Finite Group Databasebeta

← #6437214

Subgroups — #6437214

Order 512.

2260 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3170C212yesmaximal, complemented
2C2×#3167C212yesmaximal
3#36645C211yesmaximal, characteristic, complemented
4#36639C211yesmaximal, characteristic
5#30281C212yesmaximal, complemented
6#30280C211yesmaximal, characteristic, complemented
7#30276C211yesmaximal, characteristic, complemented
8#30263C211yesmaximal, characteristic, complemented
9#30262C211yesmaximal, characteristic, complemented
10#30261C212yesmaximal, complemented
11#30244C218yesmaximal, complemented
12#23797C218yesmaximal, complemented
13C2×#1744C211yesmaximal, characteristic, complemented
14C2×#533C2211yescharacteristic
15C2×#533C2212yes—
16C2×#532C2211yescharacteristic
17#3170—216no—
18#3167—216no—
19C2×#500C2212yes—
20C2×#498C2212yes—
21C2×#494C2212yes—
22C2×#492C2212yes—
23C2×#487C2212yes—
24C2×#487C2212yes—
25C2×#487C2212yes—
26C2×#486C2212yes—
27C2×#467C2218yes—
28C2×#465C2218yes—
29C2×C4×Q16C2218yes—
30C2×C4×SD16C2218yes—
31C22×C4×C8C2212yes—
32#2868C2211yescharacteristic
33#2868C2211yescharacteristic
34#2862C2211yescharacteristic
35#2862C2212yes—
36#2862C2211yescharacteristic
37#2861—22no—
38#2859—22no—
39#2858C2211yescharacteristic
40#2858C2211yescharacteristic
41#2857—22no—
42#2857C2214yes—
43#2850—22no—
44#2850C2214yes—
45C4×C4⋊Q8—22no—
46C4×C4⋊Q8C2214yes—
47C4×#124—22no—
48C4×#124C2214yes—
49#2297—22no—
50#2285—22no—
51C2×C43C2211yescharacteristic
52#2097C22116yes—
53#2096C2218yes—
54#2093C2218yes—
55#2016C2218yes—
56#2015C22116yes—
57#2013C2218yes—
58#1744C2212yes—
59#1744C2211yescharacteristic
60#1744C2211yescharacteristic
61#1744C2214yes—
62C2×#336C2214yes—
63C22×#41—21no—
64C22×Q16C2318yes—
65C22×SD16C2318yes—
66C23×C8C2314yes—
67#533—28no—
68#533—28no—
69#533—28no—
70#532—28no—
71C2×C4⋊Q8—22no—
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4⋊Q8C2312yes—
74C2×C4⋊Q8C2314yes—
75C2×C4⋊Q8C2311yescharacteristic
76C2×C4⋊Q8C2311yescharacteristic
77C2×C4⋊Q8C2312yes—
78C2×#127C2311yescharacteristic
79C2×#125—22no—
80C2×#125—21no—
81C2×#124—22no—
82C2×#124—21no—
83C2×#124C2314yes—
84C2×C22⋊Q8—21no—
85C2×C22⋊Q8—22no—
86C2×C22⋊Q8C2312yes—
87C2×C22⋊Q8C2314yes—
88C2×#121—21no—
89C2×#121C2312yes—
90C2×C4×Q8—22no—
91C2×C4×Q8—21no—
92C2×C4×Q8C2314yes—
93C2×C4×Q8C2312yes—
94C2×C4×D4—21no—
95C2×C4×D4C2312yes—
96C2×#117—22no—
97C2×#117—21no—
98C22×C42D412yes—
99C22×C42D412yes—
100C22×C42C2312yes—
101C22×C42C2311yescharacteristic
102#500—28no—
103#498—28no—
104#494—28no—
105#492—28no—
106#487—28no—
107#487—28no—
108#487—28no—
109#486—28no—
110#467—232no—
111#465—232no—
112C4×Q16—232no—
113C4×SD16—232no—
114C2×#107C2318yes—
115C2×#106C2318yes—
116C2×#103C2318yes—
117C2×#103C2318yes—
118C2×#103C2318yes—
119C2×#102C2318yes—
