Finite Group Databasebeta

← #220072

Subgroups — #220072

Order 512.

1161 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C4⋊Q8C211yesmaximal, characteristic
2#22749C212yesmaximal
3C2×#1548C212yesmaximal, complemented
4#9702C212yesmaximal, complemented
5#9135C212yesmaximal, complemented
6#9128C212yesmaximal, complemented
7#8594C211yesmaximal, characteristic, complemented
8#8342C212yesmaximal, complemented
9#8232C211yesmaximal, characteristic, complemented
10C2×D4×Q8C412yes—
11C22×C4⋊Q8C2212yes—
12C2×#431C2212yes—
13C2×#423C2212yes—
14#2868C2211yescharacteristic
15#2857C2212yes—
16#2551—22no—
17#2546C412yes—
18#2440—22no—
19#2440—22no—
20#2434C414yes—
21D4×C4⋊C4—22no—
22C4×C4⋊Q8C2211yescharacteristic
23C42×D4C2211yescharacteristic
24#1924—22no—
25#1916—22no—
26#1870—22no—
27#1831C2212yes—
28#1797C2212yes—
29#1797C2212yes—
30#1767C2212yes—
31#1753—22no—
32#1548—28no—
33C2×#333C2211yescharacteristic
34C2×#333C2212yes—
35C2×#333C2212yes—
36C2×#333C2212yes—
37C2×#333C2212yes—
38C2×#333C2212yes—
39C2×#333C2211yescharacteristic
40#1296C2212yes—
41#1296C2211yescharacteristic
42#1296C2211yescharacteristic
43C23×Q8C2×C414yes—
44C22×M16—22no—
45D4×Q8—48no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C2×C4⋊Q8—42no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8C2×C412yes—
55C2×C4⋊Q8C2×C412yes—
56C2×C4⋊Q8C2312yes—
57C2×C4⋊Q8C2311yescharacteristic
58C2×C22⋊Q8—24no—
59C2×C22⋊Q8C2×C414yes—
60C2×C4×Q8C2×C412yes—
61C2×C4×D4—22no—
62C2×C4×D4C2×C411yescharacteristic
63C2×C4×D4D412yes—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C42C2312yes—
67#431—24no—
68#431—42no—
69#423—24no—
70#423—42no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2312yes—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×#97—22no—
80C2×C4×C8—22no—
81#398—22no—
82#398—22no—
83#395—24no—
84#395—24no—
85#391C2×C412yes—
86#391C2×C414yes—
87#389—24no—
88#389—22no—
89#389—22no—
90#389—22no—
91#385—24no—
92#385—24no—
93#384—22no—
94#384—22no—
95#384C2311yescharacteristic
96#380—22no—
97#380—22no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4C2×C411yescharacteristic
100C4×C4⋊C4D412yes—
101C4×C22⋊C4C2×C412yes—
102C4×C22⋊C4D412yes—
103C4×C22⋊C4D412yes—
104C43C2311yescharacteristic
105#336—22no—
106#336—22no—
107#336—22no—
108#336—22no—
109#336—22no—
110#333—24no—
111#333—22no—
112#333—24no—
113#333—24no—
114#333—42no—
115#333—42no—
116#333D414yescomplemented
117C22×Q8—24no—
118C22×Q8—28no—
119C22×Q8—44no—
120C22×Q8—48no—
121C22×Q8—84no—
122C22×Q8C22×C414yes—
123C22×Q8C22×C412yes—
124C22×D4C22⋊C411yescharacteristic
125C23×C4—24no—
126C23×C4C22×C412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C2×M16—44no—
130C2×M16—44no—
131C2×M16—82no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—28no—
137C4⋊Q8—22no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—44no—
143C4⋊Q8—44no—
144C4⋊Q8—44no—
145C4⋊Q8—44no—
146C4⋊Q8—44no—
147C4⋊Q8—44no—
148C4⋊Q8—48no—
149C4⋊Q8—82no—
150C4⋊Q8C2×D412yes—
151C4⋊Q8#4112yes—
152C22⋊Q8—416no—
153C22⋊Q8—48no—
154C4×Q8—44no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—48no—
159C2×C4⋊C4—22no—
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—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C411yescharacteristic
176C2×C4⋊C4C22×C412yes—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C412yes—
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D412yes—
193C2×C42C2411yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—44no—
202C4⋊C8C2×D414yes—
203C22⋊C8—44no—
204C22⋊C8—44no—
205C22⋊C8—82no—
206#97—82no—
207C4×C8—44no—
208#95—24no—
209#95—24no—
210#95—24no—
211#95C22⋊C412yes—
212#95C22⋊C412yes—
213#95C22⋊C411yescharacteristic
214#95C22⋊C411yescharacteristic
215C24C2×C22⋊C412yes—
216C2×Q8—22no—
217C2×Q8—24no—
218C2×Q8—24no—
219C2×Q8—48no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—48no—
225C2×Q8—84no—
226C2×Q8—816no—
227C2×Q8—816no—
228C2×Q8C4×D414yes—
229C2×D4—24no—
230C2×D4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—48no—
250C22×C4—84no—
251C22×C4C2×C22⋊C412yes—
252C22×C4C2×C22⋊C412yes—
253C22×C4C2×C22⋊C411yescharacteristic
254C22×C4C2×C22⋊C412yes—
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C412yes—
257C22×C4#13714yes—
258C22×C4#13714yes—
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261M16—88no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—48no—
278C4⋊C4—48no—
279C4⋊C4—42no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—44no—
289C4⋊C4—48no—
290C4⋊C4—84no—
291C4⋊C4#10014yes—
292C22⋊C4—24no—
293C22⋊C4—24no—
294C22⋊C4—24no—
295C22⋊C4—48no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42—22no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—41no—
308C42—44no—
309C42C22×D411yescharacteristic
310C42C2×#4111yescharacteristic
311C42H(5,𝔽2)12yes—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23#41914yes—
319C23C22×C22⋊C411yescharacteristic
320Q8—44no—
321Q8—44no—
322Q8—816no—
323Q8—84no—
324Q8—84no—
325D4—28no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—42no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—816no—
360C2×C4C2×#10012yes—
361C2×C4C2×#10014yes—
362C2×C4#41914yes—
363C2×C4C2×C4×D411yescharacteristic
364C2×C4#51811yescharacteristic
365C2×C4D4212yes—
366C2×C4#54612yes—
367C2×C4C2×#13712yes—
368C2×C4C2×#13712yes—
369C2×C4#57712yes—
370C2×C4#57812yes—
371C8—44no—
372C8—82no—
373C8—82no—
374C8—82no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22D4×C22⋊C411yescharacteristic
383C22C22×#10011yescharacteristic
384C22#288811yescharacteristic
385C22C2×#41912yes—
386C22#289812yes—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—24no—
391C4—24no—
392C4—42no—
393C4—41no—
394C4—44no—
395C4—42no—
396C4—82no—
397C4—84no—
398C4#326612yes—
399C4#327012yes—
400C4#331212yes—
401C4#332012yes—
402C2—44no—
403C2#1774611yescharacteristic
404C2#1776012yes—
405C2#1777311yescharacteristic
406C2D4×#10011yescharacteristic
407C2#2242512yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
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
3. C2×#1548C2semidirect
4. #9702C2semidirect
5. #9135C2semidirect
6. #9128C2semidirect
7. #8594C2semidirect
8. #8342C2semidirect
9. #8232C2semidirect
116. #333D4semidirect

Maximal subgroups

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

subgroupindex
D4×C4⋊Q82
#227492
C2×#15482
#97022
#91352
#91282
#85942
#83422
#82322