Finite Group Databasebeta

← #192710

Subgroups — #192710

Order 512.

964 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2#21870C211yesmaximal, characteristic
3#21359C211yesmaximal, characteristic
4#20824C211yesmaximal, characteristic
5C2×#1485C211yesmaximal, characteristic
6#8896C212yesmaximal, complemented
7#8891C212yesmaximal, complemented
8#8888C212yesmaximal, complemented
9#8883C212yesmaximal, complemented
10#8046C211yesmaximal, characteristic
11#7904C211yesmaximal, characteristic
12C2×#533C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15C2×C4×M16C2211yescharacteristic
16#2370—22no—
17#2369—21no—
18#2365—21no—
19#2355—22no—
20#2355C2212yes—
21Q8×C4⋊C4C2212yes—
22#2329—22no—
23#2317—21no—
24C4×C22⋊Q8—22no—
25#2285C2211yescharacteristic
26#2285C2211yescharacteristic
27#2274—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29#1993—21no—
30#1960—21no—
31#1948C2212yes—
32#1948C2212yes—
33#1873—22no—
34#1853—21no—
35#1853—21no—
36#1845C2211yescharacteristic
37#1844C2211yescharacteristic
38#1834C2211yescharacteristic
39#1834C2211yescharacteristic
40#1809—22no—
41#1803C2212yes—
42#1803C2212yes—
43#1776—22no—
44C4×#106—22no—
45#1485C418yescomplemented
46C2×Q8⋊C8C2212yes—
47C2×Q8⋊C8C2212yes—
48#1296C2212yes—
49#1296C2212yes—
50#1279C2212yes—
51#1279C2212yes—
52C22×M16—21no—
53#533—22no—
54#533—22no—
55#533C2×C414yes—
56#533Q814yes—
57C2×C4⋊Q8D411yescharacteristic
58C2×C4⋊Q8D411yescharacteristic
59C2×#125—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8D412yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—21no—
64C2×C4×Q8C2312yes—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C42—21no—
71C22×C42C2311yescharacteristic
72#432—22no—
73#432—22no—
74#431—22no—
75#431C2×C414yes—
76C2×#107—22no—
77C2×#106—22no—
78C2×#106—22no—
79C2×#106—22no—
80C2×C4⋊C8C2312yes—
81C2×C4⋊C8C2312yes—
82C2×#103—22no—
83C2×#103—22no—
84C2×C22⋊C8—21no—
85C2×C22⋊C8—21no—
86C4×M16—22no—
87C4×M16C2×C414yes—
88C2×#97—21no—
89C2×#97—21no—
90C2×C4×C8—21no—
91C2×C4×C8C2311yescharacteristic
92C2×C4×C8C2311yescharacteristic
93#391—24no—
94#391—22no—
95#389—22no—
96#389—22no—
97#389—24no—
98#389—22no—
99#387—24no—
100#387—24no—
101#387—24no—
102#385—24no—
103#384—22no—
104#384C2311yescharacteristic
105#384C2311yescharacteristic
106#384C2311yescharacteristic
107#384C2311yescharacteristic
108#382—22no—
109#382—21no—
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114#380D412yes—
115C4×C4⋊C4—24no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4—22no—
119C4×C4⋊C4—22no—
120C4×C4⋊C4—22no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4D412yes—
128C4×C4⋊C4C2312yes—
129C4×C22⋊C4—24no—
130C2×#95—21no—
131#340—22no—
132#340—22no—
133#340—22no—
134#340—22no—
135#340—22no—
136#340—22no—
137Q8⋊C8C2×C418yescomplemented
138Q8⋊C8C2×C418yescomplemented
139C22×Q8—21no—
140C22×Q8C2×D412yes—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C2×M16—24no—
147C2×M16—42no—
148C2×M16—42no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8C4⋊C414yes—
156C4⋊Q8C4⋊C414yes—
157#125—24no—
158#125—24no—
159C22⋊Q8—24no—
160C22⋊Q8—24no—
161C22⋊Q8—24no—
162C22⋊Q8C4⋊C418yes—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×Q8—82no—
168C4×Q8—82no—
169C4×Q8C22×C414yes—
170C4×Q8C2×Q814yes—
171#117—24no—
172#117—24no—
173#117—44no—
174#117—82no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
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⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×D412yes—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4C2×D412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—41no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—24no—
