Finite Group Databasebeta

← #369088

Subgroups — #369088

Order 512.

2132 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2858C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic, complemented
3#21260C211yesmaximal, characteristic
4C2×#1966C211yesmaximal, characteristic, complemented
5#21203C211yesmaximal, characteristic
6C2×#1888C214yesmaximal, complemented
7#10976C214yesmaximal, complemented
8#10839C211yesmaximal, characteristic, complemented
9C4×#339C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×#127C2211yescharacteristic
12C22×#124C2214yes—
13C2×#519C2212yes—
14C2×#423C2211yescharacteristic
15C2×#418C2212yes—
16C2×#418C2214yes—
17C2×#418C2212yes—
18C22×#101C2214yes—
19C2×C4×M16C2211yescharacteristic
20#2858—24no—
21#2858C418yescomplemented
22#2365—24no—
23#2344—24no—
24#2327—22no—
25#2310C2212yes—
26C4×C4⋊Q8—22no—
27C2×#390—24no—
28#2285C2211yescharacteristic
29C4×#117C2211yescharacteristic
30C2×C43C2211yescharacteristic
31#1982—24no—
32#1966—24no—
33#1961—22no—
34#1957—22no—
35#1888—216no—
36C2×#339C2214yes—
37C2×#339C2212yes—
38C2×#339C2212yes—
39C23×Q8C2312yes—
40C23×D4—22no—
41C23×D4C2312yes—
42C22×M16C2312yes—
43C22×M16C2312yes—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8C2×C414yes—
50C2×C4⋊Q8D414yes—
51C2×#127—21no—
52C2×#127—21no—
53C2×#127—22no—
54C2×#127—22no—
55C2×#127—22no—
56C2×#127—22no—
57C2×#127C2×C414yes—
58C2×#124—28no—
59C2×#124—28no—
60C2×#124—28no—
61C2×#124—44no—
62C2×#124—44no—
63C2×#124C2×C4116yes—
64#519—28no—
65#519—44no—
66C2×C4×Q8—24no—
67C2×C4×Q8—24no—
68C2×#117—24no—
69C2×#117C2311yescharacteristic
70C2×#117C2312yes—
71C2×#117C2311yescharacteristic
72C22×C4⋊C4D414yes—
73C22×C22⋊C4—28no—
74C22×C42—22no—
75C22×C42C2312yes—
76C22×C42C2311yescharacteristic
77#423—22no—
78#423—22no—
79C2×C4⋊C8—22no—
80#418—28no—
81#418—28no—
82#418—44no—
83#418—44no—
84C2×#102—28no—
85C2×#102—24no—
86C2×#102—24no—
87C2×#101—28no—
88C2×#101—28no—
89C2×#101—44no—
90C4×M16—22no—
91C4×M16—22no—
92C2×#97—21no—
93C2×C4×C8—21no—
94#391—24no—
95#391—24no—
96#390—232no—
97#389—28no—
98#387—216no—
99#384D412yes—
100#382D412yes—
101#381—24no—
102#380—22no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4D414yes—
106C4×C22⋊C4—22no—
107#376—21no—
108C43—21no—
109C43—22no—
110C43C2×C414yes—
111#339—28no—
112#339—24no—
113#339D418yescomplemented
114C25—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—24no—
122C22×Q8—24no—
123C22×Q8C22×C414yes—
124C22×Q8C2×D414yes—
125C22×D4—28no—
126C22×D4—24no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4—22no—
130C22×D4—24no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—24no—
134C22×D4—24no—
135C22×D4—416no—
136C22×D4—42no—
137C22×D4—42no—
138C22×D4—44no—
139C22×D4C22×C414yes—
140C23×C4—22no—
141C23×C4C2×D414yes—
142C23×C4C2×D412yes—
143C23×C4C2411yescharacteristic
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—22no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—22no—
152C2×M16—24no—
153C2×M16C2×D414yes—
154C22×C8—22no—
155C22×C8—22no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—28no—
159C4⋊Q8—24no—
160C4⋊Q8—24no—
161C4⋊Q8—28no—
162#127—24no—
163#127—24no—
164#127—28no—
165#127—42no—
166#127—42no—
167#127—44no—
168#124—232no—
169#124—416no—
170#124—416no—
171#124—416no—
172C4×Q8—416no—
173C4×Q8—48no—
174C4×Q8—48no—
175#117—22no—
176#117—24no—
177#117—22no—
178#117—24no—
179#117—22no—
180#117—416no—
181#117C2×D414yes—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4C22⋊C4116yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—232no—
195C2×C22⋊C4—416no—
196C2×C22⋊C4—416no—
197C2×C42—21no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—28no—
202C2×C42—28no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—48no—
207C2×C42C22×C414yes—
208C2×C42C22×C412yes—
209C2×C42C2×D414yes—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42#4114yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C4⋊C8—44no—
218#102—416no—
219#102—48no—
220#102—48no—
221#101—416no—
222#97—42no—
223C4×C8—42no—
224#95—216no—
225#95—24no—
226#95—24no—
227C24—24no—
228C24—416no—
229C24—42no—
