Finite Group Databasebeta

← #194087

Subgroups — #194087

Order 512.

1186 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2#34374C211yesmaximal, characteristic
3#21834C211yesmaximal, characteristic
4#21202C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8975C214yesmaximal, complemented
7#8974C214yesmaximal, complemented
8#8038C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230C418yescomplemented
14C2×#500—21no—
15C2×#487—21no—
16C2×#484—22no—
17C2×#477C2214yes—
18C2×#431C2211yescharacteristic
19C2×#431C2211yescharacteristic
20C2×#419—22no—
21C2×C4×M16C2211yescharacteristic
22#2382—22no—
23#2365—21no—
24#2365—21no—
25#2355—21no—
26#2355C2212yes—
27Q8×C4⋊C4—21no—
28Q8×C4⋊C4C2212yes—
29#2323—22no—
30#2323—22no—
31#2314—22no—
32#2285C2211yescharacteristic
33#2285C2211yescharacteristic
34#1990—21no—
35#1990—21no—
36#1987C2214yes—
37#1960—22no—
38#1956C2212yes—
39#1956C2212yes—
40#1824—22no—
41#1485—24no—
42C2×Q8⋊C8C2214yes—
43#1296C2212yes—
44#1296C2212yes—
45#1279C2214yes—
46C23×Q8—21no—
47C22×M16—21no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51#533D414yes—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8C2×C414yes—
59C2×C4⋊Q8D411yescharacteristic
60C2×C4⋊Q8D411yescharacteristic
61C2×C4⋊Q8C2312yes—
62C2×#125—21no—
63C2×C22⋊Q8—21no—
64C2×C22⋊Q8D412yes—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—21no—
68C2×C4×Q8C2312yes—
69C2×#117—21no—
70C2×#117—21no—
71C2×#117D412yes—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C42C2311yescharacteristic
75#500—28no—
76#487—28no—
77#484—216no—
78#477—28no—
79#477C2×C4116yes—
80#431—22no—
81#431—22no—
82#431—22no—
83#431C2×C414yes—
84C2×#107—24no—
85C2×C4⋊C8C2312yes—
86C2×C4⋊C8C2312yes—
87#419—48no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16—22no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8—21no—
98C2×C4×C8C2312yes—
99#391—22no—
100#391—22no—
101#389—22no—
102#389—24no—
103#389—22no—
104#389—24no—
105#388—24no—
106#388—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#384—22no—
112#384D411yescharacteristic
113#384D411yescharacteristic
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#382—22no—
117#382—21no—
118#382D412yes—
119#380—21no—
120#380—21no—
121#380—21no—
122#380D412yes—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4C2312yes—
131#340—24no—
132#340—24no—
133Q8⋊C8—28no—
134C22×Q8—21no—
135C22×Q8—24no—
136C22×Q8—22no—
137C22×Q8—44no—
138C22×Q8—44no—
139C22×Q8—41no—
140C22×Q8—42no—
141C22×Q8C2×D412yes—
142C23×C4—41no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—42no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8C22⋊C414yes—
164C4⋊Q8C22⋊C414yes—
165C4⋊Q8C22×C414yes—
166#125—24no—
167#125—24no—
168C22⋊Q8—24no—
169C22⋊Q8—24no—
170C22⋊Q8—24no—
171C22⋊Q8C22⋊C418yes—
172C4×Q8—24no—
173C4×Q8—24no—
174C4×Q8—22no—
175C4×Q8—22no—
176C4×Q8—84no—
177C4×Q8—84no—
178C4×Q8C22×C414yes—
179C4×Q8C2×D414yes—
180#117—24no—
181#117—24no—
182#117—44no—
183#117—82no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C4⋊C4C2×D412yes—
211C2×C4⋊C4C2×D412yes—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—24no—
219C2×C42—24no—
220C2×C42C22×C412yes—
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4112yes—
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—216no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8C22×C414yes—
233#103—216no—
234#103—48no—
235#103—48no—
236C22⋊C8—24no—
237C22⋊C8—42no—
238#97—42no—
239C4×C8—24no—
240C4×C8—22no—
241C4×C8—22no—
242C4×C8—42no—
243#95—24no—
244#95—24no—
245#95—24no—
246#95—24no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251C24#12011yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—416no—
260C2×Q8—42no—
261C2×Q8—44no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—88no—
265C2×Q8—82no—
266C2×Q8—82no—
267C2×Q8C2×C22⋊C414yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C4×D412yes—
286C22×C4C4×D412yes—
287C22×C4#12011yescharacteristic
288C22×C4#12011yescharacteristic
289C22×C4#12011yescharacteristic
290C22×C4#12112yes—
291C22×C4#12112yes—
292C22×C4#12312yes—
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296M16—48no—
297C2×C8—24no—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—44no—
326C4⋊C4—44no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C4⋊C4—81no—
330C4⋊C4—81no—
331C4⋊C4—84no—
332C4⋊C4—84no—
333C4⋊C4C2×C22⋊C414yes—
334C22⋊C4—28no—
335C22⋊C4—82no—
336C22⋊C4—82no—
337C42—24no—
338C42—24no—
339C42—44no—
340C42—44no—
341C42—84no—
342C42—84no—
343C42C2×C22⋊C412yes—
344C42C4×D412yes—
345C42C4×D412yes—
346C42#12112yes—
347C42#12112yes—
348C42#12312yes—
349C42C23×C411yescharacteristic
350C42C22×D411yescharacteristic
351C42C2×#4111yescharacteristic
352C42C2×#4111yescharacteristic
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23#38612yes—
360C23C2×#12011yescharacteristic
361Q8—28no—
362Q8—48no—
363Q8—44no—
364Q8—48no—
365Q8—84no—
366Q8—88no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#38612yes—
402C2×C4#38612yes—
403C2×C4#38612yes—
404C2×C4#45312yes—
405C2×C4#45412yes—
406C2×C4C22×C22⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×#12111yescharacteristic
410C2×C4C2×#12111yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4#57712yes—
413C2×C4#57812yes—
414C2×C4#57812yes—
415C2×C4#57812yes—
416C8—44no—
417C8—44no—
418C8—82no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22C2×#38611yescharacteristic
427C22#300911yescharacteristic
428C22#301111yescharacteristic
429C22#301111yescharacteristic
430C22#301111yescharacteristic
431C22C2×#45311yescharacteristic
432C22C2×#45411yescharacteristic
433C4—22no—
434C4—22no—
435C4—21no—
436C4—21no—
437C4—44no—
438C4—44no—
439C4—82no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4—84no—
444C4#188314yes—
445C4#289812yes—
446C4#295012yes—
447C4#304412yes—
448C4#309712yes—
449C2—42no—
450C2#1871711yescharacteristic
451C2#1871911yescharacteristic
452C2#1873011yescharacteristic
453C2#1873211yescharacteristic
454C2#2091311yescharacteristic
455C2#2093511yescharacteristic
456C2C2×#188311yescharacteristic
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. #8975C2semidirect
7. #8974C2semidirect
13. #3230C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
#343742
#218342
#212022
C2×#14852
#89752
#89742
#80382
#78962