Finite Group Databasebeta

← #181367

Subgroups — #181367

Order 512.

1483 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60230C211yesmaximal, characteristic
2#60230C211yesmaximal, characteristic
3#34186C211yesmaximal, characteristic
4#8109C212yesmaximal, complemented
5#8108C214yesmaximal, complemented
6#8105C212yesmaximal, complemented
7#8104C214yesmaximal, complemented
8C22×C4×D4C2211yescharacteristic
9#2975—21no—
10#2975—21no—
11#2974—24no—
12#2974C418yescomplemented
13#2973—22no—
14#2973—22no—
15#2972—21no—
16#2972—21no—
17D4×M16—24no—
18D4×M16C418yescomplemented
19#2920C2212yes—
20#2920C2212yes—
21C2×#407C2214yes—
22C2×#407C2214yes—
23#2357—21no—
24#2357—22no—
25#2357—21no—
26#2354—21no—
27#2354—22no—
28#2354—21no—
29#2312C414yes—
30#2309C412yes—
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#2280C2212yes—
34#2280C2212yes—
35#2274C412yes—
36#1305C2214yes—
37#1305C2218yes—
38#1305C2214yes—
39C23×D4C2×C411yescharacteristic
40C24×C4C2312yes—
41C22×M16—24no—
42C22×M16C2×C418yes—
43C2×C4×D4C2×C414yes—
44C2×C4×D4C2×C414yes—
45C2×C4×D4D411yescharacteristic
46C2×C4×D4D412yes—
47C2×C4×D4D411yescharacteristic
48C2×C4×D4D411yescharacteristic
49C2×C4×D4D412yes—
50C2×C4×D4D411yescharacteristic
51C2×C4×D4D414yes—
52C2×C4×D4Q814yes—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117C2×C412yes—
56C2×#117C2×C412yes—
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C22⋊C4C2×C412yes—
59C22×C42C2311yescharacteristic
60#436—28no—
61#436—28no—
62#435—216no—
63#435—216no—
64C8×D4—28no—
65C8×D4—28no—
66#432—22no—
67#432—22no—
68#432—22no—
69#432C2×C414yes—
70#431—22no—
71#431—22no—
72#431—22no—
73#431C2×C414yes—
74#423—22no—
75#423C2×C414yes—
76#407—24no—
77#407—24no—
78#407—24no—
79#407—24no—
80#407C2×C418yes—
81#407C2×C418yes—
82#407C2314yes—
83#407C2314yes—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—24no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8C2×C418yes—
88C4×M16—22no—
89C4×M16C2×C414yes—
90#390—22no—
91#390—22no—
92#390—24no—
93#388—24no—
94#388—24no—
95#388—24no—
96#388—24no—
97#384—21no—
98#384—21no—
99#382—21no—
100#382—21no—
101#381—22no—
102#381—22no—
103#380—22no—
104#380—22no—
105#380C2×C412yes—
106#380C2×C412yes—
107#380C2311yescharacteristic
108#380C2312yes—
109#380C2311yescharacteristic
110#379C2×C414yes—
111#379C2×C414yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C414yes—
122C4×C22⋊C4C2×C414yes—
123C25C22×C412yes—
124C22×D4—42no—
125C22×D4—42no—
126C22×D4C4214yes—
127C22×D4C22⋊C411yescharacteristic
128C22×D4C22⋊C411yescharacteristic
129C22×D4C22⋊C412yes—
130C22×D4C4⋊C414yes—
131C23×C4—84no—
132C23×C4C22×C411yescharacteristic
133C23×C4C22×C414yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C22×C412yes—
136C23×C4C22×C414yes—
137C23×C4C2×D414yes—
138C23×C4C2×D412yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×Q814yes—
143C23×C4C2411yescharacteristic
144C2×M16—24no—
145C2×M16—28no—
146C2×M16—28no—
147C2×M16—88no—
148C2×M16—88no—
149C2×M16C22×C418yes—
150C22×C8—28no—
151C22×C8—28no—
152C4×D4C22⋊C414yes—
153C4×D4C22⋊C414yes—
154C4×D4C22⋊C418yes—
155C4×D4C22⋊C418yes—
156C4×D4C22⋊C414yes—
157C4×D4C22⋊C414yes—
158C4×D4C4⋊C414yes—
159C4×D4C4⋊C414yes—
160C4×D4C4⋊C414yes—
161C4×D4C4⋊C414yes—
162C4×D4C4⋊C418yes—
163C4×D4C4⋊C418yes—
164#117—44no—
165#117—44no—
166#117—84no—
167#117—84no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C4214yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C411yescharacteristic
178C2×C4⋊C4C22⋊C411yescharacteristic
179C2×C4⋊C4C4⋊C414yes—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4C4218yes—
185C2×C22⋊C4C22⋊C412yes—
186C2×C22⋊C4C22⋊C412yes—
187C2×C22⋊C4C22⋊C412yes—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C4⋊C418yes—
190C2×C22⋊C4C22×C414yes—
191C2×C22⋊C4C22×C412yes—
192C2×C22⋊C4C22×C412yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42C22×C412yes—
198C2×C42C22×C412yes—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×Q812yes—
205C4⋊C8—24no—
206C4⋊C8—24no—
207C4⋊C8—24no—
