Finite Group Databasebeta

← #368948

Subgroups — #368948

Order 512.

1284 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34215C211yesmaximal, characteristic
3C2×#1995C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5#14282C211yesmaximal, characteristic
6#11217C214yesmaximal
7#10899C214yesmaximal
8#10858C211yesmaximal, characteristic
9#10701C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165C418yescomplemented
14C2×#501—22no—
15C2×#500—21no—
16C2×#498—21no—
17C2×#428C2214yes—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2368—22no—
21#2368—22no—
22#2355—22no—
23#2355—22no—
24#2331—21no—
25#2331—21no—
26#2317C2212yes—
27C4×#125—22no—
28C2×#382C2211yescharacteristic
29#2274C2212yes—
30C4×#117C2211yescharacteristic
31C2×#376C2211yescharacteristic
32#1995—24no—
33#1988—22no—
34#1987—22no—
35C2×#343C2214yes—
36#1736C2212yes—
37#1736C2212yes—
38C2×#339C2214yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41#1446—22no—
42#1445—22no—
43C4⋊#95C2214yes—
44C23×Q8—21no—
45C22×M16C2312yes—
46C22×M16C2312yes—
47#533—22no—
48#533—22no—
49#533C2×C414yes—
50#533D414yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×#125D412yes—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—21no—
65C2×C4×Q8—21no—
66C2×#117C2311yescharacteristic
67C2×#117C2312yes—
68C2×#117C2311yescharacteristic
69C22×C4⋊C4—21no—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C42C2312yes—
73C22×C42C2311yescharacteristic
74#501—216no—
75#500—28no—
76#498—28no—
77#428—28no—
78#428C2×C4116yes—
79C2×#107—24no—
80C2×#106—24no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C4×M16—22no—
86C4×M16C2×C414yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89#391—24no—
90#391—24no—
91#389—22no—
92#389—22no—
93#387—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#385—24no—
98#384—22no—
99#382—22no—
100#382—21no—
101#382—21no—
102#382—21no—
103#382—21no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—24no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—21no—
113C4×C22⋊C4—21no—
114C4×C22⋊C4—24no—
115#376—21no—
116#376—22no—
117#376—22no—
118C2×#95—21no—
119C2×#95—21no—
120C2×#95—28no—
121C2×#95D412yes—
122C2×#95D412yes—
123C43—21no—
124#343—28no—
125#340—24no—
126#340—24no—
127#339—24no—
128#339—28no—
129#339Q818yescomplemented
130C22×Q8—24no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—41no—
134C22×Q8—41no—
135C22×Q8—42no—
136C22×Q8—84no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4C2×D414yes—
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C23×C4C2411yescharacteristic
143C2×M16—22no—
144C2×M16—22no—
145C2×M16—22no—
146C2×M16—22no—
147C2×M16—24no—
148C2×M16—22no—
149C2×M16—24no—
150C2×M16C22×C414yes—
151C2×M16C2×Q814yes—
152C2×M16#4114yes—
153C22×C8—22no—
154C22×C8—22no—
155C4⋊Q8—24no—
156C4⋊Q8—24no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—42no—
164C4⋊Q8—42no—
165C4⋊Q8C22⋊C414yes—
166C4⋊Q8C22⋊C414yes—
167#125—24no—
168#125C22⋊C418yes—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8—24no—
172C22⋊Q8—24no—
173C4×Q8—24no—
174C4×Q8—24no—
175C4×Q8—24no—
176C4×Q8—24no—
177#117—22no—
178#117—22no—
179#117—22no—
180#117—22no—
181#117—22no—
182#117C22×C414yes—
183#117C2×D414yes—
184#117C2×Q814yes—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4C22×C414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—21no—
219C2×C42—22no—
220C2×C42—24no—
221C2×C42—24no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42C22×C412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42C2×Q814yes—
231C2×C42#4112yes—
232C2×C42#4112yes—
233#107—216no—
234#106—216no—
235C4⋊C8—42no—
236C4⋊C8—42no—
237#97—24no—
