Finite Group Databasebeta

← #197856

Subgroups — #197856

Order 512.

1206 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3078C211yesmaximal, characteristic
2C2×#2982C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5C2×#1542C211yesmaximal, characteristic
6#9220C214yesmaximal, complemented
7#9216C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic
9#7877C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3078—24no—
13#3078C418yescomplemented
14C2×#479—22no—
15C2×#477C2212yes—
16C2×#475C2212yes—
17#2982—24no—
18#2982—24no—
19C2×C8×Q8—22no—
20C2×C8×Q8C2214yes—
21C2×#431—22no—
22C2×#431C2211yescharacteristic
23C2×#431C2211yescharacteristic
24C2×#428—21no—
25C2×#423C2211yescharacteristic
26C22×C4⋊C8C2211yescharacteristic
27C2×#419—21no—
28C22×#103—21no—
29C2×#384C2211yescharacteristic
30#1990C2212yes—
31#1956C2212yes—
32#1872—22no—
33#1803—22no—
34#1803—22no—
35#1803C2214yes—
36C2×#340—21no—
37#1542—24no—
38C2×#333C2212yes—
39C2×#332C2212yes—
40#1296C2214yes—
41#1277C2214yes—
42C23×Q8D411yescharacteristic
43C22×M16—21no—
44C23×C8—21no—
45#533—22no—
46#533—22no—
47#533—22no—
48#533C2×C414yes—
49C2×C4⋊Q8C2311yescharacteristic
50C2×C4⋊Q8C2311yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—21no—
58C2×C4×Q8—21no—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C414yes—
61C2×C4×Q8D414yes—
62C2×C4×Q8D412yes—
63C2×C4×Q8C2312yes—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#479—216no—
73#477—24no—
74#477C2×C418yescomplemented
75#475—24no—
76#475C2×C418yescomplemented
77C8×Q8—216no—
78C8×Q8—28no—
79C8×Q8—28no—
80#431—216no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431D414yes—
85#428—28no—
86C2×#107—22no—
87C2×#106—22no—
88#423—22no—
89#423C2×C414yes—
90C2×C4⋊C8—24no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—22no—
93C2×C4⋊C8—22no—
94C2×C4⋊C8—21no—
95C2×C4⋊C8D412yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×C4⋊C8C2312yes—
99#419—28no—
100C2×#103—22no—
101C2×#103—22no—
102C2×#103—24no—
103C2×#103—24no—
104C2×#103—44no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8D412yes—
107C2×C4×C8—24no—
108C2×C4×C8—21no—
109C2×C4×C8C2312yes—
110#384—21no—
111#384—21no—
112#384—21no—
113#384C2312yes—
114#340—24no—
115#340—24no—
116#333—24no—
117#332—24no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8C22⋊C414yes—
125C22×Q8SD1614yes—
126C22×Q8Q1614yes—
127C22×Q8C2×D412yes—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—41no—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4#4111yescharacteristic
134C2×M16—24no—
135C2×M16—42no—
136C2×M16—42no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—42no—
141C22×C8—42no—
142C22×C8—42no—
143C22×C8C2×D414yes—
144C4⋊Q8—21no—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8C22×C412yes—
151C4⋊Q8C22×C412yes—
152#125—24no—
153#125—42no—
154C22⋊Q8—24no—
155C22⋊Q8—24no—
156C22⋊Q8—42no—
157C22⋊Q8—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—42no—
168C4×Q8—42no—
169C4×Q8C22⋊C418yes—
170C4×Q8C22×C414yes—
171C4×Q8C2×D414yes—
172#117—24no—
173#117—42no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—82no—
196C2×C4⋊C4C22⋊C414yes—
197C2×C4⋊C4C2×D412yes—
198C2×C22⋊C4—41no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—41no—
204C2×C42C22×C412yes—
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42#4111yescharacteristic
210C2×C42#4112yes—
211C2×C42C2411yescharacteristic
212#107—28no—
213#106—28no—
214C4⋊C8—216no—
215C4⋊C8—24no—
216C4⋊C8—24no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—42no—
222C4⋊C8C22×C414yes—
223C4⋊C8C2×D414yes—
224#103—28no—
225#103—28no—
226#103—48no—
227#103—84no—
228C22⋊C8—24no—
