Finite Group Databasebeta

← #360802

Subgroups — #360802

Order 512.

1348 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34184C211yesmaximal, characteristic
2#20764C211yesmaximal, characteristic, complemented
3#20648C211yesmaximal, characteristic
4C2×#1759C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic, complemented
6#11276C214yesmaximal, complemented
7#10858C211yesmaximal, characteristic, complemented
8#10840C211yesmaximal, characteristic, complemented
9#10781C214yesmaximal
10C2×#520C2212yes—
11C2×C4×#41C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×#104C2212yes—
14C2×#417C2212yes—
15C2×C4×M16C2211yescharacteristic
16#2351—22no—
17#2349—24no—
18#2349—22no—
19#2317—22no—
20#2317C2212yes—
21#2285—21no—
22#2285C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382—21no—
25C2×#382C2211yescharacteristic
26C2×#376C2211yescharacteristic
27#1836—21no—
28#1836—21no—
29#1834—22no—
30#1803—22no—
31#1802—22no—
32#1759C418yescomplemented
33C2×#343C2214yes—
34#1736C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2212yes—
38#1442—24no—
39C4⋊#95C2214yes—
40C22×#41C2311yescharacteristic
41C22×M16C2312yes—
42C22×M16C2312yes—
43#520—28no—
44#520—44no—
45C4×#41—22no—
46C4×#41—22no—
47C4×#41C2×C414yes—
48C4×#41Q814yes—
49C2×C4×Q8—22no—
50C2×C4×Q8D412yes—
51C2×C4×D4—22no—
52C2×C4×D4—22no—
53C2×C4×D4—24no—
54C2×C4×D4D412yes—
55C2×#117—22no—
56C2×#117—21no—
57C2×#117C2312yes—
58C2×#117C2311yescharacteristic
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C42—21no—
64C22×C42C2312yes—
65C22×C42C2311yescharacteristic
66#423—22no—
67#423—22no—
68C2×C4⋊C8—22no—
69C2×#104—24no—
70C2×#104—24no—
71C2×#104C2×C418yes—
72#417—28no—
73C2×#103—22no—
74C2×#102—22no—
75C4×M16—22no—
76C4×M16C2×C414yescomplemented
77C2×#97—21no—
78C2×C4×C8—21no—
79#389—24no—
80#389—24no—
81#389—22no—
82#389—22no—
83#385—24no—
84#385—24no—
85#385—24no—
86#384—22no—
87#384—22no—
88#384—21no—
89#384—21no—
90#384D412yes—
91#382—24no—
92#382—22no—
93#382—21no—
94#382—22no—
95#382—21no—
96#382D411yescharacteristic
97#382D411yescharacteristic
98#380—24no—
99#380—22no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102#376—22no—
103#376C2×C414yescomplemented
104C2×#95—24no—
105C2×#95—22no—
106C2×#95—28no—
107C2×#95D414yes—
108#343—44no—
109#340—24no—
110#340—24no—
111#340—24no—
112#339—44no—
113#339C2×C418yescomplemented
114C2×#41—21no—
115C2×#41—21no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41C22×C412yes—
119C2×#41C2×Q812yes—
120C2×#41#4114yes—
121C2×#41#4114yes—
122C22×Q8C2×D411yescharacteristic
123C22×D4—21no—
124C22×D4C2×D411yescharacteristic
125C23×C4—21no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4C2×D414yes—
131C23×C4C2×D412yes—
132C23×C4C2411yescharacteristic
133C2×M16—22no—
134C2×M16—22no—
135C2×M16—22no—
136C2×M16—22no—
137C2×M16—22no—
138C2×M16—24no—
139C2×M16—22no—
140C2×M16—22no—
141C2×M16—24no—
142C2×M16C22×C414yes—
143C22×C8—22no—
144C22×C8—22no—
145C4×Q8—24no—
146C4×Q8—44no—
147C4×Q8—44no—
148C4×Q8C4⋊C418yes—
149C4×D4—28no—
150C4×D4—28no—
151C4×D4—24no—
152C4×D4—416no—
153C4×D4—44no—
154C4×D4—44no—
155C4×D4C4⋊C418yes—
156#117—21no—
157#117—21no—
158#117—24no—
159#117—24no—
160#117—24no—
161#117—24no—
162#117—48no—
163#117C22×C412yes—
164#117C2×Q812yes—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—21no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—22no—
190C2×C42—22no—
191C2×C42—21no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C414yes—
202C2×C42C22×C412yes—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×Q812yes—
207C2×C42#4112yes—
208C2×C42#4112yes—
209C4⋊C8—44no—
210#104—216no—
211#103—44no—
212#102—44no—
213#97—24no—
214C4×C8—24no—
215#95—24no—
216#95—24no—
217#95—28no—
218#95—22no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—88no—
223C24—21no—
224C24C22×D411yescharacteristic
225#41—24no—
226#41—24no—
227#41—24no—
228#41—24no—
229#41—24no—
230#41—24no—
231#41#11718yes—
