Finite Group Databasebeta

← #378797

Subgroups — #378797

Order 512.

834 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2090C211yesmaximal, characteristic
2#21033C212yesmaximal
3C2×C4×#106C211yesmaximal, characteristic
4#20435C211yesmaximal, characteristic
5#11366C211yesmaximal, characteristic
6#11271C211yesmaximal, characteristic
7#11238C211yesmaximal, characteristic
8#10896C212yesmaximal
9#10827C212yesmaximal
10#10822C211yesmaximal, characteristic
11#10802C212yesmaximal
12C2×#428C2212yes—
13C22×#106C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C2×#424C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C2×#405C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19#2296C2211yescharacteristic
20C2×C4×C4⋊C4C2212yes—
21#2090C418yescomplemented
22#1921—22no—
23#1920—22no—
24#1919—22no—
25#1918—22no—
26C4×#106—22no—
27C4×#106—22no—
28#1771—22no—
29C8×C4⋊C4—22no—
30C2×#343C2212yes—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2212yes—
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37#1736C2212yes—
38#1736C2212yes—
39#1736C2212yes—
40C2×#340C2212yes—
41#1729C2211yescharacteristic
42#1728C2212yes—
43#1728C2211yescharacteristic
44C22×M16C2312yes—
45C22×M16C2311yescharacteristic
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C2×#117C2311yescharacteristic
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C42—22no—
55C22×C42C2311yescharacteristic
56#428—24no—
57#428C2×C418yes—
58C2×#107—22no—
59C2×#106—22no—
60C2×#106—22no—
61C2×#106—22no—
62C2×#106—22no—
63C2×#106—24no—
64C2×#106C2×C414yes—
65#424—22no—
66#424C2×C414yes—
67C2×C4⋊C8—22no—
68#405—22no—
69#405—22no—
70#405—24no—
71C2×#97—21no—
72C2×#97—21no—
73C2×C4×C8—21no—
74C2×C4×C8—24no—
75C2×C4×C8—21no—
76C2×C4×C8D412yes—
77C2×C4×C8D412yes—
78#389D412yes—
79#389D412yes—
80#385—21no—
81#385—21no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—22no—
86C4×C4⋊C4—22no—
87C2×#95—22no—
88#343—24no—
89#343—24no—
90#343—24no—
91#343C2×C418yes—
92#341—22no—
93#341—24no—
94#341C2×C414yes—
95#340—22no—
96#340—22no—
97#340—22no—
98#340—22no—
99#340—22no—
100#340—22no—
101#340—22no—
102#340—22no—
103#336—22no—
104#336—22no—
105#336—22no—
106#336—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4#4111yescharacteristic
110C23×C4#4111yescharacteristic
111C23×C4C2411yescharacteristic
112C2×M16—21no—
113C2×M16—21no—
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×M16—22no—
129C2×M16C22×C414yes—
130C2×M16C22×C412yes—
131C2×M16#4112yes—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—21no—
140C22×C8—22no—
141C22×C8C22×C412yes—
142C22×C8C2×D412yes—
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8#4112yes—
146C22×C8#4114yes—
147#117—21no—
148#117—21no—
149#117—22no—
150#117—22no—
151#117C22×C412yes—
152#117#4112yes—
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—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22×C414yes—
166C2×C4⋊C4C2×D411yescharacteristic
167C2×C4⋊C4C2×D411yescharacteristic
168C2×C4⋊C4C2×D412yes—
169C2×C4⋊C4#4114yes—
170C2×C22⋊C4—21no—
171C2×C22⋊C4—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—21no—
176C2×C42—42no—
177C2×C42—42no—
178C2×C42—84no—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181#107—28no—
182#106—28no—
183#106—28no—
184#106—28no—
185C4⋊C8—28no—
186#97—24no—
187#97—42no—
188C4×C8—24no—
189C4×C8—24no—
190C4×C8—24no—
191C4×C8—42no—
192#95—22no—
193#95—22no—
194#95—42no—
195#95—42no—
196C24C2×#4111yescharacteristic
197C22×C4—22no—
198C22×C4—24no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—42no—
209C22×C4—42no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4—82no—
213C22×C4—82no—
214C22×C4#12112yes—
215C22×C4#12112yes—
216C22×C4C23×C411yescharacteristic
217C22×C4C22×D411yescharacteristic
218C22×C4C22×D411yescharacteristic
219C22×C4C2×#4111yescharacteristic
220C22×C4C2×#4111yescharacteristic
221C22×C4C2×#4112yes—
222C22×C4C2×#4112yes—
223C22×C4C2×#4111yescharacteristic
224C22×C42−1+412yes—
225C22×C42−1+412yes—
226M16—24no—
227M16—24no—
228M16—24no—
229M16—24no—
230M16—42no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236M16—42no—
237M16—42no—
238C2×C8—24no—
239C2×C8—22no—
240C2×C8—21no—
241C2×C8—21no—
242C2×C8—22no—
243C2×C8—24no—
244C2×C8—21no—
245C2×C8—21no—
246C2×C8—24no—
247C2×C8—24no—
248C2×C8—22no—
249C2×C8—24no—
250C2×C8—22no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8C4×D414yes—
261C2×C8#12312yes—
262C2×C8#12312yes—
263C4⋊C4—22no—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—42no—
274C4⋊C4—42no—
275C4⋊C4—44no—
276C4⋊C4C4×D414yes—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—42no—
280C42—22no—
281C42—22no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—41no—
286C42—41no—
287C42—84no—
288C42—84no—
289C23—22no—
290C23—22no—
291C23C4×#4111yescharacteristic
292C23C2×#12111yescharacteristic
293C23#52911yescharacteristic
294C23#55212yes—
295C23#55212yes—
296C2×C4—28no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—44no—
312C2×C4—84no—
313C2×C4—84no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—84no—
319C2×C4—84no—
320C2×C4#47114yes—
321C2×C4C2×C4×D411yescharacteristic
322C2×C4C2×C4×D411yescharacteristic
323C2×C4C4×#4112yes—
324C2×C4#51912yes—
325C2×C4C2×#12311yescharacteristic
326C2×C4#52511yescharacteristic
327C2×C4#52511yescharacteristic
328C2×C4#52912yes—
329C2×C4#53211yescharacteristic
330C2×C4#53611yescharacteristic
331C2×C4#53611yescharacteristic
332C2×C4#54112yes—
333C2×C4#54712yes—
334C2×C4#54812yes—
335C2×C4#55212yes—
336C2×C4#55212yes—
337C2×C4#55212yes—
338C2×C4#55612yes—
339C8—22no—
340C8—22no—
341C8—24no—
342C8—42no—
343C8—44no—
344C8—42no—
345C8—82no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—22no—
351C22—22no—
352C22C4×#12111yescharacteristic
353C22#237112yes—
354C22#248611yescharacteristic
355C22#252412yes—
356C22#253611yescharacteristic
357C22C2×#47112yes—
358C22#306712yes—
359C4—24no—
360C4—42no—
361C4—44no—
362C4—81no—
363C4—81no—
364C4—84no—
365C4—84no—
366C4C4×#12311yescharacteristic
367C4#232711yescharacteristic
368C4#236212yes—
369C4#248411yescharacteristic
370C4#251912yes—
371C4#259911yescharacteristic
372C2—22no—
373C2—22no—
374C2#1422311yescharacteristic
375C2C4×#47112yes—
376C2#2767211yescharacteristic
377C2#2767411yescharacteristic
378C2#2821812yes—
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
#210332
C2×C4×#1062
#204352
#113662
#112712
#112382
#108962
#108272
#108222
#108022