Finite Group Databasebeta

← #44559588

Subgroups — #44559588

Order 32128.

764 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 141 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16860811C211yesmaximal, characteristic, complemented
2C2×#22548738C212yesmaximal, complemented
3#30632200C211yesmaximal, characteristic, complemented
4#30631848C212yesmaximal, complemented
5#30631837C211yesmaximal, characteristic, complemented
6#30631835C212yesmaximal, complemented
7#30631826C211yesmaximal, characteristic, complemented
8#30631823C211yesmaximal, characteristic, complemented
9#30631682C212yesmaximal, complemented
10C22⋊C8×D251C211yesmaximal, characteristic, complemented
11#451×C251C211yesmaximal, characteristic, complemented
12C2×#16860811—28no—
13C2×#16860811—21no—
14C2×#16860811—21no—
15C2×#16860811C2212yes—
16C2×#16860811C2212yes—
17C2×#16860811C2211yescharacteristic, complemented
18C22×D1004—21no—
19C22×D1004C2211yescharacteristic
20C22×C4×D251—21no—
21C22×C4×D251C2211yescharacteristic
22#22548738—28no—
23C2×#16860797C2212yescomplemented
24C2×#16860797C2212yescomplemented
25C2×D2008C2212yescomplemented
26C2×C8×D251C2212yescomplemented
27#22548700—21no—
28#22548691C2211yescharacteristic, complemented
29#22548690C2211yescharacteristic, complemented
30#22548689C2211yescharacteristic
31#22548685—21no—
32#22548677—21no—
33(C22⋊D251)⋊C4—21no—
34#22548673C2211yescharacteristic, complemented
35#22548639C2212yescomplemented
36#22548637C2212yescomplemented
37#22548635C2212yescomplemented
38C22⋊(C251⋊C8)C2211yescharacteristic
39(C2×D251)⋊C8C2212yescomplemented
40C22×Q8×C251C2211yescharacteristic
41C2×Q16×C251C2212yescomplemented
42C22⋊Q8×C251C2211yescharacteristic
43#103×C251C2212yescomplemented
44C22⋊C8×C251C2211yescharacteristic
45C23×D251—21no—
46C23×D251D411yescharacteristic
47#16860811—28no—
48#16860811—28no—
49#16860811—24no—
50#16860811—24no—
51#16860811—24no—
52#16860811—24no—
53#16860811D418yescomplemented
54C2×D1004—28no—
55C2×D1004—22no—
56C2×D1004—22no—
57C2×D1004—24no—
58C2×D1004—21no—
59C2×D1004—21no—
60C2×D1004—21no—
61C2×D1004C2311yescharacteristic
62C2×D1004C2312yes—
63C2×C22⋊D251—21no—
64C22×Dic251D411yescharacteristic
65C2×C4×D251—28no—
66C2×C4×D251—22no—
67C2×C4×D251—22no—
68C2×C4×D251—24no—
69C2×C4×D251—21no—
70C2×C4×D251D411yescharacteristic
71C2×C4×D251D411yescharacteristic
72C2×C4×D251C2311yescharacteristic
73C2×C4×D251C2311yescharacteristic
74#16860797—24no—
75#16860797—24no—
76D2008—24no—
77C2×C251⋊C8C2312yes—
78C8×D251—24no—
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81(C2×D251)⋊C4—21no—
82(C2×D251)⋊C4—21no—
83(C2×D251)⋊C4—21no—
84(C2×D251)⋊C4—21no—
85C4×Dic251—21no—
86C2×Q8×C251—24no—
87C2×Q8×C251—21no—
88C2×Q8×C251C2311yescharacteristic
89C2×Q8×C251C2312yes—
90C22×C4×C251—21no—
91C22×C4×C251C2311yescharacteristic
92Q16×C251—24no—
93C2×C8×C251C2312yes—
94C4⋊C4×C251—21no—
95C4⋊C4×C251C2311yescharacteristic
96C22⋊C4×C251—21no—
97C22×D251—24no—
98C22×D251—22no—
99C22×D251—22no—
