Finite Group Databasebeta

← #374621

Subgroups — #374621

Order 512.

1642 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#424C211yesmaximal, characteristic
2C2×#2629C211yesmaximal, characteristic
3C2×#1989C212yesmaximal, complemented
4#13777C211yesmaximal, characteristic
5#11228C212yesmaximal, complemented
6#11227C212yesmaximal, complemented
7#10849C212yesmaximal, complemented
8#10722C212yesmaximal
9#10721C212yesmaximal
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#125C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C2×#424—28no—
15C2×#424—24no—
16C22×C4⋊C8C2212yes—
17C22×C4⋊C8C2212yes—
18#2629—24no—
19#2629C418yescomplemented
20C2×#398—22no—
21C2×#396—22no—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#380C2212yes—
25C2×#380C2212yes—
26C2×C4×C4⋊C4C2212yes—
27#1989—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36#1437—22no—
37#1436—22no—
38C4⋊#95C2211yescharacteristic
39C4⋊#95C2211yescharacteristic
40C24×C4C2311yescharacteristic
41C22×M16—22no—
42C22×M16—24no—
43C22×M16—24no—
44C22×M16—41no—
45C22×M16—41no—
46C23×C8—22no—
47C23×C8—24no—
48C23×C8—24no—
49C23×C8—41no—
50C23×C8—41no—
51C23×C8C2312yes—
52C23×C8C2312yes—
53C2×#125—22no—
54C2×#125—24no—
55C2×#125—24no—
56C2×#125C2×C414yes—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117C2×C414yes—
60C2×#117D414yes—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4D412yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4D412yes—
72C22×C42—22no—
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#424—232no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—24no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—24no—
81C2×C4⋊C8D418yes—
82#398—216no—
83#396—216no—
84#389—24no—
85#389C2×C418yes—
86#384—22no—
87#384C2×C414yes—
88#380—24no—
89#380—24no—
90#380C2×C418yes—
91#380Q818yes—
92C4×C4⋊C4—24no—
93C4×C4⋊C4—24no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108C25C2×D411yescharacteristic
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4C22×C412yes—
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×Q814yes—
125C23×C4#4114yes—
126C23×C4C2411yescharacteristic
127C2×M16—28no—
128C2×M16—48no—
129C2×M16—48no—
130C2×M16—44no—
131C2×M16—44no—
132C22×C8—28no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—48no—
146C22×C8—48no—
147C22×C8—44no—
148C22×C8—44no—
149C22×C8C2×D414yes—
150C22×C8C2×D414yes—
151#125—216no—
152#125—44no—
153#125—44no—
154#117—216no—
155#117—28no—
156#117—28no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—24no—
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—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4—82no—
188C2×C4⋊C4C22⋊C418yes—
189C2×C4⋊C4C22×C414yes—
190C2×C4⋊C4C22×C414yes—
191C2×C4⋊C4C2×Q814yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4—82no—
200C2×C22⋊C4—82no—
201C2×C22⋊C4C22⋊C418yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—44no—
205C2×C42—82no—
206C2×C42—82no—
207C2×C42C22×C412yes—
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D414yes—
211C2×C42C2×D412yes—
212C2×C42#4114yes—
213C4⋊C8—216no—
214C4⋊C8—216no—
215#95—28no—
216#95—28no—
217#95—28no—
218#95—44no—
219#95—82no—
220#95—82no—
221C24—22no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—22no—
227C24—22no—
228C24C2×C22⋊C412yes—
229C24C22⋊Q814yes—
230C24C22×D411yescharacteristic
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—24no—
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—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
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×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C414yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C4×Q818yes—
291C22×C4#12114yes—
292C22×C4#12114yes—
293C22×C4C22⋊Q814yes—
294C22×C4#12414yes—
295C22×C4C4⋊Q814yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C22×D412yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q812yes—
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C4H(5,𝔽2)11yescharacteristic
303C22×C4H(5,𝔽2)11yescharacteristic
304C22×C42−1+411yescharacteristic
305C22×C42−1+411yescharacteristic
306M16—416no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—416no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C4⋊C4—216no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—416no—
340C4⋊C4—88no—
341C4⋊C4—88no—
342C22⋊C4—216no—
343C22⋊C4—88no—
344C22⋊C4—88no—
345C42—24no—
346C42—22no—
347C42—22no—
348C42—88no—
349C42C4×D418yes—
350C42#12014yes—
351C42#12314yes—
352C23—22no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—24no—
359C23—24no—
360C23—24no—
361C23—22no—
362C23—42no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23#39114yes—
373C23C22×C22⋊C411yescharacteristic
374C23C2×#12112yes—
375C23C2×C22⋊Q812yes—
376C23#53711yescharacteristic
377C23#53711yescharacteristic
378C23#57612yes—
379C23#57712yes—
380C23#57712yes—
381C23#57812yes—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—28no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—24no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
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—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4#38618yes—
434C2×C4#39114yes—
435C2×C4C2×C4×D412yes—
436C2×C4C2×C4×Q812yes—
437C2×C4#52011yescharacteristic
438C2×C4#52011yescharacteristic
439C2×C4C2×#12011yescharacteristic
440C2×C4C2×#12311yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4C2×C4⋊Q811yescharacteristic
443C2×C4#53511yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#54612yes—
446C2×C4#54612yes—
447C2×C4D4×Q812yes—
448C2×C4D4×Q812yes—
449C2×C4#55212yes—
450C2×C4#55212yes—
451C2×C4#55412yes—
452C2×C4#55412yes—
453C8—44no—
454C8—44no—
455C22—24no—
456C22—28no—
457C22—44no—
458C22—44no—
459C22—44no—
460C22—44no—
461C22—42no—
462C22—42no—
463C22—44no—
464C22C2×#38611yescharacteristic
465C22C2×#39111yescharacteristic
466C22#231111yescharacteristic
467C22#241712yes—
468C22#241712yes—
469C22#289712yes—
470C22#289812yes—
471C22#306012yes—
472C22#306112yes—
473C22#306212yes—
474C22#306312yes—
475C22#308112yes—
476C22#308212yes—
477C22#308312yes—
478C22#308412yes—
479C22#325712yes—
480C22#325812yes—
481C22#325911yescharacteristic
482C22#326011yescharacteristic
483C22#326011yescharacteristic
484C22#326111yescharacteristic
485C4—24no—
486C4—28no—
487C4—44no—
488C4—84no—
489C4—84no—
490C4—84no—
491C4—84no—
492C4#236012yes—
493C4#236012yes—
494C4#243411yescharacteristic
495C4#243511yescharacteristic
496C4#250211yescharacteristic
497C4#250311yescharacteristic
498C2—44no—
499C2#1413911yescharacteristic
500C2#2097612yes—
501C2#2131512yes—
502C2#2189711yescharacteristic
503C2#2189811yescharacteristic
504C2#2484812yes—
505C2#2485012yes—
506C2#2485212yes—
507C2#2485412yes—
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
3. C2×#1989C2semidirect
5. #11228C2semidirect
6. #11227C2semidirect
7. #10849C2semidirect
19. #2629C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4242
C2×#26292
C2×#19892
#137772
#112282
#112272
#108492
#107222
#107212