Finite Group Databasebeta

← #223973

Subgroups — #223973

Order 512.

982 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36182C211yesmaximal, characteristic
2#22797C212yesmaximal, complemented
3#9906C212yesmaximal, complemented
4#9142C212yesmaximal, complemented
5#9134C212yesmaximal, complemented
6#8728C211yesmaximal, characteristic, complemented
7#8723C211yesmaximal, characteristic, complemented
8#8585C211yesmaximal, characteristic, complemented
9#8231C211yesmaximal, characteristic, complemented
10#8040C211yesmaximal, characteristic, complemented
11#8038C211yesmaximal, characteristic, complemented
12C2×#553C412yes—
13C2×#432C2212yes—
14C2×#431C2212yes—
15#2859C2211yescharacteristic
16#2856C2211yescharacteristic
17#2856C2211yescharacteristic
18#2566C412yes—
19#2512—22no—
20#2504C412yes—
21#2500C412yes—
22#2422—22no—
23#2416—22no—
24#2361—22no—
25C4×#127C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1955—22no—
30#1924—22no—
31#1923—22no—
32#1842C2212yes—
33#1797C2212yes—
34#1797C2212yes—
35#1771—22no—
36#1767C2212yes—
37C2×#331C2212yes—
38C2×#331C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#1296C2211yescharacteristic
43#1296C2212yes—
44#1296C2211yescharacteristic
45#1296C2212yes—
46#1296C2212yes—
47#553—48no—
48C2×#127C2311yescharacteristic
49C2×#126—22no—
50C2×#125C2×C412yes—
51C2×#123—22no—
52C2×#121—22no—
53C2×#121—22no—
54C2×#121—22no—
55C2×#121C2×C412yes—
56C2×#121C2×C412yes—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C412yes—
60C2×C4×D4C2×C411yescharacteristic
61C2×C4×D4D412yes—
62C2×#117C2×C412yes—
63C22×C4⋊C4—22no—
64C22×C4⋊C4C2×C412yes—
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67#432—24no—
68#432—42no—
69#431—24no—
70#431—42no—
71C2×C4⋊C8—22no—
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×C22⋊C8—22no—
81C2×#97—22no—
82C2×#97—22no—
83C2×C4×C8—22no—
84#397—24no—
85#396—24no—
86#394—24no—
87#390—22no—
88#390—22no—
89#390—22no—
90#390C2×C412yes—
91#390C2×C412yes—
92#390C2×C412yes—
93#389—22no—
94#389—22no—
95#388—24no—
96#386—24no—
97#385—24no—
98#384—22no—
99#384C2×C412yes—
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#382C2×C412yes—
104#382C2×C412yes—
105#380—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4C2×C411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C22⋊C4C2×C411yescharacteristic
110C4×C22⋊C4C2×C411yescharacteristic
111C4×C22⋊C4D412yes—
112C4×C22⋊C4D412yes—
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#331—24no—
118#331—42no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4C22⋊C411yescharacteristic
123C22×D4C22×C412yes—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C22×C411yescharacteristic
127C23×C4C22×C411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134#127—22no—
135#127—21no—
136#127—21no—
137#126—48no—
138#125—44no—
139#123—48no—
140#121—44no—
141#121—44no—
142#121—44no—
143#121—48no—
144#121—44no—
145#121—82no—
146#121—82no—
147C4×D4—24no—
148C4×D4—42no—
149C4×D4—42no—
150C4×D4—44no—
151C4×D4—48no—
152C4×D4—82no—
153C4×D4—82no—
154#117—44no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—82no—
164C2×C4⋊C4—82no—
165C2×C4⋊C4C22⋊C411yescharacteristic
166C2×C4⋊C4C22⋊C412yes—
167C2×C4⋊C4C22×C412yes—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C22⋊C4—22no—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—24no—
177C2×C22⋊C4C22⋊C411yescharacteristic
178C2×C22⋊C4C22⋊C412yes—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C411yescharacteristic
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—22no—
188C2×C42C22×C411yescharacteristic
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C22×C411yescharacteristic
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2411yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—44no—
205C4⋊C8—82no—
206C4⋊C8#4114yes—
207C22⋊C8—44no—
208C22⋊C8—44no—
209C22⋊C8—82no—
210C22⋊C8—82no—
211#97—44no—
212#97—44no—
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×D4—22no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—48no—
229C2×D4—44no—
230C2×D4—44no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—84no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
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—41no—
260C22×C4—41no—
261C22×C4—44no—
262C22×C4—44no—
263C22×C4—84no—
264C22×C4—84no—
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—
274C2×C8—24no—
275C2×C8—42no—
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—
286C4⋊C4—22no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—42no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—44no—
300C4⋊C4—84no—
301C22⋊C4—22no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C22⋊C4—44no—
305C22⋊C4—48no—
306C22⋊C4—44no—
307C22⋊C4—44no—
308C22⋊C4—44no—
309C22⋊C4—84no—
310C22⋊C4—84no—
311C22⋊C4#10014yes—
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—
327C42C2×#4112yes—
328C42H(5,𝔽2)11yescharacteristic
329C422−1+411yescharacteristic
330C23—21no—
331C23—21no—
332C23—22no—
333C23—22no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—22no—
338C23—22no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—44no—
343C23—44no—
344C23C2×#10012yes—
345C23C22×C22⋊C411yescharacteristic
346D4—44no—
347D4—82no—
348D4—82no—
349D4—88no—
350D4—82no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—164no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—21no—
364C2×C4—21no—
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—42no—
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—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—164no—
393C2×C4—164no—
394C2×C4C2×#10012yes—
395C2×C4C2×#10012yes—
396C2×C4C4×#4111yescharacteristic
397C2×C4#52011yescharacteristic
398C2×C4#54612yes—
399C2×C4#54712yes—
400C2×C4#57512yes—
401C2×C4#57512yes—
402C2×C4#57612yes—
403C2×C4#57712yes—
404C8—44no—
405C8—82no—
406C8—82no—
407C8—82no—
408C22—22no—
409C22—22no—
410C22—22no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—82no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22—84no—
423C22#233911yescharacteristic
424C22C22×#10011yescharacteristic
425C22#288911yescharacteristic
426C22#289612yes—
427C22#289712yes—
428C4—22no—
429C4—21no—
430C4—21no—
431C4—22no—
432C4—41no—
433C4—42no—
434C4—41no—
435C4—42no—
436C4—42no—
437C4—82no—
438C4—82no—
439C4—82no—
440C4#330712yes—
441C4#330912yes—
442C4#334712yes—
443C4#335012yes—
444C2—42no—
445C2—81no—
446C2—162no—
447C2#1773811yescharacteristic
448C2#1776912yes—
449C2#1777011yescharacteristic
450C2#2247911yescharacteristic
451C2#2249812yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#36182C2
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. #22797C2semidirect
3. #9906C2semidirect
4. #9142C2semidirect
5. #9134C2semidirect
6. #8728C2semidirect
7. #8723C2semidirect
8. #8585C2semidirect
9. #8231C2semidirect
10. #8040C2semidirect
11. #8038C2semidirect

Maximal subgroups

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

subgroupindex
#361822
#227972
#99062
#91422
#91342
#87282
#87232
#85852
#82312
#80402
#80382