Finite Group Databasebeta

← #240357

Subgroups — #240357

Order 512.

1084 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
1C2×#3079C211yesmaximal, characteristic
2#21373C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic, complemented
4#20648C211yesmaximal, characteristic
5C2×#1567C211yesmaximal, characteristic, complemented
6C2×#1548C211yesmaximal, characteristic, complemented
7#10483C214yesmaximal, complemented
8#10481C214yesmaximal, complemented
9#8043C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C2×C4×#41C2211yescharacteristic
12#3079—24no—
13#3079—24no—
14C2×#477C2212yes—
15C2×#475—21no—
16C2×#476C2212yes—
17C2×#474—21no—
18C2×#431C2211yescharacteristic
19C2×#428—21no—
20C2×#423C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#417—21no—
23C2×#417—21no—
24C22×C4×C8C2211yescharacteristic
25C2×#384C2211yescharacteristic
26#1990—21no—
27#1990—21no—
28#1990C2214yes—
29#1984C2212yes—
30#1954C2212yes—
31#1869—22no—
32#1803—21no—
33#1803C2212yes—
34#1802—21no—
35#1802C2212yes—
36C4×C22⋊C8—21no—
37C4×C22⋊C8—21no—
38#1736—21no—
39C2×#340—22no—
40#1567—24no—
41C2×#335C2214yes—
42#1548—24no—
43C2×#333C2214yes—
44#1296C2214yes—
45C22×#41D411yescharacteristic
46C23×Q8—21no—
47C22×M16—21no—
48C22×M16—21no—
49C23×C8D412yes—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8C2312yes—
58C4×#41—22no—
59C4×#41—22no—
60C4×#41—22no—
61C4×#41—22no—
62C2×C4×Q8D411yescharacteristic
63C2×C4×Q8C2311yescharacteristic
64C2×C4×D4—21no—
65C2×C4×D4—21no—
66C2×C4×D4D411yescharacteristic
67C2×C4×D4C2311yescharacteristic
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42—21no—
76C22×C42C2311yescharacteristic
77#477—24no—
78#477—24no—
79#476—24no—
80#476—24no—
81#475—28no—
82#474—28no—
83#431—22no—
84#431—22no—
85#428—28no—
86C2×#107—22no—
87C2×#106—22no—
88#423—22no—
89#423—22no—
90#423—22no—
91#423D414yes—
92C2×C4⋊C8—21no—
93C2×C4⋊C8—21no—
94C2×C4⋊C8C2312yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97#417—28no—
98#417—44no—
99C2×#103—22no—
100C2×#103—22no—
101C2×#102—22no—
102C2×#102—22no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C4×C8—22no—
107C2×C4×C8—22no—
108C2×C4×C8—21no—
109C2×C4×C8D411yescharacteristic
110C2×C4×C8D411yescharacteristic
111C2×C4×C8C2312yes—
112#384—21no—
113#384—21no—
114#384—21no—
115#384C2312yes—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#336—22no—
121#336—22no—
122#335—28no—
123#333—28no—
124C2×#41—22no—
125C2×#41—22no—
126C2×#41—22no—
127C2×#41—22no—
128C2×#41—22no—
129C2×#41Q1614yes—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—41no—
133C22×Q8—42no—
134C22×Q8—82no—
135C22×Q8—82no—
136C22×Q8C2×D411yescharacteristic
137C22×D4—21no—
138C22×D4C2×D411yescharacteristic
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—41no—
143C23×C4—41no—
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4#4111yescharacteristic
147C2×M16—24no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—24no—
154C22×C8—22no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—42no—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C4⋊Q8—24no—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8C2×D414yes—
171C4×Q8—22no—
172C4×Q8—22no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×D4—22no—
178C4×D4—22no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187#117—22no—
188#117—22no—
189#117—42no—
190#117—42no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42#4111yescharacteristic
228C2×C42C2411yescharacteristic
229#107—28no—
230#106—28no—
231C4⋊C8—24no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—42no—
238C4⋊C8C2×D414yes—
239#103—28no—
240#103—44no—
241#102—28no—
242#102—44no—
243C22⋊C8—44no—
244C22⋊C8—42no—
245C22⋊C8—44no—
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—22no—
250C4×C8—42no—
251C24—21no—
252C24#12111yescharacteristic
253#41—28no—
254#41—28no—
255#41—44no—
256#41—44no—
257C2×Q8—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×Q8—24no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—42no—
