Finite Group Databasebeta

← #496744

Subgroups — #496744

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2164C211yesmaximal, characteristic
2#20821C211yesmaximal, characteristic
3#20803C211yesmaximal, characteristic
4#13178C211yesmaximal, characteristic, complemented
5#13178C211yesmaximal, characteristic, complemented
6#13177C211yesmaximal, characteristic, complemented
7#13176C211yesmaximal, characteristic, complemented
8#12921C211yesmaximal, characteristic
9#12921C211yesmaximal, characteristic
10#12920C211yesmaximal, characteristic
11#12919C211yesmaximal, characteristic
12#12908C211yesmaximal, characteristic, complemented
13#12907C211yesmaximal, characteristic, complemented
14#12890C211yesmaximal, characteristic, complemented
15#12888C211yesmaximal, characteristic, complemented
16C2×#428—21no—
17C2×#427C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#405C2211yescharacteristic
20#2317—21no—
21#2314—21no—
22#2164C418yescomplemented
23C2×#367C2211yescharacteristic
24C2×#367C2211yescharacteristic
25C2×#367C2211yescharacteristic
26C2×#367C2211yescharacteristic
27C2×#366C2211yescharacteristic
28C2×#366C2211yescharacteristic
29C2×#366C2211yescharacteristic
30C2×#366C2211yescharacteristic
31C2×C4⋊C16C2211yescharacteristic
32C2×C4⋊C16C2211yescharacteristic
33C2×C22⋊C16C2211yescharacteristic
34C2×C22⋊C16C2211yescharacteristic
35C2×#346C2211yescharacteristic
36C2×C4×C16C2211yescharacteristic
37#1847—21no—
38#1847—21no—
39#1846C2211yescharacteristic
40#1846C2211yescharacteristic
41#1846C2211yescharacteristic
42#1845C2211yescharacteristic
43#1845C2211yescharacteristic
44#1845C2211yescharacteristic
45#1836—21no—
46#1809—21no—
47#1808C2211yescharacteristic
48#1808C2211yescharacteristic
49#1776—21no—
50C4×#107C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1380C2211yescharacteristic
53#1380C2211yescharacteristic
54#1380C2211yescharacteristic
55#1380C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60C22×M16D411yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#117—21no—
63C2×#117D411yescharacteristic
64C22×C4⋊C4—21no—
65C22×C16C2311yescharacteristic
66C22×C16C2311yescharacteristic
67C22×C16C2311yescharacteristic
68C22×C16C2311yescharacteristic
69#428—44no—
70#427—41no—
71#427—41no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#107—41no—
77C2×#107—41no—
78C2×#107C2311yescharacteristic
79C2×#107C2311yescharacteristic
80C2×#107C2311yescharacteristic
81C2×#107C2311yescharacteristic
82C2×#107C2311yescharacteristic
83C2×#107C2311yescharacteristic
84C2×#107C2311yescharacteristic
85C2×#107C2311yescharacteristic
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90#405C2×C414yes—
91#405Q814yes—
92C2×#97D411yescharacteristic
93C2×#97D411yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#389—41no—
97#389—41no—
98#388—41no—
99#388—41no—
100#387—41no—
101#387—41no—
102#385—41no—
103#385—41no—
104#384—21no—
105#384—21no—
106#384—21no—
107#382—21no—
108#382—21no—
109#382—21no—
110#380—21no—
111#380—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114#367—41no—
115#367—41no—
116#367—41no—
117#367—41no—
118#366—41no—
119#366—41no—
120#366—41no—
121#366—41no—
122C4⋊C16C2×C414yescomplemented
123C4⋊C16C2×C414yescomplemented
124C22⋊C16C2×C414yes—
125C22⋊C16C2×C414yes—
126#346C2×C414yescomplemented
127C4×C16C2×C414yescomplemented
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C2×M16—22no—
135C2×M16—22no—
136C2×M16C4⋊C414yes—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8C22×C412yes—
142C22×C8C2×D411yescharacteristic
143C22×C8C2×D411yescharacteristic
144C22×C8C2×D411yescharacteristic
145C22×C8C2×Q812yes—
146C22×C8C2411yescharacteristic
147#117—42no—
148#117—42no—
149#117C4⋊C414yes—
150#117Q1614yes—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—41no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—41no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4D811yescharacteristic
169C2×C4⋊C4D811yescharacteristic
170C2×C22⋊C4—41no—
171C2×C22⋊C4—41no—
172C2×C22⋊C4—41no—
173C2×C22⋊C4D811yescharacteristic
174C2×C22⋊C4D811yescharacteristic
175C2×C42—41no—
176C2×C42—41no—
177C2×C42—41no—
178C2×C42C2×D411yescharacteristic
179C2×C42C2×D411yescharacteristic
180C2×C16—21no—
181C2×C16—21no—
182C2×C16—21no—
183C2×C16—21no—
184C2×C16—21no—
