Finite Group Databasebeta

← #409272

Subgroups — #409272

Order 512.

1258 conjugacy classes of subgroups.

The subgroup lattice has 511 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1809C211yesmaximal, characteristic
2C2×#1729C211yesmaximal, characteristic
3#14264C212yesmaximal
4#11379C211yesmaximal, characteristic, complemented
5#11377C211yesmaximal, characteristic, complemented
6#11327C211yesmaximal, characteristic, complemented
7#11327C211yesmaximal, characteristic, complemented
8#11293C212yesmaximal, complemented
9#11292C212yesmaximal, complemented
10#10722C211yesmaximal, characteristic
11#10721C212yesmaximal
12C23×M16C2211yescharacteristic
13C24×C8C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C2×#389C2212yes—
18C2×#389C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21#1809—24no—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30#1729—24no—
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2211yescharacteristic
35#1444—22no—
36#1442—22no—
37#1439—22no—
38#1439—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C22×M16—24no—
43C22×M16—22no—
44C22×M16—22no—
45C22×M16—42no—
46C22×M16D414yes—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—22no—
50C23×C8—24no—
51C23×C8—42no—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C2×#117—22no—
57C2×#117—22no—
58C2×#117—42no—
59C2×#117—42no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D412yes—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—42no—
77#389—42no—
78#389—42no—
79#389—42no—
80#389—42no—
81#380—24no—
82#380—24no—
83#380—24no—
84#380Q818yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93#340—24no—
94#340—22no—
95#340—22no—
96#340—24no—
97#340—24no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105C25C2×D411yescharacteristic
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D412yes—
120C23×C4C2×Q814yes—
121C23×C4#4112yes—
122C23×C4#4114yes—
123C23×C4C2411yescharacteristic
124C2×M16—28no—
125C2×M16—28no—
126C2×M16—44no—
127C2×M16—48no—
128C2×M16—48no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—28no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—48no—
145C22×C8—48no—
146C22×C8—44no—
147C22×C8—44no—
148C22×C8C2×D412yes—
149C22×C8C2×D412yes—
150C22×C8#4112yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
153C22×C8#4112yes—
154#117—44no—
155#117—44no—
156#117—48no—
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—24no—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
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⋊C4C2×Q814yes—
189C2×C4⋊C4C2×Q814yes—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—28no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207#107—44no—
208#106—44no—
209#95—28no—
210#95—44no—
211#95—44no—
212#95—44no—
213#95—82no—
214#95—82no—
215#95—82no—
216#95—82no—
217C24—21no—
218C24—21no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—22no—
223C24—22no—
224C24#12112yes—
225C24C22⋊Q814yes—
226C24C22×D411yescharacteristic
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
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×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4#12112yes—
288C22×C4C22⋊Q814yes—
289C22×C4#12314yes—
290C22×C4#12518yes—
291C22×C4C4⋊Q814yes—
292C22×C4C22×D411yescharacteristic
293C22×C4C22×Q812yes—
294C22×C4C2×#4112yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4H(5,𝔽2)12yes—
301C22×C4H(5,𝔽2)12yes—
302M16—416no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—24no—
306C2×C8—24no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—416no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8#12314yes—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—48no—
329C4⋊C4—42no—
330C4⋊C4—42no—
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—42no—
340C4⋊C4—42no—
341C4⋊C4—88no—
342C4⋊C4—88no—
343C22⋊C4—48no—
344C22⋊C4—88no—
345C22⋊C4—88no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—42no—
351C42—42no—
352C23—24no—
353C23—24no—
354C23—24no—
355C23—24no—
356C23—24no—
357C23—24no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—24no—
362C23—41no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—42no—
371C23—42no—
372C23#39814yes—
373C23C2×#12111yescharacteristic
374C23C2×C22⋊Q812yes—
375C23#53211yescharacteristic
376C23#53211yescharacteristic
377C23#53412yes—
378C23#57612yes—
379C23#57712yes—
380C23#57712yes—
381C23#57812yes—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—28no—
391C2×C4—28no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
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—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4#39814yes—
435C2×C4C2×#12311yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4#52511yescharacteristic
438C2×C4C2×#12512yes—
439C2×C4#52911yescharacteristic
440C2×C4#52911yescharacteristic
441C2×C4C2×C4⋊Q811yescharacteristic
442C2×C4#53411yescharacteristic
443C2×C4#53511yescharacteristic
444C2×C4#53812yes—
445C2×C4#55112yes—
446C2×C4#55112yes—
447C2×C4#55112yes—
448C2×C4#55112yes—
449C2×C4#55412yes—
450C2×C4#55412yes—
451C2×C4#55412yes—
452C2×C4#55412yes—
453C8—24no—
454C8—42no—
455C8—42no—
456C8—42no—
457C22—28no—
458C22—24no—
459C22—44no—
460C22—42no—
461C22—42no—
462C22—44no—
463C22—44no—
464C22—44no—
465C22—44no—
466C22C2×#39811yescharacteristic
467C22#256412yes—
468C22#256412yes—
469C22#261711yescharacteristic
470C22#285411yescharacteristic
471C22#306212yes—
472C22#306312yes—
473C22#308112yes—
474C22#308212yes—
475C22#308312yes—
476C22#308412yes—
477C22#317311yescharacteristic
478C22#317311yescharacteristic
479C22#317611yescharacteristic
480C22#317711yescharacteristic
481C22#317812yes—
482C22#317912yes—
483C22#319812yes—
484C22#320012yes—
485C22#320112yes—
486C22#320312yes—
487C4—24no—
488C4—82no—
489C4—82no—
490C4—84no—
491C4—84no—
492C4—162no—
493C4—162no—
494C4#248211yescharacteristic
495C4#248311yescharacteristic
496C4#250211yescharacteristic
497C4#250512yes—
498C4#250512yes—
499C4#258311yescharacteristic
500C2—44no—
501C2#1501411yescharacteristic
502C2#2367112yes—
503C2#2916912yes—
504C2#2917012yes—
505C2#2917312yes—
506C2#2917412yes—
507C2#2974712yes—
508C2#3314011yescharacteristic
509C2#3314111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×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. #11379C2semidirect
5. #11377C2semidirect
6. #11327C2semidirect
7. #11327C2semidirect
8. #11293C2semidirect
9. #11292C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18092
C2×#17292
#142642
#113792
#113772
#113272
#112932
#112922
#107222
#107212