Finite Group Databasebeta

← #190313

Subgroups — #190313

Order 512.

1139 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34126C211yesmaximal, characteristic
2C2×C4×#100C211yesmaximal, characteristic
3#17364C211yesmaximal, characteristic
4#8773C214yesmaximal, complemented
5#8772C214yesmaximal, complemented
6#8068C212yesmaximal, complemented
7#7974C212yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2920C2212yes—
10C22×#100—21no—
11C2×C4×M16C2212yes—
12#2357C411yescharacteristic
13#2357C411yescharacteristic
14#2353C2212yes—
15#2353C2212yes—
16#2349C411yescharacteristic
17#2349C411yescharacteristic
18#2345—21no—
19#2342—22no—
20D4×C22⋊C4—21no—
21#2293—21no—
22#2290—22no—
23#2283C2211yescharacteristic
24#2280C412yes—
25C42×D4—21no—
26C4×#117C2211yescharacteristic
27C2×C4×C22⋊C4C412yes—
28C4×#100—22no—
29C4×#100C2214yes—
30C2×#343—21no—
31#1466C2214yes—
32#1462—22no—
33#1460—21no—
34#1460—21no—
35#1300C2214yescomplemented
36#1300C2214yescomplemented
37#1288C2218yescomplemented
38C23×D4D411yescharacteristic
39C24×C4C2×C412yes—
40C22×M16—22no—
41C2×C4×D4—21no—
42C2×C4×D4—21no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C411yescharacteristic
48C2×C4×D4C2×C411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D411yescharacteristic
52C2×C4×D4C2312yes—
53C2×#117—21no—
54C2×#117—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C22⋊C4—22no—
57C22×C22⋊C4D412yes—
58C22×C42—22no—
59C22×C42C2311yescharacteristic
60#432—22no—
61#432C2314yes—
62#431—22no—
63#431—22no—
64#423—22no—
65C2×#100—28no—
66C2×#100—24no—
67#407—22no—
68C4×M16—22no—
69C4×M16—22no—
70C4×M16—22no—
71C4×M16—22no—
72C4×M16C2314yes—
73C2×#97—22no—
74C2×C4×C8—22no—
75#390—21no—
76#390—21no—
77#389—21no—
78#389—21no—
79#388—22no—
80#388—22no—
81#388—22no—
82#388—22no—
83#387—22no—
84#387—22no—
85#386—22no—
86#386—22no—
87#385—22no—
88#385—22no—
89#383C2312yes—
90#382C2×C411yescharacteristic
91#382C2×C411yescharacteristic
92#381—21no—
93#381C2×C412yes—
94#381C2×C412yes—
95#380—22no—
96#380—21no—
97#380C2×C412yes—
98#380C2×C412yes—
99#379—22no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4—21no—
104C4×C4⋊C4C2×C411yescharacteristic
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C22⋊C4—22no—
107C4×C22⋊C4—22no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—21no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4C2×C412yes—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4C2×C411yescharacteristic
115C4×C22⋊C4C2×C412yes—
116C4×C22⋊C4C2×C411yescharacteristic
117#376—21no—
118#376C2311yescharacteristic
119#376C2311yescharacteristic
120C2×#95C2×C412yes—
121C2×#95C2×C412yes—
122C43—21no—
123#343—24no—
124#323—24no—
125#323—24no—
126C25C22⋊C412yes—
127C22×D4—24no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—42no—
131C22×D4C22⋊C411yescharacteristic
132C22×D4C22⋊C411yescharacteristic
133C22×D4C2×D412yes—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4—84no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22⋊C412yes—
144C23×C4C22×C412yes—
145C23×C4C22×C412yes—
146C23×C4C22×C411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C2×M16—28no—
150C2×M16—24no—
151C2×M16—44no—
152C2×M16—42no—
153C2×M16—42no—
154C22×C8—42no—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—24no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—44no—
168C4×D4—44no—
169#117—44no—
170#117—84no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C411yescharacteristic
180C2×C4⋊C4C22⋊C411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—21no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—48no—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C2×D412yes—
197C2×C22⋊C4C2×D412yes—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—84no—
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4112yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—42no—
218#100—216no—
219C22⋊C8—24no—
220C22⋊C8—42no—
221#97—24no—
222#97—24no—
223#97—42no—
224C4×C8—24no—
225C4×C8—42no—
226C4×C8—42no—
227#95—22no—
228#95—22no—
229#95—21no—
