Finite Group Databasebeta

← #121544

Subgroups — #121544

Order 512.

692 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21981C211yesmaximal, characteristic
2#21959C211yesmaximal, characteristic
3#20800C211yesmaximal, characteristic
4#10851C211yesmaximal, characteristic
5#10849C211yesmaximal, characteristic
6#8014C211yesmaximal, characteristic
7C2×#1283C211yesmaximal, characteristic
8C2×#431C2211yescharacteristic
9C2×#428—22no—
10C22×C4⋊C8C2211yescharacteristic
11C2×C4×M16C2211yescharacteristic
12#2285C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#384C2211yescharacteristic
15C2×C4×C4⋊C4C2211yescharacteristic
16#1948—21no—
17#1948—21no—
18#1948—21no—
19#1948—21no—
20#1948—22no—
21#1853—22no—
22#1847—22no—
23#1845—21no—
24#1844—21no—
25#1844—21no—
26#1834—21no—
27#1807—22no—
28C2×#340—22no—
29C2×#339—22no—
30#1294—22no—
31#1283C414yescomplemented
32C22×M16D412yes—
33C23×C8D412yes—
34C2×#117—21no—
35C2×#117—21no—
36C22×C4⋊C4—21no—
37C22×C4⋊C4—21no—
38C22×C4⋊C4—21no—
39C22×C4⋊C4—21no—
40C22×C42—21no—
41C22×C42C2311yescharacteristic
42#431—21no—
43#431—21no—
44#431C2×C412yes—
45#431Q812yes—
46#428—44no—
47#428—44no—
48C2×#107—24no—
49C2×#106—24no—
50C2×#106—22no—
51C2×#106—22no—
52C2×#106—24no—
53C2×C4⋊C8—21no—
54C2×C4⋊C8—21no—
55C2×C4⋊C8—21no—
56C2×C4⋊C8—21no—
57C2×C4⋊C8—21no—
58C2×C4⋊C8C2×C412yes—
59C2×C4⋊C8Q812yes—
60C2×C22⋊C8D412yes—
61C4×M16—21no—
62C4×M16—21no—
63C4×M16C2×C412yes—
64C4×M16Q812yes—
65C2×#97—21no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68#384—21no—
69#384—21no—
70#384—21no—
71#384D411yescharacteristic
72#384D411yescharacteristic
73#384D411yescharacteristic
74#384D411yescharacteristic
75#384D412yes—
76#382—21no—
77#382—21no—
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⋊C4—21no—
89C4×C4⋊C4Q812yes—
90C2×#95—21no—
91#340—22no—
92#340—22no—
93#340—24no—
94#340—44no—
95#339—44no—
96C23×C4—41no—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16C4⋊C414yes—
105C2×M16SD1614yes—
106C2×M16SD1614yes—
107C22×C8—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—22no—
111C22×C8—42no—
112C22×C8—42no—
113C22×C8C4⋊C414yes—
114#117—42no—
115#117—42no—
116#117—42no—
117#117—42no—
118C2×C4⋊C4—22no—
119C2×C4⋊C4—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—42no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—82no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—41no—
149C2×C42—22no—
150C2×C42—42no—
151C2×C42—42no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×Q811yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42C2×Q811yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167#107—48no—
168#106—48no—
169#106—84no—
170#106—82no—
171#106—82no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C22⋊C8—22no—
179C22⋊C8C4⋊C414yes—
180#97—22no—
181#97—22no—
182C4×C8—22no—
183C4×C8—22no—
184C4×C8—22no—
185C4×C8—22no—
186#95—22no—
187#95—41no—
188#95—41no—
189#95—41no—
190#95—41no—
191#95—41no—
192C24#12711yescharacteristic
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—41no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—81no—
210C22×C4—81no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—81no—
214C22×C4—81no—
215C22×C4C2×C4⋊C411yescharacteristic
216C22×C4C2×C4⋊C411yescharacteristic
217C22×C4C2×C4⋊C411yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12011yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4C4⋊Q811yescharacteristic
231C22×C4C4⋊Q811yescharacteristic
232C22×C4C4⋊Q811yescharacteristic
233C22×C4C2×SD1612yes—
234C22×C4C2×SD1612yes—
235C22×C4AGL(1,ℤ/8)12yes—
