Finite Group Databasebeta

← #6519434

Subgroups — #6519434

Order 512.

1912 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35317C211yesmaximal, characteristic
2#34368C211yesmaximal, characteristic
3#26236C214yesmaximal, complemented
4#26234C214yesmaximal, complemented
5#26233C214yesmaximal, complemented
6#26231C214yesmaximal, complemented
7#24016C214yesmaximal, complemented
8#24014C214yesmaximal, complemented
9#21804C212yesmaximal, complemented
10#21787C211yesmaximal, characteristic
11#21362C212yesmaximal, complemented
12C2×#536C2212yes—
13C2×#536C2212yes—
14C2×#532C2211yescharacteristic
15C22×C4⋊Q8C2211yescharacteristic
16C2×#484C2214yes—
17C2×#485C2218yes—
18C2×#483C2214yes—
19C2×#431C2212yes—
20#2868—21no—
21#2858—21no—
22#2658—22no—
23#2542C2214yes—
24#2542C2214yes—
25#2501—24no—
26#2379—22no—
27#2369—21no—
28#2366—21no—
29#2354C2214yes—
30D4×C4⋊C4C2214yes—
31#2324—24no—
32#2311C2214yes—
33#2285C2212yes—
34C4×#117C2211yescharacteristic
35#2036C2218yes—
36#2036C2218yes—
37#2035C2218yes—
38#2035C2218yes—
39#1981C2214yes—
40#1981C2214yes—
41#1956C2214yes—
42#1956C2214yes—
43#1954C2214yes—
44#1954C2214yes—
45#1948C2212yes—
46#1948C2212yes—
47#1947C2212yes—
48#1947C2212yes—
49#1873C2214yes—
50#1873C2214yes—
51#1872C2218yes—
52#1854C2214yes—
53#1854C2214yes—
54#1787C2218yes—
55#1780C2214yes—
56C4×#107C2214yes—
57C4×#106C2214yes—
58C22×#41C2312yes—
59C23×Q8C2312yes—
60#536—44no—
61#536—44no—
62#536—48no—
63#532—28no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8—22no—
66C2×C4⋊Q8—21no—
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—28no—
69C2×C4⋊Q8D411yescharacteristic
70C2×C4⋊Q8C2314yes—
71C2×#127D411yescharacteristic
72C2×#124—22no—
73C2×#124—22no—
74C2×#124D412yes—
75C2×#123—24no—
76C2×#123—24no—
77C2×C22⋊Q8C2314yes—
78C2×C22⋊Q8C2318yes—
79C2×C22⋊Q8C2314yes—
80C2×#121D414yes—
81C2×C4×D4—24no—
82C2×C4×D4C2314yes—
83C2×#117—22no—
84C2×#117C2311yescharacteristic
85C2×#117C2311yescharacteristic
86C2×#117C2314yes—
87C22×C4⋊C4—22no—
88C22×C4⋊C4C2314yes—
89C22×C42C2311yescharacteristic
90#485—416no—
91#484—48no—
92#483—48no—
93#431—28no—
94C2×#107C2314yes—
95C2×#107C2314yes—
96C2×#106C2314yes—
97C2×#106C2314yes—
98C2×C4⋊C8C2312yes—
99C2×C4⋊C8C2312yes—
100C2×#103C2318yes—
101C2×#103C2318yes—
102C2×#103C2318yes—
103C2×#103C2318yes—
104C2×C22⋊C8C2314yes—
105C2×C22⋊C8C2314yes—
106C2×C22⋊C8C2318yes—
107C2×C22⋊C8C2314yes—
108C2×C4×C8C2314yes—
109#399—28no—
110#399—28no—
111#391—24no—
112#390—24no—
113#390—24no—
114#390—24no—
115#390—24no—
116#389—24no—
117#388—28no—
118#386—24no—
119#386—24no—
120#385—28no—
121#384—24no—
122#384—24no—
123#384C2312yes—
124#384C2312yes—
125#382—22no—
126#380—22no—
127#380C2314yes—
128#380C2318yes—
129C4×C4⋊C4—24no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4C2314yes—
132C4×C4⋊C4C2312yes—
133C4×C4⋊C4C2312yes—
134C4×C22⋊C4—22no—
135#376—21no—
136C43—21no—
137#340C2314yes—
138#340C2314yes—
139#340C2314yes—
140#340C2314yes—
141C2×#41—28no—
142C2×#41—42no—
143C2×#41—42no—
144C2×#41—44no—
145C22×Q8—22no—
146C22×Q8—28no—
147C22×Q8—42no—
148C22×Q8—42no—
149C22×Q8—44no—
150C22×Q8C2×D412yes—
151C22×Q8C2414yes—
152C22×D4C2×D414yes—
153C22×D4C2×D414yes—
154C22×D4C2×D412yes—
155C23×C4—22no—
156C23×C4—24no—
157C23×C4C2×D412yes—
158C23×C4C2414yes—
159C23×C4C2411yescharacteristic
160C22×C8C2414yes—
161C22×C8C2414yes—
162C4⋊Q8—24no—
163C4⋊Q8—28no—
164C4⋊Q8—28no—
165C4⋊Q8—48no—
166C4⋊Q8—48no—
167C4⋊Q8—44no—
168C4⋊Q8—44no—
169#127—24no—
170#124—28no—
171#124—48no—
172#124—48no—
173#123—416no—
174#123—416no—
175C22⋊Q8—28no—
176C22⋊Q8—28no—
177C22⋊Q8—216no—
178C22⋊Q8—44no—
179C22⋊Q8—44no—
180C22⋊Q8—48no—
181#121—216no—
182C4×D4—28no—
183C4×D4—28no—
184C4×D4—88no—
185#117—22no—
186#117—24no—
187#117—22no—
188#117—48no—
189#117—84no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2414yes—
205C2×C4⋊C4C2414yes—
206C2×C4⋊C4C2411yescharacteristic
207C2×C4⋊C4C2412yes—
208C2×C4⋊C4C2411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—24no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—28no—
