋊C8—44no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C4×C8—28no—
242C4×C8—44no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251C24—24no—
252C24—22no—
253C24—42no—
254C24—42no—
255C24—42no—
256C24—42no—
257C24—42no—
258C24—42no—
259C24C2×C22⋊C411yescharacteristic
260C2×Q8—24no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—416no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8C2×M1614yes—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—22no—
271C2×D4—24no—
272C2×D4—416no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4C2×M1614yes—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—44no—
312C22×C4—44no—
313C22×C4—816no—
314C22×C4C2×C22⋊C411yescharacteristic
315C22×C4C2×C22⋊C411yescharacteristic
316C22×C4C2×C22⋊C411yescharacteristic
317C22×C4C2×C22⋊C411yescharacteristic
318C22×C4C2×C22⋊C411yescharacteristic
319C22×C4C2×C22⋊C411yescharacteristic
320C22×C4C2×C22⋊C411yescharacteristic
321C22×C4C2×C22⋊C411yescharacteristic
322C22×C4C2×C22⋊C411yescharacteristic
323C22×C4C2×C22⋊C411yescharacteristic
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4#13114yes—
326C22×C4#13514yes—
327C22×C4C23×C411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C2×C8—44no—
331C2×C8—44no—
332C2×C8—44no—
333C2×C8—44no—
334C2×C8—44no—
335C2×C8—44no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—44no—
341C4⋊C4—44no—
342C22⋊C4—28no—
343C22⋊C4—28no—
344C22⋊C4—28no—
345C22⋊C4—28no—
346C22⋊C4—44no—
347C22⋊C4—44no—
348C22⋊C4—44no—
349C22⋊C4—44no—
350C22⋊C4—44no—
351C22⋊C4—44no—
352C42—22no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C42—22no—
366C42—22no—
367C42—22no—
368C42—22no—
369C42—22no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42—41no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—44no—
379C42—44no—
380C42C2×M1614yes—
381C42H(5,𝔽2)14yes—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—24no—
387C23—48no—
388C23—48no—
389C23—48no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—84no—
397C23—84no—
398C23—84no—
399C23—84no—
400C23#40812yes—
401C23#41612yes—
402C23#42112yes—
403C23C22×C22⋊C411yescharacteristic
404Q8—28no—
405Q8—48no—
406Q8—44no—
407D4—28no—
408D4—48no—
409D4—44no—
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—28no—
423C2×C4—28no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—24no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—48no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—88no—
458C2×C4—84no—
459C2×C4—84no—
460C2×C4—88no—
461C2×C4#40712yes—
462C2×C4#40712yes—
463C2×C4#40812yes—
464C2×C4#41612yes—
465C2×C4#42112yes—
466C2×C4#51812yes—
467C2×C4#53512yes—
468C2×C4#53512yes—
469C2×C4C22×M1612yes—
470C2×C4C2×#13112yes—
471C2×C4C2×#13512yes—
472C2×C4#57512yes—
473C8—84no—
474C8—84no—
475C22—22no—
476C22—22no—
477C22—22no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—44no—
482C22—44no—
483C22—44no—
484C22—816no—
485C22—82no—
486C22—82no—
487C22—82no—
488C22—82no—
489C22#148214yes—
490C22#230911yescharacteristic
491C22C2×#40711yescharacteristic
492C22C2×#40811yescharacteristic
493C22C2×#41611yescharacteristic
494C22#289611yescharacteristic
495C22C2×#42111yescharacteristic
496C22C2×#42111yescharacteristic
497C4—22no—
498C4—22no—
499C4—22no—
500C4—22no—
501C4—22no—
502C4—22no—
503C4—44no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—41no—
508C4—41no—
509C4—41no—
510C4—82no—
511C4—82no—
512C4#297414yes—
513C4#321114yes—
514C2—24no—
515C2—84no—
516C2#1744311yescharacteristic
517C2#1745211yescharacteristic
518C2C2×#148211yescharacteristic
519C2#1746511yescharacteristic
520C2#2178811yescharacteristic
521C2#2180611yescharacteristic
522C2#2181611yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20777C2semidirect
3. #20777C2semidirect
4. #9675C2semidirect
5. #9645C2semidirect
6. #8311C2semidirect
30. #1831C22semidirect
35. C4×C4⋊C8C22semidirect
36. C4×C4⋊C8C22semidirect
40. #1744C22semidirect
41. C2×Q8⋊C8C22semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1242
#207772
#96752
#96452
#83112
#83102