Finite Group Databasebeta

← #204630

Subgroups — #204630

Order 512.

1172 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35180C211yesmaximal, characteristic
2#20533C211yesmaximal, characteristic, complemented
3C2×#1518C211yesmaximal, characteristic, complemented
4C2×#1517C211yesmaximal, characteristic
5C2×#1503C211yesmaximal, characteristic, complemented
6C2×#1502C211yesmaximal, characteristic
7#13955C211yesmaximal, characteristic, complemented
8#9681C218yesmaximal, complemented
9C2×#533C2211yescharacteristic
10C2×#532C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2850C2214yes—
14#2639—22no—
15#2592—22no—
16#2568C412yescomplemented
17#2563C412yescomplemented
18#2562—22no—
19#2548C412yescomplemented
20#2547C412yescomplemented
21#2311—22no—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C8×C4⋊C4—22no—
26C8×C4⋊C4—22no—
27C4×C22⋊C8C2214yes—
28#1518—24no—
29#1517—24no—
30#1503—24no—
31#1502—24no—
32C2×#328C2214yes—
33C2×#328C2214yes—
34C2×#327C2214yes—
35C2×#327C2214yes—
36#1442—22no—
37C4⋊#95—22no—
38#1434C2214yes—
39C22×#41—21no—
40#533—22no—
41#533—22no—
42#533—22no—
43#533—22no—
44#532—22no—
45#532—22no—
46#532—22no—
47#532—22no—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#127C2×C411yescharacteristic
52C2×#125—21no—
53C2×#124—21no—
54C2×#124C2×C414yes—
55C2×C22⋊Q8—21no—
56C2×C22⋊Q8C2312yes—
57C2×#121—21no—
58C2×#121C2312yes—
59C2×C4×Q8—21no—
60C2×C4×Q8C2×C412yes—
61C2×C4×D4—21no—
62C2×C4×D4C2×C412yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D412yes—
67C22×C42—21no—
68C22×C42C2312yes—
69C22×C42C2311yescharacteristic
70#431—22no—
71#431—22no—
72#431D414yes—
73#431D414yes—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C22⋊C8C2312yes—
77C2×C22⋊C8C2312yes—
78C2×C4×C8—21no—
79C2×C4×C8—21no—
80C2×C4×C8—24no—
81#400—24no—
82#397—24no—
83#394—24no—
84#393—24no—
85#391—24no—
86#391C2×C414yes—
87#390—24no—
88#390C2×C414yes—
89#388—28no—
90#386—28no—
91#382—22no—
92#382—22no—
93#382C2×C414yes—
94#380—24no—
95#380—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4—22no—
98C4×C4⋊C4C2×C412yes—
99C4×C4⋊C4C2×C412yes—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4D412yes—
102C2×#95—24no—
103C2×#95—21no—
104C2×#95—24no—
105C2×#95D412yes—
106C2×#95C2312yes—
107C2×#95C2312yes—
108#336—24no—
109#328—28no—
110#328—28no—
111#327—28no—
112#327—28no—
113C2×#41—42no—
114C2×#41—42no—
115C2×#41—44no—
116C2×#41—44no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8C22×C412yes—
120C22×D4—22no—
121C22×D4—21no—
122C22×D4—21no—
123C22×D4C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4C2×D414yes—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2411yescharacteristic
131C22×C8—22no—
132C22×C8—22no—
133C4⋊Q8—22no—
134C4⋊Q8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139#127—22no—
140#127—22no—
141#125—42no—
142#125—42no—
143#124—42no—
144#124—42no—
145#124—84no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8—22no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152#121—22no—
153#121—22no—
154#121—22no—
155#121—22no—
156#121—42no—
157#121—42no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—42no—
161C4×Q8—42no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—42no—
165C4×D4—42no—
166#117—42no—
167#117—42no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4C22⋊C414yes—
181C2×C4⋊C4C22⋊C414yes—
182C2×C4⋊C4C22×C412yes—
183C2×C4⋊C4C22×C412yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—28no—
189C2×C42—22no—
190C2×C42—42no—
191C2×C42—44no—
192C2×C42—82no—
