Finite Group Databasebeta

← #195658

Subgroups — #195658

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 431 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3165C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21389C211yesmaximal, characteristic
5C2×#1546C211yesmaximal, characteristic
6#9059C214yesmaximal, complemented
7#9058C214yesmaximal, complemented
8#7903C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3165—24no—
13#3165C418yescomplemented
14C2×#501—22no—
15C2×#500C2212yes—
16C2×#498C2212yes—
17C2×#431C2211yescharacteristic
18C2×#428—22no—
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21#2369—22no—
22#2365C2214yes—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2274—21no—
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#1546—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—
64C2×C4×Q8—24no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C42C2311yescharacteristic
74#501—216no—
75#500—24no—
76#500C2×C418yes—
77#498—24no—
78#498C2×C418yes—
79#431—22no—
80#431—22no—
81#428—216no—
82C2×#107—24no—
83C2×#106—24no—
84#423—22no—
85#423—22no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16C2×C414yes—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#391—28no—
99#391—24no—
100#389—24no—
101#389—24no—
102#389—24no—
103#387—28no—
104#384D412yes—
105#382—21no—
106#380—24no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—24no—
122#340—22no—
123#340—22no—
124#332—28no—
125C22×Q8—24no—
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8—21no—
131C22×Q8—82no—
132C22×Q8—82no—
133C23×C4—22no—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—28no—
138C2×M16—24no—
139C2×M16—42no—
140C2×M16—82no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—24no—
149C4⋊Q8—21no—
150C4⋊Q8—21no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8—21no—
157C4⋊Q8C22⋊C418yes—
158C4⋊Q8C22×C412yes—
159C4⋊Q8C22×C412yes—
160C4⋊Q8C22×C414yes—
161C4⋊Q8C2×D414yes—
162#125—24no—
163#125—42no—
164C22⋊Q8—24no—
165C22⋊Q8—24no—
166C22⋊Q8—42no—
167C22⋊Q8—42no—
168C4×Q8—24no—
169C4×Q8—24no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8—84no—
173C4×Q8—84no—
174#117—24no—
175#117—24no—
176#117—42no—
177#117—44no—
178#117—42no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C22⋊C418yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D414yes—
197C2×C4⋊C4C2×D414yes—
198C2×C4⋊C4C2×D412yes—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—28no—
209C2×C42—41no—
210C2×C42C22×C412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42C2411yescharacteristic
218#107—216no—
219#106—216no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—42no—
225#103—44no—
226#103—44no—
227#103—84no—
228C22⋊C8—42no—
229#97—24no—
230C4×C8—22no—
231C4×C8—42no—
232C4×C8C22×C414yes—
233#95—28no—
234#95—28no—
235#95—22no—
236#95—22no—
237#95—22no—
238#95—22no—
239C24#12011yescharacteristic
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—28no—
243C2×Q8—22no—
244C2×Q8—22no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—44no—
248C2×Q8—44no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—84no—
254C2×Q8—88no—
255C2×Q8—84no—
256C2×Q8—84no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—28no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—84no—
270C22×C4C2×C22⋊C412yes—
271C22×C4C4×D412yes—
272C22×C4C4×D412yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12011yescharacteristic
276C22×C4#12112yes—
277C22×C4#12112yes—
278C22×C4#12312yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282M16—216no—
283M16—84no—
284C2×C8—28no—
285C2×C8—24no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—44no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—22no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—44no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—88no—
314C4⋊C4C2×C22⋊C418yes—
315C4⋊C4#13714yes—
316C22⋊C4—24no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C42—24no—
321C42—42no—
322C42—44no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42—44no—
327C42—88no—
328C42C2×C22⋊C412yes—
329C42C4×D412yes—
330C42C4×D412yes—
331C42#12112yes—
332C42#12112yes—
333C42#12312yes—
334C42C23×C411yescharacteristic
335C42C22×D411yescharacteristic
336C42C2×#4111yescharacteristic
337C42C2×#4111yescharacteristic
338C23—21no—
339C23—21no—
340C23—22no—
341C23—22no—
342C23#38612yes—
343C23C2×#12011yescharacteristic
344Q8—48no—
345Q8—44no—
346Q8—44no—
347Q8—44no—
348Q8—44no—
349Q8—88no—
350Q8—82no—
351Q8—81no—
352Q8—81no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—42no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—88no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—82no—
381C2×C4#38612yes—
382C2×C4#38612yes—
383C2×C4#38612yes—
384C2×C4#45614yes—
385C2×C4C22×C22⋊C411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12111yescharacteristic
390C2×C4C2×#12311yescharacteristic
391C2×C4C2×#13712yes—
392C2×C4#57512yes—
393C2×C4#57612yes—
394C2×C4#57712yes—
395C8—28no—
396C8—82no—
397C8—82no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—42no—
402C22—42no—
403C22C2×#38611yescharacteristic
404C22#300411yescharacteristic
405C22#300511yescharacteristic
406C22#301212yes—
407C22C2×#45612yes—
408C4—22no—
409C4—22no—
410C4—21no—
411C4—21no—
412C4—44no—
413C4—48no—
414C4—82no—
415C4—84no—
416C4—82no—
417C4—82no—
418C4#188614yes—
419C4C4×#13712yes—
420C4#294612yes—
421C4#303712yes—
422C4#304111yescharacteristic
423C4#304211yescharacteristic
424C2—42no—
425C2#1859112yes—
426C2#1859312yes—
427C2#2092611yescharacteristic
428C2#2092911yescharacteristic
429C2C2×#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
13. #3165C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31652
#342162
#218942
#213892
C2×#15462
#90592
#90582
#79032
#78962