Finite Group Databasebeta

← #497502

Subgroups — #497502

Order 512.

1050 conjugacy classes of subgroups.

The subgroup lattice has 457 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36303C211yesmaximal, characteristic
2D4×#101C212yesmaximal, complemented
3#13301C212yesmaximal, complemented
4#13092C214yesmaximal, complemented
5#12860C212yesmaximal, complemented
6#12801C211yesmaximal, characteristic, complemented
7C4×#355C211yesmaximal, characteristic, complemented
8C2×#1394C212yesmaximal, complemented
9C2×D4×Q8C2211yescharacteristic
10C2×#547C411yescharacteristic
11D4×M16—22no—
12C22×#101C2214yes—
13#2862C411yescharacteristic
14#2850C412yes—
15#2568C2211yescharacteristic
16#2547C411yescharacteristic
17#2503C412yes—
18#2492—22no—
19#2488—24no—
20#2436C2212yes—
21#2339—22no—
22C4×#124C2211yescharacteristic
23C2×#391C2212yes—
24C42×D4C411yescharacteristic
25C2×#355C2212yes—
26C2×#355C2212yes—
27#1966—22no—
28#1901—24no—
29#1801C2212yes—
30#1784C2214yes—
31C4×#101C2212yes—
32#1394C2212yes—
33#1394C2214yes—
34#1394C2214yes—
35#1394C2211yescharacteristic
36#1394C2211yescharacteristic
37C22×#41—21no—
38C23×Q8C2312yes—
39C22×M16—24no—
40D4×Q8—41no—
41D4×Q8—42no—
42D4×Q8—41no—
43#547—41no—
44#547—42no—
45#547—41no—
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8C2×C411yescharacteristic
48C2×C4⋊Q8D412yes—
49C2×#124C2×C412yes—
50C2×#124C2311yescharacteristic
51C2×#123—22no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8C2×C412yes—
54C2×C22⋊Q8D414yes—
55C2×#121C2×C412yes—
56C2×C4×Q8C2311yescharacteristic
57C2×C4×D4—21no—
58C2×C4×D4C2×C411yescharacteristic
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C22×C4⋊C4C2×C412yes—
62C22×C42C2×C412yes—
63#436—42no—
64#435—44no—
65C8×D4—42no—
66#423—22no—
67C2×#101—28no—
68C2×#101—24no—
69C2×#101—24no—
70C2×#101—44no—
71C2×#101C2314yes—
72C2×#101C2312yes—
73C2×#101C2312yes—
74#407—24no—
75C4×M16—22no—
76#400—24no—
77#400—22no—
78#397—22no—
79#393—24no—
80#391—24no—
81#391—22no—
82#391—22no—
83#391C2×C412yes—
84#391C2×C412yes—
85#391C2×C412yes—
86#391C2312yes—
87#391C2311yescharacteristic
88#390C2×C411yescharacteristic
89#388—24no—
90#388—22no—
91#386—24no—
92#386—24no—
93#382—22no—
94#382C2×C412yes—
95#382C2×C411yescharacteristic
96#381—24no—
97#380—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4C2×C412yes—
103C2×#95—24no—
104C43C2×C411yescharacteristic
105#355—24no—
106#355—42no—
107#343—22no—
108#343—22no—
109C2×#41—48no—
110C2×#41—41no—
111C2×#41—42no—
112C2×#41—41no—
113C22×Q8—24no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—21no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—82no—
120C22×Q8C22×C412yes—
121C22×Q8C2×D414yes—
122C22×Q8C2411yescharacteristic
123C22×D4—22no—
124C22×D4C22⋊C411yescharacteristic
125C22×D4C22×C411yescharacteristic
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4C22×C412yes—
130C23×C4C2×D414yes—
131C2×M16—22no—
132C2×M16—24no—
133C2×M16—22no—
134C2×M16—44no—
135C2×M16—48no—
136C2×M16—84no—
137C22×C8—44no—
138C4⋊Q8—41no—
139C4⋊Q8—41no—
140C4⋊Q8—42no—
141C4⋊Q8—42no—
142C4⋊Q8—41no—
143C4⋊Q8—41no—
144#124—22no—
145#124—21no—
146#124—21no—
147#124—82no—
148#123—44no—
149#123—44no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C22⋊Q8—44no—
153C22⋊Q8—44no—
154C22⋊Q8—42no—
155C22⋊Q8—42no—
156#121—42no—
157#121—42no—
158C4×Q8—41no—
159C4×Q8—41no—
160C4×D4—22no—
161C4×D4—21no—
162C4×D4—21no—
163C4×D4—22no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—41no—
167C4×D4—41no—
168C4×D4—44no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4C22⋊C411yescharacteristic
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C414yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C411yescharacteristic
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C411yescharacteristic
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4C22⋊C412yes—
189C2×C22⋊C4C22⋊C414yes—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—22no—
