Finite Group Databasebeta

← #361107

Subgroups — #361107

Order 512.

886 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21039C212yesmaximal
2#20763C211yesmaximal, characteristic
3#20453C211yesmaximal, characteristic
4C2×#1772C211yesmaximal, characteristic
5#11340C212yesmaximal
6#11249C212yesmaximal
7#10891C212yesmaximal
8#10841C211yesmaximal, characteristic
9#10827C211yesmaximal, characteristic
10#10827C211yesmaximal, characteristic
11#10818C211yesmaximal, characteristic
12C22×#108C2211yescharacteristic
13C2×#428C2212yes—
14C2×#427C2211yescharacteristic
15C2×#405C2211yescharacteristic
16C2×#405C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18#2317C2212yes—
19C2×#384C2212yes—
20#2274C2212yes—
21#1921—22no—
22#1920—22no—
23#1919—22no—
24#1918—22no—
25#1836—22no—
26#1835—21no—
27#1834—21no—
28#1776—22no—
29C4×#107—21no—
30C4×#106—21no—
31#1772C418yescomplemented
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#341C2211yescharacteristic
35#1736C2212yes—
36#1736C2212yes—
37#1736C2212yes—
38#1735C2211yescharacteristic
39C2×#340C2211yescharacteristic
40C2×#340C2212yes—
41C2×#340C2211yescharacteristic
42C2×#339C2212yes—
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45#1729C2212yes—
46C22×M16C2312yes—
47C22×M16C2311yescharacteristic
48C22×M16C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C2×#117C2312yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4—42no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C42C2311yescharacteristic
58C2×#108—24no—
59C2×#108C2×C414yes—
60#428—24no—
61#428—24no—
62#427C2×C414yes—
63#427Q814yes—
64C2×#107—22no—
65C2×#107D411yescharacteristic
66C2×#107D411yescharacteristic
67C2×#106—22no—
68C2×#106D411yescharacteristic
69C2×#106D411yescharacteristic
70#405—22no—
71#405—22no—
72#405C2×C414yes—
73#405C2×C414yes—
74C2×#97—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78C2×C4×C8—24no—
79C2×C4×C8C2×C414yes—
80#389—22no—
81#389—22no—
82#385—42no—
83#384—22no—
84#384—24no—
85#384—24no—
86#382—22no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—22no—
91C4×C22⋊C4—42no—
92#343—24no—
93#343—24no—
94#341—22no—
95#340—21no—
96#340—21no—
97#340—22no—
98#340—22no—
99#340—21no—
100#340—21no—
101#340—22no—
102#340—22no—
103#340—22no—
104#340—22no—
105#340—22no—
106#340—22no—
107#340D411yescharacteristic
108#340D411yescharacteristic
109#340D412yes—
110#339—24no—
111#339C2×C414yes—
112#339C2×C414yes—
113#336—22no—
114#336—22no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4C2411yescharacteristic
119C2×M16—21no—
120C2×M16—22no—
121C2×M16—21no—
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×M16C22×C412yes—
131C2×M16C22×C412yes—
132C2×M16#4114yes—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8C22×C412yes—
141C22×C8C22×C412yes—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×D411yescharacteristic
146C22×C8C2×Q812yes—
147C22×C8#4112yes—
148#117—21no—
149#117—21no—
150#117—21no—
151#117—21no—
152#117—22no—
153#117—22no—
154#117—22no—
155#117—22no—
156#117C22×C412yes—
157#117C22×C412yes—
158#117C2×Q812yes—
159#117C2×Q812yes—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—48no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4—84no—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4#4114yes—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—21no—
185C2×C22⋊C4—22no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—22no—
189C2×C42—21no—
190C2×C42—22no—
191C2×C42—42no—
192C2×C42—42no—
193C2×C42C22×C412yes—
194C2×C42#4114yes—
195#108—28no—
196#107—28no—
197#107C4⋊C414yescomplemented
198#107C4⋊C414yescomplemented
199#106—28no—
200#106C4⋊C414yescomplemented
