Finite Group Databasebeta

← #233153

Subgroups — #233153

Order 512.

1632 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3206C211yesmaximal, characteristic, complemented
2C2×#3205C211yesmaximal, characteristic
3C22×#486C211yesmaximal, characteristic, complemented
4#21187C211yesmaximal, characteristic, complemented
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1545C211yesmaximal, characteristic, complemented
7#10261C218yesmaximal, complemented
8#7872C211yesmaximal, characteristic, complemented
9C2×#532C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12#3206—24no—
13#3206—24no—
14#3205—24no—
15#3205—24no—
16C2×#493—21no—
17C2×#492—21no—
18C2×#487—21no—
19C2×#486—22no—
20C2×#486—21no—
21C2×#486—24no—
22C2×#486—24no—
23C2×#486C2214yes—
24C2×#461C2214yescomplemented
25C2×#459C2214yes—
26C2×#448—22no—
27C2×#447—22no—
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C2×#423C2211yescharacteristic
31C22×#102—22no—
32C2×#408—22no—
33C22×C4×C8C2211yescharacteristic
34#1954—22no—
35#1954C2214yescomplemented
36#1549—24no—
37#1545—24no—
38C2×#332C2214yescomplemented
39C2×#331C2214yes—
40#1277C2214yescomplemented
41C22×#41—21no—
42C23×Q8—21no—
43C23×D4—21no—
44C23×D4—21no—
45C23×D4—21no—
46C22×SD16—24no—
47C22×D8—24no—
48C22×M16—21no—
49C23×C8D412yes—
50#532—22no—
51#532—22no—
52#532—22no—
53#532—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8D414yes—
61C2×C4⋊Q8D411yescharacteristic
62C2×C4⋊Q8C2312yes—
63C2×#127—22no—
64C2×#127—22no—
65C2×#127—21no—
66C2×#127—21no—
67C2×#127—22no—
68C2×#127—21no—
69C2×#127D411yescharacteristic
70C2×#127D414yes—
71C2×#127C2312yes—
72C2×#124—21no—
73C2×#121—21no—
74C2×#121D412yes—
75C2×C4×D4—21no—
76C2×C4×D4C2312yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C42C2311yescharacteristic
80#493—28no—
81#492—28no—
82#487—28no—
83#486—216no—
84#486—28no—
85#486—28no—
86#486—28no—
87#461—28no—
88#461—28no—
89#459—28no—
90#459—28no—
91#448—216no—
92#447—216no—
93#431—22no—
94#431—22no—
95#431D414yes—
96#431D414yes—
97#423—22no—
98#423—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2312yes—
101C2×C4⋊C8C2312yes—
102C2×C4⋊C8C2312yes—
103C2×#103—24no—
104C2×#102—216no—
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#332—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—42no—
128C22×Q8—41no—
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×SD16—216no—
156C2×SD16—88no—
157C2×D8—216no—
158C2×D8—88no—
159C2×M16—42no—
160C2×M16—44no—
161C22×C8—24no—
162C22×C8—24no—
163C22×C8—24no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8C2×D414yes—
168C4⋊Q8—24no—
169C4⋊Q8—22no—
170C4⋊Q8—22no—
171C4⋊Q8—24no—
172C4⋊Q8—24no—
173C4⋊Q8—22no—
174C4⋊Q8—22no—
175C4⋊Q8—22no—
176C4⋊Q8—42no—
177C4⋊Q8—42no—
178C4⋊Q8—42no—
179C4⋊Q8C2×D414yes—
180#127—24no—
181#127—24no—
182#127—22no—
183#127—22no—
184#127—24no—
185#127—22no—
186#127—22no—
187#127—22no—
188#127—42no—
189#127—42no—
190#127—42no—
191#127C2×D414yes—
192#124—42no—
193#124—42no—
194#121—24no—
195#121—24no—
196#121—42no—
197#121—42no—
198C4×D4—22no—
199C4×D4—22no—
200C4×D4—22no—
201C4×D4—22no—
202C4×D4—42no—
203C4×D4—42no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4C2×D412yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42#4111yescharacteristic
223C2×C42C2411yescharacteristic
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—22no—
228C4⋊C8—42no—
229C4⋊C8C2×D414yes—
230C4⋊C8C2×D414yes—
231#103—216no—
232#102—232no—
233#102—216no—
234C22⋊C8—24no—
235C22⋊C8—24no—
236C22⋊C8—416no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—24no—
240C4×C8—22no—
241C4×C8—22no—
242C24—22no—
243C24—42no—
244C24—44no—
245C24—84no—
246C24—84no—
247C24—84no—
248C24#12111yescharacteristic
249#41—88no—
250#41—84no—
