Finite Group Databasebeta

← #194794

Subgroups — #194794

Order 512.

1358 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
1C2×#2396C211yesmaximal, characteristic
2#34179C211yesmaximal, characteristic
3#20741C212yesmaximal, complemented
4C2×#1464C211yesmaximal, characteristic
5#9024C212yesmaximal
6#9023C212yesmaximal, complemented
7#9016C212yesmaximal
8#9015C212yesmaximal, complemented
9#8021C212yesmaximal, complemented
10C22×C4×Q8C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C2×#432C2212yes—
13#2396—24no—
14#2396C418yescomplemented
15C2×#400C2211yescharacteristic
16C2×#399C2211yescharacteristic
17C2×#398—21no—
18C2×#396—21no—
19C2×#395C2211yescharacteristic
20C2×#393C2211yescharacteristic
21#2347—24no—
22#2346—24no—
23#2316C2212yes—
24#2312—21no—
25#2312C2212yes—
26#2283—21no—
27#2283C2212yes—
28#2280—21no—
29#2274—21no—
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4—21no—
33#1824C2214yes—
34#1805—24no—
35#1787C2214yes—
36#1763—24no—
37#1464—24no—
38C2×#324C2212yes—
39C2×#323C2212yes—
40#1294C2214yes—
41#1294C2214yes—
42C23×Q8C2311yescharacteristic
43C24×C4D411yescharacteristic
44C2×C4×Q8—22no—
45C2×C4×Q8—22no—
46C2×C4×Q8—22no—
47C2×C4×Q8—24no—
48C2×C4×Q8C2×C414yes—
49C2×#117—22no—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—24no—
53C2×#117C2×C414yes—
54C2×#117D414yes—
55C22×C4⋊C4—21no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70#432—24no—
71#432—24no—
72C2×C4⋊C8—22no—
73C2×#103—28no—
74C2×C22⋊C8C2314yes—
75C2×#97—22no—
76#400—22no—
77#400C2×C414yes—
78#399—22no—
79#399C2×C414yes—
80#398—28no—
81#396—28no—
82#395—22no—
83#395C2×C414yes—
84#393—22no—
85#393C2×C414yes—
86#389—28no—
87#388—28no—
88#388—28no—
89#383—22no—
90#383—22no—
91#382—22no—
92#382D414yes—
93#381—22no—
94#380—22no—
95#380—22no—
96#380—22no—
97#380C2314yes—
98#379—22no—
99C4×C4⋊C4—28no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—28no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C414yes—
104C4×C4⋊C4D414yes—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—28no—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110#376—22no—
111#376—22no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—21no—
115C2×#95—21no—
116C2×#95C2311yescharacteristic
117C2×#95C2311yescharacteristic
118#340—28no—
119#324—24no—
120#323—24no—
121C25C2×D411yescharacteristic
122C22×Q8—21no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—44no—
126C22×Q8C22×C412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—24no—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—42no—
136C23×C4C22⋊C414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C22×C8—24no—
144C4×Q8—28no—
145C4×Q8—28no—
146C4×Q8—44no—
147C4×Q8—44no—
148#117—28no—
149#117—28no—
150#117—28no—
151#117—28no—
152#117—44no—
153#117—44no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C22⋊C414yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D418yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—44no—
188C2×C22⋊C4C22⋊C414yes—
189C2×C22⋊C4C22×C412yes—
190C2×C22⋊C4C2×D412yes—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—22no—
194C2×C42—24no—
195C2×C42—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—44no—
200C2×C42—41no—
201C2×C42—41no—
202C2×C42—42no—
203C2×C42C22⋊C414yes—
204C2×C42C22×C412yes—
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D412yes—
208C2×C42#4114yes—
209C4⋊C8—44no—
210#103—416no—
211C22⋊C8—24no—
212C22⋊C8—24no—
213#97—44no—
214#95—24no—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—21no—
224#95—21no—
225#95—24no—
226#95C22×C412yes—
227#95C22×C412yes—
228C24—21no—
229C24—22no—
230C24—21no—
231C24—22no—
232C24—42no—
233C24—42no—
234C24C2×C22⋊C412yes—
235C24C22×D411yescharacteristic
236C2×Q8—28no—
237C2×Q8—24no—
238C2×Q8—44no—
239C2×Q8—44no—
240C2×Q8—44no—
241C2×Q8—48no—
242C2×Q8—42no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—28no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—28no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—48no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12012yes—
290C22×C4#12312yes—
291C22×C4#12312yes—
292C22×C4#13714yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D412yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4112yes—
297C2×C8—48no—
298C2×C8—44no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—28no—
302C4⋊C4—24no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—48no—
306C4⋊C4—48no—
307C4⋊C4—48no—
308C4⋊C4—42no—
309C4⋊C4—48no—
310C4⋊C4—84no—
311C22⋊C4—24no—
312C22⋊C4—24no—
313C22⋊C4—28no—
314C22⋊C4—48no—
315C22⋊C4—48no—
316C22⋊C4—44no—
317C22⋊C4—44no—
318C22⋊C4—84no—
319C22⋊C4—84no—
320C42—24no—
321C42—24no—
322C42—48no—
323C42—48no—
324C42—44no—
325C42—44no—
326C42—84no—
327C42—84no—
328C42C4×D414yes—
329C42C4×D414yes—
330C42#12114yes—
331C42#12114yes—
332C23—22no—
333C23—22no—
334C23—22no—
335C23—22no—
336C23—22no—
337C23—42no—
338C23—42no—
339C23—48no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—81no—
345C23—81no—
346C23—84no—
347C23C22×C22⋊C411yescharacteristic
348C23C2×#12011yescharacteristic
349C23C2×#12311yescharacteristic
350C23C2×#13712yes—
351C23#57512yes—
352Q8—28no—
353Q8—48no—
354Q8—44no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—42no—
375C2×C4—48no—
376C2×C4—48no—
377C2×C4—42no—
378C2×C4—48no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—42no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—84no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#38612yes—
403C2×C4#38612yes—
404C2×C4#38614yes—
405C2×C4#41914yes—
406C2×C4C2×C4×D412yes—
407C2×C4C2×#12112yes—
408C8—84no—
409C22—24no—
410C22—24no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—44no—
416C22—42no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—84no—
421C22—84no—
422C22C2×#38611yescharacteristic
423C22C2×#41911yescharacteristic
424C22#289611yescharacteristic
425C22#300712yes—
426C22#301812yes—
427C22#302512yes—
428C22#309512yes—
429C4—22no—
430C4—22no—
431C4—24no—
432C4—24no—
433C4—22no—
434C4—22no—
435C4—44no—
436C4—42no—
437C4—44no—
438C4—42no—
439C4—82no—
440C4—88no—
441C4—82no—
442C2—21no—
443C2—21no—
444C2—22no—
445C2—82no—
446C2#1818514yes—
447C2#2091111yescharacteristic
448C2#2095211yescharacteristic
449C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #20741C2semidirect
6. #9023C2semidirect
8. #9015C2semidirect
9. #8021C2semidirect
14. #2396C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23962
#341792
#207412
C2×#14642
#90242
#90232
#90162
#90152
#80212