Finite Group Databasebeta

← #197968

Subgroups — #197968

Order 512.

1256 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#462C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1556C211yesmaximal, characteristic
6#9226C214yesmaximal, complemented
7#9218C214yesmaximal, complemented
8#8044C211yesmaximal, characteristic
9#7879C211yesmaximal, characteristic
10C23×Q16—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#462—24no—
14C2×#462—21no—
15C2×#462—21no—
16C2×#462C418yescomplemented
17C2×#462C2212yes—
18C2×#462C2212yes—
19#2984—24no—
20#2984—24no—
21C2×#446—22no—
22C2×#446C2212yes—
23C2×C8×Q8C2212yes—
24C2×#432—21no—
25C2×#432C2211yescharacteristic
26C2×#432C2211yescharacteristic
27C2×#431—21no—
28C2×#431C2211yescharacteristic
29C22×C4⋊C8C2211yescharacteristic
30C22×#103—21no—
31#2285C2211yescharacteristic
32#1990C2214yes—
33#1987—22no—
34#1824—21no—
35#1824—21no—
36#1823—21no—
37#1803C2212yes—
38#1803C2212yes—
39#1787—21no—
40#1763—21no—
41#1763—21no—
42#1556—24no—
43C2×#332C2214yes—
44#1296C2214yes—
45#1277C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C22×Q16—28no—
49C22×Q16—22no—
50C22×Q16—22no—
51C22×Q16—48no—
52C23×C8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8C2×C414yes—
60C2×C4⋊Q8C2312yes—
61C2×C4×Q8—21no—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—21no—
65C2×C4×Q8C2×C414yes—
66C2×C4×Q8D414yes—
67C2×C4×Q8D411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×C4×Q8C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#462—28no—
79#462—28no—
80#462—24no—
81#462—24no—
82#462C2×C418yes—
83#462C2×C418yes—
84#446—216no—
85#446—24no—
86#446—24no—
87C8×Q8—24no—
88C8×Q8—24no—
89#432—28no—
90#432—22no—
91#432—22no—
92#432—22no—
93#432D414yes—
94#431—28no—
95#431—22no—
96#431—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—22no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—22no—
101C2×C4⋊C8C2×C414yes—
102C2×C4⋊C8D412yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×C4⋊C8C2312yes—
106C2×#103—28no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×#103—22no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8D411yescharacteristic
114C2×C22⋊C8D411yescharacteristic
115C2×#97—22no—
116C2×#97—21no—
117C2×#97—21no—
118C2×C4×C8—22no—
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—
131C22×Q8—21no—
132C22×Q8—22no—
133C22×Q8—24no—
134C22×Q8—24no—
135C22×Q8—22no—
136C22×Q8—42no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—41no—
140C22×Q8—42no—
141C22×Q8C22⋊C414yes—
142C22×Q8C2×D411yescharacteristic
143C22×Q8C2×D411yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—41no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C2×Q16—28no—
151C2×Q16—28no—
152C2×Q16—432no—
153C2×Q16—48no—
154C2×Q16—48no—
155C22×C8—24no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—44no—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C4⋊Q8—24no—
163C4⋊Q8—24no—
164C4⋊Q8—24no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—22no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8C22×C414yes—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—22no—
176C4×Q8—22no—
177C4×Q8—24no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8C22⋊C414yes—
183C4×Q8C22⋊C414yes—
184C4×Q8C22×C412yes—
185C4×Q8C22×C412yes—
186C4×Q8C2×D412yes—
187C4×Q8C2×D412yes—
188#117—82no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4C22⋊C414yes—
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C22⋊C4—41no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—41no—