230#95—21no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95C22×C411yescharacteristic
238#95C22×C411yescharacteristic
239#95C22×C412yes—
240#95C22×C412yes—
241#95C22×C411yescharacteristic
242#95C22×C411yescharacteristic
243C24—22no—
244C24—22no—
245C24—24no—
246C24—42no—
247C24—44no—
248C24—82no—
249C24C2×C22⋊C411yescharacteristic
250C24C2×C22⋊C412yes—
251C24C2×C22⋊C412yes—
252C2×D4—24no—
253C2×D4—24no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C2×D4—48no—
260C2×D4—48no—
261C2×D4—42no—
262C2×D4#10414yes—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
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—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—84no—
290C22×C4—84no—
291C22×C4—84no—
292C22×C4—84no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—82no—
296C22×C4—82no—
297C22×C4—82no—
298C22×C4C2×C22⋊C411yescharacteristic
299C22×C4C2×C22⋊C412yes—
300C22×C4C2×C22⋊C412yes—
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C22⋊C411yescharacteristic
303C22×C4#11711yescharacteristic
304C22×C4#11711yescharacteristic
305C22×C4#11711yescharacteristic
306C22×C4#11711yescharacteristic
307C22×C4#12312yes—
308C22×C4#12312yes—
309C22×C4#12312yes—
310C22×C4#12312yes—
311C22×C4C23×C411yescharacteristic
312C22×C4C22×D411yescharacteristic
313C22×C4C22×D411yescharacteristic
314M16—28no—
315M16—44no—
316M16—48no—
317M16—84no—
318C2×C8—48no—
319C2×C8—44no—
320C2×C8—44no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—22no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—82no—
329C4⋊C4—82no—
330C22⋊C4—24no—
331C22⋊C4—24no—
332C22⋊C4—24no—
333C22⋊C4—24no—
334C22⋊C4—24no—
335C22⋊C4—44no—
336C22⋊C4—44no—
337C22⋊C4—44no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C22⋊C4—44no—
341C22⋊C4—82no—
342C22⋊C4—82no—
343C42—42no—
344C42—44no—
345C42—42no—
346C42—42no—
347C42—42no—
348C42—42no—
349C42—42no—
350C42—82no—
351C42—82no—
352C42#11712yes—
353C42#11712yes—
354C42#12312yes—
355C42#12312yes—
356C42#12312yes—
357C42#12312yes—
358C42C2×#4112yes—
359C42C2×#4112yes—
360C23—22no—
361C23—21no—
362C23—22no—
363C23—21no—
364C23—22no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—41no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—42no—
375C23—42no—
376C23—44no—
377C23—82no—
378C23—82no—
379C23—88no—
380C23—88no—
381C23#38111yescharacteristic
382C23#38111yescharacteristic
383C23C2×#10414yes—
384C23C22×C22⋊C411yescharacteristic
385D4—28no—
386D4—48no—
387D4—44no—
388C2×C4—24no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—41no—
425C2×C4—42no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—88no—
435C2×C4—88no—
436C2×C4#38112yes—
437C2×C4#38111yescharacteristic
438C2×C4#38112yes—
439C2×C4#38111yescharacteristic
440C2×C4C2×#10412yes—
441C2×C4#42112yes—
442C2×C4C2×#11711yescharacteristic
443C2×C4C2×#11711yescharacteristic
444C2×C4C2×#12312yes—
445C2×C4C2×#12312yes—
446C8—48no—
447C8—84no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—24no—
454C22—21no—
455C22—21no—
456C22—42no—
457C22—42no—
458C22—44no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—42no—
464C22—42no—
465C22—88no—
466C22—82no—
467C22—82no—
468C22—88no—
469C22#182712yes—
470C22C2×#38111yescharacteristic
471C22C22×#10411yescharacteristic
472C22C2×#42111yescharacteristic
473C4—22no—
474C4—22no—
475C4—22no—
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—42no—
481C4—42no—
482C4—42no—
483C4—81no—
484C4—81no—
485C4—82no—
486C4—82no—
487C4#182614yes—
488C4#182912yes—
489C2—21no—
490C2—21no—
491C2—22no—
492C2—41no—
493C2—44no—
494C2—82no—
495C2C2×#182611yescharacteristic
496C2C2×#182711yescharacteristic
497C2C2×#182911yescharacteristic
subgroupG/H
centerC2×C4#381
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#381
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8773C2semidirect
5. #8772C2semidirect
6. #8068C2semidirect
7. #7974C2semidirect
35. #1300C22semidirect
36. #1300C22semidirect
37. #1288C22semidirect

Maximal subgroups

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

subgroupindex
#341262
C2×C4×#1002
#173642
#87732
#87722
#80682
#79742