Finite Group Databasebeta

← #215585

Subgroups — #215585

Order 512.

943 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35298C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9708C212yesmaximal, complemented
5#9704C212yesmaximal, complemented
6#9704C211yesmaximal, characteristic, complemented
7#9703C211yesmaximal, characteristic, complemented
8#9696C211yesmaximal, characteristic, complemented
9#9692C211yesmaximal, characteristic, complemented
10#8345C212yesmaximal, complemented
11#8341C212yesmaximal
12C2×#533C2211yescharacteristic
13C2×#532C2211yescharacteristic
14C2×#433—21no—
15#2868C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18#2858C2211yescharacteristic
19#2620—21no—
20#2620—21no—
21#2566C414yes—
22#2545—22no—
23#2500C414yes—
24#2432—22no—
25#2326—21no—
26#2326—21no—
27C4×#117C2211yescharacteristic
28#1924—22no—
29#1917—22no—
30#1842C2212yes—
31#1842C2211yescharacteristic
32#1832—21no—
33#1831C2211yescharacteristic
34#1798—21no—
35#1797C2212yes—
36#1797C2212yes—
37#1768—21no—
38#1767C2211yescharacteristic
39#1767C2212yes—
40C4×C4⋊C8C2211yescharacteristic
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
76#433—44no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8—22no—
79C2×C4⋊C8—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C22⋊C8—22no—
85C2×C22⋊C8—22no—
86C2×#97—21no—
87C2×#97—22no—
88C2×C4×C8—21no—
89#396—24no—
90#396—24no—
91#394—24no—
92#394—24no—
93#391—21no—
94#391—21no—
95#391—22no—
96#390—21no—
97#390—21no—
98#390—22no—
99#390C2×C414yes—
100#390C2×C414yes—
101#388—24no—
102#388—24no—
103#385—24no—
104#385—24no—
105#384—21no—
106#384—21no—
107#384C2×C414yes—
108#382C2×C414yes—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4D412yes—
111C4×C22⋊C4C2×C412yes—
112C4×C22⋊C4D412yes—
113#376D411yescharacteristic
114#376D411yescharacteristic
115C43C2311yescharacteristic
116C43C2311yescharacteristic
117#336—22no—
118#336—22no—
119#336—21no—
120#336—21no—
121#336—22no—
122#336—22no—
123#332—42no—
124#332—42no—
125#332—42no—
126#332—42no—
127#332—42no—
128#332—42no—
129#331—42no—
130#331—42no—
131C2×#41—42no—
132C2×#41—42no—
133C2×#41—82no—
134C2×#41—82no—
135C22×Q8—21no—
136C22×Q8—21no—
137C22×Q8—21no—
138C22×D4—22no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4C22×C412yes—
142C23×C4—22no—
143C23×C4C22×C411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—82no—
157C4⋊Q8—82no—
158#127—22no—
159#127—22no—
160#125—44no—
161#124—44no—
162#124—42no—
163#124—42no—
164#124—42no—
165#124—42no—
166#124—82no—
167#124—82no—
168C22⋊Q8—44no—
169C22⋊Q8—42no—
170C22⋊Q8—42no—
171#121—44no—
172#121—44no—
173C4×Q8—44no—
174C4×Q8—42no—
175C4×Q8—42no—
176C4×D4—44no—
177C4×D4—44no—
178#117—22no—
179#117—22no—
180#117—42no—
181#117—44no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22⋊C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C4⋊C4C22×C412yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4C22⋊C412yes—
202C2×C22⋊C4C22⋊C412yes—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42C22×C411yescharacteristic
207C2×C42C22×C412yes—
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D412yes—
215C2×C42C2×D412yes—
216C2×C42C2×D411yescharacteristic
217C2×C42C2411yescharacteristic
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—44no—
