Finite Group Databasebeta

← #220296

Subgroups — #220296

Order 512.

909 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3336C211yesmaximal, characteristic
2#22611C212yesmaximal, complemented
3#22478C212yesmaximal
4#9782C212yesmaximal, complemented
5#9256C214yesmaximal, complemented
6#8609C211yesmaximal, characteristic
7#8382C212yesmaximal
8C8×#99C211yesmaximal, 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#1942—22no—
26#1918—24no—
27#1899—24no—
28#1823C2214yes—
29#1820—22no—
30#1808C2212yes—
31#1799C2212yes—
32C4×#107—22no—
33C4×#99C2211yescharacteristic
34C8×C22⋊C4C2212yes—
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×#107—22no—
79C2×#107C2×C411yescharacteristic
80C2×#107C2×C412yes—
81C2×#107C2×C411yescharacteristic
82C2×#107C2×C411yescharacteristic
83C2×#107D414yes—
84C2×#107D412yes—
85C2×#107D412yes—
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×C4×C8—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#342D411yescharacteristic
109#342D411yescharacteristic
110#340—22no—
111#340—24no—
112#340—24no—
113#336—22no—
114#328—44no—
115#324—21no—
116#323—21no—
117C22×Q8C22×C412yes—
118C22×D4C2×D411yescharacteristic
119C23×C4—22no—
120C23×C4C22×C411yescharacteristic
121C23×C4C2411yescharacteristic
122C2×Q16—21no—
123C2×Q16—21no—
124C2×Q16—42no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—22no—
128C22×C8—42no—
129C22×C8—42no—
130C22×C8C22×C411yescharacteristic
131C22×C8C22×C411yescharacteristic
132C22×C8C22×C411yescharacteristic
133C22×C8C22×C411yescharacteristic
134C22×C8C2×D412yes—
135C22×C8C2×D412yes—
136#126—44no—
137#126—44no—
138#125—24no—
139#125—42no—
140#125—44no—
141#125—44no—
142#123—44no—
143#123—44no—
144C22⋊Q8—24no—
145C22⋊Q8—24no—
146C22⋊Q8—24no—
147C22⋊Q8—42no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—42no—
151C4×D4—22no—
152C4×D4—22no—
153#117—22no—
154#117—22no—
155#117—44no—
156#117—44no—
157#117—42no—
158#117—48no—
159#117—84no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4#4111yescharacteristic
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4C2×D412yes—
180C2×C22⋊C4C2×D411yescharacteristic
181C2×C22⋊C4C2×D412yes—
182C2×C22⋊C4#4111yescharacteristic
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—44no—
187C2×C42C22×C411yescharacteristic
188#107—21no—
189#107—21no—
190#107—44no—
191#107C22⋊C412yes—
192#107C22⋊C411yescharacteristic
193#107C22⋊C411yescharacteristic
194#107C22⋊C412yes—
195#107C22⋊C412yes—
196#107C22⋊C414yes—
197#107C22⋊C414yes—
198#106—22no—
199#106—22no—
200#106—24no—
201C4⋊C8—22no—
202#103—24no—
203#103—22no—
204#103—22no—
205#103—24no—
206#99—24no—
207#99—82no—
208C22⋊C8—22no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—22no—
212C4×C8—21no—
213C4×C8—21no—
214C4×C8—44no—
215#95—22no—
216#95—24no—
217#95—44no—
218#95—44no—
219C24C4×D411yescharacteristic
220C24C22×D411yescharacteristic
221C2×Q8—22no—
222C2×Q8—22no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×D4—22no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—21no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—44no—
247C22×C4C4×D411yescharacteristic
248C22×C4C4×D411yescharacteristic
249C22×C4C4×D411yescharacteristic
250C22×C4C23×C411yescharacteristic
251C22×C4C22×D412yes—
252C22×C4C2×#4111yescharacteristic
253C22×C4H(5,𝔽2)12yes—
254Q16—42no—
255Q16—42no—
256C2×C8—24no—
257C2×C8—22no—
258C2×C8—21no—
259C2×C8—21no—
260C2×C8—41no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8C2×C22⋊C411yescharacteristic
270C2×C8C2×C22⋊C412yes—
271C2×C8C2×C22⋊C411yescharacteristic
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—88no—
288C4⋊C4C2×C22⋊C412yes—
289C4⋊C4C2×C22⋊C412yes—
290C4⋊C4C2×C22⋊C414yes—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—44no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4#13514yes—
303C42—21no—
304C42—21no—
305C42—42no—
306C42—42no—
307C42—42no—
308C42—42no—
309C42—42no—
310C42—42no—
311C42—88no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23C2×C4×D411yescharacteristic
320C23D4212yes—
321C23C2×#13512yes—
322C23#57611yescharacteristic
323C23#57811yescharacteristic
324Q8—42no—
325Q8—42no—
326Q8—82no—
327D4—42no—
328D4—42no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—24no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—82no—
365C2×C4—82no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4C22×C22⋊C411yescharacteristic
370C2×C4#51811yescharacteristic
371C2×C4#54612yes—
372C8—21no—
373C8—22no—
374C8—41no—
375C8—41no—
376C8—82no—
377C8#41012yes—
378C22—22no—
379C22—21no—
380C22—22no—
381C22—21no—
382C22—42no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—81no—
389C22—81no—
390C22D4×C22⋊C411yescharacteristic
391C22C4×#13511yescharacteristic
392C22#295111yescharacteristic
393C22#328012yes—
394C22#328312yes—
395C4—22no—
396C4—21no—
397C4—44no—
398C4—41no—
399C4—41no—
400C4—42no—
401C4—42no—
402C4—44no—
403C4—41no—
404C4—82no—
405C4—82no—
406C4—82no—
407C4—164no—
408C4C2×#41011yescharacteristic
409C4#288411yescharacteristic
410C2—22no—
411C2—22no—
412C2—21no—
413C2—21no—
414C2—81no—
415C2#2240611yescharacteristic
416C2#2244811yescharacteristic
417C2#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. #22611C2semidirect
4. #9782C2semidirect
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
#226112
#224782
#97822
#92562
#86092
#83822
C8×#992