Finite Group Databasebeta

← #220260

Subgroups — #220260

Order 512.

901 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic
2#22617C212yesmaximal, complemented
3#22478C212yesmaximal
4#9784C212yesmaximal, complemented
5#9256C214yesmaximal, complemented
6#8608C211yesmaximal, characteristic
7#8382C212yesmaximal
8#8160C211yesmaximal, characteristic
9C2×#552C2212yes—
10#3336—24no—
11#3336—22no—
12#3336—22no—
13C2×#499C411yescharacteristic, complemented
14C2×#485C412yescomplemented
15C2×#483C2212yes—
16C2×#479C412yescomplemented
17C2×#467C411yescharacteristic, complemented
18C2×#467C2211yescharacteristic
19C2×C4×Q16C411yescharacteristic, complemented
20C2×C8×D4C2211yescharacteristic
21C2×#427C411yescharacteristic, complemented
22C22×#107C2211yescharacteristic
23#2344C2214yes—
24#1996C2214yes—
25#1943—22no—
26#1921—24no—
27#1899—24no—
28#1823C2214yes—
29#1820—22no—
30#1807C2212yes—
31#1799C2212yes—
32#1776—22no—
33C4×#99C2211yescharacteristic
34#1752C2212yes—
35#1740—22no—
36#1520C2214yes—
37#1459C2211yescharacteristic
38#1289C2212yes—
39#1282C2212yes—
40C22×Q16C2×C411yescharacteristic
41C23×C8C2×C411yescharacteristic
42C23×C8C2311yescharacteristic
43#552—24no—
44#552—24no—
45#552—44no—
46C2×#126—22no—
47C2×#125—22no—
48C2×#125C2×C412yes—
49C2×#123—22no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2312yes—
52C2×C4×Q8C2×C412yes—
53C2×C4×D4C2311yescharacteristic
54C2×#117—22no—
55C2×#117—24no—
56C2×#117C2×C412yes—
57C22×C4⋊C4C2312yes—
58#499—22no—
59#499—22no—
60#485—24no—
61#485—24no—
62#483—24no—
63#483—24no—
64#479—24no—
65#479—24no—
66#467—22no—
67#467—22no—
68#467—22no—
69#467—22no—
70C4×Q16—22no—
71C4×Q16—21no—
72C4×Q16—21no—
73C8×D4—22no—
74C8×D4—22no—
75#427—22no—
76#427—21no—
77#427—21no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C412yes—
80C2×#107C2×C411yescharacteristic
81C2×#107C2×C411yescharacteristic
82C2×#107D414yes—
83C2×#107D412yes—
84C2×#107D412yes—
85C2×#106—22no—
86C2×#106C2×C412yes—
87C2×#106C2×C412yes—
88C2×C4⋊C8C2×C411yescharacteristic
89C2×#103C2×C412yes—
90C2×#103C2×C412yes—
91C2×#103C2×C412yes—
92C2×#99—22no—
93C2×#99D412yes—
94C2×C22⋊C8C2×C411yescharacteristic
95C2×C22⋊C8C2312yes—
96C2×C22⋊C8C2311yescharacteristic
97C2×#97—22no—
98C2×C4×C8C2×C411yescharacteristic
99#389—24no—
100#387—44no—
101#381—24no—
102#380C2314yes—
103C4×C4⋊C4—24no—
104C4×C22⋊C4C2312yes—
105#342—22no—
106#342—24no—
107#342—24no—
108#342D412yes—
109#340—22no—
110#340—24no—
111#340—24no—
112#336—22no—
113#328—44no—
114#324—21no—
115#323—21no—
116C22×Q8C22×C412yes—
117C22×D4C2×D411yescharacteristic
118C23×C4—22no—
119C23×C4C22×C411yescharacteristic
120C23×C4C2411yescharacteristic
121C2×Q16—21no—
122C2×Q16—21no—
123C2×Q16—42no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—22no—
127C22×C8—42no—
128C22×C8—42no—
129C22×C8C22×C411yescharacteristic
130C22×C8C22×C411yescharacteristic
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C411yescharacteristic
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135#126—44no—
136#126—44no—
137#125—24no—
138#125—42no—
139#125—44no—
140#125—44no—
141#123—44no—
142#123—44no—
143C22⋊Q8—24no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—42no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—42no—
150C4×D4—22no—
151C4×D4—22no—
152#117—22no—
153#117—22no—
154#117—44no—
155#117—44no—
156#117—42no—
157#117—48no—
158#117—84no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C22×C412yes—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C2×D414yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4#4111yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4C2×D414yes—
