Finite Group Databasebeta

← #192689

Subgroups — #192689

Order 512.

998 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34374C211yesmaximal, characteristic
2#21981C211yesmaximal, characteristic
3#21860C211yesmaximal, characteristic
4#20800C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8888C214yesmaximal, complemented
7#8883C214yesmaximal, complemented
8#8038C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×#428—22no—
14C2×C4×M16C2211yescharacteristic
15#2382—22no—
16#2365—21no—
17#2365—21no—
18#2355—21no—
19#2355C2212yes—
20Q8×C4⋊C4—21no—
21Q8×C4⋊C4C2212yes—
22#2323—22no—
23#2323—22no—
24#2314—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28#1990—21no—
29#1956—21no—
30#1948C2214yes—
31#1872—22no—
32#1853—22no—
33#1847—22no—
34#1845C2212yes—
35#1844—21no—
36#1844C2212yes—
37#1834—21no—
38#1803C2214yes—
39#1485C418yescomplemented
40C2×Q8⋊C8C2214yes—
41#1296C2212yes—
42#1296C2212yes—
43#1279C2214yes—
44C22×M16—21no—
45#533—22no—
46#533—22no—
47#533C2×C414yes—
48#533Q814yes—
49C2×C4⋊Q8D411yescharacteristic
50C2×C4⋊Q8D411yescharacteristic
51C2×#125—21no—
52C2×C22⋊Q8—21no—
53C2×C22⋊Q8D412yes—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8C2312yes—
58C2×#117—21no—
59C2×#117—21no—
60C2×#117D412yes—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C42C2311yescharacteristic
65#431—22no—
66#431—22no—
67#431—22no—
68#431C2×C414yes—
69#428—48no—
70C2×#107—24no—
71C2×#106—24no—
72C2×#106—24no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C4×M16—22no—
79C4×M16C2×C414yes—
80C2×#97—21no—
81C2×C4×C8—21no—
82C2×C4×C8—21no—
83C2×C4×C8C2312yes—
84#391—22no—
85#391—22no—
86#389—22no—
87#389—24no—
88#389—22no—
89#389—24no—
90#388—24no—
91#388—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#387—24no—
96#384—22no—
97#384—21no—
98#384—21no—
99#384—21no—
100#384D411yescharacteristic
101#384D411yescharacteristic
102#384C2312yes—
103#384C2311yescharacteristic
104#384C2311yescharacteristic
105#382—22no—
106#382—21no—
107#382D412yes—
108#380—21no—
109#380—21no—
110#380—21no—
111#380D412yes—
112C4×C4⋊C4—22no—
113C4×C4⋊C4—22no—
114C4×C4⋊C4—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4—21no—
119C4×C4⋊C4C2312yes—
120#340—24no—
121#340—24no—
122Q8⋊C8C2×C4116yescomplemented
123C22×Q8—21no—
124C22×Q8C2×D412yes—
125C23×C4—41no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—22no—
136C4⋊Q8—22no—
137C4⋊Q8C4⋊C414yes—
138C4⋊Q8C4⋊C414yes—
139#125—24no—
140#125—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8C4⋊C418yes—
145C4×Q8—24no—
146C4×Q8—24no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—84no—
150C4×Q8—84no—
151C4×Q8C22×C414yes—
152C4×Q8C2×Q814yes—
153#117—24no—
154#117—24no—
155#117—44no—
156#117—82no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4—41no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×D412yes—
184C2×C4⋊C4C2×D412yes—
185C2×C4⋊C4C2×D412yes—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42C22×C412yes—
195C2×C42C2×D411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D411yescharacteristic
198C2×C42C2×Q812yes—
199C2×C42#4112yes—
200C2×C42#4112yes—
201C2×C42C2411yescharacteristic
202#107—48no—
203#106—48no—
204#106—84no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—22no—
208C4⋊C8C22×C414yes—
209#103—84no—
210C22⋊C8—24no—
211C22⋊C8—42no—
212#97—24no—
213C4×C8—24no—
214C4×C8—22no—
215C4×C8—42no—
216C4×C8C22×C414yes—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225C24#12011yescharacteristic
226C2×Q8—22no—
227C2×Q8—24no—
228C2×Q8—24no—
229C2×Q8—24no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8C2×C4⋊C414yes—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—82no—
247C22×C4—82no—
248C22×C4—82no—
249C22×C4C2×C4⋊C412yes—
250C22×C4C4×D412yes—
251C22×C4C4×D412yes—
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4C22⋊Q812yes—
256C22×C4C22⋊Q812yes—
257C22×C4#12312yes—
258C22×C4C22×D411yescharacteristic
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D411yescharacteristic
261M16—216no—
262C2×C8—24no—
263C2×C8—24no—
264C2×C8—24no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8—42no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—24no—
277C4⋊C4—28no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—42no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—44no—
290C4⋊C4—44no—
291C4⋊C4—81no—
292C4⋊C4—81no—
293C4⋊C4—82no—
294C4⋊C4—82no—
295C4⋊C4—84no—
296C4⋊C4—84no—
297C4⋊C4—82no—
298C4⋊C4C2×C4⋊C414yes—
299C22⋊C4—28no—
300C22⋊C4—82no—
301C22⋊C4—82no—
302C42—24no—
303C42—24no—
304C42—44no—
305C42—44no—
306C42—84no—
307C42—84no—
308C42C2×C4⋊C412yes—
309C42C4×D412yes—
310C42C4×D412yes—
311C42C22⋊Q812yes—
312C42C22⋊Q812yes—
313C42#12312yes—
314C42C23×C411yescharacteristic
315C42C22×Q811yescharacteristic
316C42C2×#4111yescharacteristic
317C42C2×#4111yescharacteristic
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23#38512yes—
325C23C2×#12011yescharacteristic
326Q8—28no—
327Q8—44no—
328Q8—48no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—24no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—42no—
344C2×C4—42no—
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—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—164no—
363C2×C4#38512yes—
364C2×C4#38512yes—
365C2×C4#38512yes—
366C2×C4#45312yes—
367C2×C4#45612yes—
368C2×C4C22×C4⋊C411yescharacteristic
369C2×C4C2×C4×D411yescharacteristic
370C2×C4C2×C4×D411yescharacteristic
371C2×C4C2×C22⋊Q811yescharacteristic
372C2×C4C2×C22⋊Q811yescharacteristic
373C2×C4C2×#12311yescharacteristic
374C2×C4#57712yes—
375C2×C4#57712yes—
376C2×C4#57712yes—
377C2×C4#57812yes—
378C8—28no—
379C8—44no—
380C8—82no—
381C22—22no—
382C22—22no—
383C22—22no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22C2×#38511yescharacteristic
389C22#300911yescharacteristic
390C22#301011yescharacteristic
391C22#301011yescharacteristic
392C22#301111yescharacteristic
393C22C2×#45311yescharacteristic
394C22C2×#45611yescharacteristic
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—44no—
400C4—44no—
401C4—82no—
402C4—82no—
403C4—82no—
404C4—82no—
405C4—162no—
406C4#185914yes—
407C4#291212yes—
408C4#294712yes—
409C4#308212yes—
410C4#309912yes—
411C2—42no—
412C2#1871711yescharacteristic
413C2#1871811yescharacteristic
414C2#1873511yescharacteristic
415C2#1873711yescharacteristic
416C2#2084811yescharacteristic
417C2C2×#185911yescharacteristic
418C2#2087711yescharacteristic
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. #8888C2semidirect
7. #8883C2semidirect
39. #1485C4semidirect
122. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343742
#219812
#218602
#208002
C2×#14852
#88882
#88832
#80382
#78962