Finite Group Databasebeta

← #204788

Subgroups — #204788

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36639C211yesmaximal, characteristic
2C2×#1842C211yesmaximal, characteristic, complemented
3C4×#423C211yesmaximal, characteristic, complemented
4C2×#1550C212yesmaximal, complemented
5C2×#1547C212yesmaximal
6#9708C212yesmaximal, complemented
7#9707C212yesmaximal, complemented
8#9698C212yesmaximal, complemented
9#9697C212yesmaximal, complemented
10C2×#533C2211yescharacteristic
11C2×#532C2211yescharacteristic
12C2×#423C2212yes—
13C22×C4⋊C8—22no—
14C22×C4⋊C8C2212yes—
15C2×C4×M16—21no—
16#2862C2211yescharacteristic
17#2862C2211yescharacteristic
18#2861—21no—
19#2861—21no—
20#2858C2211yescharacteristic
21#2858C2211yescharacteristic
22#2850—22no—
23#2850C414yes—
24C4×#124—22no—
25C4×#124C414yes—
26#2297—21no—
27#2297—21no—
28C2×C43C2211yescharacteristic
29#1842—22no—
30#1842C2212yes—
31#1842C2212yes—
32#1767—21no—
33#1767C2212yes—
34C4×C4⋊C8—21no—
35C4×C4⋊C8C2212yes—
36#1728—21no—
37#1550—28no—
38#1547—28no—
39C2×#333C2214yes—
40C2×#332C2214yes—
41C2×#332C2214yes—
42C2×#331C2214yes—
43C22×#41—21no—
44C22×M16—22no—
45C23×C8—22no—
46#533—24no—
47#533—24no—
48#532—24no—
49#532—24no—
50C2×C4⋊Q8C2311yescharacteristic
51C2×C4⋊Q8C2311yescharacteristic
52C2×C4⋊Q8C2311yescharacteristic
53C2×#127C2311yescharacteristic
54C2×#125—21no—
55C2×#124—22no—
56C2×#124—21no—
57C2×#124C2×C414yes—
58C2×C22⋊Q8—21no—
59C2×C22⋊Q8C2×C412yes—
60C2×#121—21no—
61C2×#121C2×C412yes—
62C2×C4×Q8—21no—
63C2×C4×Q8C2×C412yes—
64C2×C4×D4—21no—
65C2×C4×D4C2×C412yes—
66C2×#117—21no—
67C22×C42C2×C412yes—
68C22×C42D412yes—
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42C2311yescharacteristic
72#423—24no—
73#423D418yes—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—24no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—24no—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—48no—
81C2×C4⋊C8C2314yes—
82C2×C4⋊C8C2314yes—
83C4×M16—44no—
84C2×#97—22no—
85C2×C4×C8—22no—
86#391—22no—
87#391—21no—
88#391—21no—
89#391—21no—
90#391—21no—
91#391C2×C414yes—
92#390—22no—
93#390—21no—
94#390—21no—
95#390—21no—
96#390—21no—
97#390C2×C414yes—
98#388—24no—
99#388—24no—
100#388—24no—
101#388—24no—
102#386—24no—
103#386—24no—
104#382—22no—
105#382C2×C414yes—
106C4×C22⋊C4—24no—
107C4×C22⋊C4—24no—
108C43C2×C412yes—
109C43D411yescharacteristic
110C43D411yescharacteristic
111C43D412yes—
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#333—28no—
117#332—28no—
118#332—28no—
119#331—28no—
120C2×#41—44no—
121C2×#41—44no—
122C2×#41—42no—
123C2×#41—42no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8C22×C412yes—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4C22×C412yes—
131C23×C4—22no—
132C23×C4C22⋊C412yes—
133C23×C4C22⋊C412yes—
134C23×C4C22×C411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C2×M16—28no—
138C2×M16—44no—
139C2×M16—44no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—48no—
143C22×C8—44no—
144C4⋊Q8—22no—
145C4⋊Q8—22no—
146C4⋊Q8—22no—
147C4⋊Q8—22no—
148C4⋊Q8—22no—
149C4⋊Q8—22no—
150#127—22no—
151#127—22no—
152#125—44no—
153#124—48no—
154#124—44no—
155#124—44no—
156#124—44no—
157C22⋊Q8—28no—
158C22⋊Q8—42no—
159C22⋊Q8—42no—
160#121—28no—
161#121—42no—
162#121—42no—
163C4×Q8—28no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×D4—28no—
167C4×D4—42no—
168C4×D4—42no—
169#117—44no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C22×C412yes—
177C2×C4⋊C4C22×C412yes—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C42—24no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—24no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—44no—
192C2×C42C22⋊C412yes—
193C2×C42C22⋊C412yes—
194C2×C42C22⋊C412yes—
195C2×C42C22⋊C412yes—
196C2×C42C22⋊C412yes—
197C2×C42C22⋊C412yes—
198C2×C42C22×C412yes—
199C2×C42C22×C412yes—
200C2×C42C22×C411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D412yes—
