Finite Group Databasebeta

← #127816

Subgroups — #127816

Order 512.

683 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21380C211yesmaximal, characteristic
2#20728C211yesmaximal, characteristic
3C4×#410C211yesmaximal, characteristic
4#7282C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic, complemented
8C2×#410—21no—
9C2×C4×M16C411yescharacteristic, complemented
10C4×#117—21no—
11#1984—21no—
12#1954—21no—
13#1870C412yescomplemented
14#1870C2212yes—
15#1822C412yescomplemented
16#1801C412yescomplemented
17#1800—21no—
18#1785—21no—
19C4×#101—21no—
20C4×#100C2212yes—
21C4×#99—21no—
22C4×#99—21no—
23C4×#99C412yes—
24C4×#99C2212yes—
25#1749—21no—
26#1749—21no—
27#1740C412yes—
28C22×#41D411yescharacteristic
29C22×M16C2×C412yes—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C2×C4×Q8C2×C411yescharacteristic
35C2×C4×Q8D411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4Q811yescharacteristic
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—
51#432—41no—
52#432—41no—
53#431—41no—
54#431—41no—
55#423—22no—
56C2×C4⋊C8—21no—
57C2×#101—22no—
58C2×#100—22no—
59#410—28no—
60C2×#99—22no—
61C2×#99—22no—
62#407—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8C2×C412yes—
66C4×M16—22no—
67C4×M16—41no—
68C4×M16—41no—
69C2×#97—21no—
70C2×#97—21no—
71C2×C4×C8—21no—
72C4×C4⋊C4—41no—
73C4×C4⋊C4—41no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—22no—
76#376—41no—
77C43—41no—
78#343—22no—
79#343C2×C414yes—
80#342—22no—
81#342C2×C414yes—
82#336—22no—
83#336—22no—
84C2×#41—24no—
85C2×#41—41no—
86C2×#41—41no—
87C2×#41—42no—
88C22×Q8C22⋊C411yescharacteristic
89C22×D4C22⋊C411yescharacteristic
90C22×D4C4⋊C411yescharacteristic
91C22×D4C2×D411yescharacteristic
92C23×C4C22⋊C411yescharacteristic
93C23×C4C22⋊C411yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C4⋊C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—24no—
102C2×M16—44no—
103C2×M16—42no—
104C22×C8—22no—
105C22×C8—22no—
106C4×Q8—41no—
107C4×Q8—41no—
108C4×Q8—41no—
109C4×Q8—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—
121C4×D4—41no—
122#117—28no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—41no—
129#117—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4C4211yescharacteristic
132C2×C4⋊C4C22⋊C411yescharacteristic
133C2×C4⋊C4C4⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C22⋊C4—24no—
138C2×C22⋊C4C4211yescharacteristic
139C2×C22⋊C4C22⋊C411yescharacteristic
140C2×C22⋊C4C4⋊C411yescharacteristic
141C2×C22⋊C4C4⋊C411yescharacteristic
142C2×C22⋊C4C22×C411yescharacteristic
143C2×C22⋊C4C2×Q811yescharacteristic
144C2×C42—21no—
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—42no—
157C2×C42—42no—
158C2×C42—42no—
159C2×C42C4211yescharacteristic
160C2×C42C4211yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172#101—44no—
173#100—44no—
174#99—28no—
175#99—28no—
176C22⋊C8—44no—
177C22⋊C8—44no—
178C22⋊C8—42no—
179C22⋊C8—42no—
180C22⋊C8—42no—
181#97—42no—
182#97—41no—
183#97—41no—
184#97—41no—
185#97—41no—
186C4×C8—42no—
187C4×C8—42no—
188#95—42no—
189#95—42no—
190C24#9511yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192C24C2×C4⋊C411yescharacteristic
193#41—48no—
194#41—48no—
195C2×Q8—41no—
196C2×Q8—41no—
197C2×Q8#10414yes—
198C2×D4—22no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—22no—
202C2×D4—22no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—22no—
212C22×C4—21no—
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—22no—
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—42no—
232C22×C4—42no—
233C22×C4—42no—
234C22×C4—44no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4#9511yescharacteristic
239C22×C4#9511yescharacteristic
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4C2×C4211yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C4⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250M16—44no—
251M16—44no—
252M16—44no—
253M16—44no—
254C2×C8—24no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—44no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—48no—
268C22⋊C4—28no—
269C22⋊C4—41no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—44no—
276C42—21no—
277C42—21no—
278C42—21no—
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—48no—
295C42—84no—
296C42—84no—
297C42#9911yescharacteristic
298C42#9911yescharacteristic
299C42#9911yescharacteristic
300C42#9911yescharacteristic
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23C2×#9511yescharacteristic
316C23#42112yes—
317Q8—44no—
318D4—44no—
319D4—44no—
320D4—44no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—48no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—84no—
363C2×C4—84no—
364C2×C4#33912yes—
365C2×C4#33912yes—
366C2×C4#34211yescharacteristic
367C2×C4#34211yescharacteristic
368C2×C4#34211yescharacteristic
369C2×C4#34211yescharacteristic
370C2×C4C2×#9911yescharacteristic
371C2×C4C2×#9911yescharacteristic
372C2×C4C2×#10412yes—
373C8—44no—
374C8—82no—
375C8—84no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—42no—
380C22—42no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—81no—
390C22—81no—
391C22—81no—
392C22—81no—
393C22C2×#33911yescharacteristic
394C22#173311yescharacteristic
395C22C2×#34211yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—22no—
400C4—21no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—41no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—41no—
410C4—88no—
411C4#212012yes—
412C2—21no—
413C2—21no—
414C2—42no—
415C2—81no—
416C2—81no—
417C2#805111yescharacteristic
418C2#1257311yescharacteristic
419C2#1258911yescharacteristic
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. #7282C2semidirect
7. C2×#432C4semidirect
9. C2×C4×M16C4semidirect
13. #1870C4semidirect
15. #1822C4semidirect
16. #1801C4semidirect

Maximal subgroups

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

subgroupindex
#213802
#207282
C4×#4102
#72822