Finite Group Databasebeta

← #106809

Subgroups — #106809

Order 512.

962 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×#2921C211yesmaximal, characteristic
2C2×#1458C211yesmaximal, characteristic
3#14340C211yesmaximal, characteristic
4#8043C212yesmaximal, complemented
5C2×#1286C212yesmaximal, complemented
6C22×#117C2211yescharacteristic
7#2921—24no—
8#2921—22no—
9#2921C414yescomplemented
10C2×#433—24no—
11C2×#431C2212yes—
12C2×#423C2212yes—
13C2×#407—21no—
14C2×C4×M16—21no—
15C2×#384C2211yescharacteristic
16C2×#382C2211yescharacteristic
17C2×#380—21no—
18#1458—22no—
19#1458C414yescomplemented
20C2×#324—21no—
21C2×#323—21no—
22#1441—21no—
23#1441C412yes—
24#1439—21no—
25#1439C412yes—
26C4⋊#95—21no—
27#1296—24no—
28#1286—24no—
29C24×C4D411yescharacteristic
30C22×M16—22no—
31C2×#117—22no—
32C2×#117—22no—
33C2×#117—22no—
34C2×#117—22no—
35C2×#117C2×C412yes—
36C2×#117C2×C412yes—
37C2×#117D412yes—
38C2×#117Q812yes—
39C22×C4⋊C4—21no—
40C22×C4⋊C4—21no—
41C22×C4⋊C4—21no—
42C22×C4⋊C4C2×C411yescharacteristic
43C22×C4⋊C4C2×C412yes—
44C22×C4⋊C4D411yescharacteristic
45C22×C22⋊C4—22no—
46C22×C22⋊C4C2×C411yescharacteristic
47C22×C22⋊C4D411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42C2311yescharacteristic
50#433—216no—
51#433—216no—
52#431—22no—
53#431—22no—
54#431—22no—
55#431C2×C414yes—
56#423—22no—
57#423—22no—
58#423C2×C414yes—
59#423D414yes—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8D414yes—
63#407—24no—
64#407—24no—
65C2×C22⋊C8—24no—
66C2×C22⋊C8—22no—
67C4×M16—24no—
68C4×M16—24no—
69C2×#97—22no—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72#384—22no—
73#384—22no—
74#382—22no—
75#382—22no—
76#380—44no—
77C2×#95—22no—
78C2×#95—21no—
79C2×#95—22no—
80C2×#95—22no—
81C2×#95C2×C412yes—
82#324—24no—
83#323—24no—
84C25C22⋊C411yescharacteristic
85C23×C4—21no—
86C23×C4—21no—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—22no—
90C23×C4—22no—
91C23×C4—22no—
92C23×C4—22no—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C412yes—
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C412yes—
100C23×C4C22×C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×M16—24no—
104C2×M16—24no—
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C22×C8—24no—
110C22×C8—42no—
111#117—24no—
112#117—24no—
113#117—24no—
114#117—24no—
115#117—24no—
116#117—24no—
117#117—24no—
118#117—24no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—24no—
124C2×C4⋊C4—41no—
125C2×C4⋊C4—41no—
126C2×C4⋊C4—42no—
127C2×C4⋊C4—41no—
128C2×C4⋊C4—41no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—82no—
134C2×C4⋊C4—82no—
135C2×C4⋊C4C4212yes—
136C2×C4⋊C4C22⋊C412yes—
137C2×C4⋊C4C4⋊C412yes—
138C2×C4⋊C4C4⋊C412yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—44no—
144C2×C22⋊C4—82no—
145C2×C22⋊C4—82no—
146C2×C22⋊C4C4212yes—
147C2×C22⋊C4C22⋊C412yes—
148C2×C22⋊C4C4⋊C412yes—
149C2×C22⋊C4C4⋊C412yes—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42C22⋊C412yes—
155C2×C42C4⋊C412yes—
156C2×C42C22×C411yescharacteristic
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D412yes—
159C2×C42C2×D412yes—
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4112yes—
163C2×C42#4112yes—
164C4⋊C8—24no—
165C4⋊C8—24no—
166C4⋊C8—24no—
167C4⋊C8—24no—
168C4⋊C8—42no—
169C4⋊C8C22⋊C418yescomplemented
170C22⋊C8—216no—
171C22⋊C8—24no—
172C22⋊C8—42no—
173#97—28no—
174C4×C8—28no—
175C4×C8—24no—
176C4×C8—42no—
177#95—24no—
178#95—44no—
179#95—42no—
180#95—82no—
181#95—82no—
182C24—21no—
183C24—21no—
184C24—21no—
185C24—21no—
186C24—21no—
187C24—21no—
188C24—41no—
189C24—41no—
190C24—41no—
191C24—41no—
