Finite Group Databasebeta

← #233160

Subgroups — #233160

Order 512.

1600 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3227C211yesmaximal, characteristic, complemented
2C2×#3205C211yesmaximal, characteristic
3C22×#486C211yesmaximal, characteristic, complemented
4#21187C211yesmaximal, characteristic, complemented
5C2×#1548C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic, complemented
7#10267C218yesmaximal, complemented
8#7872C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3227—24no—
13#3227—24no—
14#3205—24no—
15#3205—24no—
16C2×#498—21no—
17C2×#493—21no—
18C2×#486—22no—
19C2×#486—21no—
20C2×#486—24no—
21C2×#486—21no—
22C2×#486—24no—
23C2×#486C2214yes—
24C2×#481—22no—
25C2×#476C2214yescomplemented
26C2×#459C2214yes—
27C2×#447—22no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×#102—22no—
32C2×#408—22no—
33C22×C4×C8C2211yescharacteristic
34#1954—22no—
35#1954C2214yescomplemented
36#1548—24no—
37#1545—24no—
38C2×#333C2214yescomplemented
39C2×#331C2214yes—
40#1277C2214yescomplemented
41C22×#41—21no—
42C23×Q8—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4—21no—
46C22×D8—24no—
47C22×M16—21no—
48C23×C8D412yes—
49#532—22no—
50#532—22no—
51#532—22no—
52#532—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8D414yes—
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#127—22no—
63C2×#127—22no—
64C2×#127—21no—
65C2×#127—21no—
66C2×#127—22no—
67C2×#127—21no—
68C2×#127D411yescharacteristic
69C2×#127D414yes—
70C2×#127C2312yes—
71C2×#124—21no—
72C2×#121—21no—
73C2×#121D412yes—
74C2×C4×D4—21no—
75C2×C4×D4C2312yes—
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42C2311yescharacteristic
79#498—28no—
80#493—28no—
81#486—216no—
82#486—28no—
83#486—28no—
84#486—28no—
85#486—28no—
86#481—216no—
87#476—28no—
88#476—28no—
89#459—28no—
90#459—28no—
91#447—216no—
92#431—22no—
93#431—22no—
94#431D414yes—
95#431D414yes—
96C2×#106—24no—
97#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103C2×#102—216no—
104C2×#102—24no—
105C2×#102—28no—
106C2×#102—24no—
107#408—48no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×C22⋊C8—28no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8D412yes—
116C2×C4×C8C2312yes—
117#333—28no—
118#331—28no—
119C25—41no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41—44no—
123C2×#41—44no—
124C22×Q8—21no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—41no—
128C22×Q8—42no—
129C22×Q8—84no—
130C22×D4—22no—
131C22×D4—24no—
132C22×D4—24no—
133C22×D4—24no—
134C22×D4—21no—
135C22×D4—21no—
136C22×D4—22no—
137C22×D4—22no—
138C22×D4—22no—
139C22×D4—42no—
140C22×D4—42no—
141C22×D4—41no—
142C22×D4—41no—
143C22×D4—41no—
144C22×D4—42no—
145C22×D4—44no—
146C22×D4—44no—
147C22×D4—84no—
148C22×D4—84no—
149C22×D4C2×D412yes—
150C23×C4—22no—
151C23×C4—41no—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4#4111yescharacteristic
155C2×D8—216no—
156C2×D8—88no—
157C2×M16—42no—
158C2×M16—44no—
159C22×C8—24no—
160C22×C8—24no—
161C22×C8—24no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D414yes—
166C4⋊Q8—24no—
167C4⋊Q8—24no—
168C4⋊Q8—22no—
169C4⋊Q8—22no—
170C4⋊Q8—24no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—22no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8—42no—
177C4⋊Q8C2×D414yes—
178#127—24no—
179#127—24no—
180#127—22no—
181#127—22no—
182#127—24no—
183#127—22no—
184#127—22no—
185#127—22no—
186#127—42no—
187#127—42no—
188#127—42no—
189#127C2×D414yes—
190#124—42no—
191#124—42no—
192#121—24no—
193#121—24no—
194#121—42no—
195#121—42no—
196C4×D4—22no—
197C4×D4—22no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—42no—
201C4×D4—42no—
202C2×C4⋊C4—24no—
203C2×C4⋊C4—24no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42#4111yescharacteristic
221C2×C42C2411yescharacteristic
222#106—216no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228C4⋊C8C2×D414yes—
229C4⋊C8C2×D414yes—
230#102—232no—
