Finite Group Databasebeta

← #6494338

Subgroups — #6494338

Order 512.

2800 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3207C212yesmaximal, complemented
2C2×#3169C211yesmaximal, characteristic, complemented
3C2×#3147C211yesmaximal, characteristic, complemented
4#35293C212yesmaximal, complemented
5#33303C211yesmaximal, characteristic, complemented
6#33301C211yesmaximal, characteristic, complemented
7#33242C211yesmaximal, characteristic, complemented
8#33241C211yesmaximal, characteristic, complemented
9#30260C212yesmaximal, complemented
10#30259C212yesmaximal, complemented
11#28465C218yesmaximal, complemented
12#25777C218yesmaximal, complemented
13C8×#117C211yesmaximal, characteristic, complemented
14C22×#135C2214yes—
15C2×#532C2212yes—
16C2×#532C2212yes—
17#3207—216no—
18#3169—28no—
19C2×#500C2211yescharacteristic
20#3147—28no—
21C2×#495C2214yes—
22C2×#495C2218yes—
23C2×#494C2211yescharacteristic
24C2×#493C2211yescharacteristic
25C2×#493C2212yes—
26C2×#493C2211yescharacteristic
27C2×#493C2218yes—
28C2×#492C2212yes—
29C2×#492C2212yes—
30C2×#487C2212yes—
31C2×#486C2212yes—
32C2×#486C2212yes—
33C2×#466C2214yes—
34C2×#463C2218yes—
35C2×#449C2218yes—
36C2×C4×D8C2214yes—
37C2×#431C2211yescharacteristic
38C22×C4×C8C2211yescharacteristic
39#2867C2212yes—
40#2862C2212yes—
41#2858C2212yes—
42#2858C2212yes—
43#2619—24no—
44#2544—24no—
45#2544C2218yes—
46#2431—24no—
47#2431C2218yes—
48#2328—24no—
49C4×#117C2211yescharacteristic
50#2014C22116yes—
51#2012C22116yes—
52#1970C2218yes—
53#1969C2218yes—
54C8×C4⋊C4C2214yes—
55C8×C22⋊C4C2214yes—
56#1744C2211yescharacteristic
57#1744C2211yescharacteristic
58C42×C8C2211yescharacteristic
59C42×C8C2211yescharacteristic
60C22×#41—22no—
61C22×#41C2314yes—
62C2×#135—216no—
63C2×#135—416no—
64C22×Q16C2314yes—
65C22×SD16C2318yes—
66C22×SD16C2318yes—
67C22×D8C2314yes—
68C22×D8C2314yes—
69C22×D8C2318yes—
70C23×C8C2312yes—
71#532—28no—
72#532—28no—
73#532—28no—
74#532—28no—
75C2×C4⋊Q8C2312yes—
76C2×C4⋊Q8C2312yes—
77C2×#127—24no—
78C2×#127—24no—
79C2×#127C2312yes—
80C2×#127C2312yes—
81C2×#127C2318yes—
82C2×#124—22no—
83C2×#124—24no—
84C2×#124C2314yes—
85C2×#124C2318yes—
86C2×#121—22no—
87C2×#121—22no—
88C2×#121—22no—
89C2×#121C2314yes—
90C2×C4×D4—22no—
91C2×C4×D4—22no—
92C2×C4×D4—22no—
93C2×C4×D4C2314yes—
94C2×#117D412yes—
95C22×C42C2311yescharacteristic
96#500—24no—
97#495—216no—
98#495—416no—
99#494—24no—
100#493—24no—
101#493—24no—
102#493—28no—
103#493—416no—
104#492—28no—
105#492—28no—
106#487—28no—
107#486—28no—
108#486—28no—
109#466—216no—
110#463—232no—
111#449—232no—
112C4×D8—216no—
113#431D418yescomplemented
114C2×#107C2314yes—
115C2×C4⋊C8C2312yes—
116C2×#103C2318yes—
117C2×#102C2318yes—
118C2×#102C2318yes—
119C2×C22⋊C8C2312yes—
120C2×C4×C8D418yescomplemented
121C2×C4×C8C2314yes—
122C2×C4×C8C2314yes—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2312yes—
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128#394—216no—
