Finite Group Databasebeta

← #354924

Subgroups — #354924

Order 512.

1762 conjugacy classes of subgroups.

The subgroup lattice has 417 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#405C211yesmaximal, characteristic
2C2×#1808C211yesmaximal, characteristic
3C2×#1807C211yesmaximal, characteristic
4C2×#1776C212yesmaximal, complemented
5#13777C212yesmaximal
6#10863C212yesmaximal, complemented
7#10820C212yesmaximal, complemented
8#10722C212yesmaximal
9C4×#340C212yesmaximal, complemented
10C23×M16C2211yescharacteristic
11C24×C8C2211yescharacteristic
12C22×#117C2211yescharacteristic
13C22×#107C2212yes—
14C22×#106C2212yes—
15C2×#405—28no—
16C2×#405C418yescomplemented
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C2×#380C2212yes—
22C2×#380C2212yes—
23C2×C4×C4⋊C4C2214yes—
24#1808C418yescomplemented
25#1807C418yescomplemented
26#1776—28no—
27C2×#340C2214yes—
28C2×#340C2214yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31#1437—24no—
32#1436—24no—
33C4⋊#95C2214yes—
34C24×C4C2311yescharacteristic
35C22×M16—22no—
36C22×M16—22no—
37C22×M16—24no—
38C22×M16—24no—
39C22×M16C2×C414yes—
40C23×C8—24no—
41C23×C8—24no—
42C23×C8C2×C414yes—
43C23×C8C2×C414yes—
44C23×C8C2×C414yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×#117—24no—
50C2×#117—24no—
51C2×#117C2×C414yes—
52C2×#117Q814yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2314yes—
55C22×C4⋊C4C2314yes—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4D412yes—
59C22×C42—24no—
60C22×C42C2311yescharacteristic
61C22×C42C2311yescharacteristic
62C2×#107—24no—
63C2×#107—24no—
64C2×#107C2×C418yes—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106C2×C418yes—
68#405—232no—
69C2×#97—24no—
70C2×#97C2×C414yes—
71C2×#97C2×C414yes—
72C2×#97D418yes—
73C2×C4×C8—24no—
74C2×C4×C8—24no—
75C2×C4×C8C2×C414yes—
76C2×C4×C8C2×C414yes—
77#380—24no—
78#380—24no—
79#380C2×C418yes—
80#380C2×C418yes—
81C4×C4⋊C4—28no—
82C4×C4⋊C4—28no—
83C2×#95—22no—
84C2×#95—22no—
85C2×#95—24no—
86C2×#95—24no—
87C2×#95—24no—
88#340—28no—
89#340—28no—
90#340C2×C418yes—
91#340C2×C418yes—
92C25C2×D411yescharacteristic
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—24no—
98C23×C4—24no—
99C23×C4C22×C412yes—
100C23×C4C22×C412yes—
101C23×C4C22×C412yes—
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×Q812yes—
106C23×C4#4112yes—
107C23×C4#4112yes—
108C23×C4C2411yescharacteristic
109C2×M16—216no—
110C2×M16—216no—
111C2×M16—28no—
112C2×M16—28no—
113C2×M16—28no—
114C22×C8—22no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—22no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—28no—
123C22×C8—28no—
124C22×C8—216no—
125C22×C8—216no—
126C22×C8C42116yescomplemented
127C22×C8C22×C412yes—
128C22×C8C22×C412yes—
129C22×C8C22×C412yes—
130C22×C8C22×C412yes—
131C22×C8C22×C412yes—
132C22×C8C22×C412yes—
133C22×C8C22×C412yes—
134C22×C8C22×C412yes—
135C22×C8C2×D414yes—
136C22×C8C2×D414yes—
137C22×C8#4114yes—
138C22×C8#4114yes—
139#117—28no—
140#117—28no—
141#117—216no—
142C2×C4⋊C4—28no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4C4⋊C418yes—
154C2×C4⋊C4C22×C418yes—
155C2×C4⋊C4C22×C418yes—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—28no—
159C2×C22⋊C4—84no—
160C2×C22⋊C4—84no—
161C2×C22⋊C4C4⋊C418yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—48no—
165C2×C42—84no—
166C2×C42—84no—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C2×D414yes—
170C2×C42C2×Q812yes—
171C2×C42C2×Q812yes—
172C2×C42#4114yes—
173#107—216no—
174#106—216no—
175#97—28no—
176#97C22⋊C4116yescomplemented
177C4×C8—28no—
178C4×C8—28no—
179#95—28no—
180#95—28no—
181#95—48no—
182#95—84no—
183#95—84no—
184C24—22no—
185C24—22no—
186C24—22no—
187C24—24no—
188C24—22no—
189C24C2×C4⋊C412yes—
190C24C4×D412yes—
191C24C4×D412yes—
192C24C22×D411yescharacteristic
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—24no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—28no—
217C22×C4—28no—
218C22×C4—44no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—44no—
222C22×C4—44no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4C2×C4214yes—
