Finite Group Databasebeta

← #127596

Subgroups — #127596

Order 512.

694 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20629C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#7252C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#423C411yescharacteristic, complemented
8C22×C4⋊C8C411yescharacteristic, complemented
9C2×#410—21no—
10C2×#408—21no—
11C4×#117—21no—
12#1798C412yescomplemented
13#1798C2212yes—
14#1797—21no—
15#1797—21no—
16C4×#99—21no—
17C4×#99—21no—
18C4×#99C412yes—
19C4×#99C412yes—
20#1740C2212yes—
21#1729—21no—
22#1467C412yescomplemented
23#1466—21no—
24#1465—21no—
25#1465C412yescomplemented
26#1458—21no—
27#1458C2212yes—
28C22×#41D411yescharacteristic
29C22×M16—21no—
30C23×C8—21no—
31C4×#41—41no—
32C4×#41—41no—
33C4×#41—41no—
34C4×#41—41no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8Q811yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4C2×C411yescharacteristic
39C2×C4×D4C2×C411yescharacteristic
40C2×C4×D4D411yescharacteristic
41C2×C4×D4D411yescharacteristic
42C2×C4×D4Q811yescharacteristic
43C2×#117—22no—
44C2×#117C2×C411yescharacteristic
45C2×#117D411yescharacteristic
46C2×#117C2311yescharacteristic
47C22×C42C2×C411yescharacteristic
48C22×C42D411yescharacteristic
49C22×C42D411yescharacteristic
50#432—22no—
51#432—41no—
52#431—22no—
53#431—41no—
54#423—41no—
55#423—41no—
56#423—41no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8—21no—
59C2×C4⋊C8—41no—
60C2×C4⋊C8—41no—
61C2×C4⋊C8C2×C412yes—
62C2×C4⋊C8C2×C412yes—
63#410—24no—
64#410—24no—
65C2×#99—22no—
66C2×#99—22no—
67#408—44no—
68#407—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C4×M16—41no—
72C4×C4⋊C4—41no—
73C4×C4⋊C4—41no—
74C4×C22⋊C4—22no—
75C4×C22⋊C4—22no—
76#376—41no—
77C43—41no—
78#342—22no—
79#342C2×C414yes—
80#336—22no—
81#336—22no—
82#324—22no—
83#324C2×C414yes—
84#323—22no—
85#323—22no—
86C2×#41—22no—
87C2×#41—22no—
88C2×#41—22no—
89C2×#41—22no—
90C2×#41—41no—
91C2×#41—41no—
92C22×Q8C4⋊C411yescharacteristic
93C22×D4C22⋊C411yescharacteristic
94C22×D4C22⋊C411yescharacteristic
95C22×D4C4⋊C411yescharacteristic
96C23×C4C4211yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C22⋊C411yescharacteristic
100C23×C4C22⋊C411yescharacteristic
101C23×C4C22⋊C411yescharacteristic
102C23×C4C4⋊C411yescharacteristic
103C23×C4C4⋊C411yescharacteristic
104C23×C4C2×D411yescharacteristic
105C2×M16—42no—
106C2×M16—42no—
107C2×M16—42no—
108C2×M16—42no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8—42no—
114C4×Q8—41no—
115C4×Q8—41no—
116C4×Q8—41no—
117C4×Q8—41no—
118C4×D4—41no—
119C4×D4—41no—
120C4×D4—41no—
121C4×D4—41no—
122C4×D4—41no—
123C4×D4—41no—
124C4×D4—41no—
125C4×D4—41no—
126C4×D4—41no—
127C4×D4—41no—
128C4×D4—41no—
129C4×D4—41no—
130#117—24no—
131#117—24no—
132#117—41no—
133#117—41no—
134#117—41no—
135#117—41no—
136#117—41no—
137#117—41no—
138#117—42no—
139#117—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4C4211yescharacteristic
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C4⋊C411yescharacteristic
144C2×C4⋊C4C4⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C2×Q811yescharacteristic
147C2×C22⋊C4—24no—
148C2×C22⋊C4C4211yescharacteristic
149C2×C22⋊C4C22⋊C411yescharacteristic
150C2×C22⋊C4C4⋊C411yescharacteristic
151C2×C22⋊C4C4⋊C411yescharacteristic
152C2×C22⋊C4C22×C411yescharacteristic
153C2×C22⋊C4C2×D411yescharacteristic
154C2×C42—21no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—21no—
161C2×C42—21no—
162C2×C42—21no—
163C2×C42—21no—
164C2×C42—21no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42—42no—
168C2×C42—42no—
169C2×C42C4211yescharacteristic
170C2×C42C22⋊C411yescharacteristic
171C2×C42C22⋊C411yescharacteristic
172C2×C42C22⋊C411yescharacteristic
173C2×C42C4⋊C411yescharacteristic
174C2×C42C4⋊C411yescharacteristic
