Finite Group Databasebeta

← #382011

Subgroups — #382011

Order 512.

770 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C211yesmaximal, characteristic
2C2×#1944C211yesmaximal, characteristic
3#21034C212yesmaximal
4#20427C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11303C212yesmaximal
7#11270C211yesmaximal, characteristic
8#11239C211yesmaximal, characteristic
9#11216C212yesmaximal
10#10913C212yesmaximal
11#10823C211yesmaximal, characteristic
12C2×#428C2212yes—
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#424C2211yescharacteristic
17C2×C4×M16C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C2×#389C2212yes—
20C4×#117C2211yescharacteristic
21#2090C418yescomplemented
22#1944—22no—
23#1944—22no—
24#1921—22no—
25#1920—22no—
26#1919—22no—
27#1918—22no—
28#1768—22no—
29#1767—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
35C2×#341C2212yes—
36C2×#341C2211yescharacteristic
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1729C2211yescharacteristic
43#1728C2212yes—
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C42C2311yescharacteristic
56#428—24no—
57#428C2×C418yes—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—42no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—22no—
64C2×#106C2×C414yes—
65#424—22no—
66#424—22no—
67#424—22no—
68#424C2×C414yes—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C4×M16—42no—
72C2×#97—21no—
73C2×#97—42no—
74C2×#97D412yes—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#389—22no—
78#389—22no—
79#389—22no—
80#389—22no—
81C4×C4⋊C4D412yes—
82C4×C4⋊C4D412yes—
83C4×C22⋊C4—21no—
84C4×C22⋊C4—21no—
85#376—21no—
86C2×#95—22no—
87C43—21no—
88#343—24no—
89#343—42no—
90#343—42no—
91#343C2×C418yes—
92#341—22no—
93#341—42no—
94#341C2×C414yes—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#336—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4#4111yescharacteristic
109C23×C4#4111yescharacteristic
110C23×C4C2411yescharacteristic
111C2×M16—21no—
112C2×M16—21no—
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—22no—
116C2×M16—22no—
117C2×M16—22no—
118C2×M16—22no—
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×M16C22×C414yes—
129C2×M16C22×C412yes—
130C2×M16#4112yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8C22×C412yes—
142C22×C8C2×D412yes—
143C22×C8C2×D412yes—
144C22×C8#4112yes—
145#117—21no—
146#117—21no—
147#117—22no—
148#117—22no—
149#117C22×C412yes—
150#117#4112yes—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—84no—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—21no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42C2×D411yescharacteristic
177C2×C42C2×D411yescharacteristic
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180#107—28no—
181#107—44no—
182#106—28no—
183#106—28no—
184C4⋊C8—28no—
185C4⋊C8—28no—
186#97—42no—
187#97—42no—
188#97—42no—
189C4×C8—42no—
190#95—22no—
191#95—22no—
192#95—42no—
193#95—42no—
194C24C2×#4111yescharacteristic
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—24no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—82no—
208C22×C4—82no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4C23×C411yescharacteristic
216C22×C4C22×D411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C2×#4111yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C22×C4H(5,𝔽2)12yes—
222C22×C42−1+412yes—
223C22×C42−1+412yes—
224C22×C42−1+412yes—
225M16—24no—
226M16—24no—
227M16—24no—
228M16—24no—
229M16—42no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237C2×C8—24no—
238C2×C8—22no—
239C2×C8—22no—
240C2×C8—24no—
241C2×C8—22no—
242C2×C8—24no—
243C2×C8—22no—
244C2×C8—22no—
245C2×C8—22no—
246C2×C8—22no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—42no—
251C2×C8—41no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8C4×D414yes—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—24no—
262C4⋊C4—24no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—84no—
270C4⋊C4—84no—
271C4⋊C4C4×D414yes—
272C22⋊C4—22no—
273C22⋊C4—22no—
274C22⋊C4—42no—
275C42—22no—
276C42—22no—
277C42—21no—
278C42—21no—
279C42—21no—
280C42—21no—
281C42—22no—
282C42—22no—
283C42—41no—
284C42—41no—
285C42—82no—
286C42—82no—
287C23—22no—
288C23—22no—
289C23C4×#4111yescharacteristic
290C23C2×#12111yescharacteristic
291C23#52911yescharacteristic
292C23#53912yes—
293C23#54112yes—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—84no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—84no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—162no—
319C2×C4—162no—
320C2×C4C2×C4×D411yescharacteristic
321C2×C4C2×C4×D411yescharacteristic
322C2×C4#51912yes—
323C2×C4#52012yes—
324C2×C4#52511yescharacteristic
325C2×C4#53211yescharacteristic
326C2×C4#53411yescharacteristic
327C2×C4#53611yescharacteristic
328C2×C4#53612yes—
329C2×C4#53611yescharacteristic
330C2×C4#53611yescharacteristic
331C2×C4#53712yes—
332C2×C4#53912yes—
333C2×C4#53912yes—
334C2×C4#54112yes—
335C2×C4#54112yes—
336C2×C4#54112yes—
337C2×C4#54212yes—
338C8—24no—
339C8—42no—
340C8—44no—
341C8—42no—
342C8—42no—
343C8—82no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—24no—
349C22C4×#12111yescharacteristic
350C22#233612yes—
351C22#261811yescharacteristic
352C22#262111yescharacteristic
353C22#265512yes—
354C22#306812yes—
355C22#306912yes—
356C4—44no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—84no—
362C4—82no—
363C4—162no—
364C4#232211yescharacteristic
365C4#232312yes—
366C4#232711yescharacteristic
367C4#263411yescharacteristic
368C4#263811yescharacteristic
369C4#266412yes—
370C2—22no—
371C2—22no—
372C2#1431411yescharacteristic
373C2#2155712yes—
374C2#2979111yescharacteristic
375C2#2979211yescharacteristic
376C2#2988912yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C2×#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2090C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20902
C2×#19442
#210342
#204272
#113662
#113032
#112702
#112392
#112162
#109132
#108232