Finite Group Databasebeta

← #216102

Subgroups — #216102

Order 512.

983 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#540C211yesmaximal, characteristic
2#21978C211yesmaximal, characteristic, complemented
3#21430C211yesmaximal, characteristic, complemented
4#9872C211yesmaximal, characteristic, complemented
5#9871C211yesmaximal, characteristic, complemented
6#9642C211yesmaximal, characteristic, complemented
7#9638C211yesmaximal, characteristic, complemented
8#8944C212yesmaximal, complemented
9#8943C212yesmaximal, complemented
10#8565C212yesmaximal, complemented
11#8563C212yesmaximal, complemented
12C2×#540C411yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#423C2211yescharacteristic
15#2335—22no—
16#2333—22no—
17#2330C411yescharacteristic
18#2328C411yescharacteristic
19#2324C412yes—
20#2319C412yes—
21#2314C411yescharacteristic
22C4×#127C2211yescharacteristic
23C4×#125C2211yescharacteristic
24C4×#123—22no—
25C4×#121—22no—
26C4×#121C2212yes—
27C42×D4C2212yes—
28C4×#117C2211yescharacteristic
29#1940—22no—
30#1923—22no—
31#1916—22no—
32#1851—22no—
33#1842C2211yescharacteristic
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1831C2211yescharacteristic
37C4×C4⋊C8C2212yes—
38C4×C22⋊C8C2212yes—
39#1744C2211yescharacteristic
40C42×C8C2211yescharacteristic
41C2×D4⋊C8C2212yes—
42C2×D4⋊C8C2212yes—
43C2×D4⋊C8C2212yes—
44C2×D4⋊C8C2212yes—
45#1279C2212yes—
46#1279C2212yes—
47#1279C2212yes—
48#1279C2212yes—
49C22×M16—21no—
50#540—24no—
51#540—24no—
52C2×#127C2×C411yescharacteristic
53C2×#125C2×C411yescharacteristic
54C2×#123—22no—
55C2×#121—22no—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2312yes—
62C2×#117C2×C411yescharacteristic
63C2×#117C2×C411yescharacteristic
64C22×C42—22no—
65C22×C42C2311yescharacteristic
66#432—42no—
67#423—42no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—22no—
72C2×C4⋊C8C2311yescharacteristic
73C2×C4⋊C8C2311yescharacteristic
74C2×C4⋊C8C2311yescharacteristic
75C2×C4⋊C8C2311yescharacteristic
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78C2×#97—21no—
79C2×C4×C8—22no—
80C2×C4×C8C2312yes—
81C2×C4×C8C2312yes—
82#390—22no—
83#390—22no—
84#390C2×C412yes—
85#390C2×C412yes—
86#389—22no—
87#389C2×C412yes—
88#389C2×C412yes—
89#388—22no—
90#388—22no—
91#388—22no—
92#388—22no—
93#388—22no—
94#387—22no—
95#387—22no—
96#387—22no—
97#386—22no—
98#386—22no—
99#386—22no—
100#386—22no—
101#385—22no—
102#384C2×C411yescharacteristic
103#382C2×C411yescharacteristic
104#381—22no—
105#380—22no—
106#380C2×C412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4C2×C412yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C4⋊C4D412yes—
112C4×C4⋊C4C2312yes—
113C4×C22⋊C4—22no—
114C4×C22⋊C4—22no—
115C4×C22⋊C4—22no—
116C4×C22⋊C4—22no—
117C4×C22⋊C4D412yes—
118C4×C22⋊C4D412yes—
119#376C2×C411yescharacteristic
120#376C2×C411yescharacteristic
121C43C2311yescharacteristic
122C43C2311yescharacteristic
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—22no—
128D4⋊C8—42no—
129D4⋊C8—42no—
130D4⋊C8—42no—
131D4⋊C8—42no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4C22×C412yes—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4C22×C411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C2×M16—42no—
142C2×M16—82no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149#127—22no—
150#127M1614yescomplemented
151#125—22no—
152#125M1614yescomplemented
153#123—216no—
154#121—216no—
155#121—24no—
156#121—24no—
157C4×D4—24no—
158C4×D4—24no—
159C4×D4—24no—
160C4×D4—24no—
161C4×D4—22no—
162C4×D4—22no—
163C4×D4—22no—
164C4×D4—22no—
165C4×D4—82no—
166C4×D4—82no—
167C4×D4—82no—
168C4×D4—82no—
169#117—22no—
170#117—22no—
171#117—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C4⋊C4C22⋊C412yes—
178C2×C4⋊C4C22×C412yes—
179C2×C4⋊C4C22×C412yes—
180C2×C4⋊C4C22×C412yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4—22no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C22⋊C412yes—
191C2×C22⋊C4C22⋊C412yes—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—82no—
198C2×C42—82no—
199C2×C42C22×C411yescharacteristic
