Finite Group Databasebeta

← #215587

Subgroups — #215587

Order 512.

982 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#35289C211yesmaximal, characteristic
2#21926C211yesmaximal, characteristic, complemented
3#20812C211yesmaximal, characteristic, complemented
4#9709C211yesmaximal, characteristic, complemented
5#9707C212yesmaximal, complemented
6#9707C211yesmaximal, characteristic, complemented
7#9701C212yesmaximal, complemented
8#9699C211yesmaximal, characteristic, complemented
9#9697C211yesmaximal, characteristic, complemented
10#8342C212yesmaximal, complemented
11#8341C212yesmaximal
12C2×#532C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C2×#433—21no—
15#2862C2211yescharacteristic
16#2862C2211yescharacteristic
17#2858C2211yescharacteristic
18#2858C2211yescharacteristic
19#2640—21no—
20#2640—21no—
21#2563—22no—
22#2547C414yes—
23#2504—22no—
24#2436C414yes—
25#2311—21no—
26#2311—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×#333C2212yes—
42C2×#333C2212yes—
43C2×#333C2212yes—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#331C2212yes—
48C2×#331C2212yes—
49C22×#41—21no—
50C23×Q8C2×C412yes—
51#532—42no—
52#532—42no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8C2×C412yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2311yescharacteristic
61C2×#127C2311yescharacteristic
62C2×#124—21no—
63C2×#124C2×C414yes—
64C2×#124C2×C414yes—
65C2×#121—21no—
66C2×#121C2×C412yes—
67C2×C4×D4—21no—
68C2×C4×D4C2×C412yes—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#433—44no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—22no—
78C2×C4⋊C8C2312yes—
79C2×C4⋊C8C2312yes—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×C22⋊C8—22no—
83C2×C22⋊C8—22no—
84C2×#97—21no—
85C2×#97—22no—
86C2×C4×C8—21no—
87#397—24no—
88#397—24no—
89#393—24no—
90#393—24no—
91#391—21no—
92#391—21no—
93#391—21no—
94#391—21no—
95#391C2×C414yes—
96#391C2×C414yes—
97#390—22no—
98#390—22no—
99#390—21no—
100#390—21no—
101#390—21no—
102#390—21no—
103#386—24no—
104#386—24no—
105#382—22no—
106#382—22no—
107#380—24no—
108#380—24no—
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#333—42no—
124#333—42no—
125#333—42no—
126#333—42no—
127#332—42no—
128#332—42no—
129#331—42no—
130#331—42no—
131C2×#41—44no—
132C2×#41—44no—
133C2×#41—42no—
134C2×#41—42no—
135C22×Q8—22no—
136C22×Q8—21no—
137C22×Q8—44no—
138C22×Q8—84no—
139C22×Q8C22×C412yes—
140C22×Q8C22×C412yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4C22×C412yes—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4C22×C411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C4⋊Q8—22no—
155C4⋊Q8—22no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—44no—
161C4⋊Q8—42no—
162C4⋊Q8—42no—
163C4⋊Q8—44no—
164#127—22no—
165#127—22no—
166#124—44no—
167#124—84no—
168#124—84no—
169#121—44no—
170#121—42no—
171#121—42no—
172C4×D4—44no—
173C4×D4—42no—
174C4×D4—42no—
175#117—22no—
176#117—22no—
177#117—42no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4C22⋊C412yes—
189C2×C4⋊C4C22⋊C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4C22⋊C412yes—
198C2×C22⋊C4C22⋊C412yes—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42C22×C411yescharacteristic
202C2×C42C22×C412yes—
203C2×C42C22×C412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D412yes—
212C2×C42C2411yescharacteristic
213C4⋊C8—22no—
214C4⋊C8—22no—
215C4⋊C8—22no—
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—44no—
222C4⋊C8—44no—
223C4⋊C8—82no—
224C22⋊C8—44no—
225C22⋊C8—82no—
226#97—42no—
227#97—82no—
228C4×C8—42no—
229#95—24no—
230#95—24no—
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—88no—
240#41—88no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—44no—
244C2×Q8—44no—
245C2×Q8—44no—
246C2×Q8—42no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×Q8—42no—
250C2×Q8—84no—
251C2×Q8—816no—
252C2×D4—24no—
253C2×D4—44no—
254C2×D4—44no—
255C2×D4—44no—
256C2×D4—44no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—42no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—84no—
281C22×C4C2×C22⋊C412yes—
282C22×C4C2×C22⋊C412yes—
283C22×C4C2×C22⋊C412yes—
284C22×C4C2×C22⋊C412yes—
285C22×C4C2×C22⋊C411yescharacteristic
286C22×C4C2×C22⋊C411yescharacteristic
287C22×C4C2×C22⋊C411yescharacteristic
288C22×C4AGL(1,ℤ/8)12yes—
289C22×C4#13712yes—
290C22×C4C23×C411yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
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—
303C2×C8—42no—
304C2×C8—42no—
305C4⋊C4—24no—
306C4⋊C4—24no—
307C4⋊C4—42no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C22⋊C4—24no—
319C22⋊C4—42no—
320C22⋊C4—44no—
321C22⋊C4—44no—
322C22⋊C4—84no—
323C22⋊C4—84no—
324C22⋊C4—84no—
325C22⋊C4—84no—
326C42—22no—
327C42—21no—
328C42—21no—
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—
342C42H(5,𝔽2)12yes—
343C42H(5,𝔽2)12yes—
344C23—21no—
345C23—22no—
346C23—21no—
347C23—22no—
348C23—42no—
349C23—44no—
350C23—42no—
351C23—42no—
352C23—44no—
353C23#41712yes—
354C23C22×C22⋊C411yescharacteristic
355Q8—84no—
356Q8—84no—
357Q8—88no—
358Q8—82no—
359Q8—82no—
360D4—84no—
361D4—88no—
362D4—82no—
363D4—82no—
364D4—81no—
365D4—81no—
366D4—82no—
367C2×C4—21no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—24no—
371C2×C4—24no—
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—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—44no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—88no—
400C2×C4—88no—
401C2×C4#41712yes—
402C2×C4#51812yes—
403C2×C4#53512yes—
404C2×C4#53511yescharacteristic
405C2×C4#53511yescharacteristic
406C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
407C2×C4C2×#13711yescharacteristic
408C2×C4#57512yes—
409C2×C4#57712yes—
410C2×C4#57812yes—
411C8—82no—
412C8—82no—
413C8—82no—
414C8—82no—
415C22—22no—
416C22—24no—
417C22—41no—
418C22—41no—
419C22—42no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—84no—
424C22—84no—
425C22#230911yescharacteristic
426C22#288812yes—
427C22C2×#41711yescharacteristic
428C22#289611yescharacteristic
429C22#289812yes—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—42no—
435C4—42no—
436C4—42no—
437C4—42no—
438C4—82no—
439C4—82no—
440C4—82no—
441C4#321312yes—
442C4#321512yes—
443C4#322111yescharacteristic
444C4#322311yescharacteristic
445C4#322512yes—
446C2—42no—
447C2—162no—
448C2#1776412yes—
449C2#1776812yes—
450C2#2180111yescharacteristic
451C2#2180811yescharacteristic
452C2#2181411yescharacteristic
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. #9709C2semidirect
5. #9707C2semidirect
6. #9707C2semidirect
7. #9701C2semidirect
8. #9699C2semidirect
9. #9697C2semidirect
10. #8342C2semidirect

Maximal subgroups

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

subgroupindex
#352892
#219262
#208122
#97092
#97072
#97012
#96992
#96972
#83422
#83412