Finite Group Databasebeta

← #215592

Subgroups — #215592

Order 512.

919 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20786C211yesmaximal, characteristic, complemented
4#9710C212yesmaximal, complemented
5#9710C211yesmaximal, characteristic, complemented
6#9708C211yesmaximal, characteristic, complemented
7#9705C212yesmaximal, complemented
8#9700C211yesmaximal, characteristic, complemented
9#9698C211yesmaximal, characteristic, complemented
10#8345C212yesmaximal, complemented
11#8341C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14#2868C2211yescharacteristic
15#2862C2211yescharacteristic
16#2858C2211yescharacteristic
17#2858C2211yescharacteristic
18#2620—21no—
19#2620—21no—
20#2566C414yes—
21#2545—22no—
22#2500C414yes—
23#2432—22no—
24#2326—21no—
25#2326—21no—
26C4×#117C2211yescharacteristic
27#1924—22no—
28#1917—22no—
29#1842C2211yescharacteristic
30#1840—21no—
31#1831C2212yes—
32#1831C2211yescharacteristic
33#1818—21no—
34#1797C2212yes—
35#1797C2212yes—
36#1767C2211yescharacteristic
37#1767C2212yes—
38C4×C4⋊C8C2211yescharacteristic
39#1750—21no—
40#1746—21no—
41C2×#332C2212yes—
42C2×#332C2212yes—
43C2×#332C2212yes—
44C2×#332C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50#533—42no—
51#533—42no—
52#532—42no—
53#532—42no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2311yescharacteristic
56C2×C4⋊Q8C2311yescharacteristic
57C2×C4⋊Q8C2311yescharacteristic
58C2×#127C2311yescharacteristic
59C2×#125C2×C412yes—
60C2×#124—21no—
61C2×#124—21no—
62C2×#124—21no—
63C2×#124—22no—
64C2×C22⋊Q8—21no—
65C2×C22⋊Q8—21no—
66C2×#121—21no—
67C2×#121C2×C412yes—
68C2×C4×Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2×C412yes—
72C2×#117C2×C412yes—
73C2×#117D411yescharacteristic
74C2×#117D411yescharacteristic
75C22×C42C2311yescharacteristic
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C22⋊C8—22no—
82C2×C22⋊C8—22no—
83C2×#97—21no—
84C2×#97—22no—
85C2×C4×C8—21no—
86#396—24no—
87#396—24no—
88#394—24no—
89#394—24no—
90#391—21no—
91#391—21no—
92#391—22no—
93#390—21no—
94#390—21no—
95#390—22no—
96#390C2×C414yes—
97#390C2×C414yes—
98#388—24no—
99#388—24no—
100#385—24no—
101#385—24no—
102#384—21no—
103#384—21no—
104#384C2×C414yes—
105#382C2×C414yes—
106C4×C4⋊C4C2×C412yes—
107C4×C4⋊C4D412yes—
108C4×C22⋊C4C2×C412yes—
109C4×C22⋊C4D412yes—
110#376D411yescharacteristic
111#376D411yescharacteristic
112C43C2311yescharacteristic
113C43C2311yescharacteristic
114#336—22no—
115#336—22no—
116#336—22no—
117#336—21no—
118#336—21no—
119#336—22no—
120#336—22no—
121#336—22no—
122#332—42no—
123#332—42no—
124#332—42no—
125#332—42no—
126#332—42no—
127#332—42no—
128#331—42no—
129#331—42no—
130C2×#41—42no—
131C2×#41—42no—
132C2×#41—82no—
133C2×#41—82no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×D4—22no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4C22×C412yes—
141C23×C4—22no—
142C23×C4C22×C411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C4⋊Q8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—82no—
156C4⋊Q8—82no—
157#127—22no—
158#127—22no—
159#125—44no—
160#124—42no—
161#124—42no—
162#124—44no—
163#124—42no—
164#124—42no—
165#124—82no—
166#124—82no—
167C22⋊Q8—44no—
168C22⋊Q8—42no—
169C22⋊Q8—42no—
170#121—44no—
171#121—44no—
172C4×Q8—44no—
173C4×Q8—42no—
174C4×Q8—42no—
175C4×D4—44no—
176C4×D4—44no—
177#117—41no—
178#117—41no—
179#117—42no—
180#117—44no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4C22⋊C412yes—
201C2×C22⋊C4C22⋊C412yes—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42C22×C411yescharacteristic
206C2×C42C22×C412yes—
207C2×C42C22×C412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2411yescharacteristic
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—82no—
226C22⋊C8—44no—
227C22⋊C8—82no—
228#97—42no—
229#97—82no—
230C4×C8—42no—
231#95—24no—
232#95—24no—
233#95C22⋊C412yes—
234#95C22⋊C412yes—
235#95C22⋊C412yes—
236#95C22⋊C412yes—
237C24—22no—
238C24C2×C22⋊C411yescharacteristic
239#41—84no—
240#41—84no—
241#41—88no—
242C2×Q8—42no—
243C2×Q8—41no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×D4—24no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—42no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—84no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—42no—
279C22×C4—44no—
280C22×C4—84no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C411yescharacteristic
284C22×C4C2×C22⋊C411yescharacteristic
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C412yes—
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C23×C411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—84no—
315C4⋊C4—84no—
316C22⋊C4—42no—
317C22⋊C4—42no—
318C22⋊C4—44no—
319C22⋊C4—44no—
320C22⋊C4—48no—
321C22⋊C4—48no—
322C22⋊C4—44no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C42—22no—
326C42—22no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42—22no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—42no—
340C42—42no—
341C42—44no—
342C42H(5,𝔽2)12yes—
343C42H(5,𝔽2)12yes—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—21no—
348C23—22no—
349C23—42no—
350C23—44no—
351C23—42no—
352C23—42no—
353C23—44no—
354C23C22×C22⋊C411yescharacteristic
355Q8—82no—
356Q8—82no—
357Q8—81no—
358Q8—81no—
359Q8—81no—
360Q8—81no—
361Q8—82no—
362Q8—82no—
363D4—84no—
364D4—88no—
365D4—82no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—82no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—44no—
403C2×C4—88no—
404C2×C4#51812yes—
405C2×C4#53512yes—
406C2×C4#53511yescharacteristic
407C2×C4#53511yescharacteristic
408C2×C4#57511yescharacteristic
409C2×C4#57511yescharacteristic
410C2×C4#57512yes—
411C2×C4#57612yes—
412C2×C4#57712yes—
413C8—82no—
414C8—82no—
415C8—82no—
416C8—82no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—42no—
421C22—42no—
422C22—82no—
423C22—82no—
424C22—82no—
425C22—84no—
426C22—84no—
427C22#230911yescharacteristic
428C22#288912yes—
429C22#289611yescharacteristic
430C22#289611yescharacteristic
431C22#289712yes—
432C4—22no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—42no—
437C4—42no—
438C4—42no—
439C4—42no—
440C4—82no—
441C4—82no—
442C4—82no—
443C4#321912yes—
444C4#321912yes—
445C4#322011yescharacteristic
446C4#322211yescharacteristic
447C4#322412yes—
448C2—81no—
449C2—162no—
450C2#1776912yes—
451C2#1776912yes—
452C2#2180211yescharacteristic
453C2#2181211yescharacteristic
454C2#2181311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21926C2semidirect
3. #20786C2semidirect
4. #9710C2semidirect
5. #9710C2semidirect
6. #9708C2semidirect
7. #9705C2semidirect
8. #9700C2semidirect
9. #9698C2semidirect
10. #8345C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#219262
#207862
#97102
#97082
#97052
#97002
#96982
#83452
#83412