Finite Group Databasebeta

← #204405

Subgroups — #204405

Order 512.

1204 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3101C211yesmaximal, characteristic
2#21300C211yesmaximal, characteristic, complemented
3C2×#1954C211yesmaximal, characteristic
4C2×#1571C211yesmaximal, characteristic, complemented
5C2×#1481C211yesmaximal, characteristic, complemented
6C2×#1486C211yesmaximal, characteristic
7C2×#1452C211yesmaximal, characteristic, complemented
8#9666C214yesmaximal, complemented
9#9660C214yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12#3101—24no—
13#3101—24no—
14C2×#483C412yes—
15C2×#476C412yes—
16C2×#461C412yes—
17C2×#450C412yes—
18C2×C4×SD16C2214yes—
19C2×#431C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23C22×C22⋊C8—22no—
24C22×C4×C8C2211yescharacteristic
25#1984—22no—
26#1984C2212yescomplemented
27#1984C2212yescomplemented
28#1954—22no—
29#1954C2212yes—
30#1954C2212yes—
31#1729—22no—
32#1571—24no—
33C2×#335C2214yescomplemented
34#1486—24no—
35#1481—24no—
36C2×Q8⋊C8C2214yescomplemented
37C2×D4⋊C8C2214yes—
38#1452—24no—
39C2×#322C2212yescomplemented
40C2×#322C2212yescomplemented
41C23×D4C2×C411yescharacteristic
42C24×C4—21no—
43C22×SD16C2×C414yes—
44C22×M16—21no—
45C23×C8—21no—
46C23×C8D412yes—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×C4⋊Q8C2×C411yescharacteristic
53C2×#125—21no—
54C2×C22⋊Q8—21no—
55C2×C22⋊Q8C2×C412yes—
56C2×C4×Q8—21no—
57C2×C4×Q8C2312yes—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4—22no—
62C2×C4×D4—21no—
63C2×C4×D4—22no—
64C2×C4×D4C2×C412yes—
65C2×C4×D4C2312yes—
66C2×#117—21no—
67C22×C4⋊C4C2×C411yescharacteristic
68C22×C22⋊C4—21no—
69C22×C42C2311yescharacteristic
70#483—28no—
71#476—28no—
72#461—28no—
73#450—28no—
74C4×SD16—28no—
75C4×SD16—28no—
76#431—22no—
77#431D414yes—
78#431D414yes—
79#431D414yes—
80C2×#106C2×C414yes—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8C2×C412yes—
89C2×C4⋊C8C2312yes—
90C2×C4⋊C8C2312yes—
91C2×#103C2×C414yes—
92C2×#102C2×C414yes—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8—48no—
97C2×C22⋊C8C2×C412yes—
98C2×C4×C8—22no—
99C2×C4×C8—22no—
100C2×C4×C8D412yes—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2312yes—
104C2×C4×C8C2312yes—
105#336—24no—
106#336—24no—
107#335—28no—
108Q8⋊C8—28no—
109D4⋊C8—28no—
110#322—24no—
111#322—24no—
112C25—21no—
113C22×Q8—21no—
114C22×Q8C22×C412yes—
115C22×D4—24no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—42no—
120C22×D4C22×C412yes—
121C23×C4—21no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—42no—
125C23×C4—42no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4#4111yescharacteristic
131C2×SD16—28no—
132C2×SD16—28no—
133C2×SD16—44no—
134C2×M16—42no—
135C2×M16—44no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—42no—
143C22×C8—44no—
144C22×C8C22×C414yes—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151#125—42no—
152#125—42no—
153C22⋊Q8—24no—
154C22⋊Q8—24no—
155C22⋊Q8—42no—
156C22⋊Q8—42no—
157C4×Q8—22no—
158C4×Q8—22no—
159C4×Q8—22no—
160C4×Q8—22no—
161C4×Q8—42no—
162C4×Q8—42no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—22no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—22no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—42no—
173#117—42no—
174#117—42no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4C22×C412yes—
184C2×C4⋊C4C22×C412yes—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—42no—
190C2×C42—22no—
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42#4112yes—
197C2×C42#4111yescharacteristic
198C2×C42C2411yescharacteristic
199#106—28no—
200C4⋊C8—24no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—42no—
206C4⋊C8—42no—
207C4⋊C8C22⋊C418yes—
208#103—28no—
209#102—28no—
210C22⋊C8—24no—
211C22⋊C8—48no—
212C22⋊C8—48no—
