Finite Group Databasebeta

← #380366

Subgroups — #380366

Order 512.

1062 conjugacy classes of subgroups.

The subgroup lattice has 501 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20843C211yesmaximal, characteristic
2#20835C211yesmaximal, characteristic
3#20335C211yesmaximal, characteristic
4#11376C212yesmaximal, complemented
5#11317C212yesmaximal, complemented
6#11291C212yesmaximal, complemented
7#11221C212yesmaximal, complemented
8#10734C211yesmaximal, characteristic
9#10733C211yesmaximal, characteristic
10#10732C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C2×#427C2212yes—
16C2×#424C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C22×C22⋊C8C2211yescharacteristic
20#2280C2211yescharacteristic
21C2×C4×C22⋊C4C2211yescharacteristic
22#1854C414yescomplemented
23#1853C412yescomplemented
24#1852C412yescomplemented
25#1851—22no—
26#1851—22no—
27C2×#341C2212yes—
28C2×#341C2212yes—
29#1735C2212yes—
30C2×#339C2212yes—
31C2×#339C2212yes—
32C2×#339C2212yes—
33C2×#339C2212yes—
34#1729—22no—
35#1729C2212yes—
36#1729C2211yescharacteristic
37#1729C2212yes—
38#1729C2211yescharacteristic
39#1728—22no—
40C2×#336C2212yes—
41C2×#336C2212yes—
42C24×C4C2311yescharacteristic
43C22×M16—21no—
44C22×M16—22no—
45C22×M16—22no—
46C22×M16—22no—
47C22×M16—42no—
48C22×M16—42no—
49C22×M16—42no—
50C22×M16—42no—
51C22×M16D412yes—
52C22×M16D412yes—
53C22×M16Q814yes—
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2312yes—
57C23×C8—21no—
58C23×C8C2312yes—
59C23×C8C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—42no—
64C2×#117—42no—
65C2×#117C2×C412yes—
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2312yes—
69C22×C4⋊C4C2×C412yes—
70C22×C22⋊C4—21no—
71C22×C22⋊C4—21no—
72C22×C42—21no—
73C22×C42C2312yes—
74#427—42no—
75#427—42no—
76C2×#107C2×C414yes—
77C2×#106C2×C414yes—
78#424—42no—
79#424—42no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82#407—22no—
83#407—22no—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×C22⋊C8—21no—
87C2×C22⋊C8—21no—
88C2×C22⋊C8C2×C411yescharacteristic
89C2×C22⋊C8C2×C411yescharacteristic
90C2×C22⋊C8C2×C412yes—
91#381—21no—
92#380C2×C412yes—
93#380C2×C412yes—
94#379—21no—
95C4×C22⋊C4—21no—
96C4×C22⋊C4—21no—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C2×#95—21no—
100#341—42no—
101#341—42no—
102#340C2×C414yes—
103#340C2×C414yes—
104#339—42no—
105#339—42no—
106#339—42no—
107#339—42no—
108#336—22no—
109#336—22no—
110#336—24no—
111#336—24no—
112#336—42no—
113#336—42no—
114C25#4111yescharacteristic
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4C22×C411yescharacteristic
122C23×C4C22×C411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×Q812yes—
126C23×C4#4112yes—
127C23×C4#4112yes—
128C23×C4#4111yescharacteristic
129C23×C4#4112yes—
130C23×C4C2411yescharacteristic
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—22no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—42no—
148C2×M16—42no—
149C2×M16—42no—
150C2×M16—42no—
151C2×M16—48no—
152C2×M16—48no—
153C2×M16—42no—
154C2×M16—42no—
155C2×M16—48no—
156C2×M16—48no—
157C2×M16C2×Q814yes—
158C22×C8—24no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8C22×C412yes—
179C22×C8C22×C412yes—
180C22×C8C2×D412yes—
181C22×C8C2×D412yes—
182#117—22no—
183#117—22no—
184#117—22no—
185#117—22no—
186#117—22no—
187#117—22no—
188#117—22no—
189#117—22no—
190#117—44no—
191#117—44no—
192#117—42no—
193#117—42no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22×C414yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—42no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—42no—
219C2×C42—42no—
220#107—44no—
