Finite Group Databasebeta

← #220002

Subgroups — #220002

Order 512.

1139 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36149C211yesmaximal, characteristic
2#22748C212yesmaximal, complemented
3C2×#1540C212yesmaximal, complemented
4#9700C212yesmaximal, complemented
5#9132C212yesmaximal, complemented
6#9129C212yesmaximal, complemented
7#8592C211yesmaximal, characteristic, complemented
8#8341C212yesmaximal, complemented
9C4×#331C211yesmaximal, characteristic, complemented
10C2×#532C2212yes—
11C2×#431C2212yes—
12C22×C4⋊C8C2212yes—
13#2862C2211yescharacteristic
14#2857C2212yes—
15#2535C412yes—
16#2447—22no—
17#2427—22no—
18#2420C414yes—
19#2407—22no—
20#2349C412yes—
21#2339—22no—
22C4×#127C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1923—22no—
25#1916—22no—
26#1869—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30C4×C4⋊C8C2212yes—
31C8×C22⋊C4—22no—
32#1540—28no—
33C2×#332C2212yes—
34C2×#332C2212yes—
35C2×#332C2212yes—
36C2×#332C2212yes—
37C2×#331C2212yes—
38C2×#331C2211yescharacteristic
39C2×#331C2211yescharacteristic
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C22×#41—22no—
44C23×C8—22no—
45#532—24no—
46#532—24no—
47#532—42no—
48#532—42no—
49C2×C4⋊Q8C2312yes—
50C2×#127C2311yescharacteristic
51C2×#124—22no—
52C2×C22⋊Q8—22no—
53C2×#121—22no—
54C2×#121C2×C414yes—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4C2×C414yes—
58C2×C4×D4C2×C412yes—
59C2×C4×D4C2×C411yescharacteristic
60C2×C4×D4D412yes—
61C22×C42C2312yes—
62#431—24no—
63#431—42no—
64C2×C4⋊C8—24no—
65C2×C4⋊C8—22no—
66C2×C4⋊C8—42no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C4⋊C8C2312yes—
70C2×C4⋊C8C2312yes—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76#397—22no—
77#397—24no—
78#394—24no—
79#392—22no—
80#391—22no—
81#391—22no—
82#391—22no—
83#390—22no—
84#390C2×C412yes—
85#389—22no—
86#389C2×C414yes—
87#386—24no—
88#386—24no—
89#385—24no—
90#384C2311yescharacteristic
91#382C2×C412yes—
92#382C2×C412yes—
93#381—24no—
94#380—22no—
95#380—22no—
96#380C2×C414yes—
97C4×C4⋊C4C2×C411yescharacteristic
98C4×C4⋊C4D412yes—
99C4×C22⋊C4C2×C412yes—
100C4×C22⋊C4D412yes—
101C4×C22⋊C4D412yes—
102C2×#95—22no—
103C43C2311yescharacteristic
104#336—22no—
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#332—24no—
110#332—24no—
111#332—42no—
112#332—42no—
113#331—24no—
114#331—22no—
115#331D414yescomplemented
116C2×#41—44no—
117C2×#41—44no—
118C2×#41—44no—
119C2×#41—44no—
120C2×#41—82no—
121C2×#41—82no—
122C22×Q8—22no—
123C22×D4—24no—
124C22×D4—22no—
125C22×D4C22⋊C411yescharacteristic
126C22×D4C22×C412yes—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4C22×C412yes—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—44no—
137C22×C8—44no—
138C22×C8—82no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—22no—
143#127—22no—
144#127C2×D412yes—
145#127#4112yes—
146#124—44no—
147#124—44no—
148C22⋊Q8—82no—
149C22⋊Q8—82no—
150#121—28no—
151#121—44no—
152#121—44no—
153#121—44no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—28no—
157C4×D4—42no—
158C4×D4—44no—
159C4×D4—44no—
160C4×D4—44no—
161C4×D4—44no—
162C4×D4—82no—
163C4×D4—82no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22⋊C411yescharacteristic
169C2×C4⋊C4C22×C414yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—24no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4C22⋊C412yes—
177C2×C22⋊C4C22⋊C412yes—
178C2×C22⋊C4C22⋊C412yes—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C22×C411yescharacteristic
