Finite Group Databasebeta

← #127582

Subgroups — #127582

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20719C211yesmaximal, characteristic
2#20632C211yesmaximal, characteristic
3#17361C211yesmaximal, characteristic
4#7252C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6#2920—21no—
7C2×#432C411yescharacteristic, complemented
8C2×#432C411yescharacteristic, complemented
9C2×#410—21no—
10#2285—21no—
11#1818—21no—
12#1799—21no—
13#1799—21no—
14#1799C412yes—
15#1799C412yes—
16#1798C412yescomplemented
17#1798C2212yes—
18#1780—21no—
19#1749—21no—
20#1749—21no—
21#1740C2212yes—
22#1467C412yescomplemented
23#1466—21no—
24#1465—21no—
25#1465C412yescomplemented
26#1458—21no—
27#1458C2212yes—
28C22×#41D411yescharacteristic
29C4×#41—41no—
30C4×#41—41no—
31C4×#41—41no—
32C4×#41—41no—
33C2×C4×Q8C2×C411yescharacteristic
34C2×C4×Q8Q811yescharacteristic
35C2×C4×D4C2×C411yescharacteristic
36C2×C4×D4C2×C411yescharacteristic
37C2×C4×D4C2×C411yescharacteristic
38C2×C4×D4D411yescharacteristic
39C2×C4×D4D411yescharacteristic
40C2×C4×D4Q811yescharacteristic
41C2×#117—22no—
42C2×#117C2×C411yescharacteristic
43C2×#117D411yescharacteristic
44C2×#117C2311yescharacteristic
45C22×C42C2×C411yescharacteristic
46C22×C42D411yescharacteristic
47C22×C42D411yescharacteristic
48#432—22no—
49#432—41no—
50#432—41no—
51#432—41no—
52#432—41no—
53#432—41no—
54#431—22no—
55#431—41no—
56#423—41no—
57C2×C4⋊C8C2×C412yes—
58C2×C4⋊C8C2×C412yes—
59#410—44no—
60C2×#99—22no—
61C2×#99—22no—
62#407—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—22no—
67C4×M16—41no—
68C2×#97—21no—
69C2×#97—21no—
70#384—41no—
71#382—41no—
72#380—22no—
73#380—22no—
74C4×C4⋊C4—41no—
75C4×C4⋊C4—41no—
76#342—22no—
77#342C2×C414yes—
78#336—22no—
79#336—22no—
80#324—22no—
81#324C2×C414yes—
82#323—22no—
83#323—22no—
84C2×#41—22no—
85C2×#41—22no—
86C2×#41—41no—
87C2×#41—41no—
88C2×#41—42no—
89C22×Q8C4⋊C411yescharacteristic
90C22×D4C22⋊C411yescharacteristic
91C22×D4C22⋊C411yescharacteristic
92C22×D4C4⋊C411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C4⋊C411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C2×M16—42no—
103C22×C8—22no—
104C22×C8—22no—
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—41no—
122#117—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—44no—
128#117—44no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C4211yescharacteristic
133C2×C4⋊C4C22⋊C411yescharacteristic
134C2×C4⋊C4C4⋊C411yescharacteristic
135C2×C4⋊C4C4⋊C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C2×Q811yescharacteristic
138C2×C22⋊C4—24no—
139C2×C22⋊C4C4211yescharacteristic
140C2×C22⋊C4C22⋊C411yescharacteristic
141C2×C22⋊C4C4⋊C411yescharacteristic
142C2×C22⋊C4C4⋊C411yescharacteristic
143C2×C22⋊C4C22×C411yescharacteristic
144C2×C22⋊C4C2×D411yescharacteristic
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×C42C4211yescharacteristic
159C2×C42C22⋊C411yescharacteristic
160C2×C42C22⋊C411yescharacteristic
161C2×C42C22⋊C411yescharacteristic
162C2×C42C4⋊C411yescharacteristic
163C2×C42C4⋊C411yescharacteristic
164C2×C42C4⋊C411yescharacteristic
165C2×C42C4⋊C411yescharacteristic
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C4⋊C8—42no—
169C4⋊C8—42no—
170C4⋊C8—42no—
171C4⋊C8—42no—
172#99—44no—
173#99—44no—
174C22⋊C8—24no—
175C22⋊C8—44no—
176C22⋊C8—42no—
177C22⋊C8—42no—
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—
187#95—42no—
188#95—42no—
189C24#9511yescharacteristic
190C24#9511yescharacteristic
191C24C2×C22⋊C411yescharacteristic
192#41—44no—
193#41—44no—
194#41—44no—
195#41—44no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—21no—
202C2×D4—21no—
203C2×D4—22no—
204C2×D4—41no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C22×C4—21no—
211C22×C4—21no—
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—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
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—44no—
234C22×C4—81no—
235C22×C4—81no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4#9511yescharacteristic
241C22×C4#9511yescharacteristic
242C22×C4#9511yescharacteristic
243C22×C4#9511yescharacteristic
244C22×C4#9511yescharacteristic
245C22×C4#9511yescharacteristic
246C22×C4C2×C4211yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C2×C4⋊C411yescharacteristic
251C22×C4C2×C4⋊C411yescharacteristic
252C22×C4AGL(1,ℤ/8)12yes—
253C22×C4#13712yes—
254M16—84no—
255C2×C8—44no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—84no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C22⋊C4—41no—
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—84no—
298C42#9911yescharacteristic
299C42#9911yescharacteristic
300C42#9911yescharacteristic
301C42#9911yescharacteristic
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—41no—
316C23C2×#9511yescharacteristic
317C23#41811yescharacteristic
318C23#41911yescharacteristic
319Q8—44no—
320D4—44no—
321D4—42no—
322D4—42no—
323D4—42no—
324D4—42no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—84no—
370C2×C4#34211yescharacteristic
371C2×C4#34211yescharacteristic
372C2×C4#34211yescharacteristic
373C2×C4#34211yescharacteristic
374C2×C4C2×#9911yescharacteristic
375C2×C4C2×#9911yescharacteristic
376C2×C4#41712yes—
377C2×C4#42112yes—
378C2×C4#42812yes—
379C8—84no—
380C8—82no—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22#173311yescharacteristic
402C22#173611yescharacteristic
403C22C2×#34211yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—22no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—41no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—41no—
417C4—164no—
418C4#151312yes—
419C4#151312yes—
420C2—21no—
421C2—21no—
422C2—41no—
423C2—41no—
424C2—81no—
425C2—81no—
426C2#801611yescharacteristic
427C2#1257711yescharacteristic
428C2#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×#432C4semidirect
8. C2×#432C4semidirect
16. #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
#207192
#206322
#173612
#72522