231C22×C4C2×C4⋊C414yes—
232C22×C4C2×C4⋊C412yes—
233C22×C4C2×C4⋊C414yes—
234C22×C4#11714yes—
235C22×C4C4×D412yes—
236C22×C4C4×D412yes—
237C22×C4C4×Q814yes—
238C22×C4C4×Q814yes—
239C22×C4#12118yes—
240C22×C4C23×C411yescharacteristic
241C22×C4C23×C411yescharacteristic
242C22×C4C23×C411yescharacteristic
243C22×C4C22×D412yes—
244C22×C4C22×Q811yescharacteristic
245C22×C4C2×#4112yes—
246C22×C4C2×#4111yescharacteristic
247C22×C4C2×#4111yescharacteristic
248C22×C4C2×#4112yes—
249C22×C4C2×#4112yes—
250M16—232no—
251C2×C8—22no—
252C2×C8—22no—
253C2×C8—22no—
254C2×C8—22no—
255C2×C8—22no—
256C2×C8—22no—
257C2×C8—24no—
258C2×C8—24no—
259C2×C8—24no—
260C2×C8—24no—
261C2×C8—24no—
262C2×C8—24no—
263C2×C8—232no—
264C2×C8C2×C4214yes—
265C2×C8C2×C4214yes—
266C2×C8C2×C22⋊C414yes—
267C2×C8C2×C22⋊C414yes—
268C2×C8#11714yes—
269C2×C8#11714yes—
270C2×C8C4×D418yes—
271C2×C8C4×D418yes—
272C4⋊C4—216no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—28no—
276C4⋊C4—28no—
277C4⋊C4—44no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—216no—
282C22⋊C4—88no—
283C22⋊C4—88no—
284C42—22no—
285C42—22no—
286C42—24no—
287C42—816no—
288C42C2×C22⋊C414yes—
289C42#11714yes—
290C42C22⋊Q818yes—
291C23—24no—
292C23—24no—
293C23—22no—
294C23—24no—
295C23—22no—
296C23—24no—
297C23—24no—
298C23—22no—
299C23—22no—
300C23—24no—
301C23—24no—
302C23—44no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23C4×C4⋊C414yes—
307C23C22×C4⋊C411yescharacteristic
308C23C2×C4×D411yescharacteristic
309C23C2×C4×D411yescharacteristic
310C23C2×#12112yes—
311C23#53212yes—
312C23#57612yes—
313C23#57714yes—
314C23#57812yes—
315C2×C4—24no—
316C2×C4—24no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—28no—
320C2×C4—28no—
321C2×C4—24no—
322C2×C4—24no—
323C2×C4—24no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—28no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—44no—
333C2×C4—44no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—48no—
337C2×C4—48no—
338C2×C4—48no—
339C2×C4—48no—
340C2×C4—84no—
341C2×C4—84no—
342C2×C4—84no—
343C2×C4—84no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4C4×C4⋊C414yes—
349C2×C4#38018yes—
350C2×C4C22×C4211yescharacteristic
351C2×C4C22×C22⋊C411yescharacteristic
352C2×C4C2×#11711yescharacteristic
353C2×C4C2×#11711yescharacteristic
354C2×C4C2×#11711yescharacteristic
355C2×C4C2×#11711yescharacteristic
356C2×C4C2×C4×D412yes—
357C2×C4C2×C4×D412yes—
358C2×C4C2×C4×Q811yescharacteristic
359C2×C4C2×C4×Q811yescharacteristic
360C2×C4C4×#4112yes—
361C2×C4C4×#4112yes—
362C2×C4C2×C22⋊Q812yes—
363C2×C4#52512yes—
364C2×C4#52512yes—
365C2×C4#52912yes—
366C2×C4#52912yes—
367C2×C4#53312yes—
368C8—24no—
369C8—24no—
370C8—24no—
371C8C4×C22⋊C418yes—
372C22—24no—
373C22—24no—
374C22—24no—
375C22—44no—
376C22—44no—
377C22—48no—
378C22—44no—
379C22—44no—
380C22C2×C4×C4⋊C411yescharacteristic
381C22C2×#38011yescharacteristic
382C22#228511yescharacteristic
383C22C4×#12112yes—
384C22C4×#12112yes—
385C22#291212yes—
386C22#291312yes—
387C22#294811yescharacteristic
388C22#294912yes—
389C22#295011yescharacteristic
390C22#295112yes—
391C22#295212yes—
392C22#306214yes—
393C22#306314yes—
394C22#317814yes—
395C22#317914yes—
396C4—28no—
397C4—24no—
398C4—44no—
399C4—88no—
400C4—88no—
401C4C2×C4×C22⋊C411yescharacteristic
402C4C4×#11711yescharacteristic
403C4#228011yescharacteristic
404C4#228311yescharacteristic
405C4C4×C22⋊Q812yes—
406C4C4×C22⋊Q812yes—
407C2—44no—
408C2C4×#38011yescharacteristic
409C2#2045411yescharacteristic
410C2#2045511yescharacteristic
411C2#2066412yes—
412C2#2082212yes—
413C2#2152912yes—
414C2#2153012yes—
415C2#2153114yes—
subgroupG/H
center, socleC24C22×D4
commutatorC2×C4C22×C42
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1776C2semidirect
6. #10863C2semidirect
7. #10820C2semidirect
9. C4×#340C2semidirect
16. C2×#405C4semidirect
24. #1808C4semidirect
25. #1807C4semidirect
126. C22×C8C42semidirect
176. #97C22⋊C4semidirect

Maximal subgroups

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

subgroupindex
C22×#4052
C2×#18082
C2×#18072
C2×#17762
#137772
#108632
#108202
#107222
C4×#3402