219C2×C42C22×C412yes—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D412yes—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D412yes—
224C2×C42#4111yescharacteristic
225C2×C42#4112yes—
226C2×C42C2411yescharacteristic
227C4⋊C8—28no—
228C4⋊C8—28no—
229C4⋊C8—24no—
230C4⋊C8—24no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8C22×C414yes—
236C4⋊C8C2×D414yes—
237#103—28no—
238#103—28no—
239#103—44no—
240#103—44no—
241C22⋊C8—24no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—42no—
245#97—28no—
246#97—24no—
247#97—42no—
248C4×C8—28no—
249C4×C8—22no—
250C4×C8—22no—
251#95—41no—
252#95—41no—
253C24#12111yescharacteristic
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—24no—
258C2×Q8—28no—
259C2×Q8—24no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—48no—
265C2×Q8—48no—
266C2×Q8—48no—
267C2×Q8—42no—
268C2×Q8—44no—
269C2×Q8—44no—
270C2×Q8—44no—
271C2×Q8—84no—
272C2×Q8—88no—
273C2×Q8C2×C22⋊C412yes—
274C2×Q8C2×C22⋊C412yes—
275C2×Q8AGL(1,ℤ/8)14yes—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—84no—
294C22×C4C2×C22⋊C412yes—
295C22×C4C4×D412yes—
296C22×C4C4×D412yes—
297C22×C4#12012yes—
298C22×C4#12112yes—
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12312yes—
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306Q16—416no—
307Q16—416no—
308C2×C8—24no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—48no—
315C2×C8—44no—
316C2×C8—44no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C4⋊C4—22no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—44no—
338C4⋊C4—82no—
339C4⋊C4C2×C22⋊C412yes—
340C4⋊C4C2×C22⋊C412yes—
341C22⋊C4—82no—
342C42—22no—
343C42—22no—
344C42—22no—
345C42—22no—
346C42—24no—
347C42—24no—
348C42—82no—
349C42C2×C22⋊C412yes—
350C42C4×D412yes—
351C42C4×D412yes—
352C42#12012yes—
353C42#12112yes—
354C42#12312yes—
355C42C23×C411yescharacteristic
356C42C22×D411yescharacteristic
357C42C22×D411yescharacteristic
358C42C2×#4111yescharacteristic
359C23—22no—
360C23—22no—
361C23#38612yes—
362C23#47112yes—
363C23#47312yes—
364C23C2×#12111yescharacteristic
365Q8—22no—
366Q8—24no—
367Q8—22no—
368Q8—22no—
369Q8—48no—
370Q8—42no—
371Q8—44no—
372Q8—44no—
373Q8—88no—
374Q8—84no—
375Q8#41814yes—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—44no—
402C2×C4—88no—
403C2×C4—84no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4—82no—
407C2×C4#38612yes—
408C2×C4#38612yes—
409C2×C4#38612yes—
410C2×C4C22×C22⋊C411yescharacteristic
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×#12011yescharacteristic
414C2×C4C2×#12111yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C2×C4C2×AGL(1,ℤ/8)12yes—
417C2×C4#57512yes—
418C2×C4#57814yes—
419C8—44no—
420C8—42no—
421C8—42no—
422C8—44no—
423C8—82no—
424C22—24no—
425C22—24no—
426C22—22no—
427C22—22no—
428C22#190814yes—
429C22C2×#38611yescharacteristic
430C22#304412yes—
431C22#305512yes—
432C22C2×#47111yescharacteristic
433C22C2×#47311yescharacteristic
434C4—21no—
435C4—21no—
436C4—22no—
437C4—22no—
438C4—22no—
439C4—24no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—84no—
444C4—162no—
445C4C2×#41811yescharacteristic
446C4#289611yescharacteristic
447C4#295012yes—
448C4#301112yes—
449C4#305512yes—
450C2—24no—
451C2#1947412yes—
452C2#1947612yes—
453C2#2093511yescharacteristic
454C2#2097311yescharacteristic
455C2C2×#190811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
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
6. #9226C2semidirect
7. #9218C2semidirect
16. C2×#462C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4622
C2×#29842
#218342
#207402
C2×#15562
#92262
#92182
#80442
#78792