Finite Group Databasebeta

← #385216

Subgroups — #385216

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1947C212yesmaximal, complemented
2C2×#1851C211yesmaximal, characteristic
3C2×#1809C212yesmaximal
4#13876C212yesmaximal
5#11324C212yesmaximal, complemented
6#11231C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11096C211yesmaximal, characteristic
9#10908C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C22×#107C2212yes—
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2212yes—
15C22×C4⋊C8C2212yes—
16C22×C22⋊C8C2211yescharacteristic
17C2×#388—22no—
18C2×#384C2212yes—
19C2×#380C2212yes—
20C2×#380C2212yes—
21C2×C4×C4⋊C4C2212yes—
22C2×C4×C22⋊C4C2211yescharacteristic
23#1947—28no—
24#1851C418yescomplemented
25#1809—28no—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#336C2212yes—
33#1442—22no—
34#1439—22no—
35#1436—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C24×C4C2311yescharacteristic
39C22×M16—24no—
40C22×M16—24no—
41C22×M16—41no—
42C22×M16—41no—
43C22×M16C2×C414yes—
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—22no—
48C22×C4⋊C4—22no—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C22⋊C4—22no—
54C22×C22⋊C4—22no—
55C22×C22⋊C4D412yes—
56C22×C42—22no—
57C22×C42C2312yes—
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—41no—
62C2×#107—42no—
63C2×#107—42no—
64C2×#107—41no—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—42no—
68C2×C4⋊C8—24no—
69C2×C4⋊C8—24no—
70C2×C4⋊C8C2×C418yes—
71C2×C22⋊C8—41no—
72C2×C22⋊C8—41no—
73C2×C22⋊C8C2×C414yes—
74#388—48no—
75#384—24no—
76#384—42no—
77#380—24no—
78#380—24no—
79#380—42no—
80#380—42no—
81C4×C4⋊C4—24no—
82C4×C4⋊C4—42no—
83C4×C22⋊C4C2×C414yes—
84C4×C22⋊C4Q814yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95D411yescharacteristic
93C2×#95D411yescharacteristic
94#340—24no—
95#340—24no—
96#340—42no—
97#340—41no—
98#340—41no—
99#340D418yes—
100#336C2×C418yes—
101C25C2×D411yescharacteristic
102C23×C4—22no—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4—22no—
107C23×C4—21no—
108C23×C4—21no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4C22×C412yes—
115C23×C4C2×D412yes—
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4C2×Q812yes—
119C23×C4C2411yescharacteristic
120C2×M16—216no—
121C2×M16—44no—
122C2×M16—44no—
123C2×M16—48no—
124C2×M16—44no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8C22×C412yes—
138C22×C8C22×C412yes—
139C22×C8C22×C414yes—
140C22×C8C2×D414yes—
141C22×C8#4114yes—
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—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—82no—
169C2×C4⋊C4—82no—
170C2×C4⋊C4C2×D414yes—
171C2×C4⋊C4C2×D414yes—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—44no—
177C2×C22⋊C4—44no—
178C2×C22⋊C4—82no—
179C2×C22⋊C4—82no—
180C2×C22⋊C4C4⋊C418yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—44no—
186C2×C42—44no—
187C2×C42—82no—
188C2×C42C22×C414yes—
189C2×C42C2×Q814yes—
190#107—44no—
191#107—48no—
192#107—44no—
193#106—48no—
194C4⋊C8—216no—
195C22⋊C8—44no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—44no—
200#95—44no—
201#95—44no—
202#95—82no—
203#95C4⋊C414yes—
204#95C4⋊C414yes—
205C24—21no—
206C24—21no—
207C24—22no—
208C24—22no—
209C24—22no—
210C24—22no—
211C24—21no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24C2×C4⋊C412yes—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
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×C4C2×C4⋊C412yes—
278C22×C4C2×C4⋊C414yes—
279C22×C4C2×C4⋊C414yes—
280C22×C4AGL(1,ℤ/8)14yes—