185C2×C16—21no—
186C2×C16—21no—
187C2×C16—21no—
188C2×C16C22×C412yes—
189C2×C16C22×C412yes—
190C2×C16C22×C412yes—
191C2×C16C22×C412yes—
192#107—21no—
193#107—21no—
194#107—21no—
195#107—21no—
196#107—21no—
197#107—21no—
198#107—21no—
199#107—21no—
200#107—21no—
201#107—21no—
202#107—21no—
203#107—21no—
204#107—21no—
205#107—21no—
206#107—21no—
207#107—21no—
208#107—42no—
209#107—42no—
210#107—42no—
211#107—42no—
212#106—42no—
213#106—42no—
214#106—42no—
215#106—42no—
216#97C4⋊C414yes—
217#97C4⋊C414yes—
218C4×C8C22×C412yes—
219C4×C8C22×C412yes—
220C4×C8C2×Q812yes—
221C4×C8C2×Q812yes—
222#95—41no—
223#95—41no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95—41no—
228#95—41no—
229#95—41no—
230#95—41no—
231#95—41no—
232C24C2×D811yescharacteristic
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—82no—
246C22×C4—161no—
247C22×C4—161no—
248C22×C4C2×C4⋊C412yes—
249C22×C4C2×D811yescharacteristic
250C22×C4C2×D811yescharacteristic
251C22×C4C2×D811yescharacteristic
252C22×C4C2×D811yescharacteristic
253C22×C4C2×D811yescharacteristic
254C22×C4C2×Q1612yes—
255C22×C4C22×D411yescharacteristic
256M16—28no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—21no—
261C2×C8—22no—
262C2×C8—22no—
263C2×C8—22no—
264C2×C8—22no—
265C2×C8C2×C4⋊C412yes—
266C2×C8C2×C4⋊C412yes—
267C2×C8C2×C4⋊C412yes—
268C2×C8AGL(1,ℤ/8)12yes—
269C2×C8#13712yes—
270C2×C8C23×C411yescharacteristic
271C2×C8C22×Q811yescharacteristic
272C2×C8H(5,𝔽2)11yescharacteristic
273C2×C8H(5,𝔽2)11yescharacteristic
274C2×C82−1+411yescharacteristic
275C2×C82−1+411yescharacteristic
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—82no—
293C4⋊C4—161no—
294C4⋊C4—161no—
295C4⋊C4—161no—
296C4⋊C4—161no—
297C4⋊C4—161no—
298C4⋊C4—161no—
299C4⋊C4#10714yes—
300C4⋊C4#10714yes—
301C22⋊C4—82no—
302C22⋊C4—161no—
303C22⋊C4—161no—
304C22⋊C4#10714yes—
305C22⋊C4#10714yes—
306C42—82no—
307C42—161no—
308C42—161no—
309C42C2×C4⋊C412yes—
310C42C2×C4⋊C412yes—
311C42C2×Q1612yes—
312C42C2×Q1612yes—
313C16—22no—
314C16—22no—
315C16—22no—
316C16—22no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23C2×#10712yes—
324C23C22×D811yescharacteristic
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—81no—
340C2×C4—81no—
341C2×C4—81no—
342C2×C4—81no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C2×C4—81no—
349C2×C4—81no—
350C2×C4—81no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—164no—
364C2×C4C2×#10712yes—
365C2×C4C2×#10712yes—
366C2×C4C2×#10712yes—
367C2×C4C2×#10712yes—
368C2×C4C2×#10712yes—
369C2×C4C22×C4⋊C411yescharacteristic
370C2×C4#53511yescharacteristic
371C2×C4#53611yescharacteristic
372C2×C4C22×Q1611yescharacteristic
373C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
374C2×C4C2×#13711yescharacteristic
375C8—22no—
376C8—22no—
377C8#42814yes—
378C8#52011yescharacteristic
379C8#52011yescharacteristic
380C8#54411yescharacteristic
381C8#54411yescharacteristic
382C22—22no—
383C22—22no—
384C22—22no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22C22×#10711yescharacteristic
390C22#320511yescharacteristic
391C22#322611yescharacteristic
392C22#341511yescharacteristic
393C22#341611yescharacteristic
394C22#341611yescharacteristic
395C22#341711yescharacteristic
396C4—22no—
397C4—22no—
398C4—22no—
399C4—22no—
400C4—22no—
401C4—161no—
402C4—161no—
403C4—161no—
404C4—161no—
405C4—161no—
406C4—161no—
407C4—161no—
408C4—161no—
409C4#231711yescharacteristic
410C4C2×#42811yescharacteristic
411C4#320911yescharacteristic
412C4#323011yescharacteristic
413C2—42no—
414C2#2198211yescharacteristic
415C2#3341011yescharacteristic
416C2#3341111yescharacteristic
417C2#3379011yescharacteristic
418C2#3379111yescharacteristic
419C2#3380711yescharacteristic
420C2#3380811yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC24C2×D8
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #13178C2semidirect
5. #13178C2semidirect
6. #13177C2semidirect
7. #13176C2semidirect
12. #12908C2semidirect
13. #12907C2semidirect
14. #12890C2semidirect
15. #12888C2semidirect
22. #2164C4semidirect
122. C4⋊C16C2×C4semidirect
123. C4⋊C16C2×C4semidirect
126. #346C2×C4semidirect
127. C4×C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21642
#208212
#208032
#131782
#131772
#131762
#129212
#129202
#129192
#129082
#129072
#128902
#128882