Finite Group Databasebeta

← #216443

Subgroups — #216443

Order 512.

973 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
1C4×#540C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#21350C211yesmaximal, characteristic, complemented
4#9867C212yesmaximal, complemented
5#9678C212yesmaximal, complemented
6#9078C212yesmaximal, complemented
7#9076C212yesmaximal, complemented
8#8565C212yesmaximal, complemented
9#8563C212yesmaximal, complemented
10C2×#540C411yescharacteristic
11C2×#431C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2335—22no—
14#2333—22no—
15#2330C411yescharacteristic
16#2328C411yescharacteristic
17#2324C412yes—
18#2319C412yes—
19#2314C411yescharacteristic
20C4×#127C2211yescharacteristic
21C4×#125C2211yescharacteristic
22C4×#123—22no—
23C4×#121—22no—
24C4×#121C2212yes—
25C42×D4C2212yes—
26C4×#117C2211yescharacteristic
27#1940—22no—
28#1924—22no—
29#1917—22no—
30#1851—22no—
31#1842C2211yescharacteristic
32#1831C2211yescharacteristic
33#1797C2212yes—
34#1767C2212yes—
35C4×C4⋊C8C2212yes—
36C4×C4⋊C8C2212yes—
37C2×D4⋊C8C2212yes—
38C2×D4⋊C8C2212yes—
39C2×D4⋊C8C2212yes—
40C2×D4⋊C8C2212yes—
41#1279C2212yes—
42#1279C2212yes—
43#1279C2212yes—
44#1279C2212yes—
45C22×M16—21no—
46#540—28no—
47C2×#127C2×C411yescharacteristic
48C2×#125C2×C411yescharacteristic
49C2×#123—22no—
50C2×#121—22no—
51C2×#121C2×C412yes—
52C2×C4×D4—22no—
53C2×C4×D4—22no—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C412yes—
56C2×C4×D4C2312yes—
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C42—22no—
60C22×C42C2311yescharacteristic
61#431—41no—
62#431—41no—
63C2×C4⋊C8—22no—
64C2×C4⋊C8—22no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—22no—
67C2×C4⋊C8C2312yes—
68C2×C4⋊C8C2312yes—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—22no—
71C4×M16—41no—
72C4×M16—41no—
73C2×#97—21no—
74C2×#97—22no—
75C2×C4×C8C2312yes—
76C2×C4×C8C2312yes—
77#390—22no—
78#390—22no—
79#390C2×C412yes—
80#390C2×C412yes—
81#389—22no—
82#389C2×C412yes—
83#389C2×C412yes—
84#388—22no—
85#388—22no—
86#388—22no—
87#388—22no—
88#388—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#386—22no—
93#386—22no—
94#386—22no—
95#386—22no—
96#385—22no—
97#384C2×C411yescharacteristic
98#382C2×C411yescharacteristic
99#381—22no—
100#380—22no—
101#380C2×C412yes—
102C4×C4⋊C4—22no—
103C4×C4⋊C4C2×C412yes—
104C4×C4⋊C4C2×C412yes—
105C4×C4⋊C4D412yes—
106C4×C4⋊C4D412yes—
107C4×C4⋊C4C2312yes—
108C4×C22⋊C4—22no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—22no—
112C4×C22⋊C4D412yes—
113C4×C22⋊C4D412yes—
114#376C2×C411yescharacteristic
115#376C2×C411yescharacteristic
116C43C2311yescharacteristic
117C43C2311yescharacteristic
118#336—22no—
119#336—22no—
120#336—22no—
121#336—22no—
122#336—22no—
123D4⋊C8—42no—
124D4⋊C8—42no—
125D4⋊C8—42no—
126D4⋊C8—42no—
127C22×D4—22no—
128C22×D4—22no—
129C22×D4C22×C412yes—
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4C22×C411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—44no—
137C2×M16—42no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142#127—24no—
143#125—24no—
144#123—216no—
145#121—216no—
146#121—24no—
147#121—24no—
148C4×D4—24no—
149C4×D4—22no—
150C4×D4—22no—
151C4×D4—22no—
152C4×D4—22no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—82no—
156C4×D4—82no—
157C4×D4—82no—
158C4×D4—82no—
159C4×D4M1618yescomplemented
160#117—24no—
161#117—42no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4C22⋊C412yes—
181C2×C22⋊C4C22⋊C412yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—82no—
190C2×C42—82no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C22×C411yescharacteristic
196C2×C42C2×D411yescharacteristic
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D412yes—
200C2×C42C2×D412yes—
