Finite Group Databasebeta

← #197896

Subgroups — #197896

Order 512.

1294 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2C2×#2991C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic, complemented
4#20737C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic, complemented
6#9225C214yesmaximal, complemented
7#9218C214yesmaximal, complemented
8#8041C211yesmaximal, characteristic, complemented
9#7880C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×C4×#41C2211yescharacteristic
13#3039—24no—
14#3039C418yescomplemented
15C2×#462—21no—
16C2×#462C2212yes—
17C2×#461—21no—
18C2×#461C2212yes—
19#2991—24no—
20#2991—24no—
21C2×C8×Q8C2212yes—
22C2×#436—22no—
23C2×#436C2212yes—
24C2×#431—21no—
25C2×#431C2211yescharacteristic
26C2×#431C2211yescharacteristic
27C2×#423—21no—
28C2×#423C2211yescharacteristic
29C2×#423C2211yescharacteristic
30C2×#419—21no—
31#2285C2211yescharacteristic
32#1990C2214yes—
33#1987—21no—
34#1987—21no—
35#1825—21no—
36#1824—22no—
37#1803C2212yes—
38#1802C2212yes—
39#1788—21no—
40C4×#103—21no—
41C4×#102—21no—
42#1549—24no—
43C2×#332C2214yes—
44#1296C2214yes—
45#1277C2214yes—
46C22×#41D411yescharacteristic
47C23×Q8—21no—
48C2×#137—28no—
49C2×#137—48no—
50C22×Q16—22no—
51C22×SD16—22no—
52C22×M16—21no—
53C22×M16—21no—
54C22×M16—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8C2312yes—
63C4×#41—22no—
64C4×#41—22no—
65C4×#41C2×C414yes—
66C4×#41D414yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8C2311yescharacteristic
69C2×C4×D4—21no—
70C2×C4×D4—21no—
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C2×#117D411yescharacteristic
76C22×C4⋊C4—21no—
77C22×C4⋊C4—21no—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80#462—28no—
81#462—24no—
82#462C2×C418yes—
83#461—28no—
84#461—24no—
85#461C2×C418yes—
86C8×Q8—24no—
87C8×Q8—24no—
88#436—216no—
89#436—24no—
90#436—24no—
91#431—28no—
92#431—22no—
93#431—22no—
94#431—22no—
95#431—22no—
96#423—28no—
97#423—22no—
98#423—22no—
99#423C2×C414yes—
100#423D414yes—
101C2×C4⋊C8—22no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8—21no—
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8D411yescharacteristic
106C2×C4⋊C8C2312yes—
107C2×C4⋊C8C2312yes—
108C2×C4⋊C8C2312yes—
109#419—28no—
110C2×#103—22no—
111C2×#103—22no—
112C2×#103—22no—
113C2×#102—22no—
114C2×C22⋊C8—22no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8D412yes—
117C2×C4×C8—22no—
118C2×C4×C8—21no—
119C2×C4×C8C2312yes—
120#384C2312yes—
121#382—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126#340—24no—
127#340—24no—
128#340—22no—
129#340—22no—
130#332—28no—
131C2×#41—22no—
132C2×#41—24no—
133C2×#41—42no—
134C2×#41C22⋊C414yes—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—41no—
140C22×Q8—42no—
141C22×Q8C2×D411yescharacteristic
142C22×D4—21no—
143C22×D4C2×D411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—41no—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151#137—432no—
152C2×Q16—28no—
153C2×Q16—48no—
154C2×SD16—28no—
155C2×SD16—48no—
156C2×M16—24no—
157C2×M16—28no—
158C2×M16—24no—
159C2×M16—44no—
160C2×M16—84no—
161C2×M16—82no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8C2×D412yes—
166C22×C8C2×D412yes—
167C4⋊Q8—24no—
168C4⋊Q8—24no—
169C4⋊Q8—24no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172C4⋊Q8—22no—
173C4⋊Q8—42no—
174C4⋊Q8—42no—
175C4⋊Q8—42no—
176C4⋊Q8C22×C414yes—
177C4×Q8—22no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8C22⋊C414yes—
181C4×Q8C22×C412yes—
182C4×Q8C2×D412yes—
183C4×D4—24no—
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—24no—
187C4×D4—22no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C22⋊C414yes—
191C4×D4C22×C412yes—
192C4×D4C2×D412yes—
193#117—22no—
194#117—24no—
195#117—24no—
196#117—82no—
197#117C22⋊C414yes—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—41no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—41no—
223C2×C42C22×C412yes—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D412yes—
228C2×C42#4111yescharacteristic
229C2×C42#4112yes—
230C2×C42C2411yescharacteristic
