Finite Group Databasebeta

← #197967

Subgroups — #197967

Order 512.

1148 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
1C22×#477C211yesmaximal, characteristic
2C2×#2984C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4#20740C211yesmaximal, characteristic
5C2×#1558C211yesmaximal, characteristic
6#9224C214yesmaximal, complemented
7#9216C214yesmaximal, complemented
8#8044C211yesmaximal, characteristic
9#7879C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C2×#477—24no—
13C2×#477—21no—
14C2×#477—21no—
15C2×#477C418yescomplemented
16C2×#477C2212yes—
17C2×#477C2212yes—
18#2984—24no—
19#2984—24no—
20C2×#446—22no—
21C2×#446C2212yes—
22C2×C8×Q8C2212yes—
23C2×#432—21no—
24C2×#432C2211yescharacteristic
25C2×#432C2211yescharacteristic
26C2×#431—21no—
27C2×#431C2211yescharacteristic
28C22×#107—21no—
29C22×C4⋊C8C2211yescharacteristic
30C22×#103—21no—
31#2285C2211yescharacteristic
32#1990C2214yes—
33#1988—22no—
34#1823—21no—
35#1823—21no—
36#1823—21no—
37#1803C2212yes—
38#1803C2212yes—
39#1787—21no—
40#1763—21no—
41#1763—21no—
42#1558—24no—
43C2×#333C2214yes—
44#1296C2214yes—
45#1277C2214yes—
46C23×Q8—21no—
47C23×Q8D411yescharacteristic
48C23×C8—21no—
49C2×C4⋊Q8—21no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8C2312yes—
57C2×C4×Q8—21no—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—21no—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8D414yes—
63C2×C4×Q8D411yescharacteristic
64C2×C4×Q8D411yescharacteristic
65C2×C4×Q8C2311yescharacteristic
66C2×C4×Q8C2311yescharacteristic
67C2×#117—21no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C42—21no—
73C22×C42C2311yescharacteristic
74#477—28no—
75#477—28no—
76#477—24no—
77#477—24no—
78#477C2×C418yes—
79#477C2×C418yes—
80#446—216no—
81#446—24no—
82#446—24no—
83C8×Q8—24no—
84C8×Q8—24no—
85#432—28no—
86#432—22no—
87#432—22no—
88#432—22no—
89#432D414yes—
90#431—28no—
91#431—22no—
92#431—22no—
93C2×#107—28no—
94C2×#107—22no—
95C2×#107—22no—
96C2×C4⋊C8—22no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—22no—
100C2×C4⋊C8C2×C414yes—
101C2×C4⋊C8D412yes—
102C2×C4⋊C8C2312yes—
103C2×C4⋊C8C2312yes—
104C2×C4⋊C8C2312yes—
105C2×#103—28no—
106C2×#103—22no—
107C2×#103—22no—
108C2×#103—22no—
109C2×#103—22no—
110C2×C22⋊C8—21no—
111C2×C22⋊C8—21no—
112C2×C22⋊C8D411yescharacteristic
113C2×C22⋊C8D411yescharacteristic
114C2×#97—22no—
115C2×#97—21no—
116C2×#97—21no—
117C2×C4×C8—22no—
118C2×C4×C8C2312yes—
119#384C2312yes—
120#382—21no—
121#380—21no—
122#380—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125#340—24no—
126#340—24no—
127#340—22no—
128#340—22no—
129#333—28no—
130C22×Q8—21no—
131C22×Q8—22no—
132C22×Q8—24no—
133C22×Q8—24no—
134C22×Q8—22no—
135C22×Q8—42no—
136C22×Q8—41no—
137C22×Q8—42no—
138C22×Q8—82no—
139C22×Q8—82no—
140C22×Q8C22⋊C414yes—
141C22×Q8C2×D411yescharacteristic
142C22×Q8C2×D411yescharacteristic
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—41no—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4#4111yescharacteristic
149C22×C8—24no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—44no—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8C22×C414yes—
166C4×Q8—24no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—22no—
170C4×Q8—22no—
171C4×Q8—24no—
172C4×Q8—21no—
173C4×Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8C22⋊C414yes—
177C4×Q8C22⋊C414yes—
178C4×Q8C22×C412yes—
179C4×Q8C22×C412yes—
180C4×Q8C2×D412yes—
181C4×Q8C2×D412yes—
182#117—82no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C22⋊C414yes—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—41no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D412yes—
218C2×C42#4111yescharacteristic