201#106C4⋊C414yescomplemented
202#97—24no—
203#97—24no—
204C4×C8—24no—
205C4×C8—24no—
206C4×C8—28no—
207#95—42no—
208#95—42no—
209#95—42no—
210#95—42no—
211C24H(5,𝔽2)11yescharacteristic
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—24no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—82no—
230C22×C4—82no—
231C22×C4—82no—
232C22×C4—82no—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C22×Q811yescharacteristic
237C22×C4C2×#4112yes—
238C22×C4C2×#4112yes—
239C22×C4C2×#4111yescharacteristic
240C22×C4C2×#4112yes—
241C22×C42−1+412yes—
242C22×C42−1+411yescharacteristic
243M16—22no—
244M16—24no—
245M16—22no—
246M16—22no—
247M16—42no—
248M16—42no—
249M16—42no—
250M16—42no—
251M16—42no—
252M16#11714yes—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—22no—
258C2×C8—24no—
259C2×C8—22no—
260C2×C8—22no—
261C2×C8—24no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—24no—
265C2×C8—22no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—42no—
270C2×C8C2×C4⋊C412yes—
271C2×C8C2×C4⋊C412yes—
272C2×C8C4×D412yes—
273C2×C8C4×D412yes—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—24no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—24no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4—816no—
291C4⋊C4—162no—
292C4⋊C4—162no—
293C4⋊C4C2×C4⋊C412yes—
294C4⋊C4C2×C4⋊C412yes—
295C4⋊C4C2×C4⋊C412yes—
296C4⋊C4C2×C4⋊C412yes—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—22no—
300C22⋊C4—22no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C42—22no—
304C42—22no—
305C42—22no—
306C42—22no—
307C42—22no—
308C42—24no—
309C42—42no—
310C42—42no—
311C42—162no—
312C42—162no—
313C42#11714yes—
314C23—22no—
315C23—21no—
316C23—21no—
317C23#52011yescharacteristic
318C23#54212yes—
319C23D4211yescharacteristic
320C23#55112yes—
321C23#55411yescharacteristic
322C2×C4—28no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—24no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4—162no—
352C2×C4—162no—
353C2×C4C22×C4⋊C411yescharacteristic
354C2×C4C2×#11711yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×C4×D411yescharacteristic
357C2×C4C4×#4111yescharacteristic
358C2×C4#51911yescharacteristic
359C2×C4#52512yes—
360C2×C4#52912yes—
361C2×C4#53212yes—
362C2×C4#53312yes—
363C2×C4#54112yes—
364C2×C4#54712yes—
365C2×C4#54711yescharacteristic
366C2×C4#54812yes—
367C2×C4D4×Q811yescharacteristic
368C2×C4#55212yes—
369C2×C4#55212yes—
370C2×C4#55712yes—
371C8—22no—
372C8—22no—
373C8—22no—
374C8—22no—
375C8—24no—
376C8—42no—
377C8—82no—
378C22—21no—
379C22—21no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22#232011yescharacteristic
387C22#234311yescharacteristic
388C22D4×C4⋊C411yescharacteristic
389C22#246112yes—
390C22#252012yes—
391C22#329112yes—
392C22#329211yescharacteristic
393C22#329311yescharacteristic
394C4—28no—
395C4—42no—
396C4—42no—
397C4—42no—
398C4—42no—
399C4—82no—
400C4—82no—
401C4—84no—
402C4—162no—
403C4#228511yescharacteristic
404C4#229011yescharacteristic
405C4#234411yescharacteristic
406C4#235111yescharacteristic
407C4#246212yes—
408C4#251912yes—
409C2—21no—
410C2—21no—
411C2—22no—
412C2#1388911yescharacteristic
413C2#2268911yescharacteristic
414C2#2269211yescharacteristic
415C2#2684912yes—
416C2#2685012yes—
subgroupG/H
center, socleC23D42
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24H(5,𝔽2)
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
31. #1772C4semidirect
197. #107C4⋊C4semidirect
198. #107C4⋊C4semidirect
200. #106C4⋊C4semidirect
201. #106C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#210392
#207632
#204532
C2×#17722
#113402
#112492
#108912
#108412
#108272
#108182