232#41#12518yes—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8C2×C4⋊C412yes—
237C2×Q8#12412yes—
238C2×Q8#12412yes—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—24no—
242C2×D4—24no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—21no—
246C2×D4C2×C4⋊C412yes—
247C2×D4#12412yes—
248C2×D4#12412yes—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—28no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—28no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—48no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—48no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—84no—
276C22×C4—84no—
277C22×C4C4×D414yes—
278C22×C4#12012yes—
279C22×C4#12012yes—
280C22×C4C22⋊Q814yes—
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×Q811yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4C2×#4112yes—
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C22×C4H(5,𝔽2)12yes—
290C22×C42−1+412yes—
291M16—24no—
292M16—24no—
293M16—42no—
294M16—42no—
295M16—44no—
296M16—44no—
297C2×C8—24no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—22no—
304C4⋊C4—28no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—48no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—82no—
315C4⋊C4—82no—
316C4⋊C4—84no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4C2×C4⋊C414yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—28no—
324C22⋊C4—48no—
325C22⋊C4—44no—
326C42—22no—
327C42—22no—
328C42—28no—
329C42—24no—
330C42—24no—
331C42—48no—
332C42—42no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42C2×C4⋊C414yes—
337C42C4×D412yes—
338C42C4×D412yes—
339C42C22⋊Q812yes—
340C42C22⋊Q812yes—
341C42#12314yes—
342C23—22no—
343C23—22no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23C2×C4×D411yescharacteristic
350C23C2×#12011yescharacteristic
351C23C2×C22⋊Q811yescharacteristic
352C23#54612yes—
353C23#54812yes—
354Q8—22no—
355Q8—22no—
356Q8—22no—
357Q8#38214yes—
358D4—28no—
359D4—28no—
360D4—22no—
361D4—22no—
362D4—22no—
363D4#38214yes—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—28no—
372C2×C4—28no—
373C2×C4—24no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—88no—
390C2×C4—84no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—88no—
394C2×C4—88no—
395C2×C4—82no—
396C2×C4—84no—
397C2×C4—168no—
398C2×C4#38514yes—
399C2×C4#38514yes—
400C2×C4C22×C4⋊C411yescharacteristic
401C2×C4C2×#11712yes—
402C2×C4C2×C4×D411yescharacteristic
403C2×C4#52012yes—
404C2×C4C2×C22⋊Q811yescharacteristic
405C2×C4C2×#12311yescharacteristic
406C2×C4C2×#12411yescharacteristic
407C2×C4C2×#12411yescharacteristic
408C2×C4C2×#12512yes—
409C2×C4#53511yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#54412yes—
412C2×C4#54612yes—
413C2×C4#54812yes—
414C2×C4#55212yes—
415C2×C4#55212yes—
416C2×C4#55312yes—
417C2×C4#55312yes—
418C8—44no—
419C8—82no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—24no—
426C22—44no—
427C22—42no—
428C22—42no—
429C22—44no—
430C22C2×#38511yescharacteristic
431C22#235212yes—
432C22#243511yescharacteristic
433C22#243611yescharacteristic
434C22#244212yes—
435C22#302112yes—
436C22#302212yes—
437C4—24no—
438C4—24no—
439C4—24no—
440C4—44no—
441C4—84no—
442C4—82no—
443C4—84no—
444C4—82no—
445C4—162no—
446C4C2×#38211yescharacteristic
447C4#231711yescharacteristic
448C4#235212yes—
449C4#244212yes—
450C4#250311yescharacteristic
451C4#250411yescharacteristic
452C2—22no—
453C2—22no—
454C2—44no—
455C2#1388111yescharacteristic
456C2#2086512yes—
457C2#2588012yes—
458C2#2592711yescharacteristic
459C2#2592811yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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. #20764C2semidirect
5. #14348C2semidirect
6. #11276C2semidirect
7. #10858C2semidirect
8. #10840C2semidirect
32. #1759C4semidirect
76. C4×M16C2×C4semidirect
103. #376C2×C4semidirect
113. #339C2×C4semidirect

Maximal subgroups

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

subgroupindex
#341842
#207642
#206482
C2×#17592
#143482
#112762
#108582
#108402
#107812