Finite Group Databasebeta

← #360901

Subgroups — #360901

Order 512.

1118 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20628C211yesmaximal, characteristic
2#20379C211yesmaximal, characteristic
3#20347C211yesmaximal, characteristic
4#11317C212yesmaximal, complemented
5#11291C212yesmaximal, complemented
6#10887C212yesmaximal, complemented
7#10809C212yesmaximal, complemented
8#10733C211yesmaximal, characteristic
9#10733C211yesmaximal, characteristic
10#10724C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#424C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1798—21no—
22#1798—21no—
23#1797—22no—
24#1750—22no—
25#1749—21no—
26C4×C22⋊C8—21no—
27C2×#341C2212yes—
28C2×#341C2212yes—
29#1736C414yescomplemented
30#1736C414yescomplemented
31#1735C2211yescharacteristic
32#1735C2211yescharacteristic
33#1735C2211yescharacteristic
34#1735C2211yescharacteristic
35C2×#339C2212yes—
36C2×#339C2212yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39#1729C2212yes—
40#1729C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C24×C4C2311yescharacteristic
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—42no—
47C22×M16—42no—
48C22×M16C2×C412yes—
49C22×M16C2×C412yes—
50C22×M16C2312yes—
51C22×M16C2312yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C2×#117—21no—
56C2×#117—21no—
57C2×#117—21no—
58C2×#117—21no—
59C2×#117—24no—
60C2×#117—24no—
61C2×#117—42no—
62C2×#117—42no—
63C2×#117C2×C412yes—
64C2×#117C2×C412yes—
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C22×C4⋊C4C2×C412yes—
70C22×C4⋊C4C2×C412yes—
71C22×C22⋊C4—21no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—21no—
74C22×C42—21no—
75C22×C42—21no—
76C22×C42C2312yes—
77#424—24no—
78#424—42no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81#407—22no—
82#407—22no—
83#407—22no—
84#407Q814yes—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8D412yes—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94#405—24no—
95#405—42no—
96C2×#97—22no—
97C2×C4×C8—22no—
98C4×C22⋊C4—22no—
99C4×C22⋊C4—22no—
100C2×#95—21no—
101#341—42no—
102#341—42no—
103#340C2×C414yes—
104#340C2×C414yes—
105#340C2×C414yes—
106#340C2×C414yes—
107#339—24no—
108#339—24no—
109#339—42no—
110#339—42no—
111#336—22no—
112#336—22no—
113#336—22no—
114#336—22no—
115#336—42no—
116#336—42no—
117C25#4111yescharacteristic
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C22×C411yescharacteristic
126C23×C4C22×C411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×Q812yes—
130C23×C4#4112yes—
131C23×C4#4112yes—
132C23×C4#4111yescharacteristic
133C23×C4#4112yes—
134C23×C4C2411yescharacteristic
135C2×M16—24no—
136C2×M16—24no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—42no—
152C2×M16—48no—
153C2×M16—48no—
154C2×M16C2×Q814yes—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8C22×C412yes—
172C22×C8C22×C412yes—
173C22×C8C2×D412yes—
174C22×C8C2×D412yes—
175#117—22no—
176#117—22no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—24no—
182#117—24no—
183#117—44no—
184#117—44no—
185#117—44no—
186#117—44no—
187#117—42no—
188#117—42no—
189#117#4114yes—
190#117#4114yes—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C22×C412yes—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—44no—
208C2×C22⋊C4—44no—
209C2×C22⋊C4—44no—
210C2×C22⋊C4—42no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—44no—
224C2×C42—44no—
225C4⋊C8—44no—
226C4⋊C8—44no—
227C22⋊C8—24no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C22⋊C8—22no—
231C22⋊C8—22no—
232C22⋊C8—42no—
233C22⋊C8—42no—
234#97—44no—
235C4×C8—44no—
236#95—42no—
237C24—22no—
238C24—22no—
239C24—42no—
240C24—42no—
241C24C4×D411yescharacteristic
242C24C4×D411yescharacteristic
243C24#12512yes—
