Finite Group Databasebeta

← #200505

Subgroups — #200505

Order 512.

1598 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2975C211yesmaximal, characteristic, complemented
2#34936C211yesmaximal, characteristic
3#20726C211yesmaximal, characteristic, complemented
4#17717C212yesmaximal, complemented
5#9405C214yesmaximal, complemented
6#9403C214yesmaximal, complemented
7#8092C212yesmaximal, complemented
8C2×#532C2211yescharacteristic
9C2×#525—22no—
10C22×C4×D4C2211yescharacteristic
11#2975—24no—
12#2975—22no—
13#2975C2214yes—
14C2×#436—24no—
15#2920C2212yes—
16C2×#431C2212yes—
17C2×#407—22no—
18#2542C2212yes—
19#2541C2212yes—
20#2540—22no—
21#2539C414yes—
22#2501—21no—
23#2433—21no—
24#2397C414yes—
25C4×#127—22no—
26C4×#117C2211yescharacteristic
27#1821—21no—
28#1800C2214yes—
29#1783—21no—
30C4×#100—22no—
31#1553—24no—
32#1549C2214yescomplemented
33#1545C2214yescomplemented
34#1540—24no—
35#1303C2218yescomplemented
36C22×#41—21no—
37C23×D4D411yescharacteristic
38C24×C4D412yes—
39C22×M16—24no—
40#532—22no—
41#532—22no—
42#532—22no—
43#532—22no—
44C2×C4⋊Q8C2311yescharacteristic
45C2×#127—22no—
46C2×#127C2311yescharacteristic
47C2×#124—21no—
48#525—416no—
49C2×#123—24no—
50C2×C22⋊Q8—24no—
51C2×#121—24no—
52C2×#121—21no—
53C2×#121C2×C412yes—
54C2×#120—24no—
55C2×C4×D4—28no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4C2×C412yes—
62C2×C4×D4C2×C412yes—
63C2×C4×D4D412yes—
64C2×C4×D4D412yes—
65C2×C4×D4C2312yes—
66C2×#117—21no—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4C2×C412yes—
69C22×C42C2311yescharacteristic
70#436—216no—
71#436—216no—
72#432—22no—
73#432—22no—
74#431—22no—
75#431—22no—
76#431—22no—
77#431—22no—
78#431D414yescomplemented
79#431C2314yes—
80#423C2314yes—
81C2×C4⋊C8D414yescomplemented
82C2×#100—24no—
83#407—28no—
84#407—28no—
85#407—22no—
86C2×C22⋊C8—24no—
87C2×C22⋊C8—22no—
88C4×M16—22no—
89C2×C4×C8—22no—
90#399—24no—
91#399C2×C418yes—
92#392—24no—
93#392C2×C418yes—
94#390—22no—
95#390—22no—
96#390—28no—
97#384—22no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4C2312yes—
100C4×C22⋊C4—21no—
101C4×C22⋊C4C2×C412yes—
102#376C2×C412yes—
103C43—21no—
104#343—24no—
105#332—28no—
106#331—28no—
107C25C22⋊C412yes—
108C2×#41—48no—
109C2×#41—41no—
110C2×#41—41no—
111C2×#41—41no—
112C2×#41—41no—
113C22×Q8—21no—
114C22×D4—22no—
115C22×D4—28no—
116C22×D4—22no—
117C22×D4—22no—
118C22×D4—21no—
119C22×D4—21no—
120C22×D4—44no—
121C22×D4C22⋊C412yes—
122C22×D4C22×C412yes—
123C22×D4C2×D412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—48no—
130C23×C4C22⋊C412yes—
131C23×C4C22⋊C414yes—
132C23×C4C22×C411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D414yes—
136C2×M16—216no—
137C2×M16—24no—
138C2×M16—88no—
139C22×C8—24no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142#127—22no—
143#127—22no—
144#127—48no—
145#124—42no—
146#124—42no—
147#123—416no—
148C22⋊Q8—416no—
149#121—24no—
150#121—24no—
151#121—416no—
152#121—42no—
153#121—42no—
154#120—416no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—24no—
162C4×D4—24no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—432no—
166C4×D4—42no—
167C4×D4—42no—
168#117—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22×C412yes—
176C2×C4⋊C4C2×D412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4C22⋊C414yes—
185C2×C22⋊C4C22×C414yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42C22×C411yescharacteristic
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4112yes—
195C2×C42#4112yes—
196C2×C42C2411yescharacteristic
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—24no—
200C4⋊C8C2×D418yes—
201#100—48no—
202C22⋊C8—216no—
203C22⋊C8—24no—
204C22⋊C8—24no—
205C22⋊C8—42no—
206C22⋊C8—42no—
207#97—42no—
208C4×C8—24no—
209C4×C8—24no—
