Finite Group Databasebeta

← #239641

Subgroups — #239641

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2#21834C211yesmaximal, characteristic, complemented
3#21358C211yesmaximal, characteristic, complemented
4#20646C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic, complemented
6C2×#1478C211yesmaximal, characteristic
7#10421C218yesmaximal, complemented
8#7901C211yesmaximal, characteristic, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8—22no—
11C22×C4×D4C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C2×#423C2211yescharacteristic
15C2×#418—21no—
16#2542—22no—
17#2542—22no—
18#2542—22no—
19#2542C2214yes—
20C2×#399—22no—
21C2×#390—21no—
22#2285C2211yescharacteristic
23C2×#384—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25#1990C2214yescomplemented
26#1987—22no—
27#1947C2214yescomplemented
28#1852—22no—
29#1824—22no—
30#1804—22no—
31#1802C2214yes—
32C4×#107—22no—
33#1736—21no—
34#1549—24no—
35C2×#332C2214yescomplemented
36#1478—24no—
37C2×D4⋊C8C2214yes—
38#1279C2214yescomplemented
39C23×Q8—21no—
40C23×D4D411yescharacteristic
41C24×C4—21no—
42C22×M16—21no—
43C2×C4⋊Q8—21no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8—22no—
46C2×C4⋊Q8—22no—
47C2×C4⋊Q8—22no—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8D412yes—
50C2×C4⋊Q8C2312yes—
51C2×C22⋊Q8—28no—
52C2×C22⋊Q8—28no—
53C2×C22⋊Q8—416no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—22no—
60C2×C4×D4D412yes—
61C2×C4×D4C2312yes—
62C2×#117—21no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4D412yes—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4D411yescharacteristic
68C22×C22⋊C4—24no—
69C22×C22⋊C4—21no—
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#431—22no—
73#431—22no—
74#431—22no—
75#431D414yes—
76C2×#107—24no—
77#423—22no—
78#423—22no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83#418—44no—
84C2×#102—24no—
85C2×C22⋊C8—21no—
86C2×C22⋊C8—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2312yes—
89#399—28no—
90#399—28no—
91#390—24no—
92#390—24no—
93#384—24no—
94#384—24no—
95#384C2312yes—
96#382—21no—
97#380—21no—
98#380—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4—21no—
103C4×C4⋊C4D412yes—
104C4×C4⋊C4C2312yes—
105C2×#95—21no—
106#340—24no—
107#340—28no—
108#340—24no—
109#332—28no—
110D4⋊C8—28no—
111C25—21no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—48no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4Q1614yes—
124C22×D4C2×D412yes—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—48no—
131C23×C4—42no—
132C23×C4—42no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—42no—
137C2×M16—42no—
138C2×M16—42no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C4⋊Q8—24no—
143C4⋊Q8—24no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—42no—
149C4⋊Q8—42no—
150C4⋊Q8—42no—
151C4⋊Q8—42no—
152C22⋊Q8—432no—
153C22⋊Q8—416no—
154C22⋊Q8—416no—
155C4×D4—24no—
156C4×D4—24no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—42no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165#117—42no—
166#117—42no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—216no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—416no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D414yes—
186C2×C4⋊C4C2×D414yes—
187C2×C22⋊C4—216no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—416no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—41no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—42no—
197C2×C42—42no—
198C2×C42—41no—
199C2×C42C2×D412yes—
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4112yes—
204C2×C42#4112yes—
205C2×C42#4112yes—
206C2×C42C2411yescharacteristic
207#107—48no—
208C4⋊C8—22no—
209C4⋊C8—22no—
210C4⋊C8—22no—
211C4⋊C8—42no—
212C4⋊C8—42no—
213C4⋊C8C2×D414yes—
214#102—48no—
215C22⋊C8—24no—
216C22⋊C8—42no—
217C4×C8—24no—
