Finite Group Databasebeta

← #127823

Subgroups — #127823

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21851C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#7283C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#432C411yescharacteristic, complemented
7C22×C4⋊C8C411yescharacteristic, complemented
8C2×#410—21no—
9#2283—21no—
10C4×#117—21no—
11#1985C412yescomplemented
12#1955C2212yes—
13#1869—21no—
14#1869—21no—
15#1820C412yescomplemented
16#1799—21no—
17#1799C412yescomplemented
18#1797—21no—
19#1797—21no—
20#1782—21no—
21C4×#99—21no—
22C4×#99—21no—
23C4×#99—21no—
24C4×#99C412yescomplemented
25C4×#99C2212yes—
26C4×#99C2212yes—
27#1740C412yescomplemented
28C22×#41D411yescharacteristic
29C23×C8—21no—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8Q811yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4C2311yescharacteristic
42C2×#117—22no—
43C2×#117C2×C411yescharacteristic
44C2×#117C2×C411yescharacteristic
45C2×#117Q811yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D411yescharacteristic
48C22×C42D411yescharacteristic
49#432—41no—
50#432—41no—
51C2×C4⋊C8—21no—
52C2×C4⋊C8—21no—
53C2×C4⋊C8—41no—
54C2×C4⋊C8—41no—
55C2×C4⋊C8C2×C412yes—
56#410—28no—
57C2×#99—22no—
58C2×#99—22no—
59C2×#99—22no—
60C2×#99—22no—
61C2×C22⋊C8—21no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×#97C2×C412yes—
65#383—41no—
66#382—41no—
67#381—22no—
68#380—22no—
69C4×C4⋊C4—41no—
70C4×C4⋊C4—41no—
71C4×C4⋊C4—41no—
72C4×C22⋊C4—22no—
73C4×C22⋊C4—22no—
74C4×C22⋊C4—22no—
75#376—41no—
76#376—41no—
77C43—41no—
78#342—22no—
79#342—22no—
80#342C2×C414yes—
81#342C2×C414yes—
82#336—22no—
83#336—22no—
84C2×#41—24no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—42no—
88C22×Q8C4⋊C411yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C2×D411yescharacteristic
92C23×C4C4211yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—42no—
104C22×C8—82no—
105C4×Q8—41no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×D4—41no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—41no—
113C4×D4—41no—
114C4×D4—41no—
115C4×D4—41no—
116C4×D4—41no—
117C4×D4—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121#117—28no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C4211yescharacteristic
132C2×C4⋊C4C4211yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C2×Q811yescharacteristic
137C2×C22⋊C4—24no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4C22⋊C411yescharacteristic
140C2×C22⋊C4C22⋊C411yescharacteristic
141C2×C22⋊C4C4⋊C411yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C22×C411yescharacteristic
144C2×C22⋊C4C22×C411yescharacteristic
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—21no—
153C2×C42—21no—
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42C4211yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C4⋊C8—42no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176#99—28no—
177#99—28no—
178#99—44no—
179#99—44no—
180C22⋊C8—42no—
181C22⋊C8—82no—
182C22⋊C8—82no—
183#97—42no—
184#95—42no—
185#95—42no—
186#95—42no—
187#95—42no—
188#95—42no—
189#95—42no—
190C24#9511yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192C24C2×C22⋊C411yescharacteristic
193#41—48no—
194#41—48no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4#10414yes—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—41no—
232C22×C4—42no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4C2×C4211yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4C2×C4⋊C411yescharacteristic
253C22×C4C2×C4⋊C411yescharacteristic
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—84no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—48no—
268C4⋊C4—84no—
269C22⋊C4—28no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C42—21no—
280C42—21no—
281C42—21no—
282C42—21no—
283C42—21no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—21no—
288C42—21no—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—48no—
298C42—82no—
299C42—82no—
300C42—82no—
301C42—82no—
302C42—84no—
303C42#9911yescharacteristic
304C42#9911yescharacteristic
305C42#9911yescharacteristic
306C42#9911yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23#33912yes—
320C23C2×#9511yescharacteristic
321C23C2×#10412yes—
322C23#42112yes—
323Q8—44no—
324D4—44no—
325D4—44no—
326D4—44no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—48no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4#33912yes—
380C2×C4#34211yescharacteristic
381C2×C4#34211yescharacteristic
382C2×C4#34211yescharacteristic
383C2×C4#34211yescharacteristic
384C2×C4C2×#9911yescharacteristic
385C2×C4C2×#9911yescharacteristic
386C8—82no—
387C8—82no—
388C22—21no—
389C22—21no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—81no—
404C22—81no—
405C22C2×#33911yescharacteristic
406C22#173311yescharacteristic
407C22C2×#34211yescharacteristic
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—22no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—88no—
422C4—164no—
423C4#139112yes—
424C4#212811yescharacteristic
425C4#212911yescharacteristic
426C2—21no—
427C2—21no—
428C2—42no—
429C2—42no—
430C2—42no—
431C2#805111yescharacteristic
432C2C2×#139111yescharacteristic
433C2#1258311yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7283C2semidirect
6. C2×#432C4semidirect
7. C22×C4⋊C8C4semidirect
11. #1985C4semidirect
15. #1820C4semidirect
17. #1799C4semidirect
24. C4×#99C4semidirect
27. #1740C4semidirect

Maximal subgroups

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

subgroupindex
#218512
#207232
C4×#4102
#72832