201C2×C42C2×D412yes—
202C2×C42C2411yescharacteristic
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—42no—
208C4⋊C8—44no—
209C4⋊C8—44no—
210C4⋊C8—82no—
211C22⋊C8—42no—
212C22⋊C8—82no—
213#97—42no—
214#97—82no—
215C4×C8—22no—
216C4×C8—22no—
217C4×C8—22no—
218C4×C8—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95C22⋊C412yes—
227#95C22⋊C412yes—
228#95C22⋊C412yes—
229#95C22⋊C412yes—
230C24—22no—
231C24C2×C22⋊C411yescharacteristic
232C2×D4—28no—
233C2×D4—28no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—82no—
240C2×D4—82no—
241C2×D4—82no—
242C2×D4—82no—
243C2×D4C2×M1614yes—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—28no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—82no—
267C22×C4—82no—
268C22×C4—82no—
269C22×C4—82no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C412yes—
275C22×C4C2×C22⋊C411yescharacteristic
276C22×C4C2×C22⋊C412yes—
277C22×C4C23×C411yescharacteristic
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280M16—48no—
281C2×C8—24no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C2×C8—42no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—24no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—42no—
301C4⋊C4—82no—
302C4⋊C4—82no—
303C4⋊C4—82no—
304C4⋊C4C2×M1614yes—
305C22⋊C4—28no—
306C22⋊C4—28no—
307C22⋊C4—24no—
308C22⋊C4—42no—
309C22⋊C4—44no—
310C22⋊C4—42no—
311C22⋊C4—42no—
312C22⋊C4—82no—
313C22⋊C4—82no—
314C22⋊C4—82no—
315C22⋊C4—82no—
316C22⋊C4—82no—
317C42—21no—
318C42—21no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42—22no—
324C42—22no—
325C42—22no—
326C42—22no—
327C42—22no—
328C42—22no—
329C42—22no—
330C42—42no—
331C42—42no—
332C42—42no—
333C42—42no—
334C42—82no—
335C42—82no—
336C42C2×M1614yes—
337C42H(5,𝔽2)11yescharacteristic
338C42H(5,𝔽2)11yescharacteristic
339C422−1+411yescharacteristic
340C422−1+411yescharacteristic
341C23—21no—
342C23—21no—
343C23—24no—
344C23—21no—
345C23—22no—
346C23—22no—
347C23—42no—
348C23—42no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23C22×C22⋊C411yescharacteristic
354D4—24no—
355D4—24no—
356D4—42no—
357D4—42no—
358D4—88no—
359D4—88no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—88no—
396C2×C4—88no—
397C2×C4—82no—
398C2×C4—82no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4#40714yes—
402C2×C4#52011yescharacteristic
403C2×C4#52011yescharacteristic
404C2×C4#53511yescharacteristic
405C2×C4#53611yescharacteristic
406C2×C4#53711yescharacteristic
407C2×C4#53711yescharacteristic
408C2×C4C22×M1612yes—
409C2×C4#56812yes—
410C2×C4#57612yes—
411C2×C4#57712yes—
412C8—44no—
413C8—82no—
414C8—82no—
415C8—82no—
416C22—44no—
417C22—42no—
418C22—44no—
419C22—41no—
420C22—44no—
421C22—42no—
422C22—41no—
423C22—41no—
424C22—82no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22#231111yescharacteristic
429C22C2×#40711yescharacteristic
430C22#287911yescharacteristic
431C22#289712yes—
432C22#290112yes—
433C4—24no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—44no—
438C4—41no—
439C4—42no—
440C4—42no—
441C4—41no—
442C4—41no—
443C4—82no—
444C4—82no—
445C4#298912yes—
446C4#298912yes—
447C4#325712yes—
448C4#325911yescharacteristic
449C4#326011yescharacteristic
450C2—81no—
451C2—84no—
452C2#1745312yes—
453C2#1750512yes—
454C2#2185311yescharacteristic
455C2#2189711yescharacteristic
456C2#2190111yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C22×C22⋊C4
prime radicalC2×#540C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #21350C2semidirect
4. #9867C2semidirect
5. #9678C2semidirect
6. #9078C2semidirect
7. #9076C2semidirect
8. #8565C2semidirect
9. #8563C2semidirect
159. C4×D4M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#219262
#213502
#98672
#96782
#90782
#90762
#85652
#85632