Finite Group Databasebeta

← #44559624

Subgroups — #44559624

Order 32128.

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

normal subgroups only

This group has 754 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
2C2×#22548747C212yesmaximal, complemented
3C22⋊Q8×D251C211yesmaximal, characteristic, complemented
4#30631852C212yesmaximal, complemented
5#30631848C212yesmaximal, complemented
6#30631843C211yesmaximal, characteristic, complemented
7#30631840C211yesmaximal, characteristic, complemented
8#30631837C211yesmaximal, characteristic, complemented
9#30631687C212yesmaximal, complemented
10#30631649C211yesmaximal, characteristic, complemented
11#451×C251C211yesmaximal, characteristic, complemented
12C2×#16860811—28no—
13C2×#16860811—21no—
14C2×#16860811—21no—
15C2×#16860811C2212yes—
16C2×#16860811C2212yes—
17C2×Q8×D251C2211yescharacteristic, complemented
18C22×D1004—21no—
19C22×D1004C2211yescharacteristic
20C22×C4×D251—21no—
21C22×C4×D251C2211yescharacteristic
22#22548747—28no—
23C2×#16860800C2212yescomplemented
24C2×#16860799C2212yescomplemented
25C2×#16860797C2212yescomplemented
26C2×#16860793C2212yescomplemented
27#22548695C2211yescharacteristic, complemented
28#22548694C2211yescharacteristic
29#22548693—21no—
30#22548692—21no—
31#22548691C2211yescharacteristic, complemented
32C4⋊C4×D251—21no—
33C4⋊C4×D251C2211yescharacteristic, complemented
34C22⋊C4×D251—21no—
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#16860811D418yes—
53C2×Dic502—21no—
54C2×Dic502C2311yescharacteristic
55Q8×D251—24no—
56C2×D1004—28no—
57C2×D1004—22no—
58C2×D1004—22no—
59C2×D1004—24no—
60C2×D1004—21no—
61C2×D1004C2312yes—
62C22×Dic251D411yescharacteristic
63C2×C4×D251—28no—
64C2×C4×D251—22no—
65C2×C4×D251—22no—
66C2×C4×D251—24no—
67C2×C4×D251—21no—
68C2×C4×D251—21no—
69C2×C4×D251D411yescharacteristic
70C2×C4×D251D411yescharacteristic
71C2×C4×D251C2311yescharacteristic
72C2×C4×D251C2311yescharacteristic
73#16860800—24no—
74#16860799—24no—
75#16860797—24no—
76#16860793—24no—
77C2×C251⋊C8C2312yes—
78C4⋊Dic251—21no—
79C4⋊Dic251C2311yescharacteristic
80Dic251⋊C4—21no—
81Dic251⋊C4—21no—
82Dic251⋊C4—21no—
83C22⋊Dic251—21no—
84(C2×D251)⋊C4—21no—
85(C2×D251)⋊C4—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—41no—
102C22×D251—42no—
103C22×D251—42no—
104C22×D251C2×D411yescharacteristic
105C22×D251C2×D411yescharacteristic
106Dic502—22no—
107Dic502—42no—
108D1004—28no—
109D1004—24no—
110D1004—28no—
111D1004—22no—
112D1004C2×D414yes—
113C2×Dic251—22no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251C2×D411yescharacteristic
117C2×Dic251C2×D411yescharacteristic
118C4×D251—28no—
119C4×D251—22no—
120C4×D251—22no—
121C4×D251—28no—
122C4×D251—22no—
123C4×D251—42no—
124C4×D251—42no—
125C4×D251C2×D414yes—
126C251⋊C8—22no—
127C23×C251C2×D411yescharacteristic
128Q8×C251—24no—
129Q8×C251—22no—
130Q8×C251—22no—
131Q8×C251C2×D414yes—
132C2×C4×C251—24no—
133C2×C4×C251—22no—
134C2×C4×C251—22no—
135C2×C4×C251—21no—
136C2×C4×C251—21no—
137C2×C4×C251C2×D411yescharacteristic
138C2×C4×C251C2411yescharacteristic
139C8×C251—22no—
140C22×C251—21no—
141C22×C251—21no—
142C22×C251#13712yescomplemented
143C22×C251C22×D411yescharacteristic
144C2×D251—21no—
145C2×D251—21no—
146C2×D251—21no—
147C2×D251—24no—
148C2×D251—21no—
149C2×D251—21no—
150C2×D251—44no—
151C2×D251—48no—
152C2×D251—42no—
153C2×D251—42no—
154C2×D251#12012yes—
155C4×C251—24no—
156C4×C251—21no—
157C4×C251—41no—
158C4×C251—41no—
159C4×C251C22×D412yes—
160Dic251—21no—
161Dic251—41no—
162Dic251—41no—
163Dic251#12012yes—
164C2×C251—22no—
165C2×C251C2×#12011yescharacteristic
166C2×C251C2×#13711yescharacteristic
167C2×C251#57511yescharacteristic
168D251—22no—
169D251—41no—
170D251—44no—
171C251#300711yescharacteristic, complemented, Sylow
172#3007—2511nomaximal, Sylow
173C22×#41—2511no—
174C2×#137—2512no—
175C2×C22⋊Q8—2511no—
176#452—2512no—
177#452—2512no—
178#451—2511no—
179#451D25111yescharacteristic, complemented
180#450—2511no—
181#450—2511no—
182#417—2512no—
183#407—2511no—
184C2×#41—5028no—
185C2×#41—5021no—
186C2×#41—5021no—
