Finite Group Databasebeta

← #413626

Subgroups — #413626

Order 512.

864 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C212yesmaximal
2C2×#2089C212yesmaximal
3#21172C211yesmaximal, characteristic
4#21171C211yesmaximal, characteristic
5C2×#1847C211yesmaximal, characteristic
6#20427C211yesmaximal, characteristic
7#11366C212yesmaximal
8#11365C212yesmaximal
9#10858C211yesmaximal, characteristic
10#10857C211yesmaximal, characteristic
11#10845C211yesmaximal, characteristic
12C2×#428C2211yescharacteristic
13C2×#428C2212yes—
14C2×#428C2211yescharacteristic
15C22×#107C2212yes—
16C22×#106C2212yes—
17C2×#424C2212yes—
18C2×#423C2211yescharacteristic
19C2×#423C2211yescharacteristic
20C2×C4×M16C2211yescharacteristic
21C22×#97C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#382C2211yescharacteristic
24C4×#117C2211yescharacteristic
25#2090—28no—
26#2089—28no—
27#1949—22no—
28#1949—22no—
29#1948—22no—
30#1947—22no—
31#1847—22no—
32#1847—22no—
33#1768—22no—
34#1767—22no—
35C2×#343C2212yes—
36C2×#343C2212yes—
37C2×#343C2212yes—
38C2×#343C2212yes—
39C2×#341C2212yes—
40C2×#341C2212yes—
41#1736C2211yescharacteristic
42#1736C2212yes—
43#1736C2211yescharacteristic
44#1729C2212yes—
45C22×M16C2312yes—
46C22×M16C2311yescharacteristic
47C22×M16C2311yescharacteristic
48C22×M16C2312yes—
49C23×C8C2312yes—
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#428—24no—
58#428—24no—
59#428—24no—
60#428—24no—
61C2×#107—24no—
62C2×#107—24no—
63C2×#107—24no—
64C2×#107—22no—
65C2×#107—21no—
66C2×#107—21no—
67C2×#106—24no—
68C2×#106—24no—
69C2×#106—24no—
70C2×#106—22no—
71C2×#106—21no—
72C2×#106—21no—
73#424—24no—
74#424D418yes—
75#423—22no—
76#423—22no—
77#423Q814yes—
78#423Q814yes—
79C2×C4⋊C8—24no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C4×M16—22no—
83C4×M16—22no—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97—21no—
87C2×#97D412yes—
88C2×#97D412yes—
89C2×C4×C8—21no—
90#384—21no—
91#384—21no—
92#384—22no—
93#382—21no—
94#382—21no—
95#382—22no—
96C4×C4⋊C4D412yes—
97C4×C4⋊C4D412yes—
98C4×C22⋊C4—22no—
99#376—21no—
100C2×#95—22no—
101C43—21no—
102#343—24no—
103#343—24no—
104#343—24no—
105#343—24no—
106#341—24no—
107#341—24no—
108#340—24no—
109#340—22no—
110#340—22no—
111#336—24no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C2×M16—21no—
118C2×M16—21no—
119C2×M16—22no—
120C2×M16—22no—
121C2×M16—22no—
122C2×M16—22no—
123C2×M16—22no—
124C2×M16—22no—
125C2×M16—22no—
126C2×M16—22no—
127C2×M16—22no—
128C2×M16—22no—
129C2×M16—22no—
130C2×M16—22no—
131C2×M16—22no—
132C2×M16—22no—
133C2×M16C2×D414yes—
134C2×M16C2×Q812yes—
135C2×M16C2×Q812yes—
136C2×M16C2×Q814yes—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8C2×D414yes—
146C22×C8C2×D414yes—
147C22×C8#4114yes—
148#117—22no—
149#117—22no—
150#117—22no—
151#117C2×D414yes—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4C2×D412yes—
170C2×C4⋊C4C2×D411yescharacteristic
171C2×C4⋊C4C2×D411yescharacteristic
172C2×C22⋊C4—21no—
173C2×C22⋊C4—21no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42C2×D412yes—
179C2×C42C2×D411yescharacteristic
180C2×C42C2×D411yescharacteristic
181C2×C42C2×Q812yes—
182C2×C42C2×Q812yes—
183C2×C42#4112yes—
184C2×C42#4112yes—
185#107—48no—
186#107—42no—
187#107—44no—
188#107—42no—
189#106—48no—
190#106—44no—
191#106—42no—
192#106—42no—