251#41—84no—
252C2×Q8—24no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—48no—
258C2×Q8—88no—
259C2×Q8—84no—
260C2×Q8—88no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—24no—
266C2×D4—28no—
267C2×D4—44no—
268C2×D4—48no—
269C2×D4—42no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—44no—
273C2×D4—44no—
274C2×D4—48no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—44no—
279C2×D4—44no—
280C2×D4—416no—
281C2×D4—44no—
282C2×D4—44no—
283C2×D4—88no—
284C2×D4—84no—
285C2×D4—84no—
286C2×D4—88no—
287C2×D4—88no—
288C2×D4—88no—
289C2×D4—84no—
290C2×D4—84no—
291C2×D4—84no—
292C2×D4—84no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—84no—
303C22×C4#12012yes—
304C22×C4#12012yes—
305C22×C4#12012yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4#12111yescharacteristic
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12712yes—
312C22×C4#13514yes—
313C22×C4AGL(1,ℤ/8)14yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317SD16—816no—
318D8—816no—
319M16—84no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—44no—
326C2×C8—44no—
327C2×C8—48no—
328C2×C8—42no—
329C2×C8—44no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—44no—
337C4⋊C4—44no—
338C4⋊C4—42no—
339C22⋊C4—42no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C42#12012yes—
343C42#12012yes—
344C42#12012yes—
345C42#12112yes—
346C42#12112yes—
347C42#12712yes—
348C42C22×D411yescharacteristic
349C42C22×D411yescharacteristic
350C42C22×D411yescharacteristic
351C42C22×D411yescharacteristic
352C23—22no—
353C23—22no—
354C23—42no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—44no—
359C23—88no—
360C23—88no—
361C23—84no—
362C23—88no—
363C23—84no—
364C23—84no—
365C23—1616no—
366C23#39212yes—
367C23#46314yes—
368C23C2×#12111yescharacteristic
369Q8—48no—
370Q8—88no—
371Q8—84no—
372Q8—84no—
373D4—48no—
374D4—44no—
375D4—48no—
376D4—44no—
377D4—84no—
378D4—88no—
379D4—84no—
380D4—88no—
381D4—82no—
382D4—82no—
383D4—84no—
384D4—88no—
385D4—84no—
386D4—84no—
387D4—88no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—88no—
398C2×C4—84no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—84no—
402C2×C4#39212yes—
403C2×C4#39212yes—
404C2×C4#39212yes—
405C2×C4#44918yes—
406C2×C4#46314yes—
407C2×C4#47114yes—
408C2×C4#49514yes—
409C2×C4#49614yes—
410C2×C4C2×#12011yescharacteristic
411C2×C4C2×#12011yescharacteristic
412C2×C4C2×#12011yescharacteristic
413C2×C4C2×#12111yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12711yescharacteristic
416C2×C4C2×#13512yes—
417C2×C4C2×AGL(1,ℤ/8)12yes—
418C2×C4#57614yes—
419C8—44no—
420C8—44no—
421C8—44no—
422C8—82no—
423C22—22no—
424C22—22no—
425C22—28no—
426C22—82no—
427C22—84no—
428C22—82no—
429C22—84no—
430C22—84no—
431C22—82no—
432C22—88no—
433C22—1616no—
434C22—168no—
435C22#201214yes—
436C22C2×#39211yescharacteristic
437C22C2×#46312yes—
438C22#305812yes—
439C22C2×#47112yes—
440C4—22no—
441C4—21no—
442C4—21no—
443C4—84no—
444C4—82no—
445C4#201214yes—
446C4#201914yes—
447C4C2×#44912yes—
448C4#300812yes—
449C4#300812yes—
450C4C2×#49511yescharacteristic
451C4C2×#49611yescharacteristic
452C2—24no—
453C2—162no—
454C2—164no—
455C2#1919814yes—
456C2#2338711yescharacteristic
457C2C2×#201211yescharacteristic
458C2C2×#201911yescharacteristic
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×#3206C2semidirect
3. C22×#486C2semidirect
4. #21187C2semidirect
5. C2×#1549C2semidirect
6. C2×#1545C2semidirect
7. #10261C2semidirect
8. #7872C2semidirect
24. C2×#461C22semidirect
35. #1954C22semidirect
38. C2×#332C22semidirect
40. #1277C22semidirect

Maximal subgroups

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

subgroupindex
C2×#32062
C2×#32052
C22×#4862
#211872
C2×#15492
C2×#15452
#102612
#78722