Finite Group Databasebeta

← #44562110

Subgroups — #44562110

Order 32128.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#16860800C211yesmaximal, characteristic, complemented
2C2×Q16×D251C211yesmaximal, characteristic, complemented
3Q8×C22⋊D251C211yesmaximal, characteristic
4#30632350C211yesmaximal, characteristic
5#30632065C211yesmaximal, characteristic, complemented
6#30631927C211yesmaximal, characteristic, complemented
7#30631926C211yesmaximal, characteristic, complemented
8#30631896C211yesmaximal, characteristic, complemented
9#30631895C211yesmaximal, characteristic, complemented
10#30631845C211yesmaximal, characteristic, complemented
11#30631844C211yesmaximal, characteristic, complemented
12#30631841C211yesmaximal, characteristic, complemented
13#30631779C211yesmaximal, characteristic, complemented
14(C22⋊D251)⋊C8C211yesmaximal, characteristic
15#451×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C2×Q8×D251C2211yescharacteristic
18C2×Q8×D251C2211yescharacteristic
19C2×Dic1004C2211yescharacteristic
20C2×Dic1004C2211yescharacteristic
21C2×#16860800—24no—
22C2×#16860800C2211yescharacteristic
23C2×#16860800C2211yescharacteristic
24C2×#16860800C2211yescharacteristic
25C2×#16860800C2211yescharacteristic
26C2×#16860800C2211yescharacteristic
27Q16×D251—24no—
28C22×C251⋊C8C2211yescharacteristic
29C2×C8×D251C2211yescharacteristic
30#22548714—21no—
31#22548713—21no—
32#22548713C2211yescharacteristic
33#22548713C2211yescharacteristic
34#22548695C2211yescharacteristic
35#22548694—21no—
36#22548694C2211yescharacteristic
37#22548693—21no—
38#22548692—21no—
39#22548691—21no—
40#22548691C2211yescharacteristic
41Q8×Dic251C2211yescharacteristic
42#22548680C2211yescharacteristic
43C4×C22⋊D251—21no—
44C4×C22⋊D251C2211yescharacteristic
45(C22⋊D251)⋊C4—21no—
46#22548648C2211yescharacteristic
47Dic251⋊C8C2211yescharacteristic
48#22548640C2211yescharacteristic
49#22548640C2211yescharacteristic
50#22548639C2211yescharacteristic
51#22548638C2211yescharacteristic
52(C2×D251)⋊C8C2211yescharacteristic
53C8×Dic251C2211yescharacteristic
54C22×Q8×C251C2211yescharacteristic
55C2×Q16×C251C2211yescharacteristic
56C2×Q16×C251C2211yescharacteristic
57C22⋊Q8×C251C2211yescharacteristic
58#103×C251C2211yescharacteristic
59#103×C251C2211yescharacteristic
60C22⋊C8×C251C2211yescharacteristic
61C2×Dic502—21no—
62C2×Dic502—21no—
63C2×Dic502—22no—
64C2×Dic502—21no—
65C2×Dic502—41no—
66C2×Dic502C2311yescharacteristic
67C2×Dic502C2311yescharacteristic
68C2×Dic502C2311yescharacteristic
69Q8×D251—22no—
70Q8×D251—22no—
71Q8×D251—41no—
72Q8×D251—41no—
73C2×C22⋊D251D411yescharacteristic
74C22×Dic251—21no—
75C2×C4×D251—21no—
76C2×C4×D251—21no—
77C2×C4×D251C2311yescharacteristic
78Dic1004—22no—
79Dic1004—41no—
80#16860800—22no—
81#16860800—22no—
82#16860800—22no—
83#16860800—22no—
84#16860800D414yescomplemented
85C2×C251⋊C8—22no—
86C2×C251⋊C8C2311yescharacteristic
87C2×C251⋊C8C2311yescharacteristic
88C8×D251—22no—
89C4⋊Dic251—21no—
90C4⋊Dic251—21no—
91C4⋊Dic251C2311yescharacteristic
92Dic251⋊C4—21no—
93Dic251⋊C4—21no—
94Dic251⋊C4—21no—
95Dic251⋊C4—21no—
96Dic251⋊C4—21no—
97Dic251⋊C4—21no—
98Dic251⋊C4—21no—
99Dic251⋊C4D411yescharacteristic
100C22⋊Dic251—21no—
101C22⋊Dic251D411yescharacteristic
102(C2×D251)⋊C4—21no—
103(C2×D251)⋊C4—21no—
104(C2×D251)⋊C4D411yescharacteristic
105C4×Dic251—21no—
106C4×Dic251—21no—
107C4×Dic251C2311yescharacteristic
108C2×Q8×C251—21no—
109C2×Q8×C251—22no—
110C2×Q8×C251—41no—
111C2×Q8×C251C2311yescharacteristic
112C2×Q8×C251C2311yescharacteristic
113C2×Q8×C251C2311yescharacteristic
114C22×C4×C251—21no—
115C22×C4×C251C2311yescharacteristic
116Q16×C251—22no—
117Q16×C251—41no—