129#392—216no—
130#391—24no—
131#391C2318yes—
132#390—24no—
133#390—24no—
134#390—24no—
135#390C2318yes—
136#386—216no—
137#386—216no—
138#382—24no—
139C4×C4⋊C4D412yes—
140C4×C4⋊C4C2312yes—
141C4×C22⋊C4D412yes—
142C4×C22⋊C4C2312yes—
143#376D412yes—
144C43C2311yescharacteristic
145C43C2311yescharacteristic
146#336C2314yes—
147#336C2314yes—
148C2×#41—28no—
149C2×#41—28no—
150C2×#41—416no—
151C2×#41—48no—
152C2×#41—88no—
153C22×Q8—22no—
154C22×Q8C2414yes—
155C22×D4—24no—
156C22×D4—24no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—22no—
160C22×D4—22no—
161C22×D4—216no—
162C22×D4C2414yes—
163C22×D4C2414yes—
164C23×C4—24no—
165C23×C4—24no—
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2411yescharacteristic
169#135—464no—
170C2×Q16—28no—
171C2×Q16—216no—
172C2×SD16—216no—
173C2×SD16—216no—
174C2×SD16—416no—
175C2×SD16—416no—
176C2×D8—28no—
177C2×D8—28no—
178C2×D8—216no—
179C2×D8—216no—
180C2×D8—216no—
181C2×D8—416no—
182C22×C8—28no—
183C22×C8C2×D418yes—
184C22×C8C2412yes—
185C22×C8C2412yes—
186C22×C8C2412yes—
187C22×C8C2412yes—
188C4⋊Q8—24no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192#127—24no—
193#127—24no—
194#127—24no—
195#127—24no—
196#127—416no—
197#127—416no—
198#127—88no—
199#124—28no—
200#124—28no—
201#124—416no—
202#124—48no—
203#124—88no—
204#121—28no—
205#121—28no—
206#121—48no—
207#121—48no—
208#121—48no—
209C4×D4—28no—
210C4×D4—28no—
211C4×D4—48no—
212C4×D4—48no—
213C4×D4—48no—
214#117—28no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2414yes—
220C2×C22⋊C4—24no—
221C2×C22⋊C4—24no—
222C2×C22⋊C4—24no—
223C2×C22⋊C4—24no—
224C2×C22⋊C4—216no—
225C2×C22⋊C4—216no—
226C2×C22⋊C4C2×D414yes—
227C2×C42C2×D412yes—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D414yes—
233C2×C42C2×D414yes—
234C2×C42C2×D412yes—
235C2×C42C2412yes—
236C2×C42C2412yes—
237C2×C42C2411yescharacteristic
238C2×C42C2411yescharacteristic
239#107—28no—
240C4⋊C8C2×D418yes—
241#103—216no—
242#102—216no—
243#102—216no—
244C22⋊C8C2×D418yes—
245C4×C8—28no—
246C4×C8—28no—
247C4×C8C2×D418yes—
248C4×C8C2×D414yes—
249C4×C8C2×D414yes—
250C4×C8C2×D414yes—
251C4×C8C2×D414yes—
252#95—216no—
253#95C2×D414yes—
254#95C2×D414yes—
255C24—24no—
256C24—24no—
257C24C22×D411yescharacteristic
258#41—432no—
259#41—816no—
260#41—832no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—216no—
264C2×Q8—44no—
265C2×Q8—48no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—24no—
269C2×D4—24no—
270C2×D4—216no—
271C2×D4—216no—
272C2×D4—48no—
273C2×D4—48no—
274C2×D4—44no—
275C2×D4—44no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—48no—
279C2×D4—48no—
280C2×D4—432no—
281C2×D4—48no—
282C2×D4—48no—
283C2×D4—816no—
284C2×D4—816no—
285C2×D4—88no—
286C2×D4—88no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—48no—
293C22×C4—48no—
294C22×C4—88no—
295C22×C4—88no—
296C22×C4#12714yes—
297C22×C4#12714yes—
298C22×C4C22×D414yes—
