Finite Group Databasebeta

← #238143

Subgroups — #238143

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34948C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5#10357C214yesmaximal, complemented
6#10353C214yesmaximal, complemented
7#8045C211yesmaximal, characteristic
8#7901C211yesmaximal, characteristic
9#7891C211yesmaximal, characteristic
10C22×C4⋊Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#423C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C2×#419—21no—
15C2×#419—21no—
16#2571C2212yes—
17#2570—22no—
18#2569—22no—
19#2552C2212yes—
20#2442—22no—
21#2402—22no—
22C4×#125—22no—
23#2296—21no—
24#2285—21no—
25#2285C2211yescharacteristic
26C4×#117C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1990C2214yes—
29#1988—22no—
30#1988—22no—
31#1988C2212yes—
32#1988C2212yes—
33#1981—22no—
34#1981C2214yes—
35#1823—22no—
36C2×#340—22no—
37#1296C2214yes—
38#1279C2212yes—
39#1279C2214yes—
40#1279C2212yes—
41C23×Q8—21no—
42C22×M16—21no—
43C23×C8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—21no—
46C2×C4⋊Q8—21no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8Q814yes—
51C2×C4⋊Q8C2312yes—
52C2×#125—22no—
53C2×#117—21no—
54C2×#117D411yescharacteristic
55C2×#117D411yescharacteristic
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#431—22no—
62#431—22no—
63#423—22no—
64#423—22no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8C2312yes—
71C2×C4⋊C8C2312yes—
72#419—44no—
73#419—44no—
74C2×#103—24no—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C4×C8C2312yes—
78#400—24no—
79#400—24no—
80#398—24no—
81#398—24no—
82#389—22no—
83#389—22no—
84#389—22no—
85#389—22no—
86#387—24no—
87#387—24no—
88#385—24no—
89#384—22no—
90#384C2312yes—
91#382—22no—
92#382—22no—
93#382—21no—
94#380—24no—
95#380—21no—
96#380D412yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—24no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4D412yes—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4D412yes—
105C4×C4⋊C4C2311yescharacteristic
106C4×C4⋊C4C2312yes—
107C4×C4⋊C4C2311yescharacteristic
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110#376—21no—
111C2×#95—21no—
112C43—21no—
113#340—28no—
114#340—24no—
115#340—24no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—41no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8—42no—
122C22×Q8—42no—
123C22×Q8—42no—
124C23×C4—22no—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—42no—
130C2×M16—42no—
131C2×M16—42no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—44no—
136C22×C8—42no—
137C4⋊Q8—24no—
138C4⋊Q8—24no—
139C4⋊Q8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4⋊Q8C2×Q814yes—
147#125—44no—
148#125—44no—
149#117—22no—
150#117—22no—
151#117—22no—
152#117—22no—
153#117—42no—
154#117—42no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D412yes—
176C2×C4⋊C4C2×D414yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q812yes—
194C2×C42#4112yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42C2411yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8—42no—
205#103—48no—
206#103—48no—
207C22⋊C8—42no—
208C4×C8—22no—
209C4×C8—22no—
210#95—24no—
211#95—24no—
212#95—24no—
213#95—24no—
214#95—22no—
215#95—22no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220#95—22no—
221C24#12011yescharacteristic
222C2×Q8—24no—
223C2×Q8—42no—
224C2×Q8—44no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—48no—
230C2×Q8—48no—
231C2×Q8—82no—
232C2×Q8—82no—
233C2×Q8—84no—
234C2×Q8—84no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—24no—
242C22×C4—42no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4—82no—
250C22×C4#12011yescharacteristic
251C22×C4#12011yescharacteristic
252C22×C4#12011yescharacteristic
253C22×C4C22⋊Q812yes—
254C22×C4C22⋊Q812yes—
255C22×C4C22⋊Q812yes—
256C22×C4#12412yes—
257C22×C4#12412yes—
258C22×C4#12412yes—
259C22×C4#13512yes—
260C22×C4#13512yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4AGL(1,ℤ/8)12yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D411yescharacteristic
265C22×C4C22×D411yescharacteristic
266M16—84no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—44no—
274C2×C8—84no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—22no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—48no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—84no—
297C4⋊C4—84no—
298C4⋊C4#13714yes—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—44no—
302C42—41no—
303C42—41no—
304C42—41no—
305C42—41no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—42no—
310C42—42no—
311C42—44no—
312C42—82no—
313C42C22⋊Q812yes—
314C42C22⋊Q812yes—
315C42C22⋊Q812yes—
316C42#12412yes—
317C42#12412yes—
318C42#12412yes—
319C42C22×Q811yescharacteristic
320C42C2×#4111yescharacteristic
321C42C2×#4111yescharacteristic
322C42C2×#4111yescharacteristic
323C23—22no—
324C23—22no—
325C23#39312yes—
326C23#44912yes—
327C23#44912yes—
328C23C2×#12011yescharacteristic
329Q8—44no—
330Q8—44no—
331Q8—82no—
332Q8—82no—
333Q8—88no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—88no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—88no—
362C2×C4#39312yes—
363C2×C4#39312yes—
364C2×C4#39312yes—
365C2×C4#44912yes—
366C2×C4#44912yes—
367C2×C4#47814yes—
368C2×C4#47814yes—
369C2×C4#48814yes—
370C2×C4#49014yes—
371C2×C4C2×C22⋊Q811yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×C22⋊Q811yescharacteristic
374C2×C4C2×#12411yescharacteristic
375C2×C4C2×#12411yescharacteristic
376C2×C4C2×#12411yescharacteristic
377C2×C4C2×#13511yescharacteristic
378C2×C4C2×#13511yescharacteristic
379C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
380C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
381C2×C4C2×#13712yes—
382C2×C4#57512yes—
383C8—82no—
384C8—82no—
385C8—82no—
386C22—24no—
387C22—24no—
388C22—22no—
389C22—22no—
390C22#202514yes—
391C22C2×#39311yescharacteristic
392C22C2×#44911yescharacteristic
393C22C2×#44911yescharacteristic
394C22#300712yes—
395C22#302212yes—
396C4—22no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—44no—
401C4—82no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4—84no—
406C4#202514yes—
407C4C2×#47811yescharacteristic
408C4C2×#47811yescharacteristic
409C4#307812yes—
410C4C2×#48811yescharacteristic
411C4C2×#49011yescharacteristic
412C4#313911yescharacteristic
413C4#314011yescharacteristic
414C2—24no—
415C2#1880212yes—
416C2#1880412yes—
417C2C2×#202511yescharacteristic
418C2#2345511yescharacteristic
419C2#2346411yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
5. #10357C2semidirect
6. #10353C2semidirect

Maximal subgroups

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

subgroupindex
#349482
#218332
C2×#19882
#212592
#103572
#103532
#80452
#79012
#78912