Finite Group Databasebeta

← #398685

Subgroups — #398685

Order 512.

1148 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C211yesmaximal, characteristic, complemented
2#20834C211yesmaximal, characteristic
3#20661C212yesmaximal
4C2×#1807C211yesmaximal, characteristic
5C22×#341C211yesmaximal, characteristic
6#11366C211yesmaximal, characteristic, complemented
7#11330C211yesmaximal, characteristic, complemented
8#11290C212yesmaximal, complemented
9#11238C211yesmaximal, characteristic, complemented
10#11216C212yesmaximal, complemented
11#10739C212yesmaximal
12C23×M16C2212yes—
13C24×C8C2211yescharacteristic
14C2×#428C2212yes—
15C2×#428C2212yes—
16C22×#106C2211yescharacteristic
17C22×#106C2211yescharacteristic
18C2×#424C2211yescharacteristic
19C2×#424C2211yescharacteristic
20C2×#407C2211yescharacteristic
21#2280C2211yescharacteristic
22C2×#380C2212yes—
23#2090—24no—
24#1850—22no—
25#1850—22no—
26#1809—22no—
27#1809—22no—
28#1808—22no—
29#1807—22no—
30#1807—22no—
31#1807—22no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#341—24no—
37C2×#341C2212yes—
38C2×#341C2211yescharacteristic
39C2×#341C2211yescharacteristic
40#1736C2212yes—
41#1736C2212yes—
42C2×#340C2212yes—
43#1729C2211yescharacteristic
44#1729C2211yescharacteristic
45C24×C4C2311yescharacteristic
46C22×M16—22no—
47C22×M16—24no—
48C22×M16—24no—
49C22×M16—24no—
50C22×M16—42no—
51C22×M16—42no—
52C22×M16Q818yes—
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2312yes—
56C22×M16C2312yes—
57C23×C8—22no—
58C23×C8—24no—
59C23×C8—22no—
60C23×C8—22no—
61C23×C8—42no—
62C23×C8D412yes—
63C23×C8D412yes—
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71#428—24no—
72#428—24no—
73#428—42no—
74#428—42no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—22no—
82C2×#106—22no—
83C2×#106—42no—
84#424—22no—
85#424—22no—
86#424—42no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—22no—
89#407—24no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92#381—21no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380D412yes—
98#380D412yes—
99#379—21no—
100C4×C22⋊C4—21no—
101C4×C22⋊C4—21no—
102C2×#95—22no—
103#343—24no—
104#343—24no—
105#343—42no—
106#343—42no—
107#341—24no—
108#341—22no—
109#341D414yes—
110#340—24no—
111#340—22no—
112#340—22no—
113#340—22no—
114#340—22no—
115#336—22no—
116#336—22no—
117C25#4111yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×Q814yes—
131C23×C4#4111yescharacteristic
132C23×C4#4114yes—
133C23×C4C2411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—28no—
138C2×M16—22no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—22no—
142C2×M16—22no—
143C2×M16—216no—
144C2×M16—21no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—22no—
151C2×M16—21no—
152C2×M16—44no—
153C2×M16—44no—
154C2×M16—48no—
155C2×M16—48no—
156C2×M16—48no—
157C2×M16C2×D414yes—
158C2×M16C2×Q814yes—
159C2×M16C2×Q814yes—
160C2×M16#4114yes—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—28no—
165C22×C8—24no—
166C22×C8—24no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—24no—
173C22×C8—24no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—42no—
177C22×C8—42no—
178C22×C8—48no—
179C22×C8—48no—
180C22×C8—44no—
181C22×C8—44no—
182C22×C8C2×D412yes—
183C22×C8C2×D412yes—
184C22×C8C2×D412yes—
185C22×C8#4112yes—
186#117—21no—
187#117—21no—
188#117—22no—
189#117—22no—
190#117—21no—
191#117—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—84no—
213C2×C42—21no—
214C2×C42—21no—
215#107—44no—
216#107—44no—
217#106—44no—
218#106—44no—
219#106—44no—
220#106—44no—
221C4⋊C8—44no—
222C4⋊C8—44no—
223C22⋊C8—42no—
224C22⋊C8—42no—
225#95—22no—
226#95—22no—
227#95—42no—
228#95—42no—
