Finite Group Databasebeta

← #201452

Subgroups — #201452

Order 512.

1553 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34971C211yesmaximal, characteristic
2C4×#519C211yesmaximal, characteristic, complemented
3#20747C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9478C214yesmaximal, complemented
6#9476C214yesmaximal
7#7940C212yesmaximal, complemented
8C2×#519—21no—
9C2×C4×#41C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11#2920C2212yes—
12#2599—22no—
13#2581C412yescomplemented
14#2536C412yescomplemented
15#2486—21no—
16#2484C2212yes—
17#2409—21no—
18#2406C2212yes—
19#2399C412yescomplemented
20#2398C412yescomplemented
21C4×C22⋊Q8—22no—
22C4×#123—22no—
23#2283C2211yescharacteristic
24#2276—24no—
25#2274—21no—
26#2274C2214yes—
27C42×Q8—24no—
28C4×#117—22no—
29C4×#117C2212yes—
30#1826C2214yes—
31#1806—24no—
32#1789C2214yes—
33C4×#104—24no—
34#1467—22no—
35#1463C2214yes—
36#1461—22no—
37#1284C2218yes—
38C22×#41C2311yescharacteristic
39C23×Q8D411yescharacteristic
40C2×#123—22no—
41C2×C22⋊Q8—22no—
42#519—216no—
43C4×#41—22no—
44C4×#41—22no—
45C4×#41—22no—
46C4×#41—22no—
47C2×C4×Q8—28no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8C2×C412yes—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8D412yes—
57C2×C4×Q8D412yes—
58C2×C4×D4—21no—
59C2×C4×D4—21no—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4C2×C411yescharacteristic
62C2×#117—24no—
63C2×#117—22no—
64C2×#117—21no—
65C2×#117C2311yescharacteristic
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C42—21no—
69C22×C42D412yes—
70C22×C42C2311yescharacteristic
71#432—22no—
72#432—22no—
73#431—22no—
74#431—22no—
75#423—22no—
76C2×#104—28no—
77#407C2314yes—
78C4×M16—22no—
79#400C2×C414yes—
80#398—22no—
81#398C2×C414yes—
82#397—22no—
83#395—22no—
84#395C2×C414yes—
85#394C2×C414yes—
86#393—22no—
87#391—22no—
88#389—22no—
89#388—24no—
90#387—24no—
91#387—24no—
92#386—22no—
93#385—22no—
94#385—24no—
95#383—21no—
96#382C2×C412yes—
97#381—22no—
98#381C2312yes—
99#380—21no—
100#380—22no—
101C4×C4⋊C4—24no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—28no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—216no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4D414yes—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—24no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—24no—
113C4×C22⋊C4C2314yes—
114#376—24no—
115#376—22no—
116#376C2×C412yes—
117C43—24no—
118C43D414yes—
119#339—28no—
120#324—28no—
121C2×#41—22no—
122C2×#41—22no—
123C2×#41—44no—
124C22×Q8—28no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8C22⋊C412yes—
128C22×Q8C22⋊C412yes—
129C22×Q8C22×C411yescharacteristic
130C22×D4—21no—
131C22×D4C22×C411yescharacteristic
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C2×M16—24no—
141#123—48no—
142C22⋊Q8—48no—
143C4×Q8—264no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—24no—
149C4×Q8—24no—
150C4×Q8—24no—
151C4×Q8—24no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—42no—
161C4×D4—41no—
162C4×D4—41no—
163C4×D4—42no—
164C4×D4—41no—
165C4×D4—41no—
166#117—232no—
167#117—216no—
168#117—22no—
169#117—22no—
170#117—42no—
171#117—42no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4C22⋊C412yes—
184C2×C4⋊C4C22⋊C412yes—
185C2×C4⋊C4C22×C411yescharacteristic
186C2×C4⋊C4C22×C411yescharacteristic
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—21no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—21no—
193C2×C22⋊C4—21no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C42—28no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—21no—
207C2×C42—28no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42C22⋊C414yes—
211C2×C42C22⋊C414yes—
212C2×C42C22×C411yescharacteristic
