Finite Group Databasebeta

← #196926

Subgroups — #196926

Order 512.

1498 conjugacy classes of subgroups.

The subgroup lattice has 441 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35248C211yesmaximal, characteristic
2C2×#2276C211yesmaximal, characteristic
3#21820C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic
6#9145C214yesmaximal
7#9144C214yesmaximal
8#8021C211yesmaximal, characteristic
9#8013C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8—21no—
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#2855C2214yes—
16#2634—22no—
17#2581C414yescomplemented
18#2568C412yescomplemented
19#2546C412yescomplemented
20#2503—21no—
21#2484—22no—
22#2434—21no—
23#2327—22no—
24#2283C2211yescharacteristic
25#2276—24no—
26#2276—24no—
27C2×C4×C4⋊C4—22no—
28C2×C4×C4⋊C4—21no—
29C2×C4×C4⋊C4C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31C2×#376C2212yes—
32C2×#376C2211yescharacteristic
33#1985—22no—
34#1984—22no—
35#1817C2214yes—
36#1506—24no—
37#1503—24no—
38C2×#328C2214yes—
39C2×#328C2212yes—
40C2×#328C2212yes—
41#1294C2212yes—
42#1294C2212yes—
43#1294C2214yes—
44C23×Q8D411yescharacteristic
45#533—22no—
46#533—22no—
47#533—22no—
48#533—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2312yes—
55C2×C4×Q8—216no—
56C2×C4×Q8—28no—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—21no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C412yes—
63C2×C4×Q8C2×C412yes—
64C2×C4×Q8C2×C412yes—
65C2×C4×Q8D412yes—
66C2×C4×Q8D412yes—
67C2×#117—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4D412yes—
71C22×C4⋊C4D411yescharacteristic
72C22×C42—22no—
73C22×C42—21no—
74C22×C42—22no—
75C22×C42—21no—
76C22×C42C2312yes—
77C22×C42C2311yescharacteristic
78#432—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82C2×C4⋊C8—21no—
83C2×C4⋊C8—21no—
84C2×C22⋊C8C2312yes—
85C2×C22⋊C8C2312yes—
86C2×#97—21no—
87C2×C4×C8—21no—
88#400—24no—
89#400—24no—
90#398—24no—
91#395—24no—
92#391—22no—
93#391—22no—
94#391C2×C418yes—
95#389C2×C418yes—
96#387—28no—
97#387—28no—
98#385—24no—
99#383—21no—
100#382—22no—
101#382C2×C412yes—
102#381—24no—
103#381—21no—
104#380C2312yes—
105C4×C4⋊C4—28no—
106C4×C4⋊C4—216no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—24no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4D414yes—
112C4×C22⋊C4—21no—
113#376—21no—
114#376—22no—
115#376—24no—
116#376—24no—
117#376D414yes—
118C2×#95—22no—
119C2×#95—21no—
120C2×#95C2311yescharacteristic
121C2×#95C2312yes—
122C2×#95C2311yescharacteristic
123#336—28no—
124#328—28no—
125#328—24no—
126#328—24no—
127C22×Q8—28no—
128C22×Q8—21no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8C22⋊C412yes—
132C22×Q8C22⋊C412yes—
133C22×Q8C22×C412yes—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—48no—
149#125—42no—
150#125—42no—
151C22⋊Q8—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C22⋊Q8—22no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C4×Q8—264no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—24no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—24no—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—42no—
169C4×Q8—42no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—216no—
173C2×C4⋊C4—28no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—28no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4C22⋊C412yes—
187C2×C4⋊C4C22⋊C414yes—
188C2×C4⋊C4C22⋊C414yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C414yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C42—28no—
195C2×C42—216no—
196C2×C42—28no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—28no—
205C2×C42—24no—
206C2×C42—82no—
207C2×C42—84no—
208C2×C42C22×C414yes—
209C2×C42C22×C411yescharacteristic
210C2×C42C22×C411yescharacteristic
211C2×C42C2×D414yes—
212C2×C42C2×D412yes—
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4112yes—
217C2×C42#4112yes—
218C4⋊C8—42no—
219C4⋊C8—42no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224#97—42no—
225C4×C8—42no—
226#95—28no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—28no—
231#95—28no—
232#95—28no—
233#95—24no—
234#95—21no—
235#95—21no—
236#95—22no—
237#95—22no—
238#95C2×D412yes—
239#95C2×D414yes—
240#95C2×D412yes—
241C24C22×D411yescharacteristic
242C2×Q8—216no—
243C2×Q8—216no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—28no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—44no—
251C2×Q8—42no—
252C2×Q8—44no—
253C2×Q8#10418yes—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C2×C22⋊C414yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4#12012yes—
287C22×C4#12012yes—
288C22×C4#12312yes—
289C22×C4#12312yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C2×C8—44no—
297C2×C8—44no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—232no—
301C4⋊C4—216no—
302C4⋊C4—28no—
303C4⋊C4—28no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—44no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—416no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C42—232no—
321C42—216no—
322C42—24no—
323C42—24no—
324C42—24no—
325C42—24no—
326C42—44no—
327C42—44no—
328C42—84no—
329C42—88no—
330C42C4×D412yes—
331C42C4×D412yes—
332C42C4×D412yes—
333C42C4×D412yes—
334C42#12112yes—
335C42#12112yes—
336C42#12112yes—
337C42#12112yes—
338C23—22no—
339C23—22no—
340C23C22×C22⋊C411yescharacteristic
341C23C2×#12011yescharacteristic
342C23C2×#12311yescharacteristic
343C23#57712yes—
344C23#57812yes—
345Q8—216no—
346Q8—216no—
347Q8—82no—
348Q8—82no—
349Q8—84no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—28no—
368C2×C4—24no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—416no—
374C2×C4—48no—
375C2×C4—48no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4#38612yes—
393C2×C4#38612yes—
394C2×C4#38612yes—
395C2×C4#38612yes—
396C2×C4C2×#10418yes—
397C2×C4C2×#10414yes—
398C2×C4C2×C4×D411yescharacteristic
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×#12111yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C8—82no—
403C8—82no—
404C22—24no—
405C22—24no—
406C22—21no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22C2×#38611yescharacteristic
411C22#288412yes—
412C22#289812yes—
413C22C22×#10412yes—
414C22#300911yescharacteristic
415C22#301111yescharacteristic
416C22#309712yes—
417C4—22no—
418C4—22no—
419C4—22no—
420C4—28no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—48no—
426C4—44no—
427C4—44no—
428C4—44no—
429C4—84no—
430C4—82no—
431C4—82no—
432C4#189614yes—
433C2—21no—
434C2—21no—
435C2—22no—
436C2#1754914yes—
437C2#2091311yescharacteristic
438C2C2×#189611yescharacteristic
439C2#2098611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
17. #2581C4semidirect
18. #2568C4semidirect
19. #2546C4semidirect

Maximal subgroups

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

subgroupindex
#352482
C2×#22762
#218202
C2×#15062
C2×#15032
#91452
#91442
#80212
#80132