Finite Group Databasebeta

← #127396

Subgroups — #127396

Order 512.

668 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21380C211yesmaximal, characteristic
2#20723C211yesmaximal, characteristic
3#20632C211yesmaximal, characteristic
4#7237C214yesmaximal, complemented
5C2×C4×#41C2211yescharacteristic
6C2×#432C411yescharacteristic, complemented
7C2×#410—21no—
8C2×C4×M16C411yescharacteristic, complemented
9#2285—21no—
10#2283—21no—
11#1984C412yescomplemented
12#1954C2212yes—
13#1870—21no—
14#1870—21no—
15#1820C412yescomplemented
16#1799—21no—
17#1799—21no—
18#1799—21no—
19#1799C412yescomplemented
20#1799C2212yes—
21#1799C2212yes—
22#1782—21no—
23C4×#99—21no—
24C4×#99C412yescomplemented
25#1749—21no—
26#1749—21no—
27#1740C412yescomplemented
28C22×#41D411yescharacteristic
29C22×M16—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⋊C8C2×C412yes—
52#410—44no—
53C2×#99—22no—
54C2×#99—22no—
55C2×#99—22no—
56C2×#99—22no—
57C2×C22⋊C8—21no—
58C2×C22⋊C8—22no—
59C2×C22⋊C8—22no—
60C4×M16—41no—
61C4×M16—41no—
62C2×#97—21no—
63C2×#97—21no—
64C2×C4×C8C2×C412yes—
65#384—41no—
66#383—41no—
67#382—41no—
68#382—41no—
69#381—22no—
70#380—22no—
71#380—22no—
72#380—22no—
73C4×C4⋊C4—41no—
74C4×C4⋊C4—41no—
75C4×C4⋊C4—41no—
76C4×C22⋊C4—22no—
77#376—41no—
78#342—22no—
79#342—22no—
80#342C2×C414yes—
81#342C2×C414yes—
82#336—22no—
83#336—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×D4C2×D411yescharacteristic
93C23×C4C4211yescharacteristic
94C23×C4C22⋊C411yescharacteristic
95C23×C4C22⋊C411yescharacteristic
96C23×C4C22⋊C411yescharacteristic
97C23×C4C22⋊C411yescharacteristic
98C23×C4C22⋊C411yescharacteristic
99C23×C4C4⋊C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
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—41no—
123#117—41no—
124#117—41no—
125#117—41no—
126#117—41no—
127#117—41no—
128#117—44no—
129#117—44no—
130C2×C4⋊C4—42no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4—42no—
133C2×C4⋊C4—42no—
134C2×C4⋊C4C4211yescharacteristic
135C2×C4⋊C4C4211yescharacteristic
136C2×C4⋊C4C4⋊C411yescharacteristic
137C2×C4⋊C4C4⋊C411yescharacteristic
138C2×C4⋊C4C4⋊C411yescharacteristic
139C2×C4⋊C4C2×Q811yescharacteristic
140C2×C22⋊C4—24no—
141C2×C22⋊C4—24no—
142C2×C22⋊C4C22⋊C411yescharacteristic
143C2×C22⋊C4C22⋊C411yescharacteristic
144C2×C22⋊C4C4⋊C411yescharacteristic
145C2×C22⋊C4C4⋊C411yescharacteristic
146C2×C22⋊C4C22×C411yescharacteristic
147C2×C22⋊C4C22×C411yescharacteristic
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—21no—
158C2×C42—21no—
159C2×C42—21no—
160C2×C42—42no—
161C2×C42—42no—
162C2×C42C4211yescharacteristic
163C2×C42C22⋊C411yescharacteristic
164C2×C42C22⋊C411yescharacteristic
165C2×C42C22⋊C411yescharacteristic
166C2×C42C4⋊C411yescharacteristic
167C2×C42C4⋊C411yescharacteristic
168C2×C42C4⋊C411yescharacteristic
169C2×C42C4⋊C411yescharacteristic
170C2×C42C4⋊C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C4⋊C8—42no—
173#99—44no—
174#99—44no—
175#99—42no—
176#99—44no—
177#99—42no—
178C22⋊C8—44no—
179C22⋊C8—44no—
180C22⋊C8—42no—
181#97—41no—
182#97—41no—
183#97—41no—
184#97—41no—
185C4×C8—42no—
186#95—42no—
187#95—42no—
188#95—42no—
189#95—42no—
190#95—42no—
191#95—42no—
192C24#9511yescharacteristic
193C24C2×C22⋊C411yescharacteristic
194C24C2×C22⋊C411yescharacteristic
195#41—44no—
196#41—44no—
197#41—44no—
198#41—44no—
199C2×Q8—22no—
200C2×Q8—22no—
201C2×Q8—41no—
202C2×D4—22no—
203C2×D4—21no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—41no—
209C2×D4—41no—
210C2×D4—41no—
211C2×D4AGL(1,ℤ/8)12yes—
212C2×D4#13712yes—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
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—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—81no—
244C22×C4—81no—
245C22×C4—81no—
246C22×C4#9511yescharacteristic
247C22×C4#9511yescharacteristic
248C22×C4#9511yescharacteristic
249C22×C4#9511yescharacteristic
250C22×C4#9511yescharacteristic
251C22×C4#9511yescharacteristic
252C22×C4#9511yescharacteristic
253C22×C4C2×C4211yescharacteristic
254C22×C4C2×C22⋊C411yescharacteristic
255C22×C4C2×C4⋊C411yescharacteristic
256C22×C4C2×C4⋊C411yescharacteristic
257C22×C4C2×C4⋊C411yescharacteristic
258M16—48no—
259C2×C8—24no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—84no—
272C4⋊C4—84no—
273C4⋊C4—84no—
274C4⋊C4—84no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—44no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
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—21no—
298C42—21no—
299C42—21no—
300C42—21no—
301C42—21no—
302C42—21no—
303C42—84no—
304C42—84no—
305C42#9911yescharacteristic
306C42#9911yescharacteristic
307C42#9911yescharacteristic
308C42#9911yescharacteristic
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×#9511yescharacteristic
323C23#41712yes—
324C23#41811yescharacteristic
325C23#41911yescharacteristic
326Q8—44no—
327D4—42no—
328D4—42no—
329D4—44no—
330D4—42no—
331D4—42no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
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—81no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—84no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4H(3,ℤ/4)12yes—
382C2×C4#34211yescharacteristic
383C2×C4#34211yescharacteristic
384C2×C4#34211yescharacteristic
385C2×C4#34211yescharacteristic
386C2×C4C2×#9911yescharacteristic
387C2×C4C2×#9911yescharacteristic
388C2×C4#42812yes—
389C8—44no—
390C8—82no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—22no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22C2×H(3,ℤ/4)11yescharacteristic
411C22#173611yescharacteristic
412C22C2×#34211yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—41no—
427C4—164no—
428C4—164no—
429C4#212612yes—
430C2—21no—
431C2—21no—
432C2—41no—
433C2—42no—
434C2—41no—
435C2—81no—
436C2#798311yescharacteristic
437C2#1257711yescharacteristic
438C2#1258111yescharacteristic
subgroupG/H
center, socleC22C2×#342
commutatorC22×C4C2×C42
Frattini, cosocleC2×C4×D4C23
prime radicalC22×#41D4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ 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. #7237C2semidirect
6. C2×#432C4semidirect
8. C2×C4×M16C4semidirect
11. #1984C4semidirect
15. #1820C4semidirect
19. #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
#213802
#207232
#206322
#72372