Finite Group Databasebeta

← #204919

Subgroups — #204919

Order 512.

1070 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
1#35298C211yesmaximal, characteristic
2#20815C211yesmaximal, characteristic, complemented
3#20547C211yesmaximal, characteristic, complemented
4C2×#1553C211yesmaximal, characteristic
5C2×#1552C211yesmaximal, characteristic, complemented
6C2×#1544C211yesmaximal, characteristic, complemented
7C2×#1540C211yesmaximal, characteristic
8#9710C212yesmaximal, complemented
9#9708C212yesmaximal, complemented
10#9705C212yesmaximal, complemented
11#9704C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14C2×#432C2211yescharacteristic
15C2×#431C2211yescharacteristic
16C2×#431C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18#2868C2211yescharacteristic
19#2862C2211yescharacteristic
20#2858C2211yescharacteristic
21#2858C2211yescharacteristic
22#2620—22no—
23#2566C414yes—
24#2545—22no—
25#2500C414yes—
26#2432—22no—
27#2326—22no—
28C4×#117C2211yescharacteristic
29#1842C2212yes—
30#1831C2212yes—
31#1798—22no—
32#1770—22no—
33#1768—22no—
34#1767C2212yes—
35#1767C2212yes—
36#1752—22no—
37#1553—24no—
38#1552—24no—
39#1544—24no—
40#1540—24no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×C8—21no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533—22no—
55#532—22no—
56#532—22no—
57#532—22no—
58#532—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8C2311yescharacteristic
61C2×C4⋊Q8C2311yescharacteristic
62C2×C4⋊Q8C2311yescharacteristic
63C2×#127C2311yescharacteristic
64C2×#125C2×C412yes—
65C2×#124—22no—
66C2×#124—21no—
67C2×#124—22no—
68C2×C22⋊Q8—21no—
69C2×C22⋊Q8—21no—
70C2×#121—21no—
71C2×#121C2×C412yes—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×D4—21no—
75C2×C4×D4C2×C412yes—
76C2×#117C2×C412yes—
77C2×#117D412yes—
78C22×C42C2311yescharacteristic
79#432—22no—
80#432—22no—
81#431—22no—
82#431—22no—
83#431—22no—
84#431D414yes—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—24no—
88C2×C4⋊C8D414yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×C4⋊C8C2312yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×#97—22no—
98C2×#97—22no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101#396—28no—
102#394—28no—
103#391—22no—
104#391—22no—
105#390—22no—
106#390—22no—
107#390C2×C414yes—
108#390C2×C414yes—
109#388—28no—
110#385—28no—
111#384—22no—
112#384C2×C414yes—
113#382C2×C414yes—
114C4×C4⋊C4C2×C412yes—
115C4×C4⋊C4D412yes—
116C4×C22⋊C4C2×C412yes—
117C4×C22⋊C4D412yes—
118#376D412yes—
119C43C2311yescharacteristic
120C43C2311yescharacteristic
121#336—24no—
122#336—22no—
123#336—22no—
124#332—24no—
125#332—24no—
126#332—24no—
127#332—24no—
128#332—24no—
129#332—24no—
130#331—24no—
131#331—24no—
132C2×#41—42no—
133C2×#41—42no—
134C2×#41—82no—
135C2×#41—82no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×Q8—21no—
139C22×D4—22no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4C22×C412yes—
143C23×C4—22no—
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—82no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—82no—
166C4⋊Q8—82no—
167#127—21no—
168#127—21no—
169#127—21no—
170#127—21no—
171#125—24no—
172#125—24no—
173#124—48no—
174#124—42no—
175#124—42no—
176#124—82no—
177#124—82no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182#121—24no—
183#121—24no—
184#121—42no—
185#121—42no—
186C4×Q8—42no—
187C4×Q8—42no—
188C4×Q8—42no—
189C4×Q8—42no—
190C4×D4—24no—
191C4×D4—24no—
192C4×D4—42no—
193C4×D4—42no—
194#117—24no—
195#117—24no—
196#117—44no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4C22⋊C414yes—
204C2×C4⋊C4C22×C412yes—
205C2×C4⋊C4C22×C412yes—