193C4⋊C8—216no—
194C4⋊C8—28no—
195C4⋊C8—28no—
196#97—24no—
197#97—24no—
198#97—42no—
199C4×C8—42no—
200#95—22no—
201#95—22no—
202#95—42no—
203#95—42no—
204C24C2×#4111yescharacteristic
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—22no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—41no—
211C22×C4—41no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—84no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4#12112yes—
221C22×C4#12112yes—
222C22×C4#12514yes—
223C22×C4#12514yes—
224C22×C4C22×D411yescharacteristic
225C22×C4C22×D412yes—
226C22×C4C22×D411yescharacteristic
227C22×C4C22×Q811yescharacteristic
228C22×C4C22×Q811yescharacteristic
229C22×C4C2×#4112yes—
230C22×C4C2×#4111yescharacteristic
231C22×C4C2×#4111yescharacteristic
232C22×C42−1+412yes—
233C22×C42−1+412yes—
234M16—24no—
235M16—24no—
236M16—24no—
237M16—24no—
238M16—24no—
239M16—24no—
240M16—42no—
241M16—42no—
242M16—42no—
243M16—42no—
244M16—42no—
245M16—42no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—24no—
249C2×C8—24no—
250C2×C8—24no—
251C2×C8—24no—
252C2×C8—24no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—41no—
263C2×C8—41no—
264C2×C8#12118yes—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—42no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—42no—
279C4⋊C4—84no—
280C4⋊C4—84no—
281C4⋊C4#12712yes—
282C4⋊C4#12712yes—
283C22⋊C4—24no—
284C22⋊C4—42no—
285C42—24no—
286C42—42no—
287C42—82no—
288C42—82no—
289C42C22⋊Q814yes—
290C42C22⋊Q814yes—
291C42#12312yes—
292C42#12312yes—
293C42C4⋊Q812yes—
294C42C4⋊Q812yes—
295C23—22no—
296C23—22no—
297C23C2×#12111yescharacteristic
298C23C2×#12511yescharacteristic
299C23C2×#12511yescharacteristic
300C23#53212yes—
301C23#54112yes—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—44no—
309C2×C4—44no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—84no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—162no—
322C2×C4—162no—
323C2×C4—162no—
324C2×C4—162no—
325C2×C4#39814yes—
326C2×C4#39814yes—
327C2×C4C2×#12112yes—
328C2×C4C2×C22⋊Q811yescharacteristic
329C2×C4C2×C22⋊Q811yescharacteristic
330C2×C4C2×#12311yescharacteristic
331C2×C4C2×#12711yescharacteristic
332C2×C4C2×C4⋊Q811yescharacteristic
333C2×C4#53211yescharacteristic
334C2×C4#53611yescharacteristic
335C2×C4#53611yescharacteristic
336C2×C4#53612yes—
337C2×C4#54712yes—
338C2×C4#54712yes—
339C2×C4D4×Q812yes—
340C2×C4D4×Q812yes—
341C2×C4#55212yes—
342C2×C4#55212yes—
343C2×C4#55712yes—
344C2×C4#55712yes—
345C8—28no—
346C8—44no—
347C8—44no—
348C8—42no—
349C8—42no—
350C22—22no—
351C22—22no—
352C22—22no—
353C22—22no—
354C22—24no—
355C22C2×#39811yescharacteristic
356C22#250612yes—
357C22#250612yes—
358C22#262111yescharacteristic
359C22#285611yescharacteristic
360C22#306812yes—
361C22#306912yes—
362C4—24no—
363C4—24no—
364C4—42no—
365C4—42no—
366C4—82no—
367C4—162no—
368C4—161no—
369C4—161no—
370C4#248612yes—
371C4#254211yescharacteristic
372C4#254211yescharacteristic
373C4#254612yes—
374C4#256511yescharacteristic
375C4#256511yescharacteristic
376C2—22no—
377C2—22no—
378C2#1515111yescharacteristic
379C2#2367512yes—
380C2#2988912yes—
381C2#3319311yescharacteristic
382C2#3319411yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#20902
C2×#20892
#211722
#211712
C2×#18472
#204272
#113662
#113652
#108582
#108572
#108452