100C22×D251—22no—
101C22×D251—21no—
102C22×D251—41no—
103C22×D251—42no—
104C22×D251—42no—
105C22×D251C2×D411yescharacteristic
106C22×D251C2×D411yescharacteristic
107D1004—28no—
108D1004—24no—
109D1004—28no—
110D1004—22no—
111D1004—22no—
112D1004—42no—
113D1004—42no—
114D1004C2×D414yes—
115C22⋊D251—42no—
116C2×Dic251—22no—
117C2×Dic251—21no—
118C2×Dic251C2×D411yescharacteristic
119C2×Dic251C2×D411yescharacteristic
120C4×D251—28no—
121C4×D251—22no—
122C4×D251—22no—
123C4×D251—28no—
124C4×D251—22no—
125C4×D251—42no—
126C4×D251C2×D414yes—
127C251⋊C8—22no—
128C23×C251C2×D411yescharacteristic
129Q8×C251—24no—
130Q8×C251—22no—
131Q8×C251—22no—
132Q8×C251C2×D414yes—
133C2×C4×C251—24no—
134C2×C4×C251—22no—
135C2×C4×C251—22no—
136C2×C4×C251—21no—
137C2×C4×C251—21no—
138C2×C4×C251C2×D411yescharacteristic
139C2×C4×C251C2411yescharacteristic
140C8×C251—22no—
141C22×C251—21no—
142C22×C251—21no—
143C22×C251#13512yescomplemented
144C22×C251C22×D411yescharacteristic
145C2×D251—21no—
146C2×D251—21no—
147C2×D251—21no—
148C2×D251—24no—
149C2×D251—21no—
150C2×D251—21no—
151C2×D251—44no—
152C2×D251—48no—
153C2×D251—42no—
154C2×D251—42no—
155C2×D251—41no—
156C2×D251—41no—
157C2×D251—41no—
158C2×D251#12012yes—
159C4×C251—24no—
160C4×C251—21no—
161C4×C251—41no—
162C4×C251—41no—
163C4×C251C22×D412yes—
164Dic251—21no—
165Dic251—41no—
166Dic251#12012yes—
167C2×C251—22no—
168C2×C251C2×#12011yescharacteristic
169C2×C251C2×#13511yescharacteristic
170C2×C251#57511yescharacteristic
171D251—22no—
172D251—41no—
173D251—44no—
174D251—81no—
175C251#300211yescharacteristic, complemented, Sylow
176#3002—2511nomaximal, Sylow
177C22×#41—2511no—
178C2×#135—2512no—
179#525—2511no—
180#452—2512no—
181#451D25111yescharacteristic, complemented
182#450—2511no—
183#449—2512no—
184#448—2511no—
185#447—2511no—
186#416—2512no—
187C2×C22⋊C8—2511no—
188C2×#41—5028no—
189C2×#41—5021no—
190C2×#41—5021no—
191C2×#41—2511no—
192C2×#41—2512no—
193C2×#41—2512no—
194C22×Q8C2×D25111yescharacteristic
195C22×D4—5021no—
196C22×D4—2511no—
197C23×C4—5021no—
198C23×C4—2511no—
199#135—5028no—
200C2×Q16C2×D25112yescomplemented
201C2×SD16—2512no—
202C2×SD16—2512no—
203C2×D8—2512no—
204C22×C8—2512no—
205#123—5021no—
206C22⋊Q8—2511no—
207C22⋊Q8C2×D25111yescharacteristic
208#121—2511no—
209#121—2511no—
210#120—5021no—
211C4×D4—5021no—
212C4×D4—5021no—
213#117—2511no—
214#103—2512no—
215#103C2×D25112yescomplemented
216#102—2512no—
217#102—2512no—
218C22⋊C8—2511no—
219C22⋊C8—2512no—
220C22⋊C8C2×D25111yescharacteristic
221C24—5021no—
222C24—2511no—
223#41—5024no—
224#41—5028no—
225#41—5028no—
226#41—5024no—
227#41—5024no—
228#41—5024no—
229#41—2518no—
230C2×Q8—24no—
231C2×Q8—21no—
232C2×Q8C22×D25111yescharacteristic
233C2×Q8C22×D25112yes—
234C2×D4—5021no—
235C2×D4—5021no—
236C2×D4—5028no—
237C2×D4—5022no—
238C2×D4—5022no—
239C2×D4—5024no—
240C2×D4—5021no—
241C2×D4—5021no—
242C2×D4—2511no—
243C2×D4—2512no—
244C22×C4—21no—
245C22×C4—5021no—
246C22×C4—5028no—
247C22×C4—5022no—
248C22×C4—5022no—