175C2×C42C4⋊C411yescharacteristic
176C2×C42C4⋊C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C4⋊C8—42no—
180C4⋊C8—42no—
181C4⋊C8—42no—
182C4⋊C8—42no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#99—28no—
188#99—28no—
189C22⋊C8—24no—
190C22⋊C8—44no—
191C22⋊C8—42no—
192C22⋊C8—44no—
193#97—42no—
194C4×C8—42no—
195#95—42no—
196#95—42no—
197C24#9511yescharacteristic
198C24#9511yescharacteristic
199C24C2×C22⋊C411yescharacteristic
200#41—44no—
201#41—44no—
202#41—44no—
203#41—44no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—41no—
207C2×D4—22no—
208C2×D4—22no—
209C2×D4—22no—
210C2×D4—22no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—41no—
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—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—44no—
246C22×C4—81no—
247C22×C4—81no—
248C22×C4—81no—
249C22×C4#9511yescharacteristic
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4#9511yescharacteristic
254C22×C4#9511yescharacteristic
255C22×C4#10412yes—
256C22×C4#10412yes—
257C22×C4C2×C4211yescharacteristic
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C4⋊C411yescharacteristic
261C22×C4C2×C4⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4AGL(1,ℤ/8)12yes—
264C22×C4#13712yes—
265M16—84no—
266M16—84no—
267C2×C8—44no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C2×C8—42no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—84no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C22⋊C4—28no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—41no—
287C22⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—44no—
291C42—21no—
292C42—21no—
293C42—21no—
294C42—21no—
295C42—21no—
296C42—21no—
297C42—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—44no—
310C42—44no—
311C42—84no—
312C42—84no—
313C42#9911yescharacteristic
314C42#9911yescharacteristic
315C42#9911yescharacteristic
316C42#9911yescharacteristic
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#33912yes—
331C23C2×#9511yescharacteristic
332C23#41712yes—
333Q8—44no—
334D4—44no—
335D4—44no—
336D4—44no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—41no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—48no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—84no—
384C2×C4—84no—
385C2×C4#33912yes—
386C2×C4#34211yescharacteristic
387C2×C4#34211yescharacteristic
388C2×C4#34211yescharacteristic
389C2×C4#34211yescharacteristic
390C2×C4C2×#9911yescharacteristic
391C2×C4C2×#9911yescharacteristic
392C2×C4#41811yescharacteristic
393C2×C4#41911yescharacteristic
394C2×C4C2×#10411yescharacteristic
395C2×C4C2×#10411yescharacteristic
396C2×C4#42812yes—
397C8—84no—
398C8—82no—
399C8—82no—
400C22—21no—
401C22—21no—
402C22—22no—
403C22—22no—
404C22—42no—
405C22—42no—
406C22—42no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—81no—
416C22—81no—
417C22C2×#33911yescharacteristic
418C22#173611yescharacteristic
419C22C2×#34211yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—22no—
424C4—22no—
425C4—41no—
426C4—41no—
427C4—41no—
428C4—41no—
429C4—41no—
430C4—41no—
431C4—41no—
432C4—88no—
433C4#139112yes—
434C4#150711yescharacteristic
435C4#150711yescharacteristic
436C4#150811yescharacteristic
437C4#150811yescharacteristic
438C2—21no—
439C2—21no—
440C2—42no—
441C2—42no—
442C2—81no—
443C2#801911yescharacteristic
444C2C2×#139111yescharacteristic
445C2#1258911yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C2×C42
Frattini, cosocleC2×#117C23
socleC22C2×#342
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #7252C2semidirect
7. C2×#423C4semidirect
8. C22×C4⋊C8C4semidirect
12. #1798C4semidirect
22. #1467C4semidirect
25. #1465C4semidirect

Maximal subgroups

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

subgroupindex
#206292
C4×#4102
#173612
#72522