Finite Group Databasebeta

← #207659

Subgroups — #207659

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 491 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#540C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20472C211yesmaximal, characteristic, complemented
4#9872C211yesmaximal, characteristic, complemented
5#9871C211yesmaximal, characteristic, complemented
6#9642C211yesmaximal, characteristic, complemented
7#9638C211yesmaximal, characteristic, complemented
8#9078C212yesmaximal, complemented
9#9076C212yesmaximal, complemented
10#8174C212yesmaximal, complemented
11#8172C212yesmaximal, complemented
12C2×#540C411yescharacteristic
13C2×#431C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#2335—22no—
16#2333—22no—
17#2330C411yescharacteristic
18#2328C411yescharacteristic
19#2324C412yes—
20#2319C412yes—
21#2314C411yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#123—22no—
25C4×#121—22no—
26C4×#121C2212yes—
27C42×D4C2212yes—
28C4×#117C2211yescharacteristic
29#1924—22no—
30#1917—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1771—22no—
35#1767C2212yes—
36#1753—22no—
37#1744C2211yescharacteristic
38#1744C2211yescharacteristic
39C4×#97C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50#540—24no—
51#540—24no—
52C2×#127C2×C411yescharacteristic
53C2×#125C2×C411yescharacteristic
54C2×#123—22no—
55C2×#121—22no—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2312yes—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66#431—22no—
67#431—22no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—22no—
74C4×M16—22no—
75C4×M16—22no—
76C2×#97—21no—
77C2×#97—22no—
78C2×#97—22no—
79C2×C4×C8—22no—
80C2×C4×C8C2311yescharacteristic
81C2×C4×C8C2311yescharacteristic
82C2×C4×C8C2311yescharacteristic
83C2×C4×C8C2311yescharacteristic
84#390—22no—
85#390—22no—
86#390C2×C412yes—
87#390C2×C412yes—
88#389—22no—
89#389C2×C412yes—
90#389C2×C412yes—
91#388—22no—
92#388—22no—
93#388—22no—
94#388—22no—
95#388—22no—
96#387—22no—
97#387—22no—
98#387—22no—
99#386—22no—
100#386—22no—
101#386—22no—
102#386—22no—
103#385—22no—
104#384C2×C411yescharacteristic
105#382C2×C411yescharacteristic
106#381—22no—
107#380—22no—
108#380C2×C412yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C4⋊C4D412yes—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4D412yes—
120C4×C22⋊C4D412yes—
121#376C2×C411yescharacteristic
122#376C2×C411yescharacteristic
123C43C2311yescharacteristic
124C43C2311yescharacteristic
125#336—22no—
126#336—22no—
127#336—22no—
128#336—22no—
129#336—22no—
130D4⋊C8—24no—
131D4⋊C8—24no—
132D4⋊C8—24no—
133D4⋊C8—24no—
134C22×D4—22no—
135C22×D4—22no—
136C22×D4C22×C412yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—22no—
140C23×C4C22×C411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C2×M16—24no—
144C2×M16—44no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151#127—22no—
152#127M1614yescomplemented
153#125—22no—
154#125M1614yescomplemented
155#123—216no—
156#121—216no—
157#121—24no—
158#121—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—42no—
166C4×D4—42no—
167C4×D4—82no—
168C4×D4—82no—
169C4×D4—82no—
170C4×D4—82no—
171#117—22no—
172#117—22no—
173#117—41no—
174#117—41no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4C22⋊C412yes—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22×C412yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
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⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—82no—
203C2×C42—82no—
204C2×C42C22×C411yescharacteristic
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C411yescharacteristic
208C2×C42C22×C411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—24no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—82no—
222C22⋊C8—24no—
223C22⋊C8—82no—
