Finite Group Databasebeta

← #358937

Subgroups — #358937

Order 512.

1328 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1750C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11373C211yesmaximal, characteristic, complemented
5#11371C211yesmaximal, characteristic, complemented
6#11358C211yesmaximal, characteristic, complemented
7#11358C211yesmaximal, characteristic, complemented
8#11200C212yesmaximal
9#11199C212yesmaximal
10#10703C212yesmaximal, complemented
11C4×#340C212yesmaximal, complemented
12C22×#117C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17#2314—24no—
18#2314C418yescomplemented
19C2×#388—22no—
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2212yes—
27#1750—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38#1437—22no—
39#1436—22no—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2211yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C2×#117—24no—
48C2×#117—24no—
49C2×#117C2×C414yes—
50C2×#117D414yes—
51C22×C4⋊C4—22no—
52C22×C4⋊C4D412yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4D412yes—
60C22×C42—22no—
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×#97—22no—
71C2×#97—24no—
72C2×C4×C8—22no—
73C2×C4×C8—24no—
74#388—216no—
75#387—216no—
76#384—22no—
77#384C2×C414yes—
78#382—22no—
79#382C2×C414yes—
80#380—24no—
81#380—24no—
82#380—24no—
83#380C2×C418yes—
84C4×C4⋊C4—24no—
85C4×C4⋊C4—24no—
86C4×C4⋊C4—24no—
87C4×C4⋊C4C2×C418yes—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95#340—24no—
96#340—24no—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—24no—
101#340—24no—
102#340—24no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4C22×C412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8#4112yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144#117—216no—
145#117—28no—
146#117—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—28no—
170C2×C4⋊C4—82no—
171C2×C4⋊C4—82no—
172C2×C4⋊C4C22⋊C418yes—
173C2×C4⋊C4C22×C414yes—
174C2×C4⋊C4C22×C414yes—
175C2×C4⋊C4#4114yes—
176C2×C4⋊C4#4114yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—82no—
182C2×C22⋊C4—82no—
183C2×C22⋊C4—82no—
184C2×C22⋊C4—82no—
185C2×C22⋊C4C22⋊C418yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—28no—
189C2×C42—44no—
190C2×C42—82no—
191C2×C42—82no—
192C2×C42—82no—
193C2×C42—82no—
194C2×C42C22×C412yes—
195C2×C42C22×C412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42#4114yes—
199C2×C42#4114yes—
200C22⋊C8—28no—
201C22⋊C8—28no—
202#97—44no—
203C4×C8—44no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—28no—
208#95—44no—
209#95—82no—
210#95—82no—
211C24—22no—
212C24—22no—
213C24—21no—
214C24—21no—
215C24—22no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24C2×C22⋊C412yes—
220C24C22×D411yescharacteristic
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C414yes—
283C22×C4C2×C22⋊C414yes—
284C22×C4#12314yes—
285C22×C4#12314yes—
286C22×C4C23×C411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C2×#4112yes—
289C22×C4C2×#4112yes—
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4112yes—
295C22×C4C2×#4112yes—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—42no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C4⋊C4—216no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—42no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—88no—
334C22⋊C4—216no—
335C22⋊C4—88no—
336C22⋊C4—88no—
337C42—24no—
338C42—22no—
339C42—22no—
340C42—88no—
341C42—88no—
342C42#11718yes—
343C42#12314yes—
344C42#12314yes—
345C23—24no—
346C23—24no—
347C23—22no—
348C23—22no—
349C23—22no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—42no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23C22×C22⋊C411yescharacteristic
367C23C2×#12312yes—
368C23#52512yes—
369C23#53211yescharacteristic
370C23#53211yescharacteristic
371C23#57612yes—
372C23#57712yes—
373C23#57712yes—
374C23#57812yes—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—44no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4#38118yes—
428C2×C4C2×#11712yes—
429C2×C4C4×#4112yes—
430C2×C4C4×#4111yescharacteristic
431C2×C4C4×#4111yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4C2×#12311yescharacteristic
434C2×C4#52511yescharacteristic
435C2×C4#52511yescharacteristic
436C2×C4#52912yes—
437C2×C4#52912yes—
438C2×C4#52912yes—
439C2×C4#52912yes—
440C2×C4#52912yes—
441C2×C4#52912yes—
442C2×C4#52912yes—
443C2×C4#52912yes—
444C2×C4#53211yescharacteristic
445C2×C4#53211yescharacteristic
446C8—42no—
447C8—42no—
448C8—42no—
449C8—42no—
450C22—42no—
451C22—44no—
452C22—42no—
453C22—42no—
454C22—44no—
455C22—82no—
456C22—82no—
457C22—82no—
458C22—82no—
459C22C2×#38111yescharacteristic
460C22#229711yescharacteristic
461C22#229711yescharacteristic
462C22#239512yes—
463C22#239512yes—
464C22#289712yes—
465C22#289812yes—
466C22#309612yes—
467C22#309712yes—
468C22#309812yes—
469C22#309912yes—
470C22#311812yes—
471C22#312112yes—
472C22#312412yes—
473C22#312512yes—
474C22#317411yescharacteristic
475C22#317411yescharacteristic
476C22#317611yescharacteristic
477C22#317711yescharacteristic
478C22#317812yes—
479C22#317912yes—
480C4—28no—
481C4—42no—
482C4—44no—
483C4—84no—
484C4—84no—
485C4—84no—
486C4—84no—
487C4#229812yes—
488C4#229812yes—
489C4#239511yescharacteristic
490C4#239511yescharacteristic
491C4#239511yescharacteristic
492C4#239511yescharacteristic
493C2—82no—
494C2#1385311yescharacteristic
495C2#2070312yes—
496C2#2139712yes—
497C2#2141311yescharacteristic
498C2#2141411yescharacteristic
499C2#2376612yes—
500C2#2376812yes—
501C2#2377012yes—
502C2#2377212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11373C2semidirect
5. #11371C2semidirect
6. #11358C2semidirect
7. #11358C2semidirect
10. #10703C2semidirect
11. C4×#340C2semidirect
18. #2314C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#17502
#137772
#113732
#113712
#113582
#112002
#111992
#107032
C4×#3402