Finite Group Databasebeta

← #239639

Subgroups — #239639

Order 512.

1078 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic
3#21358C211yesmaximal, characteristic
4#20647C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1479C211yesmaximal, characteristic
7#10422C218yesmaximal, complemented
8#7901C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×Q8C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×#419—21no—
15#2568—22no—
16#2568—22no—
17#2568—22no—
18#2568C2214yes—
19C2×#400—22no—
20C2×#391—21no—
21C2×#391—22no—
22#2285C2211yescharacteristic
23C2×#382—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990C2214yes—
26#1987—22no—
27#1947C2214yes—
28#1852—22no—
29#1824—22no—
30#1805—22no—
31#1803C2214yes—
32C4×#107—22no—
33#1736—21no—
34#1549—24no—
35C2×#332C2214yes—
36#1479—24no—
37C2×Q8⋊C8C2214yes—
38#1279C2214yes—
39C23×Q8—21no—
40C23×Q8D411yescharacteristic
41C22×M16—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—22no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8D412yes—
49C2×C4⋊Q8C2312yes—
50C2×C4×Q8—21no—
51C2×C4×Q8—21no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8D412yes—
57C2×C4×Q8C2312yes—
58C2×#117—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4D412yes—
62C22×C4⋊C4D411yescharacteristic
63C22×C4⋊C4D412yes—
64C22×C42—21no—
65C22×C42—21no—
66C22×C42C2311yescharacteristic
67#431—22no—
68#431—22no—
69#431—22no—
70#431D414yes—
71C2×#107—24no—
72#423—22no—
73#423—22no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2312yes—
77C2×C4⋊C8C2312yes—
78#419—44no—
79C2×#103—24no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#400—28no—
85#400—28no—
86#391—28no—
87#391—24no—
88#391—24no—
89#391—28no—
90#384C2312yes—
91#382—24no—
92#382—24no—
93#382—21no—
94#380—21no—
95#380—21no—
96C4×C4⋊C4—21no—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4D412yes—
101C4×C4⋊C4C2312yes—
102C2×#95—24no—
103C2×#95—24no—
104C2×#95—21no—
105#340—24no—
106#340—28no—
107#340—24no—
108#332—28no—
109Q8⋊C8—28no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—48no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8Q1614yes—
122C22×Q8C2×D412yes—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C2×M16—42no—
131C2×M16—42no—
132C2×M16—42no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C4⋊Q8—24no—
137C4⋊Q8—24no—
138C4⋊Q8—22no—
139C4⋊Q8—22no—
140C4⋊Q8—22no—
141C4⋊Q8—22no—
142C4⋊Q8—42no—
143C4⋊Q8—42no—
144C4⋊Q8—42no—
145C4⋊Q8—42no—
146C4×Q8—24no—
147C4×Q8—24no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—42no—
153C4×Q8—42no—
154C4×Q8—42no—
155C4×Q8—42no—
156#117—42no—
157#117—42no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4C2×D414yes—
180C2×C22⋊C4—41no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—41no—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42#4112yes—
195C2×C42#4112yes—
196C2×C42#4112yes—
197C2×C42C2411yescharacteristic
198#107—48no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—42no—
203C4⋊C8—42no—
204C4⋊C8C2×D414yes—
205#103—48no—
206C22⋊C8—24no—
207C22⋊C8—42no—
208C4×C8—24no—
209C4×C8—22no—
210C4×C8—22no—
211#95—216no—
212#95—216no—
213#95—22no—
214#95—22no—
215#95—41no—
216#95—41no—
217C24#12011yescharacteristic
218C2×Q8—22no—
219C2×Q8—24no—
220C2×Q8—22no—
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—28no—
225C2×Q8—28no—
226C2×Q8—48no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—48no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—88no—
236C2×Q8—88no—
237C2×Q8C2×Q1614yes—
238C22×C4—24no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—48no—
251C22×C4—42no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12011yescharacteristic
258C22×C4#12112yes—
259C22×C4#12112yes—
260C22×C4#12112yes—
261C22×C4#12312yes—
262C22×C4#12312yes—
263C22×C4#12412yes—
264C22×C4C2×Q1612yes—
265C22×C4#13512yes—
266C22×C4AGL(1,ℤ/8)12yes—
267C22×C4#13712yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C22×D411yescharacteristic
270C22×C4C22×D411yescharacteristic
271M16—84no—
272C2×C8—24no—
273C2×C8—44no—
274C2×C8—44no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—44no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C22⋊C4—82no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—44no—
304C42—44no—
305C42—82no—
306C42#12112yes—
307C42#12112yes—
308C42#12112yes—
309C42#12312yes—
310C42#12312yes—
311C42#12412yes—
312C42C22×D411yescharacteristic
313C42C2×#4111yescharacteristic
314C42C2×#4111yescharacteristic
315C42C2×#4111yescharacteristic
316C23—22no—
317C23—22no—
318C23#39412yes—
319C23#44912yes—
320C23#45112yes—
321C23C2×#12011yescharacteristic
322Q8—24no—
323Q8—24no—
324Q8—42no—
325Q8—42no—
326Q8—48no—
327Q8—88no—
328Q8—84no—
329Q8—84no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—48no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—48no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—88no—
349C2×C4—84no—
350C2×C4—88no—
351C2×C4—82no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4#39412yes—
355C2×C4#39412yes—
356C2×C4#39412yes—
357C2×C4#44912yes—
358C2×C4#45112yes—
359C2×C4#46714yes—
360C2×C4#46814yes—
361C2×C4#48414yes—
362C2×C4#48514yes—
363C2×C4C2×#12111yescharacteristic
364C2×C4C2×#12111yescharacteristic
365C2×C4C2×#12111yescharacteristic
366C2×C4C2×#12311yescharacteristic
367C2×C4C2×#12311yescharacteristic
368C2×C4C2×#12411yescharacteristic
369C2×C4C22×Q1611yescharacteristic
370C2×C4C2×#13511yescharacteristic
371C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
372C2×C4C2×#13711yescharacteristic
373C2×C4#57514yes—
374C8—44no—
375C8—82no—
376C8—82no—
377C22—24no—
378C22—24no—
379C22—22no—
380C22—22no—
381C22#203614yes—
382C22C2×#39411yescharacteristic
383C22C2×#45111yescharacteristic
384C22C2×#44911yescharacteristic
385C22#300512yes—
386C22#302012yes—
387C4—21no—
388C4—21no—
389C4—22no—
390C4—22no—
391C4—44no—
392C4—84no—
393C4—82no—
394C4—84no—
395C4—162no—
396C4#203614yes—
397C4#304012yes—
398C4C2×#46711yescharacteristic
399C4C2×#46811yescharacteristic
400C4C2×#48511yescharacteristic
401C4C2×#48411yescharacteristic
402C4#314012yes—
403C2—24no—
404C2#1876212yes—
405C2#1877912yes—
406C2C2×#203611yescharacteristic
407C2#2349411yescharacteristic
408C2#2350611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #10422C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25682
#218342
#213582
#206472
C2×#15492
C2×#14792
#104222
#79012