192C24#9512yes—
193C24C2×C22⋊C411yescharacteristic
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—21no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
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#9512yes—
243C22×C4#9512yes—
244C22×C4#9512yes—
245C22×C4C2×C4211yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C4×D412yes—
252C22×C4C4×D412yes—
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12112yes—
256C22×C4#12112yes—
257C22×C4#12311yescharacteristic
258C22×C4#12311yescharacteristic
259C22×C4#13512yes—
260C22×C4#13512yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263M16—44no—
264M16—44no—
265M16—44no—
266M16—44no—
267C2×C8—28no—
268C2×C8—44no—
269C2×C8—44no—
270C2×C8—44no—
271C2×C8—82no—
272C2×C8—82no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—84no—
278C4⋊C4—84no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C4⋊C4—82no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—88no—
288C42—22no—
289C42—22no—
290C42—22no—
291C42—22no—
292C42C2×C22⋊C412yes—
293C42#11712yes—
294C42C4×D412yes—
295C42C4×D412yes—
296C42#12012yes—
297C42C22⋊Q812yes—
298C42C22⋊Q812yes—
299C42#12312yes—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23—42no—
310C23—42no—
311C23—42no—
312C23—42no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—81no—
322C23—81no—
323C23—81no—
324C23—81no—
325C23—81no—
326C23—81no—
327C23C2×#9511yescharacteristic
328C23#38611yescharacteristic
329C23#38611yescharacteristic
330C23#41611yescharacteristic
331C23#41611yescharacteristic
332C23#41811yescharacteristic
333C23#41911yescharacteristic
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
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×C4C4×C22⋊C412yes—
380C2×C4#37911yescharacteristic
381C2×C4#38012yes—
382C2×C4#38111yescharacteristic
383C2×C4#38512yes—
384C2×C4#38511yescharacteristic
385C2×C4#38511yescharacteristic
386C2×C4#38612yes—
387C2×C4#41612yes—
388C2×C4#41712yes—
389C2×C4#42712yes—
390C2×C4#42812yes—
391C2×C4#44912yes—
392C2×C4#44912yes—
393C2×C4#45212yes—
394C2×C4#45212yes—
395C2×C4#48212yes—
396C2×C4#48212yes—
397C2×C4#48512yes—
398C2×C4#48512yes—
399C8—48no—
400C8—84no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—21no—
405C22—21no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—44no—
414C22—42no—
415C22—42no—
416C22—42no—
417C22—82no—
418C22—82no—
419C22—82no—
420C22—81no—
421C22—81no—
422C22—81no—
423C22—81no—
424C22C22⋊#9511yescharacteristic
425C22#173511yescharacteristic
426C22#173611yescharacteristic
427C22#187311yescharacteristic
428C22#187311yescharacteristic
429C22#187511yescharacteristic
430C22#187511yescharacteristic
431C22#190911yescharacteristic
432C22#191011yescharacteristic
433C22#191411yescharacteristic
434C22#191511yescharacteristic
435C4—22no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—42no—
440C4—42no—
441C4—42no—
442C4—162no—
443C4—162no—
444C4#178812yes—
445C4#182512yes—
446C4#185412yes—
447C4#185412yes—
448C2—21no—
449C2—21no—
450C2—21no—
451C2—21no—
452C2—82no—
453C2#919211yescharacteristic
454C2#919411yescharacteristic
455C2#919611yescharacteristic
456C2#919811yescharacteristic
457C2#1073111yescharacteristic
458C2#1074211yescharacteristic
459C2#1075511yescharacteristic
subgroupG/H
center, socleC23#386
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC25C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8043C2semidirect
5. C2×#1286C2semidirect
9. #2921C4semidirect
19. #1458C4semidirect
169. C4⋊C8C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29212
C2×#14582
#143402
#80432
C2×#12862