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221#107—28no—
222#107—28no—
223C4⋊C8—28no—
224C4⋊C8—28no—
225C4⋊C8—24no—
226C4⋊C8—24no—
227C4⋊C8—22no—
228C4⋊C8—22no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8C22×C414yes—
232C4⋊C8C2×D414yes—
233#103—28no—
234#103—28no—
235#103—44no—
236#103—44no—
237C22⋊C8—24no—
238C22⋊C8—22no—
239C22⋊C8—22no—
240C22⋊C8—42no—
241#97—28no—
242#97—24no—
243#97—42no—
244C4×C8—28no—
245C4×C8—22no—
246C4×C8—22no—
247#95—41no—
248#95—41no—
249C24#12111yescharacteristic
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—28no—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—48no—
261C2×Q8—42no—
262C2×Q8—44no—
263C2×Q8—88no—
264C2×Q8—82no—
265C2×Q8—82no—
266C2×Q8—88no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8C2×C22⋊C412yes—
270C2×Q8C2×C22⋊C412yes—
271C2×Q8#13714yes—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—44no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—44no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C2×C22⋊C412yes—
292C22×C4C4×D412yes—
293C22×C4C4×D412yes—
294C22×C4#12012yes—
295C22×C4#12112yes—
296C22×C4#12111yescharacteristic
297C22×C4#12111yescharacteristic
298C22×C4#12111yescharacteristic
299C22×C4#12312yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C2×C8—24no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—48no—
310C2×C8—44no—
311C2×C8—44no—
312C2×C8—42no—
313C2×C8—42no—
314C2×C8—42no—
315C4⋊C4—22no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—24no—
323C4⋊C4—24no—
324C4⋊C4—24no—
325C4⋊C4—24no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—82no—
334C4⋊C4C2×C22⋊C412yes—
335C4⋊C4C2×C22⋊C412yes—
336C22⋊C4—82no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—22no—
341C42—24no—
342C42—24no—
343C42—82no—
344C42C2×C22⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42#12012yes—
348C42#12112yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×D411yescharacteristic
352C42C22×D411yescharacteristic
353C42C2×#4111yescharacteristic
354C23—22no—
355C23—22no—
356C23#38612yes—
357C23#47112yes—
358C23#47212yes—
359C23C2×#12111yescharacteristic
360Q8—22no—
361Q8—24no—
362Q8—22no—
363Q8—22no—
364Q8—48no—
365Q8—42no—
366Q8—84no—
367Q8—84no—
368Q8—88no—
369Q8#41914yes—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—24no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—48no—
389C2×C4—48no—
390C2×C4—42no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—88no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#38612yes—
403C2×C4#38612yes—
404C2×C4#38612yes—
405C2×C4C22×C22⋊C411yescharacteristic
406C2×C4C2×C4×D411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×#12011yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12311yescharacteristic
411C2×C4C2×#13712yes—
412C2×C4#57512yes—
413C2×C4#57814yes—
414C8—44no—
415C8—42no—
416C8—42no—
417C8—44no—
418C8—82no—
419C22—24no—
420C22—24no—
421C22—22no—
422C22—22no—
423C22#190714yes—
424C22C2×#38611yescharacteristic
425C22#304412yes—
426C22#305612yes—
427C22C2×#47111yescharacteristic
428C22C2×#47211yescharacteristic
429C4—21no—
430C4—21no—
431C4—22no—
432C4—22no—
433C4—22no—
434C4—24no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—84no—
439C4—162no—
440C4C2×#41911yescharacteristic
441C4#289611yescharacteristic
442C4#295012yes—
443C4#301112yes—
444C4#305612yes—
445C2—24no—
446C2#1947512yes—
447C2#1947712yes—
448C2#2093511yescharacteristic
449C2#2097211yescharacteristic
450C2C2×#190711yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9224C2semidirect
7. #9216C2semidirect
15. C2×#477C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4772
C2×#29842
#218332
#207402
C2×#15582
#92242
#92162
#80442
#78792