210C4×C8—42no—
211#95—22no—
212#95C22×C414yes—
213C24—22no—
214C24—22no—
215C24—22no—
216C24—44no—
217C24—44no—
218C24—82no—
219C24C2×C22⋊C411yescharacteristic
220C24C2×C22⋊C414yes—
221#41—84no—
222#41—84no—
223#41—84no—
224#41—84no—
225C2×Q8—44no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×D4—24no—
229C2×D4—24no—
230C2×D4—24no—
231C2×D4—24no—
232C2×D4—24no—
233C2×D4—44no—
234C2×D4—48no—
235C2×D4—48no—
236C2×D4—416no—
237C2×D4—44no—
238C2×D4—48no—
239C2×D4—416no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—41no—
243C2×D4—42no—
244C2×D4—41no—
245C2×D4—44no—
246C2×D4—88no—
247C2×D4—82no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—48no—
259C22×C4—48no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—84no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—81no—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C414yes—
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C4×D412yes—
274C22×C4C4×D412yes—
275C22×C4#12011yescharacteristic
276C22×C4#12011yescharacteristic
277C22×C4#12112yes—
278C22×C4#12112yes—
279C22×C4#12311yescharacteristic
280C22×C4#12311yescharacteristic
281C22×C4C23×C411yescharacteristic
282C22×C4C22×D411yescharacteristic
283C22×C4C22×D411yescharacteristic
284M16—48no—
285M16—816no—
286C2×C8—28no—
287C2×C8—48no—
288C2×C8—44no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—416no—
294C4⋊C4—42no—
295C22⋊C4—28no—
296C22⋊C4—28no—
297C22⋊C4—28no—
298C22⋊C4—416no—
299C22⋊C4—416no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C42—28no—
303C42—44no—
304C42—82no—
305C42C4×D412yes—
306C42C4×D412yes—
307C42#12012yes—
308C42#12112yes—
309C42#12112yes—
310C42#12312yes—
311C42C22×D412yes—
312C42C2×#4112yes—
313C23—22no—
314C23—22no—
315C23—24no—
316C23—24no—
317C23—24no—
318C23—41no—
319C23—44no—
320C23—44no—
321C23—44no—
322C23—44no—
323C23—44no—
324C23—44no—
325C23—44no—
326C23—44no—
327C23—44no—
328C23—42no—
329C23—82no—
330C23—82no—
331C23—816no—
332C23#38611yescharacteristic
333C23#38611yescharacteristic
334C23C22×C22⋊C411yescharacteristic
335Q8—82no—
336Q8—82no—
337D4—48no—
338D4—48no—
339D4—84no—
340D4—82no—
341D4—832no—
342D4—84no—
343D4—84no—
344D4—82no—
345D4—84no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—28no—
355C2×C4—24no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4—84no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4#38612yes—
374C2×C4#38611yescharacteristic
375C2×C4#38612yes—
376C2×C4#38611yescharacteristic
377C2×C4#45314yes—
378C2×C4C2×C4×D412yes—
379C2×C4C2×#12011yescharacteristic
380C2×C4C2×#12112yes—
381C2×C4C2×#12311yescharacteristic
382C2×C4#57514yes—
383C8—48no—
384C8—84no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—44no—
393C22—48no—
394C22—44no—
395C22—44no—
396C22—44no—
397C22—84no—
398C22—88no—
399C22—88no—
400C22—88no—
401C22—82no—
402C22—82no—
403C22—82no—
404C22#188712yes—
405C22C2×#38611yescharacteristic
406C22#289612yes—
407C4—22no—
408C4—22no—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—48no—
413C4—44no—
414C4—82no—
415C4#188712yes—
416C4C2×#45312yes—
417C4#305414yes—
418C2—22no—
419C2—22no—
420C2—41no—
421C2—84no—
422C2—162no—
423C2C2×#188711yescharacteristic
424C2#2097011yescharacteristic
425C2#2101711yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radical#34936C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2975C2semidirect
3. #20726C2semidirect
4. #17717C2semidirect
5. #9405C2semidirect
6. #9403C2semidirect
7. #8092C2semidirect
32. #1549C22semidirect
33. #1545C22semidirect
35. #1303C22semidirect
78. #431D4semidirect
81. C2×C4⋊C8D4semidirect

Maximal subgroups

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

subgroupindex
C2×#29752
#349362
#207262
#177172
#94052
#94032
#80922