Finite Group Databasebeta

← #223968

Subgroups — #223968

Order 512.

987 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36187C211yesmaximal, characteristic
2#22749C212yesmaximal, complemented
3#9905C212yesmaximal, complemented
4#9141C212yesmaximal, complemented
5#9134C212yesmaximal, complemented
6#8728C211yesmaximal, characteristic, complemented
7#8728C211yesmaximal, characteristic, complemented
8#8592C211yesmaximal, characteristic, complemented
9#8231C211yesmaximal, characteristic, complemented
10#8041C211yesmaximal, characteristic, complemented
11#8038C211yesmaximal, characteristic, complemented
12C2×#555C412yes—
13C2×#431C2212yes—
14C2×#423C2212yes—
15#2868C2211yescharacteristic
16#2856C2211yescharacteristic
17#2856C2211yescharacteristic
18#2563C412yes—
19#2537—22no—
20#2500C412yes—
21#2500C412yes—
22#2491—22no—
23#2491—22no—
24#2354—22no—
25C4×#127C2211yescharacteristic
26#2285C2211yescharacteristic
27#2285C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1924—22no—
30#1916—22no—
31#1870—22no—
32#1831C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1767C2212yes—
36#1753—22no—
37C2×#331C2212yes—
38C2×#331C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2211yescharacteristic
42#1296C2212yes—
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47C22×M16—22no—
48#555—48no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8C2×C412yes—
51C2×#127C2311yescharacteristic
52C2×#124—22no—
53C2×#124—22no—
54C2×#121—22no—
55C2×#121—22no—
56C2×#121—22no—
57C2×#121C2×C412yes—
58C2×#121C2×C412yes—
59C2×C4×D4—22no—
60C2×C4×D4C2×C412yes—
61C2×C4×D4C2×C411yescharacteristic
62C2×C4×D4D412yes—
63C2×#117—22no—
64C2×#117C2×C412yes—
65C2×#117C2×C412yes—
66C22×C42C2311yescharacteristic
67C22×C42C2311yescharacteristic
68#431—24no—
69#431—42no—
70#423—24no—
71#423—42no—
72C2×C4⋊C8—22no—
73C2×C4⋊C8C2312yes—
74C2×C4⋊C8C2312yes—
75C2×C4⋊C8C2312yes—
76C2×C4⋊C8C2312yes—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8—22no—
80C2×#97—22no—
81C2×C4×C8—22no—
82#394—24no—
83#394—24no—
84#393—24no—
85#390—22no—
86#390—22no—
87#390—22no—
88#390C2×C412yes—
89#390C2×C412yes—
90#390C2×C412yes—
91#388—24no—
92#388—24no—
93#384—22no—
94#384C2×C412yes—
95#384C2311yescharacteristic
96#384C2311yescharacteristic
97#384C2311yescharacteristic
98#382C2×C412yes—
99#382C2×C412yes—
100#380—22no—
101#380—22no—
102#380—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C411yescharacteristic
106C4×C4⋊C4D412yes—
107C4×C22⋊C4C2×C411yescharacteristic
108C4×C22⋊C4C2×C411yescharacteristic
109C4×C22⋊C4D412yes—
110C4×C22⋊C4D412yes—
111C43C2311yescharacteristic
112#336—22no—
113#336—22no—
114#336—22no—
115#336—22no—
116#336—22no—
117#331—24no—
118#331—42no—
119C22×Q8—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4C22⋊C411yescharacteristic
124C22×D4C22×C412yes—
125C23×C4—22no—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C2×M16—44no—
131C2×M16—44no—
132C2×M16—82no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C4⋊Q8—44no—
138C4⋊Q8—82no—
139C4⋊Q8—82no—
140#127—22no—
141#127—21no—
142#127—21no—
143#124—48no—
144#124—48no—
145#121—48no—
146#121—44no—
147#121—44no—
148#121—82no—
149#121—82no—
150#121—82no—
151#121—82no—
152C4×D4—24no—
153C4×D4—42no—
154C4×D4—42no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—44no—
158#117—48no—
159#117—44no—
160#117—44no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22⋊C411yescharacteristic