206C2×C42C2×D414yes—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D412yes—
210C2×C42#4114yes—
211C2×C42C2411yescharacteristic
212C4⋊C8—28no—
213C4⋊C8—24no—
214C4⋊C8—24no—
215C4⋊C8—24no—
216C4⋊C8—48no—
217C4⋊C8—44no—
218C4⋊C8—48no—
219C4⋊C8C2×D418yes—
220#97—44no—
221C4×C8—44no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226C24—22no—
227C24C2×C22⋊C411yescharacteristic
228#41—88no—
229#41—88no—
230C2×Q8—24no—
231C2×Q8—44no—
232C2×Q8—42no—
233C2×Q8—41no—
234C2×Q8—41no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—24no—
241C2×D4—44no—
242C2×D4—44no—
243C2×D4—44no—
244C2×D4—44no—
245C2×D4—42no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—41no—
253C22×C4—24no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C412yes—
272C22×C4C2×C22⋊C412yes—
273C22×C4C2×C22⋊C412yes—
274C22×C4C2×C22⋊C411yescharacteristic
275C22×C4C4×D414yes—
276C22×C4#12114yes—
277C22×C4#12412yes—
278C22×C4#12712yes—
279C22×C4#13514yes—
280C22×C4AGL(1,ℤ/8)12yes—
281C22×C4#13712yes—
282C22×C4C23×C411yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285M16—416no—
286C2×C8—28no—
287C2×C8—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—48no—
291C2×C8—88no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4—41no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C22⋊C4—48no—
303C22⋊C4—44no—
304C22⋊C4—44no—
305C22⋊C4—44no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C42—22no—
310C42—22no—
311C42—21no—
312C42—21no—
313C42—22no—
314C42—21no—
315C42—22no—
316C42—21no—
317C42—24no—
318C42—24no—
319C42—24no—
320C42—24no—
321C42—24no—
322C42—24no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42C4×D414yes—
327C42#12114yes—
328C42#12412yes—
329C42#12712yes—
330C42AGL(1,ℤ/8)14yes—
331C42#13714yes—
332C42C22×D412yes—
333C42C2×#4112yes—
334C23—22no—
335C23—22no—
336C23—44no—
337C23—42no—
338C23—44no—
339C23—42no—
340C23—42no—
341C23#39012yes—
342C23#41612yes—
343C23#41712yes—
344C23C22×C22⋊C411yescharacteristic
345Q8—84no—
346Q8—82no—
347Q8—81no—
348Q8—81no—
349Q8—82no—
350Q8—82no—
351D4—84no—
352D4—88no—
353D4—82no—
354D4—81no—
355D4—81no—
356D4—82no—
357D4—82no—
358C2×C4—216no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—44no—
375C2×C4—48no—
376C2×C4—44no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—88no—
384C2×C4#39012yes—
385C2×C4#39012yes—
386C2×C4#39012yes—
387C2×C4#41314yes—
388C2×C4#41612yes—
389C2×C4#41714yes—
390C2×C4#41714yes—
391C2×C4#41712yes—
392C2×C4#46318yes—
393C2×C4#46418yes—
394C2×C4C2×C4×D412yes—
395C2×C4C2×#12112yes—
396C2×C4C2×#12411yescharacteristic
397C2×C4C2×#12711yescharacteristic
398C2×C4C2×#13512yes—
399C2×C4C2×AGL(1,ℤ/8)12yes—
400C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
401C2×C4C2×#13712yes—
402C2×C4C2×#13711yescharacteristic
403C8—48no—
404C8—84no—
405C22—24no—
406C22—24no—
407C22—22no—
408C22—22no—
409C22—82no—
410C22—84no—
411C22—84no—
412C22—82no—
413C22—82no—
414C22#198314yes—
415C22C2×#39011yescharacteristic
416C22C2×#41312yes—
417C22C2×#41611yescharacteristic
418C22C2×#41711yescharacteristic
419C22C2×#41712yes—
420C4—24no—
421C4—24no—
422C4—24no—
423C4—22no—
424C4—22no—
425C4—22no—
426C4—42no—
427C4—82no—
428C4—82no—
429C4—82no—
430C4#198314yes—
431C4C2×#46312yes—
432C4C2×#46412yes—
433C4#314112yes—
434C4#315111yescharacteristic
435C4#315211yescharacteristic
436C2—24no—
437C2—162no—
438C2#1771912yes—
439C2#1775212yes—
440C2#2120011yescharacteristic
441C2#2121811yescharacteristic
442C2C2×#198311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1842C2semidirect
3. C4×#423C2semidirect
4. C2×#1550C2semidirect
6. #9708C2semidirect
7. #9707C2semidirect
8. #9698C2semidirect
9. #9697C2semidirect

Maximal subgroups

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

subgroupindex
#366392
C2×#18422
C4×#4232
C2×#15502
C2×#15472
#97082
#97072
#96982
#96972