249C22×C4—5024no—
250C22×C4—2511no—
251C22×C4—2511no—
252C22×C4—2511no—
253C22×C4—2511no—
254C22×C4—2511no—
255C22×C4C22×D25111yescharacteristic
256Q16—24no—
257SD16—5024no—
258SD16—5024no—
259D8—5024no—
260C2×C8—5024no—
261C2×C8—2512no—
262C2×C8C22×D25112yes—
263C4⋊C4—21no—
264C4⋊C4—5021no—
265C4⋊C4—2511no—
266C4⋊C4C22×D25111yescharacteristic
267C22⋊C4—21no—
268C22⋊C4—5021no—
269C22⋊C4—5021no—
270C22⋊C4—5021no—
271C22⋊C4—5021no—
272C42—5021no—
273C23—10042no—
274C23—10041no—
275C23—10042no—
276C23—5024no—
277C23—5022no—
278C23—5021no—
279C23—5022no—
280C23—5022no—
281C23—2511no—
282C23—2511no—
283C23D4×D25111yescharacteristic
284Q8—24no—
285Q8—22no—
286Q8—22no—
287Q8D4×D25114yes—
288D4—10042no—
289D4—10042no—
290D4—10042no—
291D4—5028no—
292D4—5024no—
293D4—5028no—
294D4—5022no—
295D4—5022no—
296D4—2514no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—10042no—
303C2×C4—5028no—
304C2×C4—5022no—
305C2×C4—5028no—
306C2×C4—5022no—
307C2×C4—5021no—
308C2×C4—5022no—
309C2×C4—5022no—
310C2×C4—2511no—
311C2×C4—2511no—
312C2×C4—2514no—
313C2×C4D4×D25111yescharacteristic
314C2×C4C23×D25111yescharacteristic
315C8—22no—
316C8—5022no—
317C22—21no—
318C22—21no—
319C22—10048no—
320C22—10044no—
321C22—10041no—
322C22—10041no—
323C22—10041no—
324C22—10042no—
325C22—10042no—
326C22—5021no—
327C22—5021no—
328C22—5021no—
329C22—5024no—
330C22—5021no—
331C22—5021no—
332C22—2512no—
333C22#2254873812yescomplemented
334C22C2×D4×D25111yescharacteristic
335C4—24no—
336C4—21no—
337C4—41no—
338C4—41no—
339C4—10041no—
340C4—5021no—
341C4—2512no—
342C4C2×D4×D25112yes—
343C2—22no—
344C2—20081no—
345C2—10044no—
346C2—10041no—
347C2—5022no—
348C2#120×D25111yescharacteristic
349C2C2×#2254873811yescharacteristic
350C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
hypercenter#451D251
hypocenterC251#3002
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
1. C22×#16860811C2semidirect
2. C2×#22548738C2semidirect
3. #30632200C2semidirect
4. #30631848C2semidirect
5. #30631837C2semidirect
6. #30631835C2semidirect
7. #30631826C2semidirect
8. #30631823C2semidirect
9. #30631682C2semidirect
10. C22⋊C8×D251C2semidirect
11. #451×C251C2semidirect
17. C2×#16860811C22semidirect
23. C2×#16860797C22semidirect
24. C2×#16860797C22semidirect
25. C2×D2008C22semidirect
26. C2×C8×D251C22semidirect
28. #22548691C22semidirect
29. #22548690C22semidirect
34. #22548673C22semidirect
35. #22548639C22semidirect
36. #22548637C22semidirect
37. #22548635C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×Q16×C251C22semidirect
43. #103×C251C22semidirect
53. #16860811D4semidirect
143. C22×C251#135semidirect
175. C251#3002semidirect
181. #451D251semidirect
200. C2×Q16C2×D251semidirect
215. #103C2×D251semidirect
333. C22#22548738semidirect

Maximal subgroups

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

subgroupindex
C22×#168608112
C2×#225487382
#306322002
#306318482
#306318372
#306318352
#306318262
#306318232
#306316822
C22⋊C8×D2512
#451×C2512
#3002251