217C2×C22⋊C4C2×D418yes—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—24no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—24no—
224C2×C42C2×D411yescharacteristic
225C2×C42C2×D411yescharacteristic
226C2×C42#4118yes—
227C2×C42C2412yes—
228#107—28no—
229#107—28no—
230#106—28no—
231#106—28no—
232C4⋊C8—24no—
233C4⋊C8—24no—
234#103—48no—
235#103—48no—
236#103—48no—
237#103—48no—
238C22⋊C8—28no—
239C22⋊C8—44no—
240C22⋊C8—44no—
241C22⋊C8—48no—
242C4×C8—28no—
243#95—22no—
244#95—22no—
245#95—24no—
246#95—28no—
247#95—24no—
248C24C22×D414yes—
249C24C22×D411yescharacteristic
250#41—88no—
251#41—88no—
252C2×Q8—24no—
253C2×Q8—28no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—28no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—44no—
260C2×Q8—44no—
261C2×Q8—44no—
262C2×Q8—88no—
263C2×Q8—88no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4—24no—
267C2×D4—28no—
268C2×D4—28no—
269C2×D4—44no—
270C2×D4—44no—
271C2×D4—42no—
272C2×D4—42no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—21no—
281C22×C4—28no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—28no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—44no—
289C22×C4—44no—
290C22×C4—44no—
291C22×C4—84no—
292C22×C4—88no—
293C22×C4#12318yes—
294C22×C4C22×D414yes—
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D412yes—
297C22×C4C22×D412yes—
298C22×C4C2×#4114yes—
299C22×C4C2511yescharacteristic
300C2×C8—24no—
301C2×C8—24no—
302C2×C8—24no—
303C2×C8—24no—
304C2×C8—24no—
305C2×C8—24no—
306C4⋊C4—28no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—416no—
315C4⋊C4—48no—
316C4⋊C4—48no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—44no—
320C4⋊C4—44no—
321C4⋊C4—84no—
322C4⋊C4—82no—
323C4⋊C4—82no—
324C4⋊C4H(5,𝔽2)12yes—
325C4⋊C4H(5,𝔽2)14yes—
326C4⋊C4H(5,𝔽2)12yes—
327C22⋊C4—24no—
328C22⋊C4—24no—
329C22⋊C4—216no—
330C22⋊C4—416no—
331C22⋊C4—48no—
332C22⋊C4—48no—
333C22⋊C4—84no—
334C22⋊C4—84no—
335C22⋊C4—88no—
336C42—24no—
337C42—24no—
338C42—44no—
339C42—48no—
340C42—44no—
341C42—84no—
342C42#12314yes—
343C42#12314yes—
344C42C2×#4114yes—
345C42C2×#4114yes—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—22no—
350C23—24no—
351C23—24no—
352C23—24no—
353C23—28no—
354C23—28no—
355C23C2×#12312yes—
356C23C23×D411yescharacteristic
357Q8—48no—
358Q8—48no—
359Q8—84no—
360Q8—88no—
361D4—416no—
362D4—84no—
363D4—88no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—216no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—48no—
382C2×C4—48no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—44no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—88no—
393C2×C4—168no—
394C2×C4C2×#12312yes—
395C2×C4C2×#12314yes—
396C2×C4C2×#12314yes—
397C2×C4#57714yes—
398C2×C4#57814yes—
399C2×C4C22×#4112yes—
400C2×C4C2×H(5,𝔽2)11yescharacteristic
401C2×C4C2×H(5,𝔽2)11yescharacteristic
402C2×C4C2×H(5,𝔽2)12yes—
403C8—44no—
404C8—44no—
405C22—24no—
406C22—22no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22—48no—
411C22—44no—
412C22—44no—
413C22—44no—
414C22—48no—
415C22C22×#12311yescharacteristic
416C22#348411yescharacteristic
417C22#348411yescharacteristic
418C22C2×#57712yes—
419C22C2×#57812yes—
420C4—24no—
421C4—22no—
422C4—42no—
423C4—42no—
424C4—44no—
425C4—44no—
426C4—84no—
427C4—84no—
428C4—84no—
429C4#309718yes—
430C4#350812yes—
431C4#350812yes—
432C2—42no—
433C2—84no—
434C2#3555411yescharacteristic
435C2C2×#309712yes—
436C2#6166411yescharacteristic
437C2#6166511yescharacteristic
438C2#6166712yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#532 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #26236C2semidirect
4. #26234C2semidirect
5. #26233C2semidirect
6. #26231C2semidirect
7. #24016C2semidirect
8. #24014C2semidirect
9. #21804C2semidirect
11. #21362C2semidirect

Maximal subgroups

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

subgroupindex
#353172
#343682
#262362
#262342
#262332
#262312
#240162
#240142
#218042
#217872
#213622