179C2×C22⋊C4C2×D411yescharacteristic
180C2×C22⋊C4#4111yescharacteristic
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—44no—
185C2×C42C22×C411yescharacteristic
186#107—21no—
187#107—21no—
188#107C22⋊C412yes—
189#107C22⋊C411yescharacteristic
190#107C22⋊C411yescharacteristic
191#107C22⋊C412yes—
192#107C22⋊C412yes—
193#107C22⋊C414yes—
194#107C22⋊C414yes—
195#106—22no—
196#106—22no—
197#106—24no—
198#106—44no—
199C4⋊C8—22no—
200#103—24no—
201#103—22no—
202#103—22no—
203#103—24no—
204#99—42no—
205#99—82no—
206C22⋊C8—22no—
207C22⋊C8—22no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210#97—44no—
211C4×C8—21no—
212C4×C8—21no—
213#95—22no—
214#95—24no—
215#95—44no—
216#95—44no—
217C24C4×D411yescharacteristic
218C24C22×D411yescharacteristic
219C2×Q8—22no—
220C2×Q8—22no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—22no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—41no—
239C22×C4—44no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4C4×D411yescharacteristic
244C22×C4C4×D411yescharacteristic
245C22×C4C4×D411yescharacteristic
246C22×C4C23×C411yescharacteristic
247C22×C4C22×D412yes—
248C22×C4C2×#4111yescharacteristic
249C22×C4H(5,𝔽2)12yes—
250Q16—42no—
251Q16—42no—
252C2×C8—24no—
253C2×C8—22no—
254C2×C8—21no—
255C2×C8—21no—
256C2×C8—41no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—41no—
261C2×C8—41no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8C2×C22⋊C411yescharacteristic
266C2×C8C2×C22⋊C412yes—
267C2×C8C2×C22⋊C411yescharacteristic
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—88no—
284C4⋊C4C2×C22⋊C412yes—
285C4⋊C4C2×C22⋊C412yes—
286C4⋊C4C2×C22⋊C414yes—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—24no—
290C22⋊C4—24no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—44no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—44no—
297C42—21no—
298C42—21no—
299C42—42no—
300C42—42no—
301C42—42no—
302C42—42no—
303C42—42no—
304C42—42no—
305C42—88no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C4×D411yescharacteristic
314C23D4212yes—
315C23#57512yes—
316C23#57712yes—
317Q8—42no—
318Q8—42no—
319Q8—82no—
320D4—44no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—24no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—84no—
356C2×C4—84no—
357C2×C4—84no—
358C2×C4C22×C22⋊C411yescharacteristic
359C2×C4#51811yescharacteristic
360C2×C4#54612yes—
361C8—21no—
362C8—22no—
363C8—41no—
364C8—41no—
365C8—82no—
366C8#41012yes—
367C22—22no—
368C22—21no—
369C22—22no—
370C22—21no—
371C22—42no—
372C22—42no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—81no—
377C22—81no—
378C22D4×C22⋊C411yescharacteristic
379C22#294311yescharacteristic
380C22#295211yescharacteristic
381C22#326714yes—
382C4—22no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—42no—
387C4—42no—
388C4—44no—
389C4—41no—
390C4—84no—
391C4—82no—
392C4—82no—
393C4—164no—
394C4C2×#41011yescharacteristic
395C4#288411yescharacteristic
396C2—24no—
397C2—21no—
398C2—21no—
399C2—81no—
400C2#2240611yescharacteristic
401C2#2243811yescharacteristic
402C2#2245711yescharacteristic
subgroupG/H
center, socleC22D4×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×D4C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22617C2semidirect
4. #9784C2semidirect
5. #9256C2semidirect
13. C2×#499C4semidirect
14. C2×#485C4semidirect
16. C2×#479C4semidirect
17. C2×#467C4semidirect
19. C2×C4×Q16C4semidirect
21. C2×#427C4semidirect

Maximal subgroups

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

subgroupindex
C2×#33362
#226172
#224782
#97842
#92562
#86082
#83822
#81602