Finite Group Databasebeta

← #195654

Subgroups — #195654

Order 512.

1306 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
1C2×#3165C211yesmaximal, characteristic
2#34209C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21389C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7903C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12C2×#519—21no—
13#3165—24no—
14#3165C418yescomplemented
15C2×#501—22no—
16C2×#500C2212yes—
17C2×#498C2212yes—
18C2×#431C2211yescharacteristic
19C2×#428—22no—
20C2×#423C2211yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2365—22no—
23#2365C2214yes—
24#2344—22no—
25#2344—22no—
26#2327—22no—
27#2310—21no—
28C4×C4⋊Q8—22no—
29#2285C2211yescharacteristic
30C4×#117C2211yescharacteristic
31#1988C2212yes—
32#1988C2212yes—
33#1981C2212yes—
34#1981C2212yes—
35#1894—22no—
36#1873—21no—
37#1873—21no—
38#1803—22no—
39C4×#103—22no—
40#1549—24no—
41C2×#332C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8—21no—
45C22×M16—21no—
46C22×M16—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8D412yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64#519—48no—
65C2×C4×Q8—24no—
66C2×C4×Q8—24no—
67C2×C4×Q8—21no—
68C2×C4×Q8—21no—
69C2×#117—24no—
70C2×#117—21no—
71C2×#117—21no—
72C2×#117D411yescharacteristic
73C2×#117D411yescharacteristic
74C22×C4⋊C4—21no—
75C22×C4⋊C4D412yes—
76C22×C42C2311yescharacteristic
77#501—216no—
78#500—24no—
79#500C2×C418yes—
80#498—24no—
81#498C2×C418yes—
82#431—22no—
83#431—22no—
84#428—216no—
85C2×#107—24no—
86C2×#106—24no—
87#423—22no—
88#423—22no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2312yes—
91C2×C4⋊C8C2312yes—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×C22⋊C8—21no—
96C4×M16—22no—
97C4×M16C2×C414yes—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8C2312yes—
101#391—24no—
102#391—24no—
103#389—24no—
104#389—24no—
105#387—28no—
106#387—28no—
107#384D412yes—
108#382—21no—
109#381—24no—
110#380—21no—
111#380D412yes—
112C4×C4⋊C4—24no—
113C4×C4⋊C4—24no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4D412yes—
116C4×C4⋊C4C2312yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C43—21no—
122#340—24no—
123#340—22no—
124#340—22no—
125#332—28no—
126C22×Q8—24no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—44no—
133C22×Q8—44no—
134C23×C4—22no—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C2×M16—28no—
139C2×M16—24no—
140C2×M16—42no—
141C2×M16—82no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—24no—
150C4⋊Q8—21no—
151C4⋊Q8—21no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8C22⋊C418yes—
159C4⋊Q8C22×C412yes—
160C4⋊Q8C22×C412yes—
161C4⋊Q8C22×C414yes—
162C4⋊Q8C2×D414yes—
163#125—24no—
164#125—42no—
165C22⋊Q8—24no—
166C22⋊Q8—24no—
167C22⋊Q8—42no—
168C22⋊Q8—42no—
169C4×Q8—24no—
170C4×Q8—24no—
171C4×Q8—416no—
172C4×Q8—48no—
173C4×Q8—48no—
174C4×Q8—42no—
175C4×Q8—42no—
176#117—24no—
177#117—24no—
178#117—416no—
179#117—42no—
180#117—44no—
181#117—42no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4C22⋊C418yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D414yes—
200C2×C4⋊C4C2×D414yes—
201C2×C4⋊C4C2×D412yes—
202C2×C22⋊C4—21no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C42—21no—
207C2×C42—28no—
208C2×C42—21no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—41no—
213C2×C42C22×C412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4112yes—
219C2×C42#4112yes—
220C2×C42C2411yescharacteristic
221#107—216no—
222#106—216no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8—42no—
228#103—44no—
229#103—44no—
230#103—84no—
231C22⋊C8—42no—
232#97—24no—
233C4×C8—22no—
234C4×C8—42no—
235C4×C8C22×C414yes—
236#95—28no—
237#95—28no—
238#95—22no—
239#95—22no—
240#95—22no—
241#95—22no—
242C24#12011yescharacteristic
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—28no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—44no—
249C2×Q8—44no—
250C2×Q8—416no—
251C2×Q8—416no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—28no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—44no—
271C22×C4—44no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4C2×C22⋊C412yes—
275C22×C4C4×D412yes—
276C22×C4C4×D412yes—
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12312yes—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C22×D411yescharacteristic
286M16—216no—
287M16—84no—
288C2×C8—28no—
289C2×C8—24no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—44no—
310C4⋊C4—416no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4C2×C22⋊C418yes—
319C4⋊C4AGL(1,ℤ/8)14yes—
320C22⋊C4—24no—
321C22⋊C4—42no—
322C22⋊C4—44no—
323C22⋊C4—44no—
324C42—24no—
325C42—42no—
326C42—416no—
327C42—44no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42—44no—
332C42C2×C22⋊C412yes—
333C42C4×D412yes—
334C42C4×D412yes—
335C42#12112yes—
336C42#12112yes—
337C42#12312yes—
338C42C23×C411yescharacteristic
339C42C22×D411yescharacteristic
340C42C2×#4111yescharacteristic
341C42C2×#4111yescharacteristic
342C23—21no—
343C23—21no—
344C23—22no—
345C23—22no—
346C23#38612yes—
347C23C2×#12011yescharacteristic
348Q8—416no—
349Q8—48no—
350Q8—48no—
351Q8—44no—
352Q8—44no—
353Q8—82no—
354Q8—81no—
355Q8—81no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—28no—
369C2×C4—416no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—82no—
385C2×C4#38612yes—
386C2×C4#38612yes—
387C2×C4#38612yes—
388C2×C4#45614yes—
389C2×C4C22×C22⋊C411yescharacteristic
390C2×C4C2×C4×D411yescharacteristic
391C2×C4C2×C4×D411yescharacteristic
392C2×C4C2×#12111yescharacteristic
393C2×C4C2×#12111yescharacteristic
394C2×C4C2×#12311yescharacteristic
395C2×C4C2×AGL(1,ℤ/8)12yes—
396C2×C4#57512yes—
397C2×C4#57712yes—
398C2×C4#57812yes—
399C8—28no—
400C8—82no—
401C8—82no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—42no—
406C22—42no—
407C22C2×#38611yescharacteristic
408C22#300311yescharacteristic
409C22#300511yescharacteristic
410C22#301312yes—
411C22C2×#45612yes—
412C4—22no—
413C4—22no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—48no—
418C4—84no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#188614yes—
423C4C4×AGL(1,ℤ/8)12yes—
424C4#294712yes—
425C4#303612yes—
426C4#304311yescharacteristic
427C4#304411yescharacteristic
428C2—42no—
429C2#1859012yes—
430C2#1859212yes—
431C2#2092511yescharacteristic
432C2#2093011yescharacteristic
433C2C2×#188611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9059C2semidirect
7. #9058C2semidirect
14. #3165C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342092
#218942
#213892
C2×#15492
#90592
#90582
#79032
#78962