Finite Group Databasebeta

← #196249

Subgroups — #196249

Order 512.

1471 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
1C2×#2972C211yesmaximal, characteristic, complemented
2#35237C211yesmaximal, characteristic
3#17478C212yesmaximal, complemented
4#17364C211yesmaximal, characteristic, complemented
5#9101C218yesmaximal, complemented
6#7858C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9#2972—24no—
10#2972—22no—
11#2972C2214yes—
12C2×C8×D4—24no—
13#2920C2212yes—
14C2×#431C2212yes—
15C22×C22⋊C8—22no—
16#2847—22no—
17#2631—22no—
18#2563C412yes—
19#2541C412yes—
20#2535—22no—
21#2504C412yes—
22#2433C412yes—
23#2404—22no—
24#2328C2214yes—
25#2283C2211yescharacteristic
26#1496—24no—
27#1492—24no—
28#1478C2218yescomplemented
29#1466—22no—
30#1462—22no—
31#1460C2214yes—
32#1274C2218yescomplemented
33C22×#41—21no—
34C23×D4C2×C411yescharacteristic
35C24×C4D412yes—
36C23×C8—24no—
37#532—22no—
38#532—22no—
39#532—22no—
40#532—22no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#127C2×C414yes—
43C2×#127C2×C411yescharacteristic
44C2×#124—21no—
45C2×#121—21no—
46C2×#121—24no—
47C2×#121C2×C412yes—
48C2×#120—24no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4D412yes—
56C2×C4×D4D412yes—
57C2×C4×D4C2312yes—
58C2×C4×D4C2312yes—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C22⋊C4D412yes—
61C22×C42C2311yescharacteristic
62C8×D4—216no—
63C8×D4—216no—
64#432—22no—
65#432—22no—
66#431—22no—
67#431—22no—
68#431—22no—
69#431D414yescomplemented
70#431C2314yes—
71#431C2314yes—
72#423—22no—
73C2×C4⋊C8—22no—
74#407—22no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—28no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—28no—
79C4×M16—22no—
80C2×C4×C8D414yescomplemented
81#397—24no—
82#393—24no—
83#392—24no—
84#392—24no—
85#390C2×C414yes—
86#390C2×C414yes—
87#386—24no—
88#386—28no—
89#386—28no—
90#383—21no—
91#382—21no—
92#382C2×C414yes—
93#381—21no—
94#380—24no—
95#380—21no—
96C4×C4⋊C4C2×C412yes—
97C4×C22⋊C4C2×C412yes—
98#376C2312yes—
99D4⋊C8—216no—
100#323—28no—
101C25C22⋊C412yes—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41—84no—
105C22×Q8—21no—
106C22×D4—28no—
107C22×D4—22no—
108C22×D4—28no—
109C22×D4—21no—
110C22×D4—21no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4C22⋊C412yes—
114C22×D4C22×C412yes—
115C22×D4C22×C412yes—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—84no—
122C23×C4C22⋊C414yes—
123C23×C4C22⋊C412yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D414yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C2×M16—42no—
129C22×C8—24no—
130C22×C8—216no—
131C22×C8—88no—
132C4⋊Q8—22no—
133C4⋊Q8—22no—
134#127—216no—
135#127—22no—
136#127—22no—
137#124—42no—
138#124—42no—
139#121—24no—
140#121—24no—
141#121—41no—
142#121—41no—
143#121—41no—
144#121—41no—
145#121—88no—
146#120—88no—
147C4×D4—24no—
148C4×D4—24no—
149C4×D4—24no—
150C4×D4—24no—
151C4×D4—24no—
152C4×D4—24no—
153C4×D4—24no—
154C4×D4—24no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—42no—
158C4×D4—42no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4C22⋊C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C22×C412yes—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—22no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—28no—
172C2×C22⋊C4—28no—
173C2×C22⋊C4C22⋊C414yes—
174C2×C22⋊C4C2×D414yes—
175C2×C42C22×C414yes—
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42#4112yes—
182C2×C42#4112yes—
183C2×C42C2411yescharacteristic
184C4⋊C8—24no—
185C4⋊C8—24no—
186C4⋊C8—24no—
187C4⋊C8—42no—
188C22⋊C8—216no—
189C22⋊C8—24no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192C22⋊C8—42no—
193#97—42no—
194C4×C8—24no—
195C4×C8—24no—