206C2×C4⋊C4C22×C412yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—28no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4C22⋊C414yes—
213C2×C42—22no—
214C2×C42C22×C412yes—
215C2×C42C22×C412yes—
216C2×C42C22×C411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42C2×D412yes—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228C4⋊C8—24no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—48no—
236C4⋊C8C2×D414yes—
237C4⋊C8C2×D414yes—
238C22⋊C8—24no—
239C22⋊C8—42no—
240C22⋊C8—42no—
241#97—42no—
242#97—82no—
243#97—82no—
244C4×C8—24no—
245C4×C8—42no—
246#95—28no—
247#95C22⋊C414yes—
248#95C22⋊C414yes—
249C24—22no—
250C24C2×C22⋊C411yescharacteristic
251#41—84no—
252#41—84no—
253#41—88no—
254C2×Q8—41no—
255C2×Q8—41no—
256C2×Q8—41no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—44no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—44no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—84no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—44no—
285C22×C4—84no—
286C22×C4C2×C22⋊C414yes—
287C22×C4C2×C22⋊C414yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×D412yes—
292C22×C4C4×D412yes—
293C22×C4#12112yes—
294C22×C4#12112yes—
295C22×C4#12412yes—
296C22×C4#12712yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C2×C8—42no—
307C2×C8—42no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—84no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—44no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—44no—
325C4⋊C4—44no—
326C4⋊C4—88no—
327C22⋊C4—44no—
328C22⋊C4—416no—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—44no—
332C22⋊C4—88no—
333C42—24no—
334C42—24no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—42no—
341C42—44no—
342C42C4×D412yes—
343C42C4×D412yes—
344C42#12112yes—
345C42#12112yes—
346C42#12412yes—
347C42#12712yes—
348C42C22×D411yescharacteristic
349C42C22×D411yescharacteristic
350C42C2×#4111yescharacteristic
351C42C2×#4111yescharacteristic
352C23—22no—
353C23—22no—
354C23—22no—
355C23—44no—
356C23—44no—
357C23—44no—
358C23—42no—
359C23#39012yes—
360C23C22×C22⋊C411yescharacteristic
361Q8—82no—
362Q8—82no—
363Q8—81no—
364Q8—81no—
365Q8—81no—
366Q8—81no—
367Q8—82no—
368Q8—82no—
369D4—82no—
370D4—82no—
371D4—88no—
372D4—82no—
373D4—82no—
374D4—81no—
375D4—81no—
376D4—82no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—88no—
401C2×C4#39012yes—
402C2×C4#39012yes—
403C2×C4#39012yes—
404C2×C4C2×C4×D411yescharacteristic
405C2×C4C2×C4×D411yescharacteristic
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12111yescharacteristic
408C2×C4C2×#12411yescharacteristic
409C2×C4C2×#12711yescharacteristic
410C2×C4#57514yes—
411C2×C4#57612yes—
412C2×C4#57712yes—
413C8—44no—
414C8—44no—
415C8—82no—
416C8—82no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—44no—
421C22—82no—
422C22—88no—
423C22—82no—
424C22—82no—
425C22C2×#39011yescharacteristic
426C22#288912yes—
427C22#289612yes—
428C22#289712yes—
429C4—22no—
430C4—22no—
431C4—21no—
432C4—21no—
433C4—22no—
434C4—44no—
435C4—44no—
436C4—82no—
437C4—82no—
438C4—82no—
439C4#304111yescharacteristic
440C4#304311yescharacteristic
441C4#304512yes—
442C4#313512yes—
443C4#315012yes—
444C2—42no—
445C2—162no—
446C2#1776914yes—
447C2#2119911yescharacteristic
448C2#2122011yescharacteristic
449C2#2126311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20815C2semidirect
3. #20547C2semidirect
5. C2×#1552C2semidirect
6. C2×#1544C2semidirect
8. #9710C2semidirect
9. #9708C2semidirect
10. #9705C2semidirect
11. #9704C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#208152
#205472
C2×#15532
C2×#15522
C2×#15442
C2×#15402
#97102
#97082
#97052
#97042