Finite Group Databasebeta

← #240292

Subgroups — #240292

Order 512.

1100 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#476C211yesmaximal, characteristic
2#21833C211yesmaximal, characteristic, complemented
3#21788C211yesmaximal, characteristic
4#20738C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic, complemented
6C2×#1558C211yesmaximal, characteristic, complemented
7#10481C214yesmaximal, complemented
8#10480C214yesmaximal, complemented
9#8044C211yesmaximal, characteristic, complemented
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#476—24no—
13C2×#476—24no—
14C2×#476—22no—
15C2×#476C2214yes—
16C2×#432C2211yescharacteristic
17C2×#432C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×#106—21no—
20C22×C4⋊C8C2211yescharacteristic
21C22×#102—21no—
22#2285C2211yescharacteristic
23#1990C2214yescomplemented
24#1988—22no—
25#1955—22no—
26#1954C2214yes—
27#1824—21no—
28#1823—21no—
29#1823—21no—
30#1802C2212yes—
31#1802C2212yes—
32#1786—21no—
33#1779—21no—
34#1779—21no—
35#1762—22no—
36#1573—24no—
37C2×#335C2212yescomplemented
38C2×#334C2212yescomplemented
39#1558—24no—
40C2×#333C2212yescomplemented
41C2×#333C2212yescomplemented
42#1296C2214yescomplemented
43C23×Q8—21no—
44C23×D4D411yescharacteristic
45C24×C4—21no—
46C23×C8—21no—
47C2×C4⋊Q8—21no—
48C2×C4⋊Q8—21no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8D414yes—
54C2×C4⋊Q8C2312yes—
55C2×C4×D4—21no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—21no—
59C2×C4×D4—22no—
60C2×C4×D4—22no—
61C2×C4×D4D412yes—
62C2×C4×D4C2312yes—
63C2×#117—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C42C2311yescharacteristic
69#476—216no—
70#476—28no—
71#476—28no—
72#432—22no—
73#432—22no—
74#432—22no—
75#432—22no—
76#431—22no—
77#431—22no—
78C2×#106—28no—
79C2×#106—24no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8D412yes—
83C2×C4⋊C8D414yes—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2312yes—
87C2×C4⋊C8C2312yes—
88C2×#102—28no—
89C2×#102—24no—
90C2×#102—22no—
91C2×#102—22no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—24no—
95C2×C22⋊C8—24no—
96C2×C22⋊C8D411yescharacteristic
97C2×C22⋊C8D411yescharacteristic
98C2×#97—21no—
99C2×#97—21no—
100C2×C4×C8C2312yes—
101#384C2312yes—
102#382—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—22no—
111#335—24no—
112#334—24no—
113#333—24no—
114#333—24no—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—41no—
119C22×Q8—42no—
120C22×Q8—82no—
121C22×Q8—82no—
122C22×D4—24no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4—42no—
127C22×D4C2×D412yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—84no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4#4111yescharacteristic
138C22×C8—24no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—44no—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—42no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8C2×D414yes—
156C4×D4—24no—
157C4×D4—24no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—22no—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—42no—
165C4×D4—42no—
166#117—82no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—42no—
181C2×C4⋊C4C2×D412yes—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—41no—
187C2×C42—41no—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42#4112yes—
193C2×C42#4112yes—
194C2×C42#4111yescharacteristic
195C2×C42C2411yescharacteristic
196#106—216no—
197C4⋊C8—24no—
198C4⋊C8—24no—
199C4⋊C8—22no—
200C4⋊C8—21no—
201C4⋊C8—21no—
202C4⋊C8—22no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—22no—
206C4⋊C8C2×D414yes—
207#102—216no—
208#102—44no—
209#102—44no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C22⋊C8—42no—
213C22⋊C8—42no—
214C22⋊C8—84no—
215C22⋊C8—84no—
216#97—42no—
217#97—42no—