196C4×C8C2×D418yes—
197#95—22no—
198#95—22no—
199#95—28no—
200#95—22no—
201#95—22no—
202C24—22no—
203C24—22no—
204C24—28no—
205C24—22no—
206C24—44no—
207C24—82no—
208C24C2×C22⋊C411yescharacteristic
209C24C2×C22⋊C414yes—
210#41—88no—
211#41—88no—
212C2×Q8—42no—
213C2×Q8—82no—
214C2×D4—216no—
215C2×D4—216no—
216C2×D4—28no—
217C2×D4—24no—
218C2×D4—24no—
219C2×D4—24no—
220C2×D4—28no—
221C2×D4—24no—
222C2×D4—44no—
223C2×D4—48no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—416no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—82no—
231C2×D4—88no—
232C2×D4—88no—
233C2×D4#13118yes—
234C22×C4—24no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
246C22×C4—48no—
247C22×C4—44no—
248C22×C4—84no—
249C22×C4—82no—
250C22×C4—84no—
251C22×C4—84no—
252C22×C4—84no—
253C22×C4—84no—
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C22⋊C411yescharacteristic
256C22×C4C2×C22⋊C414yes—
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C4×D412yes—
259C22×C4C4×D412yes—
260C22×C4#12011yescharacteristic
261C22×C4#12011yescharacteristic
262C22×C4#12112yes—
263C22×C4#12112yes—
264C22×C4#12311yescharacteristic
265C22×C4#12311yescharacteristic
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D411yescharacteristic
268C22×C4C22×D411yescharacteristic
269M16—84no—
270C2×C8—28no—
271C2×C8—48no—
272C2×C8—44no—
273C2×C8—816no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—44no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—28no—
281C22⋊C4—28no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—42no—
285C22⋊C4—44no—
286C22⋊C4—88no—
287C22⋊C4—88no—
288C42—28no—
289C42C4×D412yes—
290C42C4×D412yes—
291C42#12012yes—
292C42#12112yes—
293C42#12112yes—
294C42#12312yes—
295C42C22×D412yes—
296C42C2×#4112yes—
297C23—28no—
298C23—28no—
299C23—22no—
300C23—22no—
301C23—24no—
302C23—24no—
303C23—41no—
304C23—44no—
305C23—44no—
306C23—44no—
307C23—44no—
308C23—44no—
309C23—44no—
310C23—44no—
311C23—42no—
312C23—82no—
313C23—82no—
314C23—816no—
315C23#38611yescharacteristic
316C23#38611yescharacteristic
317C23#40818yes—
318C23C22×C22⋊C411yescharacteristic
319Q8—84no—
320D4—216no—
321D4—216no—
322D4—48no—
323D4—82no—
324D4—84no—
325D4—84no—
326D4—84no—
327D4—82no—
328D4—1616no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—48no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—84no—
353C2×C4—84no—
354C2×C4—88no—
355C2×C4—82no—
356C2×C4#38612yes—
357C2×C4#38611yescharacteristic
358C2×C4#38612yes—
359C2×C4#38611yescharacteristic
360C2×C4C2×C4×D412yes—
361C2×C4C2×#12011yescharacteristic
362C2×C4C2×#12112yes—
363C2×C4C2×#12311yescharacteristic
364C2×C4C2×#13114yes—
365C8—48no—
366C8—84no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—28no—
375C22—44no—
376C22—48no—
377C22—416no—
378C22—44no—
379C22—44no—
380C22—84no—
381C22—88no—
382C22—88no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22#191012yes—
387C22C2×#38611yescharacteristic
388C22C2×#40812yes—
389C4—22no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—22no—
395C4—44no—
396C4—84no—
397C4—82no—
398C4#191112yes—
399C4#292914yes—
400C2—22no—
401C2—22no—
402C2—41no—
403C2—48no—
404C2—162no—
405C2#2089511yescharacteristic
406C2C2×#191011yescharacteristic
407C2C2×#191111yescharacteristic
subgroupG/H
center, socleC22C2×#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#35237C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2972C2semidirect
3. #17478C2semidirect
4. #17364C2semidirect
5. #9101C2semidirect
6. #7858C2semidirect
28. #1478C22semidirect
32. #1274C22semidirect
69. #431D4semidirect
80. C2×C4×C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29722
#352372
#174782
#173642
#91012
#78582