193C2×C42—82no—
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C22×C412yes—
197C2×C42C22×C411yescharacteristic
198C2×C42C2×D412yes—
199C4⋊C8—42no—
200C4⋊C8—42no—
201#101—22no—
202#101—22no—
203#101—24no—
204#101—24no—
205#101—22no—
206#101—48no—
207#101—84no—
208#101#4114yescomplemented
209C22⋊C8—44no—
210C22⋊C8—44no—
211#97—42no—
212C4×C8—42no—
213#95—24no—
214#95—24no—
215#95—21no—
216#95—22no—
217#95—22no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—24no—
222C24—21no—
223C24C2×C22⋊C412yes—
224#41—88no—
225#41—84no—
226#41—84no—
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—24no—
231C2×Q8—41no—
232C2×Q8—41no—
233C2×Q8—44no—
234C2×Q8—42no—
235C2×Q8—44no—
236C2×Q8—42no—
237C2×Q8—44no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—82no—
241C2×Q8—84no—
242C2×Q8—82no—
243C2×Q8—84no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8C2×#4112yes—
247C2×Q8H(5,𝔽2)11yescharacteristic
248C2×Q82−1+411yescharacteristic
249C2×D4—24no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—48no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4#9912yes—
258C2×D4#9911yescharacteristic
259C2×D4#9911yescharacteristic
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—48no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C412yes—
288C22×C4C2×C22⋊C411yescharacteristic
289C22×C4C23×C411yescharacteristic
290C22×C4C22×D412yes—
291M16—24no—
292M16—42no—
293M16—42no—
294M16—44no—
295M16—44no—
296M16—42no—
297M16—88no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—44no—
301C2×C8—88no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C22⋊C4—24no—
316C22⋊C4—24no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—44no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—42no—
324C22⋊C4—42no—
325C22⋊C4—84no—
326C42—22no—
327C42—22no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42—41no—
333C42—42no—
334C42—42no—
335C42—42no—
336C42—82no—
337C23—22no—
338C23—22no—
339C23—24no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23C2×#9912yes—
347C23C2×#9912yes—
348C23C22×C22⋊C411yescharacteristic
349Q8—42no—
350Q8—41no—
351Q8—41no—
352Q8—84no—
353Q8—84no—
354Q8—82no—
355Q8—82no—
356Q8—82no—
357Q8—82no—
358D4—24no—
359D4—22no—
360D4—81no—
361D4—81no—
362D4—84no—
363D4—84no—
364D4#35514yescomplemented
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—24no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—44no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—82no—
402C2×C4—88no—
403C2×C4—84no—
404C2×C4—84no—
405C2×C4—82no—
406C2×C4C2×#9911yescharacteristic
407C2×C4C2×#9911yescharacteristic
408C2×C4C4×#4111yescharacteristic
409C2×C4#52011yescharacteristic
410C2×C4#54612yes—
411C2×C4#54712yes—
412C8—44no—
413C8—84no—
414C8—82no—
415C22—21no—
416C22—21no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—44no—
423C22—44no—
424C22—82no—
425C22—82no—
426C22—81no—
427C22—81no—
428C22—81no—
429C22C2×#35514yes—
430C22#233911yescharacteristic
431C22C22×#9911yescharacteristic
432C22#288311yescharacteristic
433C4—21no—
434C4—22no—
435C4—22no—
436C4—21no—
437C4—21no—
438C4—42no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—44no—
443C4—81no—
444C4—82no—
445C4—81no—
446C4—82no—
447C4C2×#35512yes—
448C2—21no—
449C2—21no—
450C2—24no—
451C2—42no—
452C2—161no—
453C2#2247611yescharacteristic
454C2C22×#35511yescharacteristic
455C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC2×#547C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×D4 ◁ C2×D4×Q8 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. D4×#101C2semidirect
3. #13301C2semidirect
4. #13092C2semidirect
5. #12860C2semidirect
6. #12801C2semidirect
7. C4×#355C2semidirect
8. C2×#1394C2semidirect
208. #101#41semidirect
364. D4#355semidirect

Maximal subgroups

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

subgroupindex
#363032
D4×#1012
#133012
#130922
#128602
#128012
C4×#3552
C2×#13942