Finite Group Databasebeta

← #110633

Subgroups — #110633

Order 512.

711 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21385C211yesmaximal, characteristic
2#21306C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4#17314C211yesmaximal, characteristic
5#10851C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7#7981C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×C4×C4⋊C4C2211yescharacteristic
15#1987C412yescomplemented
16#1986—22no—
17#1986C414yescomplemented
18#1981C412yescomplemented
19#1872—22no—
20C4×#103—22no—
21#1733—22no—
22C2×#339—22no—
23#1520—22no—
24#1518C414yescomplemented
25#1503—22no—
26#1450—22no—
27C8×M16—22no—
28C8×M16—22no—
29#1294—22no—
30#1289—22no—
31C22×M16D412yes—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431C2×C412yes—
60#431D412yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8C2×C412yes—
63C2×#103—24no—
64C2×C22⋊C8D412yes—
65C2×C22⋊C8D412yes—
66C4×M16—21no—
67C4×M16—21no—
68C4×M16C2×C412yes—
69C4×M16D412yes—
70C2×#97—21no—
71C2×C4×C8—21no—
72C2×C4×C8—21no—
73C2×C4×C8D412yes—
74#384—21no—
75#384—21no—
76#382—21no—
77#382C2×C412yes—
78#380—21no—
79#380—21no—
80#380—21no—
81#380—21no—
82C4×C4⋊C4—21no—
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4C2×C412yes—
89C4×C4⋊C4C2×C412yes—
90C2×#95—21no—
91#340—24no—
92#339—28no—
93#339—44no—
94#328—28no—
95#322—24no—
96C82—24no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4—41no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16C22⋊C414yes—
107C2×M16SD1614yes—
108C2×M16Q1614yes—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8C22⋊C414yes—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8C4212yes—
119C4⋊Q8C4212yes—
120#125—22no—
121#125—22no—
122#125—41no—
123#125—41no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140#117—22no—
141#117—22no—
142#117—41no—
143#117—41no—
144#117—42no—
145#117—42no—
146#117—42no—
147#117—42no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—44no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—41no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C22⋊C4—41no—
164C2×C22⋊C4—41no—
165C2×C22⋊C4—41no—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—41no—
169C2×C42—42no—
170C2×C42—42no—
171C2×C42—41no—
172C2×C42—41no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42#4111yescharacteristic
180C2×C42#4111yescharacteristic
181C2×C42#4111yescharacteristic
182C2×C42#4111yescharacteristic
183C2×C42#4111yescharacteristic
184C2×C42#4111yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8—22no—
188C4⋊C8—22no—
189#103—48no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8C22⋊C414yes—
194#97—22no—
195#97—22no—
196C4×C8—22no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—41no—
207#95—41no—
208C24#12711yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—82no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—42no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—81no—
223C22×C4—82no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4#12111yescharacteristic
238C22×C4#12111yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4#12411yescharacteristic
242C22×C4#12411yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4C2×SD1612yes—
245C22×C4C2×Q1612yes—
246C22×C4#13512yes—
247C22×C4#13512yes—
248M16—24no—
249M16—24no—
250M16—24no—
251M16#10318yescomplemented
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—24no—
257C2×C8—44no—
258C2×C8—82no—
259C4⋊C4—24no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—44no—
275C4⋊C4—82no—
276C4⋊C4—82no—
277C4⋊C4—82no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C42—42no—
287C42—42no—
288C42—44no—
289C42—44no—
290C42—82no—
291C42—82no—
292C42C2×C4211yescharacteristic
293C42#11711yescharacteristic
294C42#11711yescharacteristic
295C42#11711yescharacteristic
296C42C4×D411yescharacteristic
297C42C4×D411yescharacteristic
298C42C4×D411yescharacteristic
299C42C4×D411yescharacteristic
300C42C4×D411yescharacteristic
301C42C4×D411yescharacteristic
302C42#12011yescharacteristic
303C42#12311yescharacteristic
304C42#12311yescharacteristic
305C42#12311yescharacteristic
306C42#12611yescharacteristic
307C42#12611yescharacteristic
308C23—41no—
309C23—41no—
310C23#39011yescharacteristic
311C23#39011yescharacteristic
312C23#39011yescharacteristic
313C23#49211yescharacteristic
314C23#49411yescharacteristic
315C23#49511yescharacteristic
316C23#49511yescharacteristic
317Q8—44no—
318Q8—82no—
319Q8—82no—
320Q8—82no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4—82no—
342C2×C4—82no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4C4×C22⋊C411yescharacteristic
349C2×C4C4×C22⋊C411yescharacteristic
350C2×C4C4×C22⋊C411yescharacteristic
351C2×C4#38611yescharacteristic
352C2×C4#38611yescharacteristic
353C2×C4#38611yescharacteristic
354C2×C4#38811yescharacteristic
355C2×C4#38811yescharacteristic
356C2×C4#38811yescharacteristic
357C2×C4#38811yescharacteristic
358C2×C4#38811yescharacteristic
359C2×C4#38811yescharacteristic
360C2×C4C2×#10312yes—
361C2×C4#41612yes—
362C2×C4C4×SD1612yes—
363C2×C4C4×SD1612yes—
364C2×C4C4×Q1612yes—
365C2×C4C4×Q1612yes—
366C2×C4#46512yes—
367C2×C4#46512yes—
368C2×C4#46712yes—
369C2×C4#46712yes—
370C2×C4#48712yes—
371C2×C4#48812yes—
372C8—28no—
373C8—44no—
374C8—44no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22#143711yescharacteristic
386C22#195611yescharacteristic
387C22#195811yescharacteristic
388C22#196811yescharacteristic
389C22#196811yescharacteristic
390C22#196911yescharacteristic
391C22#196911yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—84no—
400C4—82no—
401C4—82no—
402C4—162no—
403C4—162no—
404C4C4×#10312yes—
405C4#188412yes—
406C4#189212yes—
407C4#193312yes—
408C4#193312yes—
409C2—21no—
410C2—21no—
411C2—21no—
412C2—21no—
413C2—21no—
414C2—21no—
415C2#1092611yescharacteristic
416C2#1095111yescharacteristic
417C2#1095911yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
socleC22#1437
prime radicalC24#127
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
15. #1987C4semidirect
17. #1986C4semidirect
18. #1981C4semidirect
24. #1518C4semidirect
251. M16#103semidirect

Maximal subgroups

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

subgroupindex
#213852
#213062
#175752
#173142
#108512
#80142
#79812