193C2×C42C22×C414yes—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D414yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42#4112yes—
202C2×C42#4112yes—
203C4⋊C8—24no—
204C4⋊C8—24no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8C2×D414yes—
208C22⋊C8C2×D414yes—
209C4×C8—24no—
210C4×C8—24no—
211C4×C8—84no—
212#95—28no—
213#95—28no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—48no—
219#95—42no—
220#95—84no—
221#95C22⋊C414yes—
222#95C22⋊C414yes—
223C24—22no—
224C24C22×D411yescharacteristic
225#41—88no—
226#41—84no—
227#41—84no—
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—82no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—44no—
247C2×D4—82no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4C2×C22⋊C418yes—
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C411yescharacteristic
277C22×C4#12412yes—
278C22×C4#12412yes—
279C22×C4#12712yes—
280C22×C4#12712yes—
281C22×C4#13514yes—
282C22×C4#13514yes—
283C22×C4C23×C411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4111yescharacteristic
288C22×C4C2×#4111yescharacteristic
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—48no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—84no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—88no—
316C22⋊C4—88no—
317C42—44no—
318C42—44no—
319C42—44no—
320C42—84no—
321C42C4×D412yes—
322C42C4×D412yes—
323C42C4×D412yes—
324C42C4×D412yes—
325C42#12112yes—
326C42#12112yes—
327C42#12112yes—
328C42#12112yes—
329C23—21no—
330C23—21no—
331C23—22no—
332C23—42no—
333C23—42no—
334C23—44no—
335C23—44no—
336C23—42no—
337C23C22×C22⋊C411yescharacteristic
338C23C2×#12411yescharacteristic
339C23C2×#12711yescharacteristic
340C23C2×#13512yes—
341C23C2×#13512yes—
342Q8—82no—
343Q8—82no—
344Q8—84no—
345Q8—84no—
346D4—82no—
347D4—82no—
348D4—84no—
349D4—84no—
350D4—82no—
351D4—82no—
352D4—84no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—82no—
381C2×C4—82no—
382C2×C4—82no—
383C2×C4—82no—
384C2×C4—88no—
385C2×C4—82no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—82no—
389C2×C4—82no—
390C2×C4#39012yes—
391C2×C4#39012yes—
392C2×C4#39012yes—
393C2×C4#39012yes—
394C2×C4#41614yes—
395C2×C4#41614yes—
396C2×C4#48818yes—
397C2×C4#49514yes—
398C2×C4#49514yes—
399C2×C4C2×C4×D411yescharacteristic
400C2×C4C2×C4×D411yescharacteristic
401C2×C4C2×#12111yescharacteristic
402C2×C4C2×#12111yescharacteristic
403C8—44no—
404C8—44no—
405C22—21no—
406C22—21no—
407C22—22no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—82no—
414C22—82no—
415C22—84no—
416C22—84no—
417C22—82no—
418C22C2×#39011yescharacteristic
419C22#288311yescharacteristic
420C22#288411yescharacteristic
421C22C2×#41612yes—
422C22#290112yes—
423C22C2×#48812yes—
424C22C2×#49511yescharacteristic
425C22C2×#49511yescharacteristic
426C4—22no—
427C4—22no—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—48no—
433C4—84no—
434C4—82no—
435C4—82no—
436C4—82no—
437C4—162no—
438C4#195814yes—
439C4#195814yes—
440C2—21no—
441C2—21no—
442C2—22no—
443C2—162no—
444C2#1761612yes—
445C2#1761712yes—
446C2C2×#195811yescharacteristic
447C2#2121411yescharacteristic
448C2#2125311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20533C2semidirect
3. C2×#1518C2semidirect
5. C2×#1503C2semidirect
7. #13955C2semidirect
8. #9681C2semidirect
16. #2568C4semidirect
17. #2563C4semidirect
19. #2548C4semidirect
20. #2547C4semidirect

Maximal subgroups

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

subgroupindex
#351802
#205332
C2×#15182
C2×#15172
C2×#15032
C2×#15022
#139552
#96812