281C22×C4#13714yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D412yes—
284C22×C4C22×Q811yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)11yescharacteristic
287C22×C4H(5,𝔽2)11yescharacteristic
288C22×C4H(5,𝔽2)12yes—
289C22×C42−1+411yescharacteristic
290C22×C42−1+411yescharacteristic
291C22×C42−1+412yes—
292M16—416no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—24no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—24no—
303C2×C8—41no—
304C2×C8—42no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C2×C8—41no—
309C2×C8C4×D418yes—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
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—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—88no—
327C4⋊C4—88no—
328C22⋊C4—216no—
329C22⋊C4—88no—
330C22⋊C4—88no—
331C42—24no—
332C42—24no—
333C42—24no—
334C42—24no—
335C42—88no—
336C23—22no—
337C23—24no—
338C23—24no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—42no—
345C23—42no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—42no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23C22×C4⋊C411yescharacteristic
360C23#53711yescharacteristic
361C23#53711yescharacteristic
362C23D4212yes—
363C23#54712yes—
364C23C2×AGL(1,ℤ/8)12yes—
365C23C2×#13712yes—
366C23#57611yescharacteristic
367C23#57611yescharacteristic
368C23#57811yescharacteristic
369C23#57811yescharacteristic
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
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—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4#42818yes—
423C2×C4C2×C4×D412yes—
424C2×C4#51811yescharacteristic
425C2×C4#51911yescharacteristic
426C2×C4#52012yes—
427C2×C4#53612yes—
428C2×C4#53712yes—
429C2×C4#54112yes—
430C2×C4#54212yes—
431C2×C4#54411yescharacteristic
432C2×C4#54411yescharacteristic
433C2×C4D4211yescharacteristic
434C2×C4#54612yes—
435C2×C4D4×Q812yes—
436C2×C4#55111yescharacteristic
437C2×C4#55412yes—
438C2×C4#56011yescharacteristic
439C2×C4#56212yes—
440C2×C4#56311yescharacteristic
441C8—28no—
442C8—42no—
443C8—42no—
444C22—42no—
445C22—44no—
446C22—44no—
447C22—42no—
448C22—42no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—82no—
454C22—82no—
455C22#231611yescharacteristic
456C22D4×C4⋊C412yes—
457C22#267212yes—
458C22#267511yescharacteristic
459C22#272211yescharacteristic
460C22C2×#42812yes—
461C22#291311yescharacteristic
462C22#291311yescharacteristic
463C22#325311yescharacteristic
464C22#325412yes—
465C22#325511yescharacteristic
466C22#325911yescharacteristic
467C22#326111yescharacteristic
468C22#326211yescharacteristic
469C22#326211yescharacteristic
470C22#326412yes—
471C22#327012yes—
472C22#327412yes—
473C22#327612yes—
474C22#334312yes—
475C22#334612yes—
476C22#335312yes—
477C22#335712yes—
478C4—28no—
479C4—42no—
480C4—44no—
481C4—84no—
482C4—84no—
483C4—162no—
484C4—162no—
485C4#232312yes—
486C4#235811yescharacteristic
487C4#237611yescharacteristic
488C4#267711yescharacteristic
489C4#270412yes—
490C4#274211yescharacteristic
491C2—82no—
492C2#1440211yescharacteristic
493C2#2196411yescharacteristic
494C2#2196811yescharacteristic
495C2#2266112yes—
496C2#2266512yes—
497C2#3068112yes—
498C2#3069212yes—
499C2#3085711yescharacteristic
500C2#3086411yescharacteristic
501C2#3242011yescharacteristic
502C2#3242411yescharacteristic
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
1. C2×#1947C2semidirect
5. #11324C2semidirect
6. #11231C2semidirect
9. #10908C2semidirect
24. #1851C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19472
C2×#18512
C2×#18092
#138762
#113242
#112312
#110962
#109082