Finite Group Databasebeta

← #181365

Subgroups — #181365

Order 512.

1663 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic
3#34181C211yesmaximal, characteristic
4#8109C212yesmaximal, complemented
5#8107C214yesmaximal, complemented
6#8105C212yesmaximal, complemented
7C2×#1305C214yesmaximal, complemented
8C2×#518C412yes—
9C22×C4×D4C2211yescharacteristic
10#2975—21no—
11#2975—21no—
12#2974—24no—
13#2974C418yescomplemented
14#2973—22no—
15#2973—22no—
16#2972—21no—
17#2972—21no—
18D4×M16—24no—
19D4×M16C418yescomplemented
20#2920C2212yes—
21#2920C2212yes—
22C2×#407C2214yes—
23C2×#407C2214yes—
24#2349—21no—
25#2349—22no—
26#2349—21no—
27D4×C4⋊C4—21no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4—21no—
30#2316C412yes—
31#2315C414yes—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34C2×#380C2212yes—
35C2×#380C2212yes—
36#1305C2214yes—
37#1305C2218yes—
38#1305C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2312yes—
41C22×M16—24no—
42C22×M16C2×C418yescomplemented
43#518—48no—
44C2×C4×D4—24no—
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4C2×C414yes—
49C2×C4×D4D411yescharacteristic
50C2×C4×D4D412yes—
51C2×C4×D4D411yescharacteristic
52C2×C4×D4D411yescharacteristic
53C2×C4×D4D412yes—
54C2×C4×D4D411yescharacteristic
55C2×C4×D4D414yes—
56C2×C4×D4Q814yes—
57C2×#117C2×C412yes—
58C2×#117C2×C412yes—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4C2×C414yes—
63C22×C22⋊C4C2×C414yes—
64C22×C22⋊C4C2×C412yes—
65C22×C42C2311yescharacteristic
66#436—28no—
67#436—28no—
68#435—216no—
69#435—216no—
70C8×D4—28no—
71C8×D4—28no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432C2×C414yescomplemented
76#431—22no—
77#431—22no—
78#431—22no—
79#431C2×C414yescomplemented
80#423—22no—
81#423C2×C414yescomplemented
82#407—24no—
83#407—24no—
84#407—24no—
85#407—24no—
86#407C2×C418yescomplemented
87#407C2×C418yescomplemented
88#407C2314yes—
89#407C2314yes—
90C2×C22⋊C8—24no—
91C2×C22⋊C8—24no—
92C2×C22⋊C8—24no—
93C2×C22⋊C8C2×C418yescomplemented
94C4×M16—22no—
95C4×M16C2×C414yescomplemented
96#389—24no—
97#389—22no—
98#389—22no—
99#385—24no—
100#385—24no—
101#385—24no—
102#385—24no—
103#384—21no—
104#384—21no—
105#382—21no—
106#382—21no—
107#380—22no—
108#380—22no—
109#380—22no—
110#380—22no—
111#380C2×C412yes—
112#380C2×C412yes—
113#380C2×C414yes—
114#380C2×C414yes—
115#380C2311yescharacteristic
116#380C2312yes—
117#380C2311yescharacteristic
118C4×C4⋊C4—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C2×#95—22no—
123C2×#95—22no—
124C25C22×C412yes—
125C22×D4—24no—
126C22×D4—24no—
127C22×D4C4214yes—
128C22×D4C22⋊C412yes—
129C22×D4C22⋊C411yescharacteristic
130C22×D4C22⋊C411yescharacteristic
131C22×D4C4⋊C414yes—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—84no—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C414yes—
137C23×C4C22×C411yescharacteristic
138C23×C4C22×C412yes—
139C23×C4C22×C414yes—
140C23×C4C2×D414yes—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q814yes—
146C23×C4C2411yescharacteristic
147C2×M16—24no—
148C2×M16—28no—
149C2×M16—28no—
150C2×M16—88no—
151C2×M16—88no—
152C2×M16C22×C418yes—
153C22×C8—28no—
154C22×C8—28no—
155C4×D4—416no—
156C4×D4—48no—
157C4×D4—48no—
158C4×D4C22⋊C414yes—
159C4×D4C22⋊C414yes—
160C4×D4C22⋊C418yes—
161C4×D4C22⋊C418yes—
162C4×D4C22⋊C414yes—
163C4×D4C22⋊C414yes—
164C4×D4C4⋊C418yes—
165C4×D4C4⋊C414yes—
166C4×D4C4⋊C418yes—
167C4×D4C4⋊C414yes—
168C4×D4C4⋊C414yes—
169C4×D4C4⋊C414yes—
170#117—44no—
171#117—44no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—84no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4C4214yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C4⋊C414yes—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—48no—
191C2×C22⋊C4—48no—
192C2×C22⋊C4C4218yes—
193C2×C22⋊C4C22⋊C412yes—
194C2×C22⋊C4C22⋊C412yes—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22⋊C412yes—
197C2×C22⋊C4C4⋊C418yes—
198C2×C22⋊C4C22×C412yes—
199C2×C22⋊C4C22×C414yes—
200C2×C22⋊C4C22×C412yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42C22×C412yes—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×Q812yes—
211C4⋊C8—24no—
212C4⋊C8—24no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C22⋊C8—28no—
216C22⋊C8—28no—
217C22⋊C8—28no—
218C22⋊C8—28no—