238C4×C8—24no—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—42no—
250#95—42no—
251#95—88no—
252C24C22×D411yescharacteristic
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—84no—
260C2×Q8—84no—
261C2×Q8—88no—
262C2×Q8—88no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—28no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—44no—
274C22×C4—48no—
275C22×C4—44no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—44no—
281C22×C4—44no—
282C22×C4C4×D412yes—
283C22×C4C4×D412yes—
284C22×C4#12011yescharacteristic
285C22×C4#12011yescharacteristic
286C22×C4#12011yescharacteristic
287C22×C4#12011yescharacteristic
288C22×C4#12112yes—
289C22×C4#12112yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×Q812yes—
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4112yes—
297C22×C4C2×#4111yescharacteristic
298C22×C42−1+412yes—
299C22×C42−1+412yes—
300M16—24no—
301M16—24no—
302M16—24no—
303M16—44no—
304M16—44no—
305M16C4×Q818yes—
306C2×C8—24no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—42no—
310C2×C8—42no—
311C2×C8—42no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—28no—
316C4⋊C4—28no—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—44no—
328C4⋊C4—44no—
329C4⋊C4—44no—
330C4⋊C4—84no—
331C4⋊C4—84no—
332C4⋊C4C2×C22⋊C414yes—
333C4⋊C4C2×C22⋊C414yes—
334C4⋊C4C4⋊Q814yes—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—24no—
338C22⋊C4—42no—
339C42—24no—
340C42—24no—
341C42—24no—
342C42—24no—
343C42—24no—
344C42—42no—
345C42—42no—
346C42—42no—
347C42—82no—
348C42—82no—
349C42—84no—
350C42—84no—
351C42C2×C22⋊C412yes—
352C42C2×C22⋊C412yes—
353C42C4×D412yes—
354C42C4×D412yes—
355C42#12112yes—
356C42#12112yes—
357C42#12312yes—
358C42#12312yes—
359C23—22no—
360C23—22no—
361C23C2×C4×D411yescharacteristic
362C23C2×#12011yescharacteristic
363C23C2×#12111yescharacteristic
364C23#54712yes—
365C23D4×Q812yes—
366Q8—84no—
367Q8—84no—
368Q8—88no—
369Q8—88no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—48no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—42no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—84no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—84no—
401C2×C4—84no—
402C2×C4—88no—
403C2×C4#38612yes—
404C2×C4#38612yes—
405C2×C4#38612yes—
406C2×C4#38612yes—
407C2×C4C22×C22⋊C411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C4×Q812yes—
410C2×C4#51912yes—
411C2×C4C2×#12111yescharacteristic
412C2×C4C2×#12311yescharacteristic
413C2×C4C2×C4⋊Q811yescharacteristic
414C2×C4#53211yescharacteristic
415C2×C4#53312yes—
416C2×C4#53611yescharacteristic
417C2×C4#53611yescharacteristic
418C2×C4#54112yes—
419C2×C4#54812yes—
420C2×C4#54812yes—
421C2×C4D4×Q812yes—
422C2×C4D4×Q812yes—
423C2×C4#55612yes—
424C2×C4#55712yes—
425C8—28no—
426C8—82no—
427C22—24no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—24no—
433C22C2×#38611yescharacteristic
434C22#236512yes—
435C22#242912yes—
436C22#243211yescharacteristic
437C22#243411yescharacteristic
438C22#302112yes—
439C22#302212yes—
440C4—22no—
441C4—22no—
442C4—22no—
443C4—22no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4—84no—
448C4—82no—
449C4—84no—
450C4—84no—
451C4C4×C4⋊Q811yescharacteristic
452C4#232711yescharacteristic
453C4Q8×C22⋊C412yes—
454C4#244612yes—
455C4#254511yescharacteristic
456C4#254611yescharacteristic
457C2—22no—
458C2—22no—
459C2#1402411yescharacteristic
460C2#2094411yescharacteristic
461C2#2094511yescharacteristic
462C2#2569212yes—
463C2#2583312yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #3165C4semidirect
129. #339Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342152
C2×#19952
#213132
#142822
#112172
#108992
#108582
#107012