120C2×C4×C8D4116yes—
121C2×C4×C8C2312yes—
122C2×C4×C8C2312yes—
123C2×C4×C8C2314yes—
124#391—24no—
125#391—22no—
126#391—22no—
127#391—22no—
128#391C2318yes—
129#391C2314yes—
130#390—22no—
131#390—22no—
132#390—22no—
133#390C2314yes—
134#389—28no—
135#389—28no—
136#388—28no—
137#388—28no—
138#386—28no—
139#384—22no—
140#384—22no—
141#384—22no—
142#384C2314yes—
143#382—22no—
144#382—24no—
145#382C2314yes—
146#380—28no—
147C4×C4⋊C4—28no—
148C4×C22⋊C4—28no—
149C43D412yes—
150C43D412yes—
151C43C2312yes—
152C43C2311yescharacteristic
153C43C2311yescharacteristic
154#336C2314yes—
155#336C2314yes—
156#336C2318yes—
157C2×#41—44no—
158C2×#41—48no—
159C22×Q8—21no—
160C22×Q8—22no—
161C22×Q8—21no—
162C22×Q8C2412yes—
163C22×Q8C2414yes—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—21no—
167C22×D4C2412yes—
168C23×C4—22no—
169C23×C4C2×D414yes—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C2×Q16—216no—
174C2×Q16—416no—
175C2×SD16—216no—
176C2×SD16—416no—
177C22×C8—48no—
178C22×C8C2×D4116yes—
179C22×C8C2414yes—
180C22×C8C2414yes—
181C4⋊Q8—24no—
182C4⋊Q8—24no—
183C4⋊Q8—24no—
184C4⋊Q8—24no—
185C4⋊Q8—24no—
186C4⋊Q8—24no—
187C4⋊Q8—24no—
188C4⋊Q8—48no—
189C4⋊Q8—48no—
190#127—24no—
191#125—48no—
192#125—44no—
193#124—48no—
194#124—44no—
195#124—48no—
196C22⋊Q8—28no—
197C22⋊Q8—28no—
198C22⋊Q8—28no—
199C22⋊Q8—44no—
200C22⋊Q8—48no—
201#121—28no—
202#121—44no—
203C4×Q8—28no—
204C4×Q8—28no—
205C4×Q8—28no—
206C4×Q8—48no—
207C4×Q8—44no—
208C4×D4—28no—
209C4×D4—44no—
210#117—48no—
211#117—44no—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—28no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4C2412yes—
226C2×C4⋊C4C2412yes—
227C2×C4⋊C4C2414yes—
228C2×C22⋊C4—24no—
229C2×C22⋊C4—22no—
230C2×C22⋊C4—22no—
231C2×C22⋊C4—22no—
232C2×C22⋊C4—28no—
233C2×C22⋊C4—28no—
234C2×C42—28no—
235C2×C42—28no—
236C2×C42—22no—
237C2×C42—24no—
238C2×C42—44no—
239C2×C42C2×D412yes—
240C2×C42C2×D412yes—
241C2×C42C2×D414yes—
242C2×C42C2×D412yes—
243C2×C42C2×D414yes—
244C2×C42C2×D414yes—
245C2×C42C2×D411yescharacteristic
246C2×C42C2×D411yescharacteristic
247C2×C42C2×D418yes—
248C2×C42C2×D412yes—
249C2×C42C2412yes—
250C2×C42C2412yes—
251C2×C42C2411yescharacteristic
252C2×C42C2411yescharacteristic
253#107—216no—
254#106—216no—
255#103—216no—
256#103—216no—
257#103—216no—
258#102—216no—
259C4×C8C2×D4116yes—
260C4×C8C2×D418yes—
261C4×C8C2×D418yes—
262#95—28no—
263#95—28no—
264#95—216no—
265C24—22no—
266C24C22×D411yescharacteristic
267#41—816no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—28no—
272C2×Q8—216no—
273C2×Q8—44no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—44no—
277C2×Q8—48no—
278C2×Q8—44no—
279C2×D4—24no—
280C2×D4—28no—
281C2×D4—44no—
282C2×D4—48no—
283C2×D4—42no—
284C2×D4—42no—
285C2×D4—44no—
286C2×D4—44no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—28no—