219C22⋊C8—24no—
220C22⋊C8—24no—
221C22⋊C8—24no—
222C22⋊C8C22×C418yes—
223#97—24no—
224#97—24no—
225C4×C8—24no—
226C4×C8—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—21no—
232#95—22no—
233#95—21no—
234C24—24no—
235C24—24no—
236C24—82no—
237C24—82no—
238C24C2×C4214yes—
239C24C2×C22⋊C412yes—
240C24C2×C22⋊C412yes—
241C24C2×C4⋊C414yes—
242C24C23×C411yescharacteristic
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—216no—
246C2×D4—216no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4#9518yes—
250C2×D4#9514yes—
251C2×D4#9514yes—
252C2×D4#10014yes—
253C2×D4#10114yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—82no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—84no—
271C22×C4—81no—
272C22×C4—84no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4C2×C4212yes—
277C22×C4C2×C4212yes—
278C22×C4C2×C4214yes—
279C22×C4C2×C22⋊C414yes—
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C414yes—
283C22×C4C2×C22⋊C414yes—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C4⋊C412yes—
290C22×C4C2×C4⋊C414yes—
291C22×C4C2×C4⋊C414yes—
292C22×C4C2×C4⋊C414yes—
293C22×C4C2×C4⋊C414yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C414yes—
296C22×C4C23×C411yescharacteristic
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×Q811yescharacteristic
302M16—48no—
303M16—48no—
304M16—88no—
305M16—88no—
306C2×C8—28no—
307C2×C8—28no—
308C2×C8—88no—
309C2×C8—88no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—82no—
316C4⋊C4—82no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4#9518yes—
320C4⋊C4#9514yes—
321C4⋊C4#9514yes—
322C22⋊C4—48no—
323C22⋊C4—48no—
324C22⋊C4—42no—
325C22⋊C4—44no—
326C22⋊C4—42no—
327C22⋊C4#9518yes—
328C22⋊C4#9518yes—
329C22⋊C4#9518yes—
330C22⋊C4#9518yes—
331C42—44no—
332C42—44no—
333C42C2×C22⋊C412yes—
334C42C2×C22⋊C412yes—
335C42C2×C22⋊C412yes—
336C42C2×C22⋊C412yes—
337C42C2×C4⋊C412yes—
338C42C2×C4⋊C412yes—
339C42C2×C4⋊C412yes—
340C42C2×C4⋊C412yes—
341C23—21no—
342C23—22no—
343C23—21no—
344C23—22no—
345C23—28no—
346C23—28no—
347C23—42no—
348C23—42no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—44no—
357C23—88no—
358C23—88no—
359C23C2×#9514yes—
360C23C2×#9514yes—
361C23C2×#10014yes—
362C23C2×#10114yes—
363C23C22×C4211yescharacteristic
364C23C22×C22⋊C411yescharacteristic
365C23C22×C4⋊C411yescharacteristic
366D4—216no—
367D4—28no—
368D4#343116yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—28no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—28no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—168no—
405C2×C4—168no—
406C2×C4C2×#9512yes—
407C2×C4C2×#9512yes—
408C2×C4C2×#9512yes—
409C2×C4C2×#9514yes—
410C2×C4C2×#9514yes—
411C2×C4C2×#9512yes—
412C2×C4C2×#10012yes—
413C2×C4C2×#10112yes—
414C2×C4C22×C22⋊C411yescharacteristic
415C2×C4C22×C22⋊C411yescharacteristic
416C2×C4C22×C4⋊C411yescharacteristic
417C2×C4C22×C4⋊C411yescharacteristic
418C8—48no—
419C8—48no—
420C22—21no—
421C22—24no—
422C22—21no—
423C22—24no—
424C22—21no—
425C22—21no—
426C22—24no—
427C22—21no—
428C22—21no—
429C22—24no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22—44no—
434C22—44no—
435C22—44no—
436C22—44no—
437C22—42no—
438C22—42no—
439C22—88no—
440C22—88no—
441C22C2×#34318yes—
442C22C22×#9511yescharacteristic
443C22C22×#10011yescharacteristic
444C22C22×#10111yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—24no—
449C4—24no—
450C4—22no—
451C4—22no—
452C4—22no—
453C4—84no—
454C4—82no—
455C4—82no—
456C4C2×#34314yes—
457C4#288312yes—
458C2—21no—
459C2—21no—
460C2—28no—
461C2—44no—
462C2—41no—
463C2—41no—
464C2#2035411yescharacteristic
465C2C22×#34311yescharacteristic
466C2#2036011yescharacteristic
subgroupG/H
center, socleC22C22×#95
commutatorC23C22×C42
Frattini, cosocleC23×C4C24
prime radicalC23×D4C2×C4
LCSG ▷ C23 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8109C2semidirect
5. #8107C2semidirect
6. #8105C2semidirect
7. C2×#1305C2semidirect
13. #2974C4semidirect
19. D4×M16C4semidirect
42. C22×M16C2×C4semidirect
75. #432C2×C4semidirect
79. #431C2×C4semidirect
81. #423C2×C4semidirect
86. #407C2×C4semidirect
87. #407C2×C4semidirect
93. C2×C22⋊C8C2×C4semidirect
95. C4×M16C2×C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341812
#81092
#81072
#81052
C2×#13052