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42C2×D412yes—
189C2×C42C2×D412yes—
190C2×C42C2411yescharacteristic
191C4⋊C8—22no—
192C4⋊C8—22no—
193C4⋊C8—22no—
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—44no—
199C4⋊C8C2×D414yes—
200C22⋊C8—44no—
201C22⋊C8—44no—
202C22⋊C8—82no—
203C4×C8—44no—
204C4×C8—82no—
205#95—24no—
206#95—22no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95C22⋊C411yescharacteristic
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C411yescharacteristic
214C24—22no—
215C24C2×C22⋊C412yes—
216#41—88no—
217#41—88no—
218#41—88no—
219#41—88no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×D4—24no—
227C2×D4—22no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—41no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—84no—
240C2×D4C4×D414yes—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—48no—
259C22×C4—44no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C411yescharacteristic
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C411yescharacteristic
267C22×C4C2×C22⋊C412yes—
268C22×C4#13514yes—
269C22×C4#13514yes—
270C22×C4C23×C411yescharacteristic
271C22×C4C22×D412yes—
272C2×C8—24no—
273C2×C8—42no—
274C2×C8—42no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—88no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—41no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—84no—
295C22⋊C4—24no—
296C22⋊C4—24no—
297C22⋊C4—24no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—44no—
301C22⋊C4—84no—
302C22⋊C4—84no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—22no—
310C42—22no—
311C42—22no—
312C42—22no—
313C42—22no—
314C42—41no—
315C42C22×D411yescharacteristic
316C42C2×#4111yescharacteristic
317C42H(5,𝔽2)12yes—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—42no—
326C23—42no—
327C23—44no—
328C23—44no—
329C23#41614yes—
330C23C22×C22⋊C411yescharacteristic
331Q8—82no—
332Q8—82no—
333Q8—82no—
334Q8—82no—
335D4—44no—
336D4—44no—
337D4—44no—
338D4—44no—
339D4—44no—
340D4—44no—
341D4—44no—
342D4—816no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—816no—
375C2×C4—164no—
376C2×C4#41312yes—
377C2×C4#41614yes—
378C2×C4C2×C4×D411yescharacteristic
379C2×C4#51811yescharacteristic
380C2×C4D4212yes—
381C2×C4#54612yes—
382C2×C4C2×#13512yes—
383C2×C4C2×#13512yes—
384C2×C4#57612yes—
385C2×C4#57712yes—
386C8—44no—
387C8—82no—
388C8—82no—
389C8—82no—
390C22—24no—
391C22—24no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—44no—
398C22—48no—
399C22—82no—
400C22—84no—
401C22—82no—
402C22D4×C22⋊C411yescharacteristic
403C22C2×#41311yescharacteristic
404C22#288911yescharacteristic
405C22C2×#41612yes—
406C22#289712yes—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—24no—
411C4—41no—
412C4—42no—
413C4—42no—
414C4—42no—
415C4—82no—
416C4—84no—
417C4#328012yes—
418C4#328212yes—
419C4#329612yes—
420C4#329712yes—
421C2—44no—
422C2—84no—
423C2#1770011yescharacteristic
424C2#1772712yes—
425C2#1777111yescharacteristic
426C2#2241111yescharacteristic
427C2#2241412yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36149C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22748C2semidirect
3. C2×#1540C2semidirect
4. #9700C2semidirect
5. #9132C2semidirect
6. #9129C2semidirect
7. #8592C2semidirect
8. #8341C2semidirect
9. C4×#331C2semidirect
115. #331D4semidirect

Maximal subgroups

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

subgroupindex
#361492
#227482
C2×#15402
#97002
#91322
#91292
#85922
#83412
C4×#3312