227C4⋊C8—44no—
228C4⋊C8—82no—
229C22⋊C8—44no—
230C22⋊C8—82no—
231#97—42no—
232#97—82no—
233C4×C8—42no—
234#95—24no—
235#95—24no—
236#95C22⋊C412yes—
237#95C22⋊C412yes—
238#95C22⋊C412yes—
239#95C22⋊C412yes—
240C24—22no—
241C24C2×C22⋊C411yescharacteristic
242#41—84no—
243#41—84no—
244#41—88no—
245C2×Q8—42no—
246C2×Q8—41no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×D4—24no—
258C2×D4—44no—
259C2×D4—44no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—84no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—44no—
283C22×C4—84no—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C412yes—
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4C2×C22⋊C412yes—
289C22×C4C2×C22⋊C411yescharacteristic
290C22×C4C2×C22⋊C412yes—
291C22×C4C23×C411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C22×D411yescharacteristic
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—
303C2×C8—42no—
304C2×C8—42no—
305C2×C8—42no—
306C4⋊C4—24no—
307C4⋊C4—24no—
308C4⋊C4—24no—
309C4⋊C4—24no—
310C4⋊C4—42no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C22⋊C4—24no—
320C22⋊C4—42no—
321C22⋊C4—44no—
322C22⋊C4—44no—
323C22⋊C4—48no—
324C22⋊C4—44no—
325C22⋊C4—48no—
326C22⋊C4—84no—
327C22⋊C4—84no—
328C42—22no—
329C42—22no—
330C42—21no—
331C42—21no—
332C42—21no—
333C42—21no—
334C42—22no—
335C42—22no—
336C42—22no—
337C42—22no—
338C42—22no—
339C42—22no—
340C42—42no—
341C42—42no—
342C42—44no—
343C42AGL(1,ℤ/8)12yes—
344C42#13712yes—
345C42H(5,𝔽2)12yes—
346C42H(5,𝔽2)12yes—
347C23—21no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23—22no—
352C23—42no—
353C23—44no—
354C23—42no—
355C23—42no—
356C23—44no—
357C23C22×C22⋊C411yescharacteristic
358Q8—82no—
359Q8—82no—
360Q8—81no—
361Q8—81no—
362Q8—81no—
363Q8—81no—
364Q8—82no—
365Q8—82no—
366D4—84no—
367D4—88no—
368D4—82no—
369D4—82no—
370D4—81no—
371D4—81no—
372D4—82no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—88no—
406C2×C4#41714yes—
407C2×C4#51812yes—
408C2×C4#53512yes—
409C2×C4#53511yescharacteristic
410C2×C4#53511yescharacteristic
411C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
412C2×C4C2×#13711yescharacteristic
413C2×C4#57512yes—
414C2×C4#57612yes—
415C2×C4#57712yes—
416C8—82no—
417C8—82no—
418C8—82no—
419C8—82no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—42no—
424C22—42no—
425C22—82no—
426C22—82no—
427C22—82no—
428C22—84no—
429C22—84no—
430C22#230911yescharacteristic
431C22#288912yes—
432C22C2×#41711yescharacteristic
433C22#289611yescharacteristic
434C22#289712yes—
435C4—22no—
436C4—24no—
437C4—41no—
438C4—41no—
439C4—41no—
440C4—42no—
441C4—42no—
442C4—42no—
443C4—82no—
444C4—82no—
445C4—82no—
446C4#321412yes—
447C4#321612yes—
448C4#322011yescharacteristic
449C4#322211yescharacteristic
450C4#322412yes—
451C2—81no—
452C2—162no—
453C2#1776512yes—
454C2#1776912yes—
455C2#2180211yescharacteristic
456C2#2180911yescharacteristic
457C2#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. #20812C2semidirect
4. #9708C2semidirect
5. #9704C2semidirect
6. #9704C2semidirect
7. #9703C2semidirect
8. #9696C2semidirect
9. #9692C2semidirect
10. #8345C2semidirect

Maximal subgroups

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

subgroupindex
#352982
#219262
#208122
#97082
#97042
#97032
#96962
#96922
#83452
#83412