264C2×Q8—44no—
265C2×Q8—88no—
266C2×Q8—88no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8—82no—
270C2×Q8—82no—
271C2×Q8C2×Q1612yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—42no—
280C2×D4C2×Q1612yes—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—44no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—82no—
302C22×C4—82no—
303C22×C4—82no—
304C22×C4—82no—
305C22×C4—82no—
306C22×C4#12012yes—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12312yes—
313C22×C4#12312yes—
314C22×C4#12412yes—
315C22×C4C2×Q1612yes—
316C22×C4#13512yes—
317C22×C4AGL(1,ℤ/8)12yes—
318C22×C4#13712yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C2×#4111yescharacteristic
322M16—48no—
323M16—84no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—48no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8#13714yes—
340C4⋊C4—24no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—24no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—44no—
351C4⋊C4—44no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—44no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C22⋊C4—42no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—42no—
364C42—41no—
365C42—41no—
366C42—42no—
367C42—42no—
368C42#12012yes—
369C42#12112yes—
370C42#12112yes—
371C42#12312yes—
372C42#12312yes—
373C42#12412yes—
374C42C22×D411yescharacteristic
375C42C22×D411yescharacteristic
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—22no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23#39412yes—
390C23#46712yes—
391C23#46812yes—
392C23C2×#12111yescharacteristic
393Q8—22no—
394Q8—22no—
395Q8—41no—
396Q8—41no—
397Q8—84no—
398Q8—84no—
399Q8—88no—
400D4—22no—
401D4—22no—
402D4—48no—
403D4—41no—
404D4—41no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—48no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—44no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—88no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—84no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—164no—
434C2×C4#39412yes—
435C2×C4#39412yes—
436C2×C4#39412yes—
437C2×C4#44914yes—
438C2×C4#45114yes—
439C2×C4#46712yes—
440C2×C4#46812yes—
441C2×C4#47112yes—
442C2×C4#47212yes—
443C2×C4#48414yes—
444C2×C4#48514yes—
445C2×C4C2×#12011yescharacteristic
446C2×C4C2×#12111yescharacteristic
447C2×C4C2×#12111yescharacteristic
448C2×C4C2×#12311yescharacteristic
449C2×C4C2×#12311yescharacteristic
450C2×C4C2×#12411yescharacteristic
451C2×C4C22×Q1611yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
454C2×C4C2×#13711yescharacteristic
455C2×C4C2×#13712yes—
456C2×C4#57512yes—
457C8—24no—
458C8—44no—
459C8—42no—
460C8—82no—
461C8—82no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—22no—
467C22—22no—
468C22—42no—
469C22—42no—
470C22—42no—
471C22—81no—
472C22—81no—
473C22—81no—
474C22—81no—
475C22#203614yes—
476C22C2×#39411yescharacteristic
477C22#303911yescharacteristic
478C22#304011yescharacteristic
479C22C2×#46711yescharacteristic
480C22C2×#46811yescharacteristic
481C22C2×#47111yescharacteristic
482C22C2×#47211yescharacteristic
483C4—21no—
484C4—21no—
485C4—22no—
486C4—42no—
487C4—81no—
488C4—84no—
489C4—81no—
490C4—162no—
491C4#203614yes—
492C4#205114yes—
493C4C2×#45111yescharacteristic
494C4C2×#44911yescharacteristic
495C4#303712yes—
496C4#305612yes—
497C4C2×#48511yescharacteristic
498C4C2×#48411yescharacteristic
499C2—24no—
500C2—82no—
501C2#1927211yescharacteristic
502C2#1927711yescharacteristic
503C2#1929511yescharacteristic
504C2#1929911yescharacteristic
505C2C2×#203611yescharacteristic
506C2C2×#205111yescharacteristic
507C2#2354711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1990C2semidirect
5. C2×#1567C2semidirect
6. C2×#1548C2semidirect
7. #10483C2semidirect
8. #10481C2semidirect
9. #8043C2semidirect

Maximal subgroups

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

subgroupindex
C2×#30792
#213732
C2×#19902
#206482
C2×#15672
C2×#15482
#104832
#104812
#80432