213C2×C42C22×C411yescharacteristic
214C2×C42C2×D418yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4112yes—
220C2×C42#4112yes—
221C4⋊C8—42no—
222C4⋊C8—42no—
223#104—416no—
224C22⋊C8—24no—
225C22⋊C8—24no—
226#97—42no—
227C4×C8—42no—
228#95—28no—
229#95—22no—
230#95—24no—
231#95—22no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95—22no—
238#95C22×C414yes—
239C24—21no—
240C24C22×D411yescharacteristic
241#41—28no—
242#41—44no—
243#41—44no—
244#41—44no—
245C2×Q8—216no—
246C2×Q8—216no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—28no—
250C2×Q8—24no—
251C2×Q8—22no—
252C2×Q8—42no—
253C2×Q8#10418yescomplemented
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—28no—
263C22×C4—24no—
264C22×C4—21no—
265C22×C4—28no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—24no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—44no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—44no—
282C22×C4C2×C22⋊C411yescharacteristic
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C414yes—
285C22×C4C2×C22⋊C414yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4#12012yes—
289C22×C4#12011yescharacteristic
290C22×C4#12011yescharacteristic
291C22×C4#12312yes—
292C22×C4#12311yescharacteristic
293C22×C4#12311yescharacteristic
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D412yes—
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4112yes—
298M16—48no—
299C2×C8—44no—
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—48no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—48no—
314C22⋊C4—216no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—41no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—42no—
322C22⋊C4—48no—
323C42—232no—
324C42—216no—
325C42—24no—
326C42—24no—
327C42—24no—
328C42—24no—
329C42—48no—
330C42—48no—
331C42—44no—
332C42—42no—
333C42—41no—
334C42—42no—
335C42—41no—
336C42—88no—
337C42C4×D412yes—
338C42C4×D412yes—
339C42C4×D412yes—
340C42C4×D412yes—
341C42#12112yes—
342C42#12112yes—
343C42#12112yes—
344C42#12112yes—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—42no—
351C23—42no—
352C23—41no—
353C23C22×C22⋊C411yescharacteristic
354C23C2×#12011yescharacteristic
355C23C2×#12311yescharacteristic
356Q8—216no—
357Q8—216no—
358Q8—24no—
359Q8—42no—
360D4—24no—
361D4—42no—
362D4—44no—
363D4—44no—
364C2×C4—28no—
365C2×C4—28no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—216no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—42no—
384C2×C4—48no—
385C2×C4—48no—
386C2×C4—41no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—82no—
402C2×C4#38611yescharacteristic
403C2×C4#38611yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38612yes—
406C2×C4#38612yes—
407C2×C4#38611yescharacteristic
408C2×C4C2×#10418yes—
409C2×C4C2×#10414yes—
410C2×C4C2×C4×D412yes—
411C2×C4C2×#12112yes—
412C8—84no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—44no—
422C22—42no—
423C22—82no—
424C22—81no—
425C22—81no—
426C22C2×#38611yescharacteristic
427C22C22×#10412yes—
428C22#302512yes—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—22no—
435C4—28no—
436C4—416no—
437C4—42no—
438C4—44no—
439C4—84no—
440C4—81no—
441C4—81no—
442C4#189614yes—
443C4#191212yes—
444C2—22no—
445C2—22no—
446C2—41no—
447C2—82no—
448C2#2095911yescharacteristic
449C2C2×#189611yescharacteristic
450C2C2×#191211yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#519C2semidirect
4. #17365C2semidirect
5. #9478C2semidirect
7. #7940C2semidirect
13. #2581C4semidirect
14. #2536C4semidirect
19. #2399C4semidirect
20. #2398C4semidirect
253. C2×Q8#104semidirect

Maximal subgroups

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

subgroupindex
#349712
C4×#5192
#207472
#173652
#94782
#94762
#79402