213C22⋊C8C22⋊C418yes—
214C4×C8—24no—
215C4×C8—21no—
216C4×C8—21no—
217C4×C8—22no—
218C4×C8—21no—
219C4×C8—21no—
220C4×C8C2×D414yes—
221C4×C8C2×D414yes—
222C4×C8C2×D414yes—
223C24—24no—
224C24—22no—
225C24—42no—
226C24—44no—
227C24—42no—
228C24—44no—
229C24C4×D411yescharacteristic
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—44no—
235C2×Q8—42no—
236C2×D4—24no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—48no—
243C2×D4—48no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4#13114yes—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—88no—
262C22×C4—84no—
263C22×C4—84no—
264C22×C4C2×C22⋊C412yes—
265C22×C4C4×D412yes—
266C22×C4C4×D411yescharacteristic
267C22×C4C4×D411yescharacteristic
268C22×C4C4×D411yescharacteristic
269C22×C4#12112yes—
270C22×C4#12112yes—
271C22×C4#12412yes—
272C22×C4#12712yes—
273C22×C4AGL(1,ℤ/8)14yes—
274C22×C4#13714yes—
275C22×C4C23×C411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278SD16—416no—
279M16—84no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—44no—
290C2×C8—48no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—44no—
294C2×C8—44no—
295C2×C8—84no—
296C2×C8C2×C22⋊C418yes—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—42no—
308C22⋊C4—44no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C42—42no—
313C42—42no—
314C42C2×C22⋊C412yes—
315C42C4×D412yes—
316C42#12112yes—
317C42#12112yes—
318C42#12412yes—
319C42#12712yes—
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C22×D411yescharacteristic
323C42C2×#4111yescharacteristic
324C23—21no—
325C23—21no—
326C23—22no—
327C23—24no—
328C23—48no—
329C23—416no—
330C23—42no—
331C23—44no—
332C23—44no—
333C23—42no—
334C23—88no—
335C23—84no—
336C23—84no—
337C23#39012yes—
338C23#44014yes—
339C23C2×C4×D411yescharacteristic
340Q8—44no—
341Q8—44no—
342Q8—84no—
343D4—24no—
344D4—24no—
345D4—48no—
346D4—42no—
347D4—42no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—84no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—88no—
367C2×C4—84no—
368C2×C4#39012yes—
369C2×C4#39012yes—
370C2×C4#39012yes—
371C2×C4#44014yes—
372C2×C4#46818yes—
373C2×C4#49614yes—
374C2×C4#49714yes—
375C2×C4C22×C22⋊C411yescharacteristic
376C2×C4C2×C4×D411yescharacteristic
377C2×C4C2×#12111yescharacteristic
378C2×C4C2×#12111yescharacteristic
379C2×C4C2×#12411yescharacteristic
380C2×C4C2×#12711yescharacteristic
381C2×C4C2×#13112yes—
382C2×C4#56812yes—
383C2×C4C2×AGL(1,ℤ/8)12yes—
384C2×C4C2×#13712yes—
385C8—28no—
386C8—44no—
387C8—42no—
388C8—42no—
389C8—82no—
390C8—82no—
391C22—24no—
392C22—22no—
393C22—24no—
394C22—22no—
395C22—44no—
396C22—48no—
397C22—816no—
398C22—84no—
399C22—82no—
400C22—82no—
401C22#197314yes—
402C22C2×#39011yescharacteristic
403C22#292911yescharacteristic
404C22#293011yescharacteristic
405C22C2×#44012yes—
406C22#295512yes—
407C4—22no—
408C4—21no—
409C4—21no—
410C4—22no—
411C4—82no—
412C4—82no—
413C4—82no—
414C4#197314yes—
415C4#288012yes—
416C4#293312yes—
417C4C2×#46812yes—
418C4C2×#49611yescharacteristic
419C4C2×#49711yescharacteristic
420C2—24no—
421C2—84no—
422C2#1790512yes—
423C2#1793312yes—
424C2#2119111yescharacteristic
425C2C2×#197311yescharacteristic
426C2#2124211yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC23×D4C2×C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21300C2semidirect
4. C2×#1571C2semidirect
5. C2×#1481C2semidirect
7. C2×#1452C2semidirect
8. #9666C2semidirect
9. #9660C2semidirect
26. #1984C22semidirect
27. #1984C22semidirect
33. C2×#335C22semidirect
36. C2×Q8⋊C8C22semidirect
39. C2×#322C22semidirect
40. C2×#322C22semidirect

Maximal subgroups

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

subgroupindex
C2×#31012
#213002
C2×#19542
C2×#15712
C2×#14812
C2×#14862
C2×#14522
#96662
#96602