Finite Group Databasebeta

← #495402

Subgroups — #495402

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35574C211yesmaximal, characteristic
2#20634C211yesmaximal, characteristic
3#20397C211yesmaximal, characteristic
4#13294C214yesmaximal, complemented
5#13293C214yesmaximal, complemented
6#12575C212yesmaximal
7#12573C212yesmaximal
8C22×C4⋊Q8C411yescharacteristic, complemented
9C2×#520C2211yescharacteristic
10C2×#413C2212yes—
11C2×#410C2212yes—
12#2873—21no—
13#2579C412yescomplemented
14#2579C2212yes—
15#2452—22no—
16#2451—22no—
17#2440—21no—
18#2440—21no—
19#2439—21no—
20#2439—21no—
21#2409C412yescomplemented
22#2409C2212yes—
23#2317C411yescharacteristic, complemented
24#2316—21no—
25#2316C411yescharacteristic, complemented
26#2311C2211yescharacteristic
27#2297C2211yescharacteristic
28#2296C411yescharacteristic, complemented
29#1799—22no—
30#1799C2214yes—
31C4×#101—22no—
32C4×#100C2214yes—
33C2×#343—21no—
34#1740—21no—
35#1390C2214yes—
36#1390C2214yes—
37#1390C2218yes—
38C22×#41C2311yescharacteristic
39C23×Q8C2×C412yes—
40C22×M16—22no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—44no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47#520—44no—
48C2×C4×Q8D412yes—
49C2×C4×D4C2×C412yes—
50C2×C4×D4D412yes—
51C2×C4×D4C2312yes—
52C2×#117—22no—
53C2×#117C2×C411yescharacteristic
54C2×#117C2×C411yescharacteristic
55C2×#117C2×C411yescharacteristic
56C2×#117D411yescharacteristic
57C2×#117D411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C412yes—
61C22×C4⋊C4D411yescharacteristic
62C22×C4⋊C4D411yescharacteristic
63C22×C42C2×C411yescharacteristic
64#413—24no—
65#413Q818yescomplemented
66C2×#101—22no—
67C2×#100C2314yes—
68#410—28no—
69C2×#99—22no—
70C2×#99C2314yes—
71#405—44no—
72C4×M16—24no—
73#398—22no—
74#398—22no—
75#396—22no—
76#396—22no—
77#395—22no—
78#395—22no—
79#393—22no—
80#393—22no—
81#391—21no—
82#391—21no—
83#391—22no—
84#391—22no—
85#390—21no—
86#390—21no—
87#389—21no—
88#389—22no—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—22no—
93#389—44no—
94#387—22no—
95#387—22no—
96#386C2×C412yes—
97#386C2×C411yescharacteristic
98#386C2×C411yescharacteristic
99#386C2312yes—
100#386C2311yescharacteristic
101#386C2311yescharacteristic
102#385—22no—
103#385—22no—
104#385C2×C412yes—
105#385C2×C412yes—
106#385C2×C412yes—
107#385C2×C412yes—
108#384—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112#380—24no—
113#380C2×C412yes—
114#380C2×C412yes—
115C4×C4⋊C4—22no—
116C4×C22⋊C4—21no—
117C4×C22⋊C4—21no—
118#343—24no—
119#342—24no—
120C2×#41—42no—
121C2×#41—42no—
122C2×#41C2×Q814yes—
123C2×#41#4114yes—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—44no—
127C22×Q8—84no—
128C22×Q8C2×D411yescharacteristic
129C22×D4C22×C411yescharacteristic
130C22×D4C2×D411yescharacteristic
131C22×D4C2411yescharacteristic
132C23×C4C22⋊C412yes—
133C23×C4C22×C411yescharacteristic
134C23×C4C22×C412yes—
135C23×C4C22×C411yescharacteristic
136C23×C4C22×C411yescharacteristic
137C23×C4C22×C412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C2×M16—28no—
142C2×M16—24no—
143C2×M16—44no—
144C2×M16—42no—
145C22×C8—42no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C4⋊Q8—44no—
149C22⋊Q8—84no—
150C22⋊Q8—84no—
151C4×Q8—44no—
152C4×D4—44no—
153C4×D4—44no—
154C4×D4—44no—
155#117—24no—
156#117—42no—
157#117—42no—
158#117—42no—
159#117—42no—
160#117—44no—
161#117—44no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C22⋊C411yescharacteristic
182C2×C4⋊C4C22⋊C412yes—
183C2×C4⋊C4C22⋊C411yescharacteristic