229C24—21no—
230C24—21no—
231C24—22no—
232C24—24no—
233C24—42no—
234C24#12111yescharacteristic
235C24#12111yescharacteristic
236C24#12514yes—
237C24C2×#4111yescharacteristic
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4C22⋊Q814yes—
278C22×C4C22⋊Q814yes—
279C22×C4#12312yes—
280C22×C4#12312yes—
281C22×C4#12514yes—
282C22×C4C4⋊Q812yes—
283C22×C4C4⋊Q812yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×Q812yes—
288C22×C4C2×#4111yescharacteristic
289C22×C4C2×#4112yes—
290C22×C4C2×#4112yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C4H(5,𝔽2)11yescharacteristic
293C22×C42−1+411yescharacteristic
294M16—24no—
295M16—24no—
296M16—24no—
297M16—24no—
298M16—24no—
299M16—416no—
300M16—42no—
301M16—42no—
302M16—42no—
303M16—42no—
304M16—42no—
305M16—42no—
306M16C22⋊Q818yes—
307C2×C8—24no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—22no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—48no—
319C2×C8—48no—
320C2×C8—416no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8#12114yes—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C22⋊C4—44no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—42no—
348C22⋊C4—44no—
349C22⋊C4—84no—
350C22⋊C4—84no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42—41no—
355C23—21no—
356C23—21no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—24no—
361C23—28no—
362C23—21no—
363C23—24no—
364C23—21no—
365C23—22no—
366C23—22no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—44no—
372C23#39812yes—
373C23#39812yes—
374C23#47114yes—
375C23C2×#12111yescharacteristic
376C23C2×#12512yes—
377C23#54811yescharacteristic
378C23#55311yescharacteristic
379C23#55312yes—
380C2×C4—24no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—24no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—162no—
411C2×C4—162no—
412C2×C4—162no—
413C2×C4—162no—
414C2×C4#39812yes—
415C2×C4#39812yes—
416C2×C4C2×#12111yescharacteristic
417C2×C4C2×C22⋊Q812yes—
418C2×C4C2×C22⋊Q812yes—
419C2×C4C2×#12311yescharacteristic
420C2×C4C2×C4⋊Q811yescharacteristic
421C2×C4#53211yescharacteristic
422C2×C4#53312yes—
423C2×C4#53411yescharacteristic
424C2×C4#53711yescharacteristic
425C2×C4#54312yes—
426C2×C4#54412yes—
427C2×C4D4211yescharacteristic
428C2×C4#54612yes—
429C2×C4#54612yes—
430C2×C4#54711yescharacteristic
431C2×C4#55112yes—
432C2×C4#55112yes—
433C2×C4#55212yes—
434C8—24no—
435C8—28no—
436C8—42no—
437C8—44no—
438C8—81no—
439C8—81no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—22no—
446C22—24no—
447C22—22no—
448C22—22no—
449C22—24no—
450C22—22no—
451C22—22no—
452C22—22no—
453C22—48no—
454C22—48no—
455C22—42no—
456C22—42no—
457C22#209014yescomplemented
458C22C2×#39811yescharacteristic
459C22#241611yescharacteristic
460C22#244212yes—
461C22#254311yescharacteristic
462C22#256912yes—
463C22C2×#47112yes—
464C22#306712yes—
465C4—22no—
466C4—22no—
467C4—82no—
468C4—82no—
469C4—162no—
470C4—162no—
471C4#241211yescharacteristic
472C4#243712yes—
473C4#243812yes—
474C4#249811yescharacteristic
475C4#265911yescharacteristic
476C4#285711yescharacteristic
477C2—22no—
478C2—24no—
479C2—22no—
480C2—42no—
481C2#1476611yescharacteristic
482C2C2×#209011yescharacteristic
483C2#2367411yescharacteristic
484C2#2480212yes—
485C2#2859312yes—
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2090C2semidirect
6. #11366C2semidirect
7. #11330C2semidirect
8. #11290C2semidirect
9. #11238C2semidirect
10. #11216C2semidirect
457. C22#2090semidirect

Maximal subgroups

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

subgroupindex
C2×#20902
#208342
#206612
C2×#18072
C22×#3412
#113662
#113302
#112902
#112382
#112162
#107392