231C4⋊C8—28no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—22no—
237C4⋊C8—22no—
238C4⋊C8—22no—
239C4⋊C8—22no—
240C4⋊C8C22×C414yes—
241C4⋊C8C2×D414yes—
242#103—28no—
243#103—28no—
244#103—44no—
245#102—44no—
246C22⋊C8—28no—
247C22⋊C8—24no—
248C22⋊C8—42no—
249C4×C8—28no—
250C4×C8—22no—
251C4×C8—22no—
252C4×C8—42no—
253#95—41no—
254#95—41no—
255C24—21no—
256C24#12111yescharacteristic
257#41—28no—
258#41—44no—
259#41—48no—
260C2×Q8—21no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—48no—
264C2×Q8—48no—
265C2×Q8—42no—
266C2×Q8—44no—
267C2×Q8—44no—
268C2×Q8—44no—
269C2×Q8—84no—
270C2×Q8—88no—
271C2×Q8C2×C22⋊C412yes—
272C2×Q8AGL(1,ℤ/8)14yes—
273C2×D4—24no—
274C2×D4—21no—
275C2×D4—22no—
276C2×D4—22no—
277C2×D4—44no—
278C2×D4C2×C22⋊C412yes—
279C22×C4—21no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—44no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—44no—
299C22×C4—42no—
300C22×C4—84no—
301C22×C4C2×C22⋊C412yes—
302C22×C4C4×D412yes—
303C22×C4C4×D412yes—
304C22×C4#12012yes—
305C22×C4#12112yes—
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4#12111yescharacteristic
309C22×C4#12312yes—
310C22×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4C2×#4111yescharacteristic
313Q16—416no—
314SD16—416no—
315M16—48no—
316M16—84no—
317M16—88no—
318C2×C8—24no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—44no—
325C2×C8—44no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—44no—
339C4⋊C4—44no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—82no—
345C4⋊C4C2×C22⋊C412yes—
346C4⋊C4C2×C22⋊C412yes—
347C22⋊C4—24no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—24no—
351C22⋊C4—82no—
352C42—22no—
353C42—22no—
354C42—24no—
355C42—24no—
356C42—82no—
357C42C2×C22⋊C412yes—
358C42C4×D412yes—
359C42C4×D412yes—
360C42#12012yes—
361C42#12112yes—
362C42#12312yes—
363C42C23×C411yescharacteristic
364C42C22×D411yescharacteristic
365C42C22×D411yescharacteristic
366C42C2×#4111yescharacteristic
367C23—22no—
368C23—22no—
369C23—21no—
370C23—21no—
371C23—22no—
372C23—42no—
373C23—42no—
374C23—41no—
375C23—41no—
376C23#38612yes—
377C23C2×#12111yescharacteristic
378Q8—22no—
379Q8—24no—
380Q8—44no—
381Q8—44no—
382Q8—88no—
383Q8—84no—
384Q8#41814yes—
385D4—22no—
386D4—22no—
387D4—48no—
388D4—42no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—24no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—24no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—48no—
410C2×C4—44no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—88no—
415C2×C4—84no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4#38612yes—
420C2×C4#38612yes—
421C2×C4#38612yes—
422C2×C4#47112yes—
423C2×C4#47312yes—
424C2×C4C22×C22⋊C411yescharacteristic
425C2×C4C2×C4×D411yescharacteristic
426C2×C4C2×C4×D411yescharacteristic
427C2×C4C2×#12011yescharacteristic
428C2×C4C2×#12111yescharacteristic
429C2×C4C2×#12311yescharacteristic
430C2×C4C2×AGL(1,ℤ/8)12yes—
431C2×C4#57512yes—
432C2×C4#57712yes—
433C2×C4#57812yes—
434C8—44no—
435C8—42no—
436C8—44no—
437C8—42no—
438C8—82no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—42no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—82no—
448C22—82no—
449C22C2×#38611yescharacteristic
450C22#304612yes—
451C22#305512yes—
452C22C2×#47111yescharacteristic
453C22C2×#47311yescharacteristic
454C4—21no—
455C4—21no—
456C4—22no—
457C4—22no—
458C4—24no—
459C4—42no—
460C4—42no—
461C4—84no—
462C4—162no—
463C4#190814yes—
464C4C2×#41811yescharacteristic
465C4#289611yescharacteristic
466C4#294712yes—
467C4#300911yescharacteristic
468C4#301111yescharacteristic
469C4#305512yes—
470C2—42no—
471C2—82no—
472C2#1950112yes—
473C2#1950212yes—
474C2#2093111yescharacteristic
475C2#2097311yescharacteristic
476C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21834C2semidirect
5. C2×#1549C2semidirect
6. #9225C2semidirect
7. #9218C2semidirect
8. #8041C2semidirect
9. #7880C2semidirect
14. #3039C4semidirect

Maximal subgroups

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

subgroupindex
C2×#30392
C2×#29912
#218342
#207372
C2×#15492
#92252
#92182
#80412
#78802