218C4×C8—22no—
219C4×C8—22no—
220#95—22no—
221#95—22no—
222#95—41no—
223#95—41no—
224C24—22no—
225C24—24no—
226C24—44no—
227C24—44no—
228C24—42no—
229C24—42no—
230C24#12011yescharacteristic
231C2×Q8—48no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—48no—
235C2×Q8—44no—
236C2×Q8—88no—
237C2×Q8—88no—
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—28no—
245C2×D4—28no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—44no—
249C2×D4—44no—
250C2×D4C2×Q1614yes—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—42no—
257C22×C4—48no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—88no—
270C22×C4—88no—
271C22×C4#12011yescharacteristic
272C22×C4#12011yescharacteristic
273C22×C4#12011yescharacteristic
274C22×C4#12112yes—
275C22×C4#12112yes—
276C22×C4#12112yes—
277C22×C4#12312yes—
278C22×C4#12312yes—
279C22×C4#12412yes—
280C22×C4C2×Q1612yes—
281C22×C4#13512yes—
282C22×C4AGL(1,ℤ/8)12yes—
283C22×C4#13712yes—
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—84no—
288C2×C8—24no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C4⋊C4—28no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—432no—
302C4⋊C4—44no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—44no—
311C4⋊C4AGL(1,ℤ/8)18yes—
312C22⋊C4—432no—
313C22⋊C4—42no—
314C22⋊C4—42no—
315C22⋊C4—44no—
316C22⋊C4—82no—
317C42—42no—
318C42—42no—
319C42—44no—
320C42—82no—
321C42#12112yes—
322C42#12112yes—
323C42#12112yes—
324C42#12312yes—
325C42#12312yes—
326C42#12412yes—
327C42C22×D411yescharacteristic
328C42C2×#4111yescharacteristic
329C42C2×#4111yescharacteristic
330C42C2×#4111yescharacteristic
331C23—22no—
332C23—22no—
333C23—24no—
334C23—416no—
335C23—42no—
336C23—42no—
337C23—44no—
338C23—44no—
339C23—48no—
340C23—84no—
341C23—88no—
342C23—84no—
343C23#39412yes—
344C23#44912yes—
345C23#45112yes—
346C23C2×#12011yescharacteristic
347Q8—88no—
348Q8—84no—
349Q8—84no—
350D4—24no—
351D4—24no—
352D4—42no—
353D4—42no—
354D4—48no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—416no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—44no—
370C2×C4—88no—
371C2×C4—84no—
372C2×C4—88no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4#39412yes—
379C2×C4#39412yes—
380C2×C4#39412yes—
381C2×C4#44912yes—
382C2×C4#45112yes—
383C2×C4#46714yes—
384C2×C4#46814yes—
385C2×C4#48414yes—
386C2×C4#48514yes—
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×#12311yescharacteristic
392C2×C4C2×#12411yescharacteristic
393C2×C4C22×Q1611yescharacteristic
394C2×C4C2×#13511yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
396C2×C4C2×AGL(1,ℤ/8)14yes—
397C2×C4C2×#13711yescharacteristic
398C8—44no—
399C8—82no—
400C8—82no—
401C22—24no—
402C22—24no—
403C22—22no—
404C22—22no—
405C22—48no—
406C22—44no—
407C22—816no—
408C22—84no—
409C22—82no—
410C22—82no—
411C22#203614yes—
412C22C2×#39411yescharacteristic
413C22C2×#45111yescharacteristic
414C22C2×#44911yescharacteristic
415C22#300312yes—
416C22#302012yes—
417C4—21no—
418C4—21no—
419C4—22no—
420C4—22no—
421C4—48no—
422C4—82no—
423C4—84no—
424C4—162no—
425C4#203614yes—
426C4#303812yes—
427C4C2×#46711yescharacteristic
428C4C2×#46811yescharacteristic
429C4C2×#48511yescharacteristic
430C4C2×#48411yescharacteristic
431C4#313812yes—
432C2—24no—
433C2—84no—
434C2#1875912yes—
435C2#1878212yes—
436C2C2×#203611yescharacteristic
437C2#2349411yescharacteristic
438C2#2350511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC23×D4D4
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
2. #21834C2semidirect
3. #21358C2semidirect
5. C2×#1549C2semidirect
7. #10421C2semidirect
8. #7901C2semidirect
25. #1990C22semidirect
27. #1947C22semidirect
35. C2×#332C22semidirect
38. #1279C22semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
#218342
#213582
#206462
C2×#15492
C2×#14782
#104212
#79012