290C22×C4—28no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—44no—
297C22×C4—44no—
298C22×C4—48no—
299C22×C4#12714yes—
300C22×C4#12714yes—
301C22×C4#13718yes—
302C22×C4C22×D414yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C22×D414yes—
306C22×C4C22×D414yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2511yescharacteristic
311Q16—432no—
312SD16—432no—
313C2×C8—28no—
314C2×C8—28no—
315C2×C8—416no—
316C2×C8C22×D418yes—
317C2×C8C22×D418yes—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—416no—
323C4⋊C4—44no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—48no—
330C4⋊C4—44no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—416no—
335C22⋊C4—416no—
336C22⋊C4—48no—
337C22⋊C4—44no—
338C22⋊C4—44no—
339C22⋊C4—44no—
340C22⋊C4—416no—
341C42—28no—
342C42—24no—
343C42—24no—
344C42—28no—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—44no—
349C42—44no—
350C42—48no—
351C42#12714yes—
352C42#12714yes—
353C42#13518yes—
354C42C22×D414yes—
355C42C22×D414yes—
356C23—22no—
357C23—22no—
358C23—42no—
359C23—44no—
360C23—44no—
361C23—44no—
362C23C2×#12712yes—
363C23C2×#13714yes—
364C23C23×D411yescharacteristic
365Q8—48no—
366Q8—48no—
367Q8—48no—
368Q8—84no—
369Q8—88no—
370Q8—84no—
371D4—48no—
372D4—88no—
373D4—84no—
374D4—84no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—28no—
380C2×C4—28no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—416no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—44no—
393C2×C4—48no—
394C2×C4—44no—
395C2×C4—88no—
396C2×C4#497116yes—
397C2×C4C2×#12712yes—
398C2×C4C2×#12712yes—
399C2×C4C2×#12712yes—
400C2×C4C2×#12712yes—
401C2×C4C2×#12712yes—
402C2×C4C2×#13518yes—
403C2×C4C2×#13514yes—
404C2×C4C2×#13714yes—
405C2×C4C2×#13714yes—
406C2×C4C23×D412yes—
407C2×C4#58514yes—
408C8—28no—
409C8—28no—
410C22—28no—
411C22—22no—
412C22—22no—
413C22—88no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22C2×#49714yes—
418C22C22×#12711yescharacteristic
419C22#344912yes—
420C22C22×#13512yes—
421C22C22×#13712yes—
422C4—28no—
423C4—22no—
424C4—22no—
425C4—44no—
426C4—82no—
427C4—84no—
428C4—82no—
429C4—84no—
430C4—82no—
431C4#314714yes—
432C4C2×#49714yes—
433C4C2×#49714yes—
434C4#344914yes—
435C2—24no—
436C2—162no—
437C2#3464111yescharacteristic
438C2C2×#314711yescharacteristic
439C2C22×#49711yescharacteristic
440C2#6094714yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#3170C2semidirect
3. #36645C2semidirect
5. #30281C2semidirect
6. #30280C2semidirect
7. #30276C2semidirect
8. #30263C2semidirect
9. #30262C2semidirect
10. #30261C2semidirect
11. #30244C2semidirect
12. #23797C2semidirect
13. C2×#1744C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31702
C2×#31672
#366452
#366392
#302812
#302802
#302762
#302632
#302622
#302612
#302442
#237972
C2×#17442