187C2×#41—2512no—
188C2×#41—2512no—
189C22×Q8—2511no—
190C22×Q8C2×D25111yescharacteristic
191C22×D4—5021no—
192C22×D4—2511no—
193C23×C4—5021no—
194C23×C4—2511no—
195#137—5028no—
196C2×Q16—2512no—
197C2×Q16C2×D25112yescomplemented
198C2×SD16—2512no—
199C2×SD16—2512no—
200C2×M16—2512no—
201C22⋊Q8—5021no—
202C22⋊Q8—5021no—
203C22⋊Q8—2511no—
204C22⋊Q8—2511no—
205C22⋊Q8—2511no—
206C22⋊Q8C2×D25111yescharacteristic
207C2×C4⋊C4—5021no—
208C2×C4⋊C4—2511no—
209C2×C22⋊C4—5021no—
210#103—2512no—
211#103C2×D25112yescomplemented
212#102—2512no—
213#102—2512no—
214C22⋊C8—2511no—
215C22⋊C8—2512no—
216C22⋊C8C2×D25111yescharacteristic
217C24—5021no—
218C24—2511no—
219#41—5028no—
220#41—5028no—
221#41—5024no—
222#41—5024no—
223#41—5024no—
224#41—2518no—
225C2×Q8—24no—
226C2×Q8—21no—
227C2×Q8—5021no—
228C2×Q8—5024no—
229C2×Q8—2511no—
230C2×Q8C22×D25111yescharacteristic
231C2×Q8C22×D25112yes—
232C2×D4—5028no—
233C2×D4—5022no—
234C2×D4—5022no—
235C2×D4—5024no—
236C2×D4—5021no—
237C2×D4—2512no—
238C22×C4—21no—
239C22×C4—5021no—
240C22×C4—5021no—
241C22×C4—5028no—
242C22×C4—5022no—
243C22×C4—5022no—
244C22×C4—5024no—
245C22×C4—2511no—
246C22×C4—2511no—
247C22×C4—2511no—
248C22×C4—2511no—
249C22×C4—2511no—
250C22×C4C22×D25111yescharacteristic
251Q16—24no—
252Q16—5024no—
253SD16—5024no—
254SD16—5024no—
255M16—5024no—
256C2×C8—2512no—
257C2×C8C22×D25112yes—
258C4⋊C4—21no—
259C4⋊C4—5021no—
260C4⋊C4—5021no—
261C4⋊C4—5021no—
262C4⋊C4—5021no—
263C4⋊C4—2511no—
264C4⋊C4C22×D25111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—5021no—
267C22⋊C4—5021no—
268C22⋊C4—5021no—
269C23—10042no—
270C23—10041no—
271C23—10042no—
272C23—5024no—
273C23—5022no—
274C23—5022no—
275C23—5022no—
276C23—2511no—
277C23—2511no—
278C23D4×D25111yescharacteristic
279Q8—24no—
280Q8—22no—
281Q8—22no—
282Q8—10042no—
283Q8—5022no—
284Q8D4×D25114yes—
285D4—5028no—
286D4—5024no—
287D4—5028no—
288D4—5022no—
289D4—2514no—
290C2×C4—24no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—10042no—
296C2×C4—10042no—
297C2×C4—5028no—
298C2×C4—5022no—
299C2×C4—5028no—
300C2×C4—5022no—
301C2×C4—5021no—
302C2×C4—5021no—
303C2×C4—5022no—
304C2×C4—5022no—
305C2×C4—2511no—
306C2×C4—2511no—
307C2×C4—2514no—
308C2×C4D4×D25111yescharacteristic
309C2×C4C23×D25111yescharacteristic
310C8—22no—
311C8—5022no—
312C22—21no—
313C22—21no—
314C22—10048no—
315C22—10044no—
316C22—10042no—
317C22—10042no—
318C22—5021no—
319C22—5021no—
320C22—5021no—
321C22—5024no—
322C22—5021no—
323C22—5021no—
324C22—2512no—
325C22#2254874712yescomplemented
326C22C2×D4×D25111yescharacteristic
327C4—24no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—10041no—
332C4—10041no—
333C4—5021no—
334C4—2512no—
335C4C2×D4×D25112yes—
336C2—22no—
337C2—10044no—
338C2—10041no—
339C2—5022no—
340C2#120×D25111yescharacteristic
341C2C2×#2254874711yescharacteristic
342C2#3063248611yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radicalC22×#16860811C2
hypercenter#451D251
hypocenterC251#3007
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
2. C2×#22548747C2semidirect
3. C22⋊Q8×D251C2semidirect
4. #30631852C2semidirect
5. #30631848C2semidirect
6. #30631843C2semidirect
7. #30631840C2semidirect
8. #30631837C2semidirect
9. #30631687C2semidirect
10. #30631649C2semidirect
11. #451×C251C2semidirect
17. C2×Q8×D251C22semidirect
23. C2×#16860800C22semidirect
24. C2×#16860799C22semidirect
25. C2×#16860797C22semidirect
26. C2×#16860793C22semidirect
27. #22548695C22semidirect
31. #22548691C22semidirect
33. C4⋊C4×D251C22semidirect
35. #22548639C22semidirect
36. #22548637C22semidirect
37. #22548635C22semidirect
39. (C2×D251)⋊C8C22semidirect
41. C2×Q16×C251C22semidirect
43. #103×C251C22semidirect
142. C22×C251#137semidirect
171. C251#3007semidirect
179. #451D251semidirect
197. C2×Q16C2×D251semidirect
211. #103C2×D251semidirect
325. C22#22548747semidirect

Maximal subgroups

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

subgroupindex
C22×#168608112
C2×#225487472
C22⋊Q8×D2512
#306318522
#306318482
#306318432
#306318402
#306318372
#306316872
#306316492
#451×C2512
#3007251