200C2×C42C22×C411yescharacteristic
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C411yescharacteristic
203C2×C42C22×C411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D412yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D412yes—
209C2×C42C2×D412yes—
210C2×C42C2411yescharacteristic
211C4⋊C8—22no—
212C4⋊C8—22no—
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—42no—
216C4⋊C8—44no—
217C4⋊C8—82no—
218C4⋊C8—82no—
219C22⋊C8—44no—
220C22⋊C8—42no—
221#97—42no—
222C4×C8—22no—
223C4×C8—22no—
224C4×C8—22no—
225C4×C8—22no—
226C4×C8—44no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238C24—22no—
239C24C2×C22⋊C411yescharacteristic
240C2×D4—28no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—28no—
247C2×D4—82no—
248C2×D4—82no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4C2×M1614yes—
252C22×C4—22no—
253C22×C4—28no—
254C22×C4—22no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—82no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4—82no—
277C22×C4C2×C22⋊C411yescharacteristic
278C22×C4C2×C22⋊C411yescharacteristic
279C22×C4C2×C22⋊C412yes—
280C22×C4C2×C22⋊C412yes—
281C22×C4C2×C22⋊C411yescharacteristic
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C23×C411yescharacteristic
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287M16—84no—
288C2×C8—24no—
289C2×C8—42no—
290C2×C8—42no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—41no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—28no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—82no—
311C4⋊C4—82no—
312C4⋊C4—82no—
313C4⋊C4C2×M1614yes—
314C22⋊C4—28no—
315C22⋊C4—28no—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—42no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—82no—
322C22⋊C4—82no—
323C22⋊C4—82no—
324C22⋊C4—82no—
325C22⋊C4—82no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—22no—
330C42—21no—
331C42—21no—
332C42—22no—
333C42—22no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—42no—
338C42—42no—
339C42—42no—
340C42—42no—
341C42—82no—
342C42—82no—
343C42C2×M1612yes—
344C42C2×M1612yes—
345C42#13114yes—
346C42H(5,𝔽2)12yes—
347C422−1+412yes—
348C23—21no—
349C23—21no—
350C23—24no—
351C23—21no—
352C23—22no—
353C23—22no—
354C23—42no—
355C23—42no—
356C23—42no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23C22×C22⋊C411yescharacteristic
361D4—24no—
362D4—24no—
363D4—42no—
364D4—42no—
365D4—88no—
366D4—88no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—82no—
400C2×C4—88no—
401C2×C4—88no—
402C2×C4—82no—
403C2×C4—82no—
404C2×C4—82no—
405C2×C4—82no—
406C2×C4#40714yes—
407C2×C4#40814yes—
408C2×C4#52012yes—
409C2×C4#53511yescharacteristic
410C2×C4#53611yescharacteristic
411C2×C4#53712yes—
412C2×C4C22×M1611yescharacteristic
413C2×C4C22×M1611yescharacteristic
414C2×C4C2×#13112yes—
415C2×C4#57612yes—
416C2×C4#57712yes—
417C8—44no—
418C8—82no—
419C8—81no—
420C8—81no—
421C8—81no—
422C8—81no—
423C22—44no—
424C22—42no—
425C22—44no—
426C22—42no—
427C22—41no—
428C22—44no—
429C22—41no—
430C22—41no—
431C22—82no—
432C22—82no—
433C22—82no—
434C22—82no—
435C22#231111yescharacteristic
436C22C2×#40711yescharacteristic
437C22C2×#40811yescharacteristic
438C22#289712yes—
439C22#290112yes—
440C4—24no—
441C4—24no—
442C4—22no—
443C4—42no—
444C4—44no—
445C4—42no—
446C4—41no—
447C4—41no—
448C4—41no—
449C4—82no—
450C4—82no—
451C4#299012yes—
452C4#299112yes—
453C4#322011yescharacteristic
454C4#322111yescharacteristic
455C4#324112yes—
456C2—81no—
457C2—84no—
458C2#1745312yes—
459C2#1748312yes—
460C2#2184911yescharacteristic
461C2#2185811yescharacteristic
462C2#2187211yescharacteristic
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. #21978C2semidirect
3. #21430C2semidirect
4. #9872C2semidirect
5. #9871C2semidirect
6. #9642C2semidirect
7. #9638C2semidirect
8. #8944C2semidirect
9. #8943C2semidirect
10. #8565C2semidirect
11. #8563C2semidirect
150. #127M16semidirect
152. #125M16semidirect

Maximal subgroups

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

subgroupindex
C4×#5402
#219782
#214302
#98722
#98712
#96422
#96382
#89442
#89432
#85652
#85632