230C24—42no—
231C24—44no—
232C24—44no—
233C24C22×D411yescharacteristic
234C2×Q8—28no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—28no—
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—28no—
241C2×Q8—28no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—44no—
245C2×Q8—42no—
246C2×Q8—42no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×D4—28no—
253C2×D4—24no—
254C2×D4—24no—
255C2×D4—28no—
256C2×D4—432no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—44no—
261C2×D4—42no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—44no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—48no—
272C2×D4—48no—
273C2×D4—88no—
274C2×D4—816no—
275C2×D4—88no—
276C22×C4—28no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—24no—
282C22×C4—28no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—48no—
288C22×C4C2×C22⋊C418yes—
289C22×C4C2×C22⋊C414yes—
290C22×C4#12012yes—
291C22×C4#12012yes—
292C22×C4#12314yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C22×D411yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4H(5,𝔽2)12yes—
302C22×C4H(5,𝔽2)12yes—
303M16—24no—
304M16—24no—
305M16—42no—
306M16—42no—
307M16—44no—
308M16—44no—
309C2×C8—24no—
310C2×C8—42no—
311C2×C8—42no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—22no—
316C4⋊C4—216no—
317C4⋊C4—216no—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—28no—
321C4⋊C4—416no—
322C4⋊C4—48no—
323C4⋊C4#12714yes—
324C22⋊C4—24no—
325C22⋊C4—464no—
326C22⋊C4—44no—
327C22⋊C4—42no—
328C42—24no—
329C42—24no—
330C42—24no—
331C42—24no—
332C42—416no—
333C42—416no—
334C42—42no—
335C42—42no—
336C42—42no—
337C42—48no—
338C42C4×D418yes—
339C42C4×D414yes—
340C42C4×D412yes—
341C42C4×D412yes—
342C42#12112yes—
343C42#12112yes—
344C42#12114yes—
345C23—24no—
346C23—416no—
347C23—48no—
348C23—44no—
349C23—88no—
350C23—816no—
351C23—816no—
352C23—82no—
353C23—82no—
354C23—84no—
355C23—88no—
356C23C22×C22⋊C411yescharacteristic
357C23C2×#12011yescharacteristic
358C23C2×#12311yescharacteristic
359C23D4212yes—
360C23#54612yes—
361Q8—44no—
362Q8—44no—
363Q8—48no—
364Q8—48no—
365Q8—44no—
366Q8—44no—
367D4—44no—
368D4—44no—
369D4—48no—
370D4—816no—
371D4—816no—
372D4—84no—
373D4—82no—
374D4—82no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—28no—
378C2×C4—28no—
379C2×C4—216no—
380C2×C4—24no—
381C2×C4—28no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—48no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4#38614yes—
394C2×C4#38614yes—
395C2×C4#45418yes—
396C2×C4C2×C4×D411yescharacteristic
397C2×C4C2×C4×D412yes—
398C2×C4C2×C4×D411yescharacteristic
399C2×C4#51812yes—
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12711yescharacteristic
403C2×C4#53212yes—
404C2×C4#53211yescharacteristic
405C2×C4#53511yescharacteristic
406C2×C4#53511yescharacteristic
407C2×C4#54012yes—
408C2×C4D4212yes—
409C2×C4D4212yes—
410C2×C4#54612yes—
411C2×C4#55012yes—
412C2×C4#55012yes—
413C2×C4#55912yes—
414C8—44no—
415C8—82no—
416C22—24no—
417C22—22no—
418C22—22no—
419C22—24no—
420C22—816no—
421C22—84no—
422C22—88no—
423C22—168no—
424C22—164no—
425C22—164no—
426C22C2×#38611yescharacteristic
427C22D4×C22⋊C412yes—
428C22#243111yescharacteristic
429C22#243311yescharacteristic
430C22#244512yes—
431C22C2×#45414yes—
432C4—24no—
433C4—48no—
434C4—44no—
435C4—48no—
436C4—84no—
437C4—82no—
438C4#188818yescomplemented
439C4C4×#12711yescharacteristic
440C4#232811yescharacteristic
441C4#236612yes—
442C4#242812yes—
443C4#254111yescharacteristic
444C4#254411yescharacteristic
445C2—22no—
446C2—22no—
447C2—164no—
448C2#1402511yescharacteristic
449C2C2×#188812yes—
450C2#2559112yes—
451C2#2574512yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#127C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34209C2semidirect
4. C2×#1966C2semidirect
6. C2×#1888C2semidirect
7. #10976C2semidirect
8. #10839C2semidirect
9. C4×#339C2semidirect
21. #2858C4semidirect
113. #339D4semidirect
438. C4#1888semidirect

Maximal subgroups

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

subgroupindex
C2×#28582
#342092
#212602
C2×#19662
#212032
C2×#18882
#109762
#108392
C4×#3392