Finite Group Databasebeta

← #388098

Subgroups — #388098

Order 512.

1146 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1850C211yesmaximal, characteristic
2#13875C212yesmaximal
3#11268C212yesmaximal, complemented
4#11231C212yesmaximal, complemented
5#11095C211yesmaximal, characteristic
6#11095C211yesmaximal, characteristic
7#10908C212yesmaximal, complemented
8#10850C212yesmaximal, complemented
9#10721C212yesmaximal
10C24×C8C2211yescharacteristic
11C22×#106C2211yescharacteristic
12C22×#106C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C2×#382C2212yes—
16C2×#380C2212yes—
17C2×#380C2212yes—
18C2×C4×C4⋊C4C2212yes—
19C2×C4×C22⋊C4C2211yescharacteristic
20#1850C418yescomplemented
21C2×#340C2212yes—
22C2×#340C2211yescharacteristic
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2211yescharacteristic
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#336C2212yes—
30#1443—22no—
31#1440—22no—
32#1437—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35#1434—22no—
36C24×C4C2311yescharacteristic
37C23×C8—41no—
38C23×C8—41no—
39C23×C8—42no—
40C23×C8—42no—
41C23×C8C2×C414yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D411yescharacteristic
52C22×C22⋊C4D411yescharacteristic
53C22×C42—22no—
54C22×C42C2312yes—
55C2×#106—24no—
56C2×#106—41no—
57C2×#106—41no—
58C2×#106—42no—
59C2×C4⋊C8—42no—
60C2×C4⋊C8—42no—
61C2×C4⋊C8C2×C418yes—
62C2×C22⋊C8—41no—
63C2×C22⋊C8—41no—
64C2×C22⋊C8C2×C414yes—
65#382—24no—
66#382—42no—
67#380—24no—
68#380—24no—
69#380—42no—
70#380—42no—
71C4×C4⋊C4—24no—
72C4×C4⋊C4—42no—
73C4×C22⋊C4C2×C414yes—
74C4×C22⋊C4Q814yes—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95D412yes—
85#340—24no—
86#340—41no—
87#340—42no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—41no—
92#340—42no—
93#336C2×C418yes—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4C22×C412yes—
108C23×C4C2×D412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×Q812yes—
112C23×C4C2411yescharacteristic
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—22no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—48no—
128C22×C8—44no—
129C22×C8—48no—
130C22×C8—44no—
131C22×C8—44no—
132C22×C8C22×C414yes—
133C22×C8C22×C412yes—
134C22×C8C22×C412yes—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
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#4114yes—
158C2×C4⋊C4#4114yes—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—42no—
161C2×C22⋊C4—44no—
162C2×C22⋊C4—44no—
163C2×C22⋊C4—44no—
164C2×C22⋊C4—44no—
165C2×C22⋊C4—82no—
166C2×C22⋊C4—82no—
167C2×C22⋊C4C4⋊C414yes—
168C2×C22⋊C4C4⋊C414yes—
169C2×C42—22no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—44no—
174C2×C42—44no—
175C2×C42—82no—
176C2×C42C22×C414yes—
177C2×C42C2×Q814yes—
178#106—44no—
179#106—44no—
180C4⋊C8—48no—
181C22⋊C8—44no—
182#95—44no—
183#95—44no—
184#95—44no—
185#95—44no—
186#95—44no—
187#95—44no—
188#95—82no—
189#95—82no—
190#95—82no—
191#95C4⋊C418yes—
192C24—21no—
193C24—21no—
194C24—22no—
195C24—22no—
196C24—22no—
197C24—22no—
198C24—22no—
199C24C2×C4⋊C412yes—
200C24#13514yes—
201C24C22×D411yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—24no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
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×C4C2×C4⋊C412yes—
265C22×C4C2×C4⋊C414yes—
266C22×C4C2×C4⋊C414yes—
267C22×C4C23×C411yescharacteristic
268C22×C4C22×Q811yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4H(5,𝔽2)12yes—
272C22×C4H(5,𝔽2)12yes—
273C22×C4H(5,𝔽2)11yescharacteristic
274C22×C4H(5,𝔽2)11yescharacteristic
275C22×C42−1+411yescharacteristic
276C22×C42−1+411yescharacteristic
277C2×C8—24no—
278C2×C8—22no—
279C2×C8—22no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—416no—
285C2×C8—41no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—42no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C22⋊C4—28no—
312C22⋊C4—44no—
313C22⋊C4—88no—
314C22⋊C4—88no—
315C42—24no—
316C42—24no—
317C42—24no—
318C42—24no—
319C42—88no—
320C23—22no—
321C23—24no—
322C23—22no—
323C23—24no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—24no—
328C23—24no—
329C23—24no—
330C23—42no—
331C23—41no—
332C23—41no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23#42714yes—
341C23C22×C4⋊C411yescharacteristic
342C23#53711yescharacteristic
343C23#53711yescharacteristic
344C23#54612yes—
345C23#54612yes—
346C23C2×#13512yes—
347C23#57512yes—
348C23#57712yes—
349C23#57712yes—
350C2×C4—22no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—82no—
387C2×C4—82no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—82no—
394C2×C4—82no—
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×C4C4×#4112yes—
404C2×C4#51812yes—
405C2×C4#51811yescharacteristic
406C2×C4#51911yescharacteristic
407C2×C4#53912yes—
408C2×C4#53912yes—
409C2×C4#54212yes—
410C2×C4#54212yes—
411C2×C4#54411yescharacteristic
412C2×C4#54411yescharacteristic
413C2×C4#55111yescharacteristic
414C2×C4#55312yes—
415C2×C4#55412yes—
416C2×C4#55412yes—
417C2×C4#55911yescharacteristic
418C2×C4#56012yes—
419C2×C4#56011yescharacteristic
420C2×C4H(3,𝔽4)11yescharacteristic
421C8—42no—
422C8—42no—
423C8—44no—
424C22—28no—
425C22—24no—
426C22—44no—
427C22—44no—
428C22—44no—
429C22—44no—
430C22—42no—
431C22—44no—
432C22—42no—
433C22#231611yescharacteristic
434C22#234912yes—
435C22#269112yes—
436C22#269311yescharacteristic
437C22#271211yescharacteristic
438C22C2×#42711yescharacteristic
439C22#291111yescharacteristic
440C22#291212yes—
441C22#325012yes—
442C22#325012yes—
443C22#326012yes—
444C22#326311yescharacteristic
445C22#326311yescharacteristic
446C22#329612yes—
447C22#329612yes—
448C22#330112yes—
449C22#330112yes—
450C22#330712yes—
451C22#331112yes—
452C22#331712yes—
453C22#332112yes—
454C4—42no—
455C4—44no—
456C4—44no—
457C4—84no—
458C4—84no—
459C4—162no—
460C4—162no—
461C4#233212yes—
462C4#237411yescharacteristic
463C4#237511yescharacteristic
464C4#276311yescharacteristic
465C4#279012yes—
466C4#280711yescharacteristic
467C2—44no—
468C2#1451411yescharacteristic
469C2#2196011yescharacteristic
470C2#2196711yescharacteristic
471C2#2270612yes—
472C2#2271012yes—
473C2#3148912yes—
474C2#3149112yes—
475C2#3155312yes—
476C2#3211312yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #11268C2semidirect
4. #11231C2semidirect
7. #10908C2semidirect
8. #10850C2semidirect
20. #1850C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18502
#138752
#112682
#112312
#110952
#109082
#108502
#107212