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—24no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C22⋊C411yescharacteristic
180C2×C22⋊C4C22⋊C411yescharacteristic
181C2×C22⋊C4C22⋊C412yes—
182C2×C22⋊C4C22⋊C412yes—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42C22×C411yescharacteristic
192C2×C42C22×C411yescharacteristic
193C2×C42C22×C411yescharacteristic
194C2×C42C22×C411yescharacteristic
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D412yes—
199C2×C42C2411yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—44no—
208C4⋊C8#4114yes—
209C22⋊C8—44no—
210C22⋊C8—44no—
211C22⋊C8—82no—
212#97—82no—
213C4×C8—44no—
214#95—24no—
215#95—24no—
216#95C22⋊C412yes—
217#95C22⋊C412yes—
218#95C22⋊C412yes—
219C24—22no—
220C24C2×C22⋊C411yescharacteristic
221C24C2×C22⋊C411yescharacteristic
222C2×Q8—44no—
223C2×Q8—84no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—84no—
239C2×D4—84no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—48no—
264C22×C4—44no—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4C2×C22⋊C412yes—
268C22×C4C2×C22⋊C411yescharacteristic
269C22×C4C2×C22⋊C411yescharacteristic
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D412yes—
274M16—88no—
275C2×C8—24no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—44no—
298C4⋊C4—84no—
299C4⋊C4#10014yes—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—48no—
305C22⋊C4—44no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—44no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42—21no—
316C42—22no—
317C42—22no—
318C42—22no—
319C42—22no—
320C42—22no—
321C42—22no—
322C42—22no—
323C42—22no—
324C42—22no—
325C42—41no—
326C42—44no—
327C42—44no—
328C42C2×#4112yes—
329C42H(5,𝔽2)11yescharacteristic
330C422−1+411yescharacteristic
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—42no—
344C23—44no—
345C23—44no—
346C23C22×C22⋊C411yescharacteristic
347Q8—164no—
348D4—44no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—82no—
355D4—164no—
356D4—164no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—88no—
394C2×C4C2×#10012yes—
395C2×C4C2×#10012yes—
396C2×C4C2×#10012yes—
397C2×C4C4×#4111yescharacteristic
398C2×C4#52011yescharacteristic
399C2×C4#54612yes—
400C2×C4#54712yes—
401C2×C4#57512yes—
402C2×C4#57512yes—
403C2×C4#57612yes—
404C2×C4#57712yes—
405C8—44no—
406C8—82no—
407C8—82no—
408C8—82no—
409C22—41no—
410C22—41no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—42no—
419C22—82no—
420C22—84no—
421C22—82no—
422C22—82no—
423C22—84no—
424C22#233911yescharacteristic
425C22C22×#10011yescharacteristic
426C22#288911yescharacteristic
427C22#289612yes—
428C22#289712yes—
429C4—22no—
430C4—21no—
431C4—21no—
432C4—22no—
433C4—22no—
434C4—42no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4#331912yes—
442C4#332112yes—
443C4#334712yes—
444C4#334812yes—
445C2—81no—
446C2—81no—
447C2—162no—
448C2#1773811yescharacteristic
449C2#1776912yes—
450C2#1778211yescharacteristic
451C2#2247911yescharacteristic
452C2#2249212yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36187C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22749C2semidirect
3. #9905C2semidirect
4. #9141C2semidirect
5. #9134C2semidirect
6. #8728C2semidirect
7. #8728C2semidirect
8. #8592C2semidirect
9. #8231C2semidirect
10. #8041C2semidirect
11. #8038C2semidirect

Maximal subgroups

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

subgroupindex
#361872
#227492
#99052
#91412
#91342
#87282
#85922
#82312
#80412
#80382