224#97—24no—
225#97—44no—
226#97—82no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—44no—
232C4×C8#4112yes—
233C4×C8#4112yes—
234C4×C8#4112yes—
235C4×C8#4112yes—
236#95—22no—
237#95—22no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95C22⋊C412yes—
244#95C22⋊C412yes—
245#95C22⋊C412yes—
246#95C22⋊C412yes—
247C24—22no—
248C24C2×C22⋊C411yescharacteristic
249C2×D4—28no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—28no—
256C2×D4—82no—
257C2×D4—82no—
258C2×D4—82no—
259C2×D4—82no—
260C2×D4C2×M1614yes—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—42no—
278C22×C4—44no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—82no—
288C22×C4—82no—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C2×C22⋊C412yes—
292C22×C4C2×C22⋊C412yes—
293C22×C4C2×C22⋊C412yes—
294C22×C4C2×C22⋊C411yescharacteristic
295C22×C4C2×C22⋊C412yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299M16—48no—
300C2×C8—24no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C4⋊C4—22no—
315C4⋊C4—28no—
316C4⋊C4—24no—
317C4⋊C4—22no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4C2×M1614yes—
326C22⋊C4—28no—
327C22⋊C4—28no—
328C22⋊C4—24no—
329C22⋊C4—42no—
330C22⋊C4—44no—
331C22⋊C4—42no—
332C22⋊C4—42no—
333C22⋊C4—82no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C22⋊C4—82no—
338C42—21no—
339C42—21no—
340C42—21no—
341C42—21no—
342C42—21no—
343C42—21no—
344C42—21no—
345C42—21no—
346C42—21no—
347C42—21no—
348C42—21no—
349C42—21no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—22no—
354C42—42no—
355C42—42no—
356C42—42no—
357C42—42no—
358C42—82no—
359C42—82no—
360C42C2×M1612yes—
361C42C2×M1612yes—
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C42C2×#4111yescharacteristic
365C42C2×#4111yescharacteristic
366C23—21no—
367C23—24no—
368C23—21no—
369C23—22no—
370C23—22no—
371C23—21no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23C22×C22⋊C411yescharacteristic
379D4—24no—
380D4—24no—
381D4—42no—
382D4—42no—
383D4—88no—
384D4—88no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—88no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4#40714yes—
429C2×C4C4×#4111yescharacteristic
430C2×C4C4×#4111yescharacteristic
431C2×C4#52511yescharacteristic
432C2×C4#52511yescharacteristic
433C2×C4#53211yescharacteristic
434C2×C4#53211yescharacteristic
435C2×C4C22×M1611yescharacteristic
436C2×C4C22×M1611yescharacteristic
437C2×C4#56811yescharacteristic
438C2×C4#56811yescharacteristic
439C2×C4#57612yes—
440C2×C4#57712yes—
441C8—24no—
442C8—42no—
443C8—42no—
444C8—42no—
445C8—82no—
446C8—82no—
447C22—44no—
448C22—42no—
449C22—44no—
450C22—42no—
451C22—41no—
452C22—44no—
453C22—41no—
454C22—41no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22#229711yescharacteristic
460C22C2×#40711yescharacteristic
461C22#287911yescharacteristic
462C22#289712yes—
463C22#290112yes—
464C4—24no—
465C4—21no—
466C4—21no—
467C4—42no—
468C4—44no—
469C4—42no—
470C4—42no—
471C4—41no—
472C4—41no—
473C4—41no—
474C4—82no—
475C4—82no—
476C4#297011yescharacteristic
477C4#297011yescharacteristic
478C4#297111yescharacteristic
479C4#297111yescharacteristic
480C4#317411yescharacteristic
481C4#317611yescharacteristic
482C4#317912yes—
483C2—81no—
484C2—84no—
485C2#1745312yes—
486C2#1750512yes—
487C2#2138011yescharacteristic
488C2#2141411yescharacteristic
489C2#2141911yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#540C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20472C2semidirect
4. #9872C2semidirect
5. #9871C2semidirect
6. #9642C2semidirect
7. #9638C2semidirect
8. #9078C2semidirect
9. #9076C2semidirect
10. #8174C2semidirect
11. #8172C2semidirect
152. #127M16semidirect
154. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#219262
#204722
#98722
#98712
#96422
#96382
#90782
#90762
#81742
#81722