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22⋊C411yescharacteristic
186C2×C4⋊C4C22⋊C411yescharacteristic
187C2×C4⋊C4C2×D412yes—
188C2×C22⋊C4—21no—
189C2×C22⋊C4—21no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—21no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C22⋊C412yes—
196C2×C22⋊C4C22×C412yes—
197C2×C22⋊C4C22×C411yescharacteristic
198C2×C22⋊C4C22×C411yescharacteristic
199C2×C22⋊C4C22×C412yes—
200C2×C22⋊C4C2×D412yes—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C22⋊C412yes—
208C2×C42C22⋊C412yes—
209C2×C42C2×D412yes—
210#101—28no—
211#100—24no—
212#100C2×Q818yes—
213#99—28no—
214#99—44no—
215#97—44no—
216C4×C8—44no—
217#95—21no—
218#95—21no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—44no—
230C24C2×C22⋊C411yescharacteristic
231C24C23×C411yescharacteristic
232C24C22×D411yescharacteristic
233#41—44no—
234#41—44no—
235#41—44no—
236#41—44no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—44no—
242C2×Q8—44no—
243C2×Q8—82no—
244C2×Q8—82no—
245C2×Q8—88no—
246C2×Q8—88no—
247C2×Q8C22⋊Q814yes—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4C4×Q814yes—
255C2×D4#12412yes—
256C2×D4C4⋊Q812yes—
257C2×D4C22×Q812yes—
258C2×D4C2×#4112yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C412yes—
290C22×C4C2×C22⋊C411yescharacteristic
291C22×C4C4×Q814yes—
292C22×C4C22⋊Q814yes—
293C22×C4#12412yes—
294C22×C4C4⋊Q812yes—
295C22×C4C22×D411yescharacteristic
296M16—28no—
297M16—44no—
298M16—48no—
299C2×C8—44no—
300C2×C8—48no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—84no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4—164no—
321C4⋊C4#9914yes—
322C22⋊C4—24no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—42no—
326C22⋊C4—42no—
327C22⋊C4—42no—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—44no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C22⋊C4—82no—
336C42—24no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—42no—
342C42—42no—
343C42—84no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23#39112yes—
357C23C22×C22⋊C411yescharacteristic
358C23C2×C4×Q812yes—
359C23C2×#12411yescharacteristic
360C23C2×C4⋊Q811yescharacteristic
361Q8—44no—
362Q8—88no—
363Q8—88no—
364D4—44no—
365D4—42no—
366D4—42no—
367D4—42no—
368D4—42no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—84no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—162no—
417C2×C4#39112yes—
418C2×C4#39112yes—
419C2×C4#39112yes—
420C2×C4C2×#9914yes—
421C2×C4C2×#9912yes—
422C2×C4#41012yes—
423C2×C4C2×C22⋊Q812yes—
424C8—48no—
425C22—21no—
426C22—21no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—42no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—81no—
443C22—81no—
444C22C2×#39111yescharacteristic
445C22C22×#9911yescharacteristic
446C22C2×#41011yescharacteristic
447C4—24no—
448C4—22no—
449C4—22no—
450C4—22no—
451C4—42no—
452C4—42no—
453C4—42no—
454C4—42no—
455C4—42no—
456C4—81no—
457C4—81no—
458C4—82no—
459C4—82no—
460C4—164no—
461C4#199614yes—
462C2—21no—
463C2—21no—
464C2—41no—
465C2—42no—
466C2—41no—
467C2—42no—
468C2C2×#199611yescharacteristic
469C2#3334811yescharacteristic
470C2#3334911yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13294C2semidirect
5. #13293C2semidirect
8. C22×C4⋊Q8C4semidirect
13. #2579C4semidirect
21. #2409C4semidirect
23. #2317C4semidirect
25. #2316C4semidirect
28. #2296C4semidirect
65. #413Q8semidirect

Maximal subgroups

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

subgroupindex
#355742
#206342
#203972
#132942
#132932
#125752
#125732