Finite Group Databasebeta

← #380322

Subgroups — #380322

Order 512.

1308 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2314C211yesmaximal, characteristic
2C2×#1949C212yesmaximal, complemented
3C2×#1798C211yesmaximal, characteristic
4#14340C211yesmaximal, characteristic
5#11325C212yesmaximal, complemented
6#11267C212yesmaximal, complemented
7#11096C212yesmaximal
8#11095C212yesmaximal
9#10849C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16#2314—28no—
17C2×#388—22no—
18C2×#387—22no—
19C2×#384C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#382C2211yescharacteristic
22C2×#382C2211yescharacteristic
23C2×#380C2212yes—
24C2×#380C2212yes—
25C2×C4×C4⋊C4C2212yes—
26#1949—28no—
27#1798C418yescomplemented
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35#1441—22no—
36#1439—22no—
37C4⋊#95C2212yes—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#117—24no—
42C2×#117—24no—
43C2×#117C2×C414yes—
44C2×#117Q814yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4D412yes—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C22⋊C4—22no—
53C22×C22⋊C4—22no—
54C22×C22⋊C4D412yes—
55C22×C42—22no—
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—42no—
60C2×#107—42no—
61C2×#106—24no—
62C2×#106—42no—
63C2×#106—42no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—24no—
66C2×C4⋊C8C2×C418yes—
67C2×C22⋊C8—41no—
68C2×C22⋊C8—41no—
69C2×C22⋊C8—41no—
70C2×C22⋊C8—41no—
71C2×C22⋊C8C2×C414yes—
72C2×C22⋊C8C2×C414yes—
73#388—216no—
74#387—216no—
75#384—24no—
76#384—24no—
77#382—24no—
78#382—24no—
79#380—24no—
80#380—24no—
81#380—42no—
82#380—42no—
83C4×C4⋊C4—24no—
84C4×C4⋊C4D418yes—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92#340—24no—
93#340—24no—
94#340—42no—
95#340—42no—
96#340—42no—
97#340—42no—
98#336C2×C418yes—
99C25C2×D411yescharacteristic
100C23×C4—22no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—21no—
104C23×C4—21no—
105C23×C4—22no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2×D411yescharacteristic
115C23×C4C2×D412yes—
116C23×C4C2×Q812yes—
117C23×C4C2411yescharacteristic
118C22×C8—22no—
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—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8C22×C414yes—
131C22×C8C22×C414yes—
132#117—216no—
133#117—216no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—24no—
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—28no—
159C2×C4⋊C4—44no—
160C2×C4⋊C4—82no—
161C2×C4⋊C4—82no—
162C2×C4⋊C4—82no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4C4⋊C418yes—
165C2×C4⋊C4C2×D414yes—
166C2×C4⋊C4C2×D414yes—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—28no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—82no—
172C2×C22⋊C4—82no—
173C2×C22⋊C4—82no—
174C2×C22⋊C4—82no—
175C2×C22⋊C4C4⋊C418yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—28no—
179C2×C42—82no—
180C2×C42—82no—
181C2×C42C22×C412yes—
182C2×C42C22×C412yes—
183C2×C42C2×D414yes—
184C2×C42C2×Q812yes—
185C2×C42C2×Q812yes—
186C2×C42#4114yes—
187#107—48no—
188#106—48no—
189C4⋊C8—216no—
190C22⋊C8—44no—
191C22⋊C8—44no—
192#95—28no—
193#95—28no—
194#95—28no—
195#95—44no—
196#95—44no—
197#95—82no—
198#95—82no—
199C24—21no—
200C24—21no—
201C24—21no—
202C24—21no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—22no—
208C24—22no—
209C24C2×C4⋊C412yes—
210C24C22×D411yescharacteristic
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—22no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—42no—
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×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4C2×C4⋊C414yes—
272C22×C4C2×C4⋊C412yes—
273C22×C4C2×C4⋊C414yes—
274C22×C4#12014yes—
275C22×C4#12314yes—
276C22×C4C23×C411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C22×Q811yescharacteristic
279C22×C4C2×#4112yes—
280C22×C4H(5,𝔽2)11yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C4H(5,𝔽2)12yes—
283C22×C42−1+411yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+412yes—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—216no—
301C4⋊C4—24no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—42no—
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—88no—
318C4⋊C4—88no—
319C22⋊C4—216no—
320C22⋊C4—88no—
321C22⋊C4—88no—
322C42—24no—
323C42—24no—
324C42—88no—
325C42C4×D418yes—
326C42C22⋊Q818yes—
327C23—22no—
328C23—22no—
329C23—24no—
330C23—24no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—42no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23C22×C4⋊C411yescharacteristic
351C23C2×#12011yescharacteristic
352C23C2×#12311yescharacteristic
353C23#53511yescharacteristic
354C23#53611yescharacteristic
355C23#53711yescharacteristic
356C23#53711yescharacteristic
357C23#57612yes—
358C23#57712yes—
359C23#57712yes—
360C23#57812yes—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
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×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#38518yes—
414C2×C4C2×C4×D412yes—
415C2×C4#51811yescharacteristic
416C2×C4#51911yescharacteristic
417C2×C4#52012yes—
418C2×C4C2×C22⋊Q812yes—
419C2×C4#53512yes—
420C2×C4#53612yes—
421C2×C4#53712yes—
422C2×C4#53712yes—
423C2×C4#54012yes—
424C2×C4#54112yes—
425C2×C4#54212yes—
426C2×C4#54212yes—
427C2×C4#54411yescharacteristic
428C2×C4#54412yes—
429C2×C4#54411yescharacteristic
430C8—44no—
431C8—44no—
432C22—44no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—44no—
437C22—81no—
438C22—81no—
439C22—81no—
440C22—81no—
441C22—82no—
442C22—82no—
443C22C2×#38511yescharacteristic
444C22#231611yescharacteristic
445C22#231711yescharacteristic
446C22#261511yescharacteristic
447C22#261611yescharacteristic
448C22#262411yescharacteristic
449C22#262511yescharacteristic
450C22#291212yes—
451C22#291312yes—
452C22#301212yes—
453C22#301312yes—
454C22#309612yes—
455C22#309712yes—
456C22#322412yes—
457C22#322512yes—
458C22#324112yes—
459C22#324212yes—
460C22#325911yescharacteristic
461C22#326011yescharacteristic
462C22#326011yescharacteristic
463C22#326111yescharacteristic
464C22#326212yes—
465C22#326312yes—
466C4—28no—
467C4—44no—
468C4—42no—
469C4—84no—
470C4—84no—
471C4—162no—
472C4—162no—
473C4#232312yes—
474C4#232412yes—
475C4#262812yes—
476C4#262912yes—
477C2—82no—
478C2#1426411yescharacteristic
479C2#2084912yes—
480C2#2196711yescharacteristic
481C2#2196811yescharacteristic
482C2#2199212yes—
483C2#2967712yes—
484C2#2970011yescharacteristic
485C2#2970111yescharacteristic
486C2#2992911yescharacteristic
487C2#2993011yescharacteristic
488C2#2994712yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1949C2semidirect
5. #11325C2semidirect
6. #11267C2semidirect
9. #10849C2semidirect
27. #1798C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23142
C2×#19492
C2×#17982
#143402
#113252
#112672
#110962
#110952
#108492