Finite Group Databasebeta

← #192644

Subgroups — #192644

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34187C211yesmaximal, characteristic
2#21988C211yesmaximal, characteristic
3#20824C211yesmaximal, characteristic
4#20734C211yesmaximal, characteristic
5C2×#1491C211yesmaximal, characteristic
6#8895C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8887C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic
11#7896C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2356—22no—
17#2356—22no—
18#2355—21no—
19#2355—21no—
20#2355C2212yes—
21#2355C2212yes—
22#2313—22no—
23#2310—21no—
24#2285C2211yescharacteristic
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2283—22no—
28#2274—21no—
29#1948C2212yes—
30#1947C2212yes—
31#1854—22no—
32#1847—21no—
33#1845C2211yescharacteristic
34#1844C2211yescharacteristic
35#1836—21no—
36#1834C2211yescharacteristic
37#1834C2211yescharacteristic
38#1824—21no—
39#1809—22no—
40#1803C2212yes—
41#1803C2212yes—
42#1787—21no—
43#1776—22no—
44C4×#103—22no—
45#1491C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C23×Q8D411yescharacteristic
53C22×M16—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8—21no—
56C2×C4×Q8—22no—
57C2×C4×Q8—22no—
58C2×C4×Q8C2×C414yes—
59C2×C4×Q8D412yes—
60C2×C4×Q8Q814yes—
61C2×C4×Q8C2312yes—
62C2×#117—21no—
63C2×#117—21no—
64C2×#117—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4D411yescharacteristic
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69#432—22no—
70#432C2×C414yes—
71#431—22no—
72#431—22no—
73C2×#107—22no—
74C2×#107—22no—
75C2×#106—22no—
76C2×#106—22no—
77C2×C4⋊C8C2312yes—
78C2×C4⋊C8C2312yes—
79C2×#103—22no—
80C2×#103—22no—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—22no—
84C4×M16C2×C414yes—
85C2×#97—21no—
86C2×#97—21no—
87C2×C4×C8—21no—
88C2×C4×C8C2311yescharacteristic
89C2×C4×C8C2311yescharacteristic
90#391—22no—
91#391—22no—
92#391—22no—
93#391—22no—
94#387—24no—
95#387—24no—
96#387—24no—
97#387—24no—
98#384D411yescharacteristic
99#384D411yescharacteristic
100#384C2311yescharacteristic
101#384C2311yescharacteristic
102#384C2311yescharacteristic
103#384C2311yescharacteristic
104#383—24no—
105#382—22no—
106#382—22no—
107#382—21no—
108#382—21no—
109#382—21no—
110#381—24no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380—21no—
115#380D412yes—
116#380D412yes—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—22no—
122C4×C4⋊C4—22no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130#376—24no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137Q8⋊C8C2×C418yes—
138Q8⋊C8C2×C418yes—
139C22×Q8—21no—
140C22×Q8—22no—
141C22×Q8—42no—
142C22×Q8—42no—
143C22×Q8C4⋊C414yes—
144C22×Q8C2×D412yes—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C2×M16—24no—
151C2×M16—42no—
152C2×M16—42no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C4×Q8—24no—
158C4×Q8—24no—
159C4×Q8—24no—
160C4×Q8—24no—
161C4×Q8—24no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8C4⋊C418yes—
165C4×Q8C22×C414yes—
166C4×Q8C2×Q814yes—
167#117—44no—
168#117—44no—
169#117—82no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4C4⋊C414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—41no—
199C2×C42—24no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—41no—
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×Q812yes—
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216#107—44no—
217#107—82no—
218#106—44no—
219#106—82no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8C22×C414yes—
224#103—44no—
225#103—44no—
226C22⋊C8—24no—
227C22⋊C8—42no—
228#97—24no—
229#97—24no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—42no—
233C4×C8C22×C412yes—
234C4×C8C22×C412yes—
235#95—24no—
236#95—24no—
237#95—24no—
238#95—24no—
239#95—24no—
240#95—24no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—41no—
246#95—41no—
247C24#12011yescharacteristic
248C2×Q8—22no—
249C2×Q8—24no—
250C2×Q8—22no—
251C2×Q8—22no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—48no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—48no—
258C2×Q8C2×C4⋊C414yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4C2×C4⋊C412yes—
280C22×C4C4×D412yes—
281C22×C4C4×D412yes—
282C22×C4#12011yescharacteristic
283C22×C4#12011yescharacteristic
284C22×C4#12011yescharacteristic
285C22×C4C22⋊Q812yes—
286C22×C4C22⋊Q812yes—
287C22×C4#12312yes—
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291M16—48no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—24no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C2×C8—42no—
303C2×C8—42no—
304C4⋊C4—24no—
305C4⋊C4—24no—
306C4⋊C4—22no—
307C4⋊C4—28no—
308C4⋊C4—24no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—24no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—42no—
326C4⋊C4—81no—
327C4⋊C4—81no—
328C4⋊C4C2×C4⋊C414yes—
329C22⋊C4—44no—
330C22⋊C4—44no—
331C22⋊C4—82no—
332C42—24no—
333C42—24no—
334C42—28no—
335C42—44no—
336C42—44no—
337C42—82no—
338C42—84no—
339C42—84no—
340C42C2×C4⋊C412yes—
341C42C4×D412yes—
342C42C4×D412yes—
343C42C22⋊Q812yes—
344C42C22⋊Q812yes—
345C42#12312yes—
346C42C23×C411yescharacteristic
347C42C22×Q811yescharacteristic
348C42C2×#4111yescharacteristic
349C42C2×#4111yescharacteristic
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23#38512yes—
357C23C2×#12011yescharacteristic
358Q8—24no—
359Q8—24no—
360Q8—48no—
361Q8—42no—
362Q8—42no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—48no—
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—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—84no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—81no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4#38512yes—
404C2×C4#38512yes—
405C2×C4#38512yes—
406C2×C4C22×C4⋊C411yescharacteristic
407C2×C4C2×C4×D411yescharacteristic
408C2×C4C2×C4×D411yescharacteristic
409C2×C4C2×C22⋊Q811yescharacteristic
410C2×C4C2×C22⋊Q811yescharacteristic
411C2×C4C2×#12311yescharacteristic
412C2×C4#57512yes—
413C2×C4#57512yes—
414C2×C4#57712yes—
415C2×C4#57812yes—
416C8—24no—
417C8—42no—
418C8—44no—
419C8—82no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22C2×#38511yescharacteristic
428C22#300511yescharacteristic
429C22#300511yescharacteristic
430C22#301011yescharacteristic
431C22#301111yescharacteristic
432C22#301811yescharacteristic
433C22#301811yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—22no—
439C4—44no—
440C4—44no—
441C4—82no—
442C4—82no—
443C4—82no—
444C4—82no—
445C4—161no—
446C4—161no—
447C4#291112yes—
448C4#294712yes—
449C4#308212yes—
450C4#309312yes—
451C2—42no—
452C2#1860211yescharacteristic
453C2#1860311yescharacteristic
454C2#1860411yescharacteristic
455C2#1860511yescharacteristic
456C2#2084511yescharacteristic
457C2#2086311yescharacteristic
458C2#2087711yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8895C2semidirect
7. #8891C2semidirect
8. #8887C2semidirect
9. #8883C2semidirect
45. #1491C4semidirect

Maximal subgroups

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

subgroupindex
#341872
#219882
#208242
#207342
C2×#14912
#88952
#88912
#88872
#88832
#80462
#78962