244C24#12612yes—
245C24C2×#4111yescharacteristic
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—44no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4#11712yes—
286C22×C4#11712yes—
287C22×C4C4×D411yescharacteristic
288C22×C4C4×D411yescharacteristic
289C22×C4C4×Q812yes—
290C22×C4C4×Q812yes—
291C22×C4C22⋊Q812yes—
292C22×C4C22⋊Q812yes—
293C22×C4#12312yes—
294C22×C4#12312yes—
295C22×C4#12512yes—
296C22×C4#12612yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×Q811yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4112yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4H(5,𝔽2)12yes—
306C22×C4H(5,𝔽2)12yes—
307M16—24no—
308M16—24no—
309M16—48no—
310M16—48no—
311M16—42no—
312M16—42no—
313M16—42no—
314M16—42no—
315M16—42no—
316M16—42no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—22no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—42no—
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—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C4⋊C4—22no—
344C4⋊C4—22no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—44no—
350C4⋊C4—44no—
351C4⋊C4#11714yes—
352C4⋊C4#11714yes—
353C22⋊C4—24no—
354C22⋊C4—24no—
355C22⋊C4—44no—
356C22⋊C4—44no—
357C22⋊C4—48no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C42—24no—
361C42—24no—
362C42—44no—
363C42—44no—
364C42—42no—
365C42—42no—
366C42—42no—
367C42—42no—
368C42—42no—
369C42—42no—
370C23—22no—
371C23—22no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—22no—
377C23—22no—
378C23—22no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23#38712yes—
391C23#38712yes—
392C23C2×C4×D411yescharacteristic
393C23C2×#12511yescharacteristic
394C23C2×#12611yescharacteristic
395C23#55112yes—
396C23#55112yes—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—24no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—88no—
431C2×C4—88no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4#38712yes—
437C2×C4#38712yes—
438C2×C4C2×#11711yescharacteristic
439C2×C4C2×#11711yescharacteristic
440C2×C4C2×#11711yescharacteristic
441C2×C4C2×C4×Q811yescharacteristic
442C2×C4#51811yescharacteristic
443C2×C4#51811yescharacteristic
444C2×C4C2×C22⋊Q811yescharacteristic
445C2×C4C2×#12311yescharacteristic
446C2×C4#52912yes—
447C2×C4#52912yes—
448C2×C4#53812yes—
449C2×C4#53812yes—
450C2×C4#54612yes—
451C2×C4#54612yes—
452C2×C4#55312yes—
453C2×C4#55312yes—
454C2×C4#55412yes—
455C2×C4#55412yes—
456C8—42no—
457C8—42no—
458C8—42no—
459C8—42no—
460C8—44no—
461C8—82no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—21no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—22no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—42no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—84no—
481C22—84no—
482C22C2×#38711yescharacteristic
483C22#234211yescharacteristic
484C22#234211yescharacteristic
485C22#250512yes—
486C22#251012yes—
487C22#295512yes—
488C22#295612yes—
489C4—22no—
490C4—22no—
491C4—44no—
492C4—44no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4#228311yescharacteristic
497C4#231211yescharacteristic
498C4#234511yescharacteristic
499C4#234511yescharacteristic
500C4#244412yes—
501C4#245012yes—
502C2—22no—
503C2—22no—
504C2—42no—
505C2—42no—
506C2#1388311yescharacteristic
507C2#2104111yescharacteristic
508C2#2104211yescharacteristic
509C2#2258812yes—
510C2#2258812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11317C2semidirect
5. #11291C2semidirect
6. #10887C2semidirect
7. #10809C2semidirect
29. #1736C4semidirect
30. #1736C4semidirect

Maximal subgroups

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

subgroupindex
#206282
#203792
#203472
#113172
#112912
#108872
#108092
#107332
#107242
#107152