Finite Group Databasebeta

← #388099

Subgroups — #388099

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1851C211yesmaximal, characteristic
2#13876C212yesmaximal
3#11267C212yesmaximal, complemented
4#11232C212yesmaximal, complemented
5#11096C211yesmaximal, characteristic
6#11096C211yesmaximal, characteristic
7#10907C212yesmaximal, complemented
8#10849C212yesmaximal, complemented
9#10722C212yesmaximal
10C23×M16C2211yescharacteristic
11C22×#107C2211yescharacteristic
12C22×#107C2211yescharacteristic
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C2×#388—22no—
16C2×#384C2212yes—
17C2×#380C2212yes—
18C2×#380C2212yes—
19C2×C4×C4⋊C4C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1851C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2211yescharacteristic
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2211yescharacteristic
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31#1442—22no—
32#1439—22no—
33#1436—22no—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C22×M16—41no—
38C22×M16—41no—
39C22×M16—42no—
40C22×M16—42no—
41C22×M16C2×C414yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C22⋊C4—22no—
52C22×C22⋊C4—22no—
53C22×C22⋊C4D412yes—
54C22×C42—22no—
55C22×C42C2312yes—
56C2×#107—24no—
57C2×#107—41no—
58C2×#107—41no—
59C2×#107—42no—
60C2×C4⋊C8—42no—
61C2×C4⋊C8—42no—
62C2×C4⋊C8C2×C418yes—
63C2×C22⋊C8—41no—
64C2×C22⋊C8—41no—
65C2×C22⋊C8C2×C414yes—
66#388—48no—
67#384—24no—
68#384—42no—
69#380—24no—
70#380—24no—
71#380—42no—
72#380—42no—
73C4×C4⋊C4—24no—
74C4×C4⋊C4—42no—
75C4×C22⋊C4C2×C414yes—
76C4×C22⋊C4Q814yes—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83C2×#95—22no—
84C2×#95D411yescharacteristic
85C2×#95D411yescharacteristic
86#340—24no—
87#340—41no—
88#340—42no—
89#340—42no—
90#340—42no—
91#340—42no—
92#340—41no—
93#340—42no—
94#336C2×C418yes—
95C25C2×D411yescharacteristic
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—21no—
101C23×C4—21no—
102C23×C4—21no—
103C23×C4—21no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4C22×C412yes—
109C23×C4C2×D412yes—
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D412yes—
112C23×C4C2×Q812yes—
113C23×C4C2411yescharacteristic
114C2×M16—48no—
115C2×M16—44no—
116C2×M16—48no—
117C2×M16—44no—
118C2×M16—44no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—22no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—22no—
133C22×C8C22×C414yes—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
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—22no—
158C2×C4⋊C4—44no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4#4114yes—
165C2×C4⋊C4#4114yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4C4⋊C418yes—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—44no—
180C2×C42—44no—
181C2×C42—82no—
182C2×C42C22×C414yes—
183C2×C42C2×Q814yes—
184#107—44no—
185#107—44no—
186C4⋊C8—48no—
187C22⋊C8—44no—
188#95—44no—
189#95—44no—
190#95—44no—
191#95—44no—
192#95—44no—
193#95—44no—
194#95—82no—
195#95C4⋊C414yes—
196#95C4⋊C414yes—
197C24—21no—
198C24—21no—
199C24—22no—
200C24—22no—
201C24—22no—
202C24—22no—
203C24—21no—
204C24—21no—
205C24—21no—
206C24—21no—
207C24C2×C4⋊C412yes—
208C24C22×D411yescharacteristic
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—41no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
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×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4C2×C4⋊C412yes—
270C22×C4C2×C4⋊C414yes—
271C22×C4C2×C4⋊C414yes—
272C22×C4AGL(1,ℤ/8)14yes—
273C22×C4#13714yes—
274C22×C4C23×C411yescharacteristic
275C22×C4C22×Q811yescharacteristic
276C22×C4C2×#4112yes—
277C22×C4C2×#4112yes—
278C22×C4H(5,𝔽2)12yes—
279C22×C4H(5,𝔽2)12yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+411yescharacteristic
283C22×C42−1+411yescharacteristic
284M16—416no—
285C2×C8—24no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—41no—
293C2×C8—42no—
294C2×C8—41no—
295C2×C8—42no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—41no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—88no—
319C4⋊C4—88no—
320C22⋊C4—216no—
321C22⋊C4—88no—
322C22⋊C4—88no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—24no—
327C42—88no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23—22no—
332C23—22no—
333C23—42no—
334C23—42no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—42no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23C22×C4⋊C411yescharacteristic
352C23#53711yescharacteristic
353C23#53711yescharacteristic
354C23#54612yes—
355C23#54612yes—
356C23C2×AGL(1,ℤ/8)12yes—
357C23C2×#13712yes—
358C23#57611yescharacteristic
359C23#57611yescharacteristic
360C23#57811yescharacteristic
361C23#57811yescharacteristic
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
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×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#42818yes—
415C2×C4C4×#4112yes—
416C2×C4#51812yes—
417C2×C4#51811yescharacteristic
418C2×C4#51911yescharacteristic
419C2×C4#53912yes—
420C2×C4#53912yes—
421C2×C4#54212yes—
422C2×C4#54212yes—
423C2×C4#54411yescharacteristic
424C2×C4#54411yescharacteristic
425C2×C4#55111yescharacteristic
426C2×C4#55312yes—
427C2×C4#55412yes—
428C2×C4#55412yes—
429C2×C4#55911yescharacteristic
430C2×C4#56012yes—
431C2×C4#56011yescharacteristic
432C2×C4H(3,𝔽4)11yescharacteristic
433C8—42no—
434C8—42no—
435C8—44no—
436C22—42no—
437C22—44no—
438C22—44no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#231611yescharacteristic
448C22#234912yes—
449C22#269112yes—
450C22#269311yescharacteristic
451C22#271211yescharacteristic
452C22C2×#42812yes—
453C22#291311yescharacteristic
454C22#291311yescharacteristic
455C22#325311yescharacteristic
456C22#325412yes—
457C22#325511yescharacteristic
458C22#325911yescharacteristic
459C22#326111yescharacteristic
460C22#326211yescharacteristic
461C22#326211yescharacteristic
462C22#330812yes—
463C22#331212yes—
464C22#331412yes—
465C22#331812yes—
466C22#332512yes—
467C22#332812yes—
468C22#333012yes—
469C22#333312yes—
470C4—28no—
471C4—42no—
472C4—44no—
473C4—84no—
474C4—84no—
475C4—162no—
476C4—162no—
477C4#233212yes—
478C4#237411yescharacteristic
479C4#237511yescharacteristic
480C4#276311yescharacteristic
481C4#279012yes—
482C4#280711yescharacteristic
483C2—82no—
484C2#1451411yescharacteristic
485C2#2196411yescharacteristic
486C2#2196811yescharacteristic
487C2#2270912yes—
488C2#2271312yes—
489C2#3151012yes—
490C2#3151612yes—
491C2#3156411yescharacteristic
492C2#3156811yescharacteristic
493C2#3212411yescharacteristic
494C2#3212811yescharacteristic
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. #11267C2semidirect
4. #11232C2semidirect
7. #10907C2semidirect
8. #10849C2semidirect
21. #1851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18512
#138762
#112672
#112322
#110962
#109072
#108492
#107222