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D412yes—
303C22×C4C22×D412yes—
304C22×C4C22×D414yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2511yescharacteristic
308Q16—416no—
309SD16—432no—
310SD16—816no—
311D8—416no—
312D8—416no—
313D8—816no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—24no—
317C2×C8—216no—
318C2×C8C22×D414yes—
319C2×C8C22×D414yes—
320C2×C8C22×D414yes—
321C2×C8C22×D414yes—
322C2×C8H(5,𝔽2)18yes—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—44no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C22⋊C4—28no—
330C22⋊C4—48no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—48no—
334C22⋊C4—816no—
335C22⋊C4—816no—
336C42—24no—
337C42—24no—
338C42—24no—
339C42—24no—
340C42—24no—
341C42#12714yes—
342C42#12714yes—
343C42#13518yescomplemented
344C42C22×D412yes—
345C42C22×D412yes—
346C42C22×D412yes—
347C42C22×D412yes—
348C23—22no—
349C23—22no—
350C23—22no—
351C23—44no—
352C23—44no—
353C23—48no—
354C23—48no—
355C23—48no—
356C23—48no—
357C23—48no—
358C23—48no—
359C23C2×#12712yes—
360C23C23×D411yescharacteristic
361Q8—48no—
362Q8—48no—
363Q8—88no—
364D4—48no—
365D4—48no—
366D4—48no—
367D4—48no—
368D4—816no—
369D4—816no—
370D4—88no—
371D4—88no—
372D4—88no—
373D4—88no—
374D4—1632no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—22no—
380C2×C4—28no—
381C2×C4—28no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—48no—
391C2×C4—816no—
392C2×C4—816no—
393C2×C4C2×#12712yes—
394C2×C4C2×#12712yes—
395C2×C4C2×#12712yes—
396C2×C4C2×#12712yes—
397C2×C4C2×#12712yes—
398C2×C4C2×#13518yes—
399C2×C4C2×#13514yes—
400C2×C4#57614yes—
401C2×C4C23×D411yescharacteristic
402C2×C4C23×D411yescharacteristic
403C2×C4C2×H(5,𝔽2)14yes—
404C8—24no—
405C8—24no—
406C8—24no—
407C8#53518yes—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—44no—
412C22—84no—
413C22—84no—
414C22—816no—
415C22—816no—
416C22—84no—
417C22—84no—
418C22—84no—
419C22—84no—
420C22C22×#12711yescharacteristic
421C22#349812yes—
422C22C22×#13512yes—
423C22C2×#57612yes—
424C4—22no—
425C4—28no—
426C4—22no—
427C4—22no—
428C4—44no—
429C4—84no—
430C4—84no—
431C4#314714yes—
432C4#315514yes—
433C4C2×#53512yes—
434C4#349812yes—
435C2—42no—
436C2—164no—
437C2—164no—
438C2#3527411yescharacteristic
439C2C2×#314711yescharacteristic
440C2C2×#315511yescharacteristic
441C2#6241714yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3207C2semidirect
2. C2×#3169C2semidirect
3. C2×#3147C2semidirect
4. #35293C2semidirect
5. #33303C2semidirect
6. #33301C2semidirect
7. #33242C2semidirect
8. #33241C2semidirect
9. #30260C2semidirect
10. #30259C2semidirect
11. #28465C2semidirect
12. #25777C2semidirect
13. C8×#117C2semidirect
113. #431D4semidirect
120. C2×C4×C8D4semidirect
343. C42#135semidirect

Maximal subgroups

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

subgroupindex
C2×#32072
C2×#31692
C2×#31472
#352932
#333032
#333012
#332422
#332412
#302602
#302592
#284652
#257772
C8×#1172