221#106—44no—
222C4⋊C8—44no—
223C4⋊C8—44no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228C22⋊C8—42no—
229#95—22no—
230#95—22no—
231#95—42no—
232C24—22no—
233C24—22no—
234C24—42no—
235C24—42no—
236C24C4×D411yescharacteristic
237C24C4×D411yescharacteristic
238C24#12512yes—
239C24#12612yes—
240C24C2×#4111yescharacteristic
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4#11712yes—
277C22×C4#11712yes—
278C22×C4C4×D411yescharacteristic
279C22×C4C4×D411yescharacteristic
280C22×C4C4×Q812yes—
281C22×C4C4×Q812yes—
282C22×C4C22⋊Q812yes—
283C22×C4C22⋊Q812yes—
284C22×C4#12312yes—
285C22×C4#12312yes—
286C22×C4#12512yes—
287C22×C4#12612yes—
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×Q811yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4H(5,𝔽2)12yes—
295C22×C4H(5,𝔽2)12yes—
296C22×C4H(5,𝔽2)12yes—
297C22×C42−1+412yes—
298M16—24no—
299M16—24no—
300M16—48no—
301M16—48no—
302M16—48no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16—42no—
307M16—42no—
308M16—42no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—48no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—44no—
344C22⋊C4—44no—
345C22⋊C4—48no—
346C22⋊C4—42no—
347C22⋊C4—42no—
348C22⋊C4—42no—
349C22⋊C4—42no—
350C22⋊C4—82no—
351C22⋊C4—82no—
352C42—44no—
353C42—41no—
354C42—41no—
355C42—42no—
356C42—42no—
357C42—41no—
358C42—41no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C23—22no—
364C23—22no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—22no—
370C23—22no—
371C23—22no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23#38712yes—
384C23#38712yes—
385C23C2×C4×D411yescharacteristic
386C23C2×#12511yescharacteristic
387C23C2×#12611yescharacteristic
388C23#53812yes—
389C23#53912yes—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—44no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—88no—
424C2×C4—82no—
425C2×C4#38712yes—
426C2×C4#38712yes—
427C2×C4C2×#11711yescharacteristic
428C2×C4C2×C4×Q811yescharacteristic
429C2×C4#51812yes—
430C2×C4#52012yes—
431C2×C4C2×C22⋊Q811yescharacteristic
432C2×C4C2×#12311yescharacteristic
433C2×C4#53411yescharacteristic
434C2×C4#53411yescharacteristic
435C2×C4#53511yescharacteristic
436C2×C4#53611yescharacteristic
437C2×C4#53812yes—
438C2×C4#53812yes—
439C2×C4#53812yes—
440C2×C4#53912yes—
441C2×C4#53912yes—
442C2×C4#53912yes—
443C2×C4#54312yes—
444C2×C4#54412yes—
445C8—42no—
446C8—42no—
447C8—42no—
448C8—42no—
449C8—44no—
450C8—81no—
451C8—81no—
452C22—22no—
453C22—22no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22—42no—
467C22—42no—
468C22—42no—
469C22—42no—
470C22—84no—
471C22—84no—
472C22C2×#38711yescharacteristic
473C22#232111yescharacteristic
474C22#232211yescharacteristic
475C22#264312yes—
476C22#264612yes—
477C22#295612yes—
478C22#295712yes—
479C4—22no—
480C4—22no—
481C4—84no—
482C4—82no—
483C4—84no—
484C4—82no—
485C4#231812yes—
486C4#232912yes—
487C4#262211yescharacteristic
488C4#262311yescharacteristic
489C4#262611yescharacteristic
490C4#262711yescharacteristic
491C2—22no—
492C2—22no—
493C2—42no—
494C2—42no—
495C2#1426511yescharacteristic
496C2#2104211yescharacteristic
497C2#2104311yescharacteristic
498C2#2206212yes—
499C2#2209712yes—
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 ◁ C23×M16 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11376C2semidirect
5. #11317C2semidirect
6. #11291C2semidirect
7. #11221C2semidirect
22. #1854C4semidirect
23. #1853C4semidirect
24. #1852C4semidirect

Maximal subgroups

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

subgroupindex
#208432
#208352
#203352
#113762
#113172
#112912
#112212
#107342
#107332
#107322
#107152