218C4×C8—22no—
219C4×C8—22no—
220#95—41no—
221#95—41no—
222C24—22no—
223C24—44no—
224C24—42no—
225C24—42no—
226C24—42no—
227C24—82no—
228C24#12111yescharacteristic
229C2×Q8—24no—
230C2×Q8—42no—
231C2×Q8—44no—
232C2×Q8—88no—
233C2×Q8—82no—
234C2×Q8—82no—
235C2×Q8—88no—
236C2×Q8—82no—
237C2×Q8—82no—
238C2×D4—28no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—48no—
245C2×D4—44no—
246C2×D4—44no—
247C2×D4—42no—
248C2×D4#13714yes—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—82no—
265C22×C4—82no—
266C22×C4—88no—
267C22×C4—88no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4—82no—
271C22×C4#12012yes—
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12111yescharacteristic
275C22×C4#12111yescharacteristic
276C22×C4#12111yescharacteristic
277C22×C4#12312yes—
278C22×C4#12312yes—
279C22×C4#12412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C2×C8—24no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—48no—
289C2×C8—44no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—44no—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—24no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—44no—
304C4⋊C4—44no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—82no—
309C22⋊C4—42no—
310C22⋊C4—42no—
311C22⋊C4—44no—
312C22⋊C4—82no—
313C42—82no—
314C42#12012yes—
315C42#12112yes—
316C42#12112yes—
317C42#12312yes—
318C42#12312yes—
319C42#12412yes—
320C42C22×D411yescharacteristic
321C42C22×D411yescharacteristic
322C42C2×#4111yescharacteristic
323C42C2×#4111yescharacteristic
324C23—22no—
325C23—22no—
326C23—24no—
327C23—48no—
328C23—42no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—42no—
334C23—88no—
335C23—88no—
336C23—84no—
337C23—84no—
338C23#39412yes—
339C23#47112yes—
340C23#47212yes—
341C23C2×#12111yescharacteristic
342Q8—84no—
343Q8—84no—
344Q8—88no—
345D4—24no—
346D4—24no—
347D4—48no—
348D4—42no—
349D4—42no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—24no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—88no—
368C2×C4—84no—
369C2×C4—88no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—84no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#39412yes—
378C2×C4#39412yes—
379C2×C4#39412yes—
380C2×C4C2×#12011yescharacteristic
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12311yescharacteristic
384C2×C4C2×#12311yescharacteristic
385C2×C4C2×#12411yescharacteristic
386C2×C4C2×#13712yes—
387C2×C4#57512yes—
388C2×C4#57714yes—
389C8—44no—
390C8—42no—
391C8—42no—
392C8—81no—
393C8—81no—
394C8—82no—
395C22—24no—
396C22—24no—
397C22—22no—
398C22—22no—
399C22—44no—
400C22—44no—
401C22—44no—
402C22—816no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—82no—
407C22#205114yes—
408C22C2×#39411yescharacteristic
409C22#304212yes—
410C22#305612yes—
411C22C2×#47111yescharacteristic
412C22C2×#47211yescharacteristic
413C4—22no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—22no—
418C4—84no—
419C4—82no—
420C4—162no—
421C4#300411yescharacteristic
422C4#300511yescharacteristic
423C4#304312yes—
424C4#305712yes—
425C4#309512yes—
426C2—24no—
427C2—84no—
428C2#1946312yes—
429C2#1946512yes—
430C2#2348111yescharacteristic
431C2C2×#205111yescharacteristic
432C2#2354111yescharacteristic
subgroupG/H
center, socleC23C2×#121
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. #21833C2semidirect
5. C2×#1573C2semidirect
6. C2×#1558C2semidirect
7. #10481C2semidirect
8. #10480C2semidirect
9. #8044C2semidirect
23. #1990C22semidirect
37. C2×#335C22semidirect
38. C2×#334C22semidirect
40. C2×#333C22semidirect
41. C2×#333C22semidirect
42. #1296C22semidirect

Maximal subgroups

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

subgroupindex
C22×#4762
#218332
#217882
#207382
C2×#15732
C2×#15582
#104812
#104802
#80442