208C4⋊C8—24no—
209C22⋊C8—28no—
210C22⋊C8—28no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213C22⋊C8—24no—
214C22⋊C8—24no—
215C22⋊C8—24no—
216C22⋊C8C22×C418yes—
217#97—24no—
218#97—24no—
219C4×C8—24no—
220C4×C8—24no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—21no—
226#95—22no—
227#95—21no—
228C24—42no—
229C24—42no—
230C24—82no—
231C24—82no—
232C24C2×C4214yes—
233C24C2×C22⋊C412yes—
234C24C2×C22⋊C412yes—
235C24C2×C4⋊C414yes—
236C24C23×C411yescharacteristic
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—48no—
242C2×D4—48no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4#9514yes—
246C2×D4#9514yes—
247C2×D4#9518yes—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—44no—
254C22×C4—44no—
255C22×C4—48no—
256C22×C4—48no—
257C22×C4—42no—
258C22×C4—82no—
259C22×C4—84no—
260C22×C4—82no—
261C22×C4—82no—
262C22×C4—84no—
263C22×C4—81no—
264C22×C4C2×C4212yes—
265C22×C4C2×C4212yes—
266C22×C4C2×C4214yes—
267C22×C4C2×C22⋊C414yes—
268C22×C4C2×C22⋊C414yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C414yes—
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4C2×C4⋊C412yes—
278C22×C4C2×C4⋊C414yes—
279C22×C4C2×C4⋊C414yes—
280C22×C4C2×C4⋊C414yes—
281C22×C4C2×C4⋊C414yes—
282C22×C4C2×C4⋊C412yes—
283C22×C4C2×C4⋊C414yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C23×C411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×Q811yescharacteristic
290M16—48no—
291M16—48no—
292M16—88no—
293M16—88no—
294C2×C8—28no—
295C2×C8—28no—
296C2×C8—88no—
297C2×C8—88no—
298C4⋊C4—44no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—42no—
302C4⋊C4—44no—
303C4⋊C4—82no—
304C4⋊C4—82no—
305C4⋊C4#9518yes—
306C4⋊C4#9514yes—
307C4⋊C4#9514yes—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C22⋊C4—84no—
314C22⋊C4—84no—
315C22⋊C4#9518yes—
316C22⋊C4#9518yes—
317C22⋊C4#9518yes—
318C22⋊C4#9518yes—
319C42—44no—
320C42—44no—
321C42—84no—
322C42—84no—
323C42C2×C22⋊C412yes—
324C42C2×C22⋊C412yes—
325C42C2×C22⋊C412yes—
326C42C2×C22⋊C412yes—
327C42C2×C4⋊C412yes—
328C42C2×C4⋊C412yes—
329C42C2×C4⋊C412yes—
330C42C2×C4⋊C412yes—
331C23—21no—
332C23—21no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—41no—
338C23—41no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—44no—
344C23—44no—
345C23—44no—
346C23—44no—
347C23—44no—
348C23—44no—
349C23—88no—
350C23—88no—
351C23C2×#9514yes—
352C23C2×#9514yes—
353C23C22×C4211yescharacteristic
354C23C22×C22⋊C411yescharacteristic
355C23C22×C4⋊C411yescharacteristic
356D4—28no—
357D4—28no—
358D4—48no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—28no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—44no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4C2×#9512yes—
395C2×C4C2×#9512yes—
396C2×C4C2×#9512yes—
397C2×C4C2×#9514yes—
398C2×C4C2×#9514yes—
399C2×C4C2×#9512yes—
400C2×C4#41312yes—
401C2×C4#41312yes—
402C2×C4C22×C22⋊C411yescharacteristic
403C2×C4C22×C22⋊C411yescharacteristic
404C2×C4C22×C4⋊C411yescharacteristic
405C2×C4C22×C4⋊C411yescharacteristic
406C8—48no—
407C8—48no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—21no—
412C22—24no—
413C22—24no—
414C22—21no—
415C22—21no—
416C22—24no—
417C22—24no—
418C22—21no—
419C22—21no—
420C22—24no—
421C22—24no—
422C22—42no—
423C22—42no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—88no—
429C22—88no—
430C22C22×#9511yescharacteristic
431C22C2×#41311yescharacteristic
432C22C2×#41311yescharacteristic
433C4—21no—
434C4—21no—
435C4—21no—
436C4—24no—
437C4—24no—
438C4—22no—
439C4—22no—
440C4—22no—
441C4—84no—
442C4—82no—
443C4—82no—
444C4#174214yes—
445C4#288312yes—
446C2—21no—
447C2—21no—
448C2—41no—
449C2—41no—
450C2—44no—
451C2—44no—
452C2#2035411yescharacteristic
453C2C2×#174211yescharacteristic
454C2#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. #8108C2semidirect
6. #8105C2semidirect
7. #8104C2semidirect
12. #2974C4semidirect
18. D4×M16C4semidirect

Maximal subgroups

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

subgroupindex
#602302
#341862
#81092
#81082
#81052
#81042