Finite Group Databasebeta

← #381393

Subgroups — #381393

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 461 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1798C211yesmaximal, characteristic
2#14343C211yesmaximal, characteristic
3#13777C211yesmaximal, characteristic
4#11232C212yesmaximal, complemented
5#11231C212yesmaximal, complemented
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10C22×#117C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15C2×#380C2211yescharacteristic
16C2×#380C2211yescharacteristic
17C2×#380C2212yes—
18C2×C4×C4⋊C4C2212yes—
19#1798C418yescomplemented
20C2×#340C2212yes—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#336C2211yescharacteristic
27C2×#336C2211yescharacteristic
28#1445—22no—
29#1440—22no—
30#1437—22no—
31#1436—22no—
32C4⋊#95C2212yes—
33C4⋊#95C2211yescharacteristic
34C4⋊#95C2211yescharacteristic
35C4⋊#95C2212yes—
36C24×C4C2311yescharacteristic
37C23×C8C2312yes—
38C23×C8C2312yes—
39C2×#117—42no—
40C2×#117—42no—
41C2×#117C2×C414yes—
42C2×#117Q814yes—
43C22×C4⋊C4D412yes—
44C22×C4⋊C4C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C22⋊C4—21no—
49C22×C22⋊C4—21no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42—22no—
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#107—24no—
56C2×#107—42no—
57C2×#107—42no—
58C2×#106—24no—
59C2×#106—42no—
60C2×#106—42no—
61C2×C4⋊C8—42no—
62C2×C4⋊C8—42no—
63C2×C4⋊C8C2×C418yes—
64C2×C22⋊C8—42no—
65C2×C22⋊C8—42no—
66C2×C22⋊C8C2×C418yes—
67#380—24no—
68#380—24no—
69#380—42no—
70#380—42no—
71C4×C4⋊C4—42no—
72C4×C4⋊C4—42no—
73C2×#95—21no—
74C2×#95—21no—
75C2×#95—22no—
76C2×#95—22no—
77C2×#95—22no—
78C2×#95—22no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—22no—
83#340—24no—
84#340—24no—
85#340—42no—
86#340—42no—
87#340—42no—
88#340—42no—
89#336C2×C414yes—
90#336C2×C414yes—
91C25C2×D411yescharacteristic
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—22no—
98C23×C4—22no—
99C23×C4—22no—
100C23×C4—22no—
101C23×C4—22no—
102C23×C4C22×C412yes—
103C23×C4C2×D412yes—
104C23×C4C2×D412yes—
105C23×C4C2×D411yescharacteristic
106C23×C4C2×Q812yes—
107C23×C4C2411yescharacteristic
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—22no—
112C22×C8—22no—
113C22×C8—22no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8C22×C414yes—
121C22×C8C22×C414yes—
122#117—48no—
123#117—44no—
124#117—44no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
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—42no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C4⋊C418yes—
150C2×C4⋊C4#4114yes—
151C2×C4⋊C4#4114yes—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—44no—
157C2×C22⋊C4—82no—
158C2×C22⋊C4—82no—
159C2×C22⋊C4—82no—
160C2×C22⋊C4—82no—
161C2×C22⋊C4C4⋊C418yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—22no—
166C2×C42—44no—
167C2×C42—82no—
168C2×C42—82no—
169C2×C42C22×C412yes—
170C2×C42C22×C412yes—
171C2×C42C2×Q812yes—
172C2×C42C2×Q812yes—
173#107—48no—
174#106—48no—
175C4⋊C8—48no—
176C22⋊C8—48no—
177#95—24no—
178#95—24no—
179#95—44no—
180#95—44no—
181#95—82no—
182#95—82no—
183#95—82no—
184#95—82no—
185#95—82no—
186#95—82no—
187C24—21no—
188C24—21no—
189C24—22no—
190C24—22no—
191C24—22no—
192C24—22no—
193C24—22no—
194C24—22no—
195C24C2×C4⋊C412yes—
196C24C22×D411yescharacteristic
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
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—24no—
214C22×C4—22no—
215C22×C4—24no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
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×C4C2×C4⋊C414yes—
258C22×C4C2×C4⋊C414yes—
259C22×C4C2×C4⋊C412yes—
260C22×C4C23×C411yescharacteristic
261C22×C4C22×Q811yescharacteristic
262C22×C4C2×#4112yes—
263C22×C4C2×#4112yes—
264C22×C4H(5,𝔽2)12yes—
265C22×C4H(5,𝔽2)12yes—
266C22×C4H(5,𝔽2)12yes—
267C22×C42−1+412yes—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C2×C8—24no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—48no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C22⋊C4—48no—
300C22⋊C4—84no—
301C22⋊C4—84no—
302C22⋊C4—88no—
303C42—24no—
304C42—24no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—88no—
310C23—22no—
311C23—22no—
312C23—24no—
313C23—24no—
314C23—22no—
315C23—41no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23C22×C4⋊C411yescharacteristic
332C23#52511yescharacteristic
333C23#52511yescharacteristic
334C23#53412yes—
335C23#53712yes—
336C23#57612yes—
337C23#57712yes—
338C23#57712yes—
339C23#57812yes—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
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×C4C4×#4112yes—
392C2×C4#51812yes—
393C2×C4#51811yescharacteristic
394C2×C4#51911yescharacteristic
395C2×C4#53312yes—
396C2×C4#53812yes—
397C2×C4#53812yes—
398C2×C4#53912yes—
399C2×C4#53912yes—
400C2×C4#54012yes—
401C2×C4#54012yes—
402C2×C4#54212yes—
403C2×C4#54212yes—
404C2×C4#54311yescharacteristic
405C2×C4#54311yescharacteristic
406C2×C4#54412yes—
407C8—44no—
408C8—44no—
409C22—42no—
410C22—42no—
411C22—44no—
412C22—44no—
413C22—42no—
414C22—82no—
415C22—82no—
416C22—82no—
417C22—82no—
418C22#229611yescharacteristic
419C22#231511yescharacteristic
420C22#231611yescharacteristic
421C22#263012yes—
422C22#263012yes—
423C22#291212yes—
424C22#291312yes—
425C22#312211yescharacteristic
426C22#312311yescharacteristic
427C22#312412yes—
428C22#312512yes—
429C22#312611yescharacteristic
430C22#312711yescharacteristic
431C22#319812yes—
432C22#319812yes—
433C22#320112yes—
434C22#320112yes—
435C22#325712yes—
436C22#325712yes—
437C22#325812yes—
438C22#325812yes—
439C4—44no—
440C4—44no—
441C4—42no—
442C4—84no—
443C4—84no—
444C4—162no—
445C4—162no—
446C4#233212yes—
447C4#233312yes—
448C4#266512yes—
449C4#266712yes—
450C2—82no—
451C2#1429711yescharacteristic
452C2#2136811yescharacteristic
453C2#2136911yescharacteristic
454C2#2194512yes—
455C2#2196612yes—
456C2#3002312yes—
457C2#3002512yes—
458C2#3002712yes—
459C2#3002912yes—
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
4. #11232C2semidirect
5. #11231C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
19. #1798C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17982
#143432
#137772
#112322
#112312
#110962
#110952
#109082
#109072