229C22⋊C8—24no—
230C4×C8—216no—
231C4×C8—24no—
232C4×C8—22no—
233C4×C8—22no—
234C24#12111yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—28no—
241C2×Q8—24no—
242C2×Q8—28no—
243C2×Q8—24no—
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—81no—
251C2×Q8—81no—
252C2×Q8#103116yescomplemented
253C2×Q8C2×C22⋊C414yes—
254C2×Q8C2×SD1614yes—
255C2×Q8C2×Q1614yes—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—81no—
270C22×C4—81no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4—82no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C4×D412yes—
276C22×C4C4×D412yes—
277C22×C4#12012yes—
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4#12112yes—
281C22×C4#12111yescharacteristic
282C22×C4#12312yes—
283C22×C4C2×SD1612yes—
284C22×C4C2×Q1612yes—
285C22×C4#13512yes—
286C22×C4#13512yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290M16—48no—
291C2×C8—24no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—24no—
295C2×C8—24no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—84no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—24no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C4⋊C4C2×C22⋊C414yes—
327C22⋊C4—44no—
328C42—24no—
329C42—24no—
330C42—28no—
331C42—44no—
332C42C2×C22⋊C412yes—
333C42C4×D412yes—
334C42C4×D412yes—
335C42#12012yes—
336C42#12112yes—
337C42#12312yes—
338C42C23×C411yescharacteristic
339C42C22×D411yescharacteristic
340C42C22×D411yescharacteristic
341C42C2×#4111yescharacteristic
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23#38612yes—
347C23#46512yes—
348C23#46712yes—
349C23C2×#12111yescharacteristic
350Q8—24no—
351Q8—216no—
352Q8—24no—
353Q8—24no—
354Q8—44no—
355Q8—82no—
356Q8—82no—
357Q8—82no—
358Q8C2×#10318yes—
359C2×C4—216no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4—82no—
386C2×C4—164no—
387C2×C4#38612yes—
388C2×C4#38612yes—
389C2×C4#38612yes—
390C2×C4C2×#10314yes—
391C2×C4#41614yes—
392C2×C4#46512yes—
393C2×C4#46514yes—
394C2×C4#46712yes—
395C2×C4#46714yes—
396C2×C4C22×C22⋊C411yescharacteristic
397C2×C4C2×C4×D411yescharacteristic
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×#12011yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12311yescharacteristic
402C2×C4C22×SD1611yescharacteristic
403C2×C4C22×Q1611yescharacteristic
404C2×C4C2×#13511yescharacteristic
405C2×C4C2×#13511yescharacteristic
406C2×C4#57712yes—
407C2×C4#57812yes—
408C8—44no—
409C8—44no—
410C8—44no—
411C8—82no—
412C22—22no—
413C22—22no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22#189214yes—
419C22C2×#38611yescharacteristic
420C22#304311yescharacteristic
421C22#304411yescharacteristic
422C22C2×#46511yescharacteristic
423C22C2×#46711yescharacteristic
424C22#306811yescharacteristic
425C22#306911yescharacteristic
426C4—21no—
427C4—21no—
428C4—22no—
429C4—22no—
430C4—28no—
431C4—44no—
432C4—82no—
433C4—82no—
434C4—162no—
435C4#189214yes—
436C4C22×#10311yescharacteristic
437C4C2×#41611yescharacteristic
438C4#294712yes—
439C4#301312yes—
440C4C2×#46511yescharacteristic
441C4C2×#46711yescharacteristic
442C2—24no—
443C2#1939611yescharacteristic
444C2#1939911yescharacteristic
445C2#1940211yescharacteristic
446C2#1940511yescharacteristic
447C2#2093011yescharacteristic
448C2C2×#189211yescharacteristic
449C2#2100411yescharacteristic
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. #9220C2semidirect
7. #9216C2semidirect
13. #3078C4semidirect
74. #477C2×C4semidirect
76. #475C2×C4semidirect
252. C2×Q8#103semidirect

Maximal subgroups

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

subgroupindex
C2×#30782
C2×#29822
#218602
C2×#18032
C2×#15422
#92202
#92162
#80432
#78772