231#102—216no—
232#102—216no—
233C22⋊C8—24no—
234C22⋊C8—24no—
235C22⋊C8—416no—
236C4×C8—24no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—22no—
240C4×C8—22no—
241C24—22no—
242C24—42no—
243C24—44no—
244C24—84no—
245C24—84no—
246C24—84no—
247C24#12111yescharacteristic
248#41—88no—
249#41—84no—
250#41—84no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—88no—
257C2×Q8—88no—
258C2×Q8—84no—
259C2×Q8—84no—
260C2×D4—22no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—24no—
265C2×D4—28no—
266C2×D4—44no—
267C2×D4—48no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—44no—
272C2×D4—44no—
273C2×D4—48no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—416no—
280C2×D4—44no—
281C2×D4—44no—
282C2×D4—88no—
283C2×D4—84no—
284C2×D4—84no—
285C2×D4—88no—
286C2×D4—88no—
287C2×D4—88no—
288C2×D4—84no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—42no—
296C22×C4—44no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—84no—
302C22×C4#12012yes—
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4#12112yes—
306C22×C4#12112yes—
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12712yes—
311C22×C4#13514yes—
312C22×C4AGL(1,ℤ/8)14yes—
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316D8—816no—
317M16—84no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—44no—
324C2×C8—44no—
325C2×C8—48no—
326C2×C8—42no—
327C2×C8—44no—
328C4⋊C4—24no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—42no—
339C22⋊C4—44no—
340C42#12012yes—
341C42#12012yes—
342C42#12012yes—
343C42#12112yes—
344C42#12112yes—
345C42#12712yes—
346C42C22×D411yescharacteristic
347C42C22×D411yescharacteristic
348C42C22×D411yescharacteristic
349C42C22×D411yescharacteristic
350C23—22no—
351C23—22no—
352C23—42no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—88no—
358C23—88no—
359C23—84no—
360C23—88no—
361C23—84no—
362C23—84no—
363C23—1616no—
364C23#39212yes—
365C23#46314yes—
366C23C2×#12111yescharacteristic
367Q8—84no—
368Q8—84no—
369Q8—84no—
370Q8—88no—
371D4—48no—
372D4—44no—
373D4—48no—
374D4—44no—
375D4—84no—
376D4—88no—
377D4—84no—
378D4—88no—
379D4—82no—
380D4—82no—
381D4—84no—
382D4—88no—
383D4—84no—
384D4—84no—
385D4—88no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—88no—
396C2×C4—84no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—84no—
400C2×C4#39212yes—
401C2×C4#39212yes—
402C2×C4#39212yes—
403C2×C4#44918yes—
404C2×C4#46314yes—
405C2×C4#47214yes—
406C2×C4#49514yes—
407C2×C4#49614yes—
408C2×C4C2×#12011yescharacteristic
409C2×C4C2×#12011yescharacteristic
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×#12111yescharacteristic
412C2×C4C2×#12111yescharacteristic
413C2×C4C2×#12711yescharacteristic
414C2×C4C2×#13512yes—
415C2×C4C2×AGL(1,ℤ/8)12yes—
416C2×C4#57714yes—
417C8—44no—
418C8—44no—
419C8—44no—
420C8—82no—
421C22—22no—
422C22—22no—
423C22—28no—
424C22—82no—
425C22—84no—
426C22—82no—
427C22—84no—
428C22—84no—
429C22—82no—
430C22—88no—
431C22—1616no—
432C22—168no—
433C22#201214yes—
434C22C2×#39211yescharacteristic
435C22C2×#46312yes—
436C22#305712yes—
437C22C2×#47212yes—
438C4—22no—
439C4—21no—
440C4—21no—
441C4—82no—
442C4—84no—
443C4#201214yes—
444C4#201814yes—
445C4C2×#44912yes—
446C4#300912yes—
447C4#301012yes—
448C4C2×#49511yescharacteristic
449C4C2×#49611yescharacteristic
450C2—24no—
451C2—162no—
452C2—164no—
453C2#1919614yes—
454C2#2338811yescharacteristic
455C2C2×#201211yescharacteristic
456C2C2×#201811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#3205C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3227C2semidirect
3. C22×#486C2semidirect
4. #21187C2semidirect
5. C2×#1548C2semidirect
6. C2×#1545C2semidirect
7. #10267C2semidirect
8. #7872C2semidirect
25. C2×#476C22semidirect
35. #1954C22semidirect
38. C2×#333C22semidirect
40. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32272
C2×#32052
C22×#4862
#211872
C2×#15482
C2×#15452
#102672
#78722