236C22×C4#13712yes—
237M16—24no—
238M16—24no—
239M16—24no—
240M16#10618yescomplemented
241C2×C8—24no—
242C2×C8—24no—
243C2×C8—24no—
244C2×C8—24no—
245C2×C8—24no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—82no—
252C4⋊C4—24no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—44no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—81no—
274C4⋊C4—81no—
275C4⋊C4—82no—
276C4⋊C4—81no—
277C4⋊C4—81no—
278C4⋊C4—82no—
279C22⋊C4—81no—
280C22⋊C4—81no—
281C22⋊C4—81no—
282C22⋊C4—81no—
283C42—44no—
284C42—82no—
285C42—82no—
286C42—82no—
287C42C4×D411yescharacteristic
288C42C4×D411yescharacteristic
289C42C4×D411yescharacteristic
290C42C4×Q811yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42C22⋊Q811yescharacteristic
296C42C22⋊Q811yescharacteristic
297C42#12511yescharacteristic
298C42#12511yescharacteristic
299C42#12511yescharacteristic
300C42C4⋊Q811yescharacteristic
301C42C4⋊Q811yescharacteristic
302C42C4⋊Q811yescharacteristic
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23#38411yescharacteristic
308C23#39211yescharacteristic
309C23#39911yescharacteristic
310C23#49211yescharacteristic
311C23#49211yescharacteristic
312C23#49611yescharacteristic
313C23#49711yescharacteristic
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—44no—
333C2×C4—82no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4—82no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—82no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—162no—
355C2×C4—162no—
356C2×C4#38511yescharacteristic
357C2×C4#38511yescharacteristic
358C2×C4#38511yescharacteristic
359C2×C4#38911yescharacteristic
360C2×C4#38911yescharacteristic
361C2×C4#38911yescharacteristic
362C2×C4#39511yescharacteristic
363C2×C4#39511yescharacteristic
364C2×C4#39511yescharacteristic
365C2×C4#39811yescharacteristic
366C2×C4#39811yescharacteristic
367C2×C4#39811yescharacteristic
368C2×C4C2×#10612yes—
369C2×C4#42812yes—
370C2×C4#44812yes—
371C2×C4#45012yes—
372C2×C4#45311yescharacteristic
373C2×C4#45411yescharacteristic
374C2×C4#45511yescharacteristic
375C2×C4#45611yescharacteristic
376C2×C4#46012yes—
377C2×C4#46112yes—
378C2×C4#47512yes—
379C2×C4#47612yes—
380C2×C4#48112yes—
381C2×C4#48312yes—
382C2×C4#49812yes—
383C2×C4#50112yes—
384C8—28no—
385C8—44no—
386C8—44no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22#144411yescharacteristic
400C22#184411yescharacteristic
401C22#184711yescharacteristic
402C22#200611yescharacteristic
403C22#200711yescharacteristic
404C22#200811yescharacteristic
405C22#200911yescharacteristic
406C22#201311yescharacteristic
407C22#201411yescharacteristic
408C22#209411yescharacteristic
409C22#209611yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—84no—
417C4—161no—
418C4—162no—
419C4—161no—
420C4—161no—
421C4—161no—
422C4#185312yes—
423C4#185611yescharacteristic
424C4#185711yescharacteristic
425C4#194812yes—
426C4#205812yes—
427C4#206011yescharacteristic
428C4#206111yescharacteristic
429C4#208412yes—
430C2—21no—
431C2—21no—
432C2—21no—
433C2—21no—
434C2#1019411yescharacteristic
435C2#1019711yescharacteristic
436C2#1020111yescharacteristic
437C2#1020611yescharacteristic
438C2#1132311yescharacteristic
439C2#1133311yescharacteristic
440C2#1134211yescharacteristic
subgroupG/H
center, socleC23#392
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#127
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1283C4semidirect
240. M16#106semidirect

Maximal subgroups

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

subgroupindex
#219812
#219592
#208002
#108512
#108492
#80142
C2×#12832