Finite Group Databasebeta

← #235775

Subgroups — #235775

Order 512.

1410 conjugacy classes of subgroups.

The subgroup lattice has 425 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3229C211yesmaximal, characteristic
2C22×#487C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic
4#21259C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic
7#10073C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3229—24no—
13#3229—24no—
14C2×#498—21no—
15C2×#487—24no—
16C2×#487—21no—
17C2×#487—22no—
18C2×#487—24no—
19C2×#487C2214yes—
20C2×#483—22no—
21C2×#475C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C2×#419—22no—
25C22×#103—22no—
26C22×C4×C8C2211yescharacteristic
27#2868—21no—
28#2862—21no—
29#2660—22no—
30#2567C2214yes—
31#2546—22no—
32#2502—22no—
33#2434—22no—
34#2317—22no—
35C4×#117C2211yescharacteristic
36#1981—22no—
37#1981C2214yes—
38#1548—24no—
39C2×#333C2214yes—
40#1493—24no—
41C2×Q8⋊C8C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×Q8D412yes—
45C22×M16—21no—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D412yes—
66C2×C4⋊Q8D414yes—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8D412yes—
73C2×C4×Q8—21no—
74C2×C4×Q8C2312yes—
75C2×#117—21no—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C42C2311yescharacteristic
82#498—28no—
83#487—216no—
84#487—28no—
85#487—28no—
86#487—28no—
87#483—216no—
88#475—28no—
89#475—28no—
90#431—22no—
91#431D414yes—
92C2×#106—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98#419—48no—
99C2×#103—216no—
100C2×#103—24no—
101C2×#103—28no—
102C2×#103—28no—
103C2×C22⋊C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—22no—
106C2×C4×C8—21no—
107C2×C4×C8D414yes—
108C2×C4×C8C2312yes—
109#398—28no—
110#395—28no—
111#391—24no—
112#391—24no—
113#389—28no—
114#385—28no—
115#384—24no—
116#384—24no—
117#382—24no—
118C4×C4⋊C4D412yes—
119C4×C4⋊C4C2312yes—
120C4×C22⋊C4—21no—
121C4×C22⋊C4—21no—
122#376—21no—
123C43—21no—
124#333—28no—
125Q8⋊C8—28no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—24no—
131C22×Q8—22no—
132C22×Q8—48no—
133C22×Q8—41no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8—44no—
137C22×Q8C2×D412yes—
138C22×Q8C2×D414yes—
139C23×C4—22no—
140C23×C4—41no—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C2×M16—42no—
145C2×M16—44no—
146C22×C8—24no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—44no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—24no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—88no—
175C4⋊Q8C2×D414yes—
176#125—42no—
177#125—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—42no—
187C4×Q8—42no—
188#117—24no—
189#117—42no—
190#117—42no—
191#117SD1618yes—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230#106—216no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—42no—
235C4⋊C8C2×D414yes—
236#103—232no—
237#103—216no—
238#103—416no—
239C22⋊C8—24no—
240C4×C8—24no—
241C4×C8—24no—
242C4×C8—22no—
243C4×C8C2×D414yes—
244#95—216no—
245#95—22no—
246#95—22no—
247C24#12011yescharacteristic
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—24no—
253C2×Q8—28no—
254C2×Q8—48no—
255C2×Q8—416no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—44no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—48no—
262C2×Q8—44no—
263C2×Q8—44no—
264C2×Q8—416no—
265C2×Q8—88no—
266C2×Q8—84no—
267C2×Q8—84no—
268C2×Q8—84no—
269C22×C4—24no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—84no—
279C22×C4—84no—
280C22×C4#12012yes—
281C22×C4#12012yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4#12112yes—
286C22×C4#12112yes—
287C22×C4#12112yes—
288C22×C4#12712yes—
289C22×C4C2×SD1614yes—
290C22×C4#13714yes—
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
294M16—84no—
295C2×C8—24no—
296C2×C8—24no—
297C2×C8—44no—
298C2×C8—48no—
299C2×C8—44no—
300C2×C8—42no—
301C2×C8—44no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—44no—
316C4⋊C4—44no—
317C4⋊C4—44no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—42no—
323C4⋊C4—816no—
324C4⋊C4C2×SD1618yes—
325C22⋊C4—28no—
326C22⋊C4—44no—
327C42—28no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42—44no—
332C42#12012yes—
333C42#12012yes—
334C42#12112yes—
335C42#12112yes—
336C42#12112yes—
337C42#12712yes—
338C42C22×D411yescharacteristic
339C42C22×D411yescharacteristic
340C42C22×D411yescharacteristic
341C42C2×#4111yescharacteristic
342C23—22no—
343C23—22no—
344C23#39212yes—
345C23#44814yes—
346C23C2×#12011yescharacteristic
347Q8—48no—
348Q8—44no—
349Q8—48no—
350Q8—44no—
351Q8—84no—
352Q8—82no—
353Q8—82no—
354Q8—84no—
355Q8—88no—
356Q8—88no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—28no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—48no—
373C2×C4—88no—
374C2×C4—84no—
375C2×C4—88no—
376C2×C4#39212yes—
377C2×C4#39212yes—
378C2×C4#39212yes—
379C2×C4#44814yes—
380C2×C4#45514yes—
381C2×C4#46118yes—
382C2×C4#49214yes—
383C2×C4#49714yes—
384C2×C4C2×#12011yescharacteristic
385C2×C4C2×#12011yescharacteristic
386C2×C4C2×#12111yescharacteristic
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12711yescharacteristic
390C2×C4C22×SD1612yes—
391C2×C4C2×#13712yes—
392C2×C4#57814yes—
393C8—44no—
394C8—44no—
395C8—82no—
396C22—22no—
397C22—22no—
398C22—28no—
399C22#201314yes—
400C22C2×#39211yescharacteristic
401C22C2×#44812yes—
402C22#301112yes—
403C22C2×#45512yes—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—48no—
408C4—82no—
409C4—84no—
410C4—82no—
411C4—84no—
412C4#200414yes—
413C4#201314yes—
414C4#301112yes—
415C4C2×#46112yes—
416C4#305912yes—
417C4C2×#49211yescharacteristic
418C4C2×#49711yescharacteristic
419C2—24no—
420C2#1863214yes—
421C2#2338911yescharacteristic
422C2C2×#200411yescharacteristic
423C2C2×#201311yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10073C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32292
C22×#4872
#352812
#212592
C2×#15482
C2×#14932
#100732
#78902