Finite Group Databasebeta

← #391346

Subgroups — #391346

Order 512.

1216 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2#14697C212yesmaximal
3#11327C212yesmaximal, complemented
4#11316C212yesmaximal, complemented
5#11293C212yesmaximal, complemented
6#11219C212yesmaximal, complemented
7#11199C212yesmaximal
8#11095C211yesmaximal, characteristic
9#10712C211yesmaximal, characteristic
10C23×C42C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C22⋊C8C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15C2×#389C2212yes—
16C2×#389C2212yes—
17C2×#385—22no—
18C2×#380C2212yes—
19C2×#380C2212yes—
20#2075C418yescomplemented
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2211yescharacteristic
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2211yescharacteristic
28C2×#340C2212yes—
29C2×#336C2212yes—
30C2×#336C2212yes—
31#1436—22no—
32#1436—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C22⋊#95—22no—
36C24×C4D412yes—
37C24×C4C2311yescharacteristic
38C23×C8C2312yes—
39C23×C8C2311yescharacteristic
40C23×C8C2311yescharacteristic
41C22×C4⋊C4—22no—
42C22×C4⋊C4—22no—
43C22×C4⋊C4C2312yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C22⋊C4—22no—
48C22×C22⋊C4—22no—
49C22×C42—42no—
50C22×C42—42no—
51C22×C42—41no—
52C22×C42—41no—
53C22×C42D411yescharacteristic
54C22×C42D411yescharacteristic
55C22×C42C2312yes—
56C2×#107—41no—
57C2×#107—41no—
58C2×#107C2×C414yescomplemented
59C2×#106—42no—
60C2×#106—41no—
61C2×C22⋊C8—22no—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8—41no—
66C2×C22⋊C8C2×C414yes—
67#389—42no—
68#389—42no—
69#389—42no—
70#389—42no—
71#385—48no—
72#380—24no—
73#380—42no—
74#380—42no—
75#380C2×C418yes—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84#340—42no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—41no—
89#340—42no—
90#340C2×C414yescomplemented
91#340C2×C418yescomplemented
92#336—42no—
93#336—42no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—21no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—24no—
107C23×C4—24no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—42no—
111C23×C4—42no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4C22×C412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D412yes—
117C23×C4C2×D412yes—
118C23×C4#4112yes—
119C23×C4C2411yescharacteristic
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C22×C412yes—
136C22×C8C22×C412yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C22×C414yes—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—44no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—41no—
180C2×C42—42no—
181C2×C42—42no—
182C2×C42—41no—
183C2×C42—48no—
184C2×C42—44no—
185C2×C42—44no—
186#107—44no—
187#106—44no—
188C22⋊C8—44no—
189C22⋊C8—44no—
190#95—28no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—44no—
195#95—44no—
196#95—44no—
197#95—44no—
198C24—22no—
199C24—21no—
200C24—22no—
201C24—21no—
202C24—22no—
203C24—22no—
204C24—22no—
205C24C4×D412yes—
206C24AGL(1,ℤ/8)14yes—
207C24C22×D411yescharacteristic
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—48no—
235C22×C4—48no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—48no—
240C22×C4—48no—
241C22×C4—42no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C4×D412yes—
278C22×C4#13514yes—
279C22×C4#13714yes—
280C22×C4C23×C411yescharacteristic
281C22×C4C2×#4111yescharacteristic
282C22×C4H(5,𝔽2)11yescharacteristic
283C22×C4H(5,𝔽2)11yescharacteristic
284C22×C4H(5,𝔽2)11yescharacteristic
285C22×C4H(5,𝔽2)12yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288C22×C42−1+412yes—
289C22×C42−1+411yescharacteristic
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—88no—
324C4⋊C4—88no—
325C22⋊C4—416no—
326C22⋊C4—88no—
327C42—44no—
328C42—44no—
329C42—42no—
330C42—42no—
331C42—42no—
332C42—42no—
333C23—24no—
334C23—22no—
335C23—24no—
336C23—24no—
337C23—24no—
338C23—22no—
339C23—24no—
340C23—24no—
341C23—24no—
342C23—24no—
343C23—41no—
344C23—42no—
345C23—42no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—42no—
353C23#44214yes—
354C23C2×C4×D411yescharacteristic
355C23#53411yescharacteristic
356C23#53412yes—
357C23#53612yes—
358C23#53711yescharacteristic
359C23C2×#13512yes—
360C23C2×AGL(1,ℤ/8)12yes—
361C23C2×#13712yes—
362C23#57512yes—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—432no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4#51811yescharacteristic
418C2×C4#52012yes—
419C2×C4#52012yes—
420C2×C4#52011yescharacteristic
421C2×C4#53811yescharacteristic
422C2×C4#53912yes—
423C2×C4#53912yes—
424C2×C4#53911yescharacteristic
425C2×C4#56012yes—
426C2×C4#56011yescharacteristic
427C2×C4H(3,𝔽4)11yescharacteristic
428C2×C4#56212yes—
429C2×C4#56212yes—
430C2×C4#56312yes—
431C2×C4#56311yescharacteristic
432C2×C4#56312yes—
433C2×C4#56311yescharacteristic
434C2×C4#56312yes—
435C8—42no—
436C8—42no—
437C8—82no—
438C22—28no—
439C22—24no—
440C22—44no—
441C22—42no—
442C22—44no—
443C22—42no—
444C22—44no—
445C22—44no—
446C22—44no—
447C22#232212yes—
448C22#232511yescharacteristic
449C22#281211yescharacteristic
450C22#281412yes—
451C22#281711yescharacteristic
452C22C4×#13511yescharacteristic
453C22C2×#44211yescharacteristic
454C22C4×#13712yes—
455C22#318812yes—
456C22#319012yes—
457C22#319112yes—
458C22#319312yes—
459C22#319412yes—
460C22#319612yes—
461C22#323112yes—
462C22#323312yes—
463C22#323412yes—
464C22#323812yes—
465C22#324712yes—
466C22#325511yescharacteristic
467C22#325611yescharacteristic
468C4—42no—
469C4—44no—
470C4—44no—
471C4—84no—
472C4—84no—
473C4—162no—
474C4—162no—
475C4#237611yescharacteristic
476C4#237911yescharacteristic
477C4#238312yes—
478C4#282712yes—
479C4#283411yescharacteristic
480C4#283511yescharacteristic
481C2—44no—
482C2#1462711yescharacteristic
483C2#2207712yes—
484C2#2208112yes—
485C2#2218411yescharacteristic
486C2#2218911yescharacteristic
487C2#3270912yes—
488C2#3274512yes—
489C2#3274712yes—
490C2#3278712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11327C2semidirect
4. #11316C2semidirect
5. #11293C2semidirect
6. #11219C2semidirect
20. #2075C4semidirect
58. C2×#107C2×C4semidirect
90. #340C2×C4semidirect
91. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
#146972
#113272
#113162
#112932
#112192
#111992
#110952
#107122