208C2×C42—24no—
209C2×C42—22no—
210C2×C42—41no—
211C2×C42—82no—
212C2×C42C22×C412yes—
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×Q812yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219C2×C42C2411yescharacteristic
220#107—44no—
221#106—44no—
222#106—44no—
223#106—44no—
224C4⋊C8—22no—
225C4⋊C8—22no—
226C4⋊C8—22no—
227C4⋊C8C22×C414yes—
228#103—82no—
229#103—82no—
230C22⋊C8—24no—
231C22⋊C8—42no—
232#97—24no—
233#97—42no—
234C4×C8—24no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8C22×C412yes—
238C4×C8C22×C412yes—
239#95—24no—
240#95—24no—
241#95—24no—
242#95—24no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—41no—
248#95—41no—
249C24#12011yescharacteristic
250C2×Q8—22no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—22no—
256C2×Q8—81no—
257C2×Q8—81no—
258C2×Q8C2×C4⋊C414yes—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—81no—
276C22×C4—81no—
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—24no—
296C2×C8—22no—
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—24no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—24no—
312C4⋊C4—28no—
313C4⋊C4—22no—
314C4⋊C4—22no—
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—42no—
327C4⋊C4—42no—
328C4⋊C4—84no—
329C4⋊C4—84no—
330C4⋊C4C2×C4⋊C414yes—
331C4⋊C4#13714yes—
332C22⋊C4—28no—
333C22⋊C4—44no—
334C22⋊C4—82no—
335C42—24no—
336C42—24no—
337C42—44no—
338C42—42no—
339C42—42no—
340C42—82no—
341C42—82no—
342C42—84no—
343C42—84no—
344C42C2×C4⋊C412yes—
345C42C4×D412yes—
346C42C4×D412yes—
347C42C22⋊Q812yes—
348C42C22⋊Q812yes—
349C42#12312yes—
350C42C23×C411yescharacteristic
351C42C22×Q811yescharacteristic
352C42C2×#4111yescharacteristic
353C42C2×#4111yescharacteristic
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23#38512yes—
361C23C2×#12011yescharacteristic
362Q8—24no—
363Q8—24no—
364Q8—42no—
365Q8—42no—
366Q8—84no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—24no—
376C2×C4—24no—
377C2×C4—24no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
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—42no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—84no—
400C2×C4—84no—
401C2×C4—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4#38512yes—
408C2×C4#38512yes—
409C2×C4#38512yes—
410C2×C4C22×C4⋊C411yescharacteristic
411C2×C4C2×C4×D411yescharacteristic
412C2×C4C2×C4×D411yescharacteristic
413C2×C4C2×C22⋊Q811yescharacteristic
414C2×C4C2×C22⋊Q811yescharacteristic
415C2×C4C2×#12311yescharacteristic
416C2×C4C2×#13712yes—
417C2×C4#57512yes—
418C2×C4#57712yes—
419C2×C4#57712yes—
420C8—24no—
421C8—42no—
422C8—44no—
423C8—82no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22C2×#38511yescharacteristic
432C22#300411yescharacteristic
433C22#300511yescharacteristic
434C22#300911yescharacteristic
435C22#301011yescharacteristic
436C22#301811yescharacteristic
437C22#301811yescharacteristic
438C4—22no—
439C4—21no—
440C4—21no—
441C4—22no—
442C4—22no—
443C4—44no—
444C4—44no—
445C4—44no—
446C4—82no—
447C4—82no—
448C4—82no—
449C4—161no—
450C4—161no—
451C4#291212yes—
452C4C4×#13712yes—
453C4#307812yes—
454C4#309912yes—
455C2—42no—
456C2#1860111yescharacteristic
457C2#1860211yescharacteristic
458C2#1861211yescharacteristic
459C2#1861511yescharacteristic
460C2#2084811yescharacteristic
461C2#2086311yescharacteristic
462C2#2087411yescharacteristic
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. #8896C2semidirect
7. #8891C2semidirect
8. #8888C2semidirect
9. #8883C2semidirect
45. #1485C4semidirect
137. Q8⋊C8C2×C4semidirect
138. Q8⋊C8C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342132
#218702
#213592
#208242
C2×#14852
#88962
#88912
#88882
#88832
#80462
#79042