118C2×C8×C251C2311yescharacteristic
119C2×C8×C251C2311yescharacteristic
120C4⋊C4×C251—21no—
121C4⋊C4×C251C2311yescharacteristic
122C22⋊C4×C251—21no—
123C22×D251C2×D411yescharacteristic
124Dic502—21no—
125Dic502—21no—
126Dic502—21no—
127Dic502—21no—
128Dic502—21no—
129Dic502—42no—
130Dic502—42no—
131Dic502—42no—
132Dic502C2×D412yes—
133C22⋊D251Q1614yescomplemented
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—42no—
139C2×Dic251C2×D411yescharacteristic
140C2×Dic251C2×D411yescharacteristic
141C2×Dic251C2×D411yescharacteristic
142C4×D251—21no—
143C4×D251—21no—
144C4×D251—42no—
145C4×D251—42no—
146C251⋊C8—21no—
147C251⋊C8C2×D412yes—
148C23×C251C2×D411yescharacteristic
149Q8×C251—21no—
150Q8×C251—21no—
151Q8×C251—21no—
152Q8×C251—21no—
153Q8×C251—21no—
154Q8×C251—42no—
155Q8×C251C2×D412yes—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—42no—
163C2×C4×C251C2×D411yescharacteristic
164C2×C4×C251C2411yescharacteristic
165C8×C251—21no—
166C8×C251—21no—
167C22×C251—21no—
168C22×C251—21no—
169C22×C251C2×Q1612yescomplemented
170C22×C251C22×D411yescharacteristic
171C2×D251—21no—
172C2×D251—21no—
173C2×D251C2×Q1612yescomplemented
174C4×C251—21no—
175C4×C251—22no—
176C4×C251—41no—
177C4×C251—41no—
178C4×C251—41no—
179C4×C251C22×D411yescharacteristic
180C4×C251H(5,𝔽2)11yescharacteristic
181Dic251—21no—
182Dic251—22no—
183Dic251—21no—
184Dic251—41no—
185Dic251—41no—
186Dic251—41no—
187Dic251C2×Q1612yescomplemented
188C2×C251—22no—
189C2×C251D4211yescharacteristic
190C2×C251C22×Q1611yescharacteristic
191C2×C251#57811yescharacteristic
192D251—22no—
193C251D4×Q1611yescharacteristic, complemented, Sylow
194D4×Q16—2511nomaximal, Sylow
195C22×Q16—2511no—
196C22×Q16—2511no—
197D4×Q8—2511no—
198D4×Q8—2511no—
199#494—2511no—
200#467—2511no—
201#467—2511no—
202#462—2511no—
203#462—2511no—
204#451—2511no—
205#451—2511no—
206#451—2511no—
207#451D25111yescharacteristic, complemented
208C4×Q16—2511no—
209C8×D4—2511no—
210C22×Q8—2511no—
211C22×Q8—2511no—
212C22×Q8—2511no—
213C22×Q8C2×D25111yescharacteristic
214C2×Q16—5024no—
215C2×Q16—5024no—
216C2×Q16—2511no—
217C2×Q16—2511no—
218C2×Q16—2511no—
219C2×Q16—2511no—
220C2×Q16—2511no—
221C2×Q16—2511no—
222C2×Q16—2511no—
223C2×Q16C2×D25111yescharacteristic
224C2×Q16C2×D25111yescharacteristic
225C22×C8—2511no—
226C22×C8—2511no—
227C4⋊Q8—5021no—
228C4⋊Q8—5021no—
229C4⋊Q8—2511no—
230C4⋊Q8—2511no—
231C22⋊Q8—5021no—
232C22⋊Q8—5021no—
233C22⋊Q8—5021no—
234C22⋊Q8—5021no—
235C22⋊Q8—2511no—
236C22⋊Q8—2511no—
237C22⋊Q8—2511no—
238C22⋊Q8C2×D25111yescharacteristic
239C4×Q8—2511no—
240C4×Q8—2511no—
241C4×D4—5021no—
242C4×D4—5021no—
243C4×D4—2511no—
244#107—2511no—
245C4⋊C8—2511no—
246#103—2511no—
247#103—2511no—
248#103—2511no—
249#103—2511no—
250#103C2×D25111yescharacteristic
251#103C2×D25111yescharacteristic
252C22⋊C8—2511no—
253C22⋊C8C2×D25111yescharacteristic
254C4×C8—2511no—
255C2×Q8—21no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—10041no—
259C2×Q8—10041no—
260C2×Q8—10041no—
261C2×Q8—5022no—
262C2×Q8—5021no—
263C2×Q8—5021no—
264C2×Q8—5022no—
265C2×Q8—5022no—
266C2×Q8—5021no—
267C2×Q8—2511no—
268C2×Q8—2511no—
269C2×Q8—2511no—
270C2×Q8C22×D25111yescharacteristic
271C2×Q8C22×D25111yescharacteristic
272C2×Q8C22×D25111yescharacteristic
273C2×D4—2511no—
274C22×C4—21no—
275C22×C4—5021no—
276C22×C4—5021no—
277C22×C4—5021no—
278C22×C4—2511no—
279C22×C4C22×D25111yescharacteristic
280Q16—22no—
281Q16—41no—
282Q16—10041no—
283Q16—5022no—
284Q16—5022no—
285Q16—5022no—
286Q16—5022no—
287Q16—5022no—
288Q16—2514no—
289C2×C8—5022no—
290C2×C8—5022no—
291C2×C8—2511no—
292C2×C8—2511no—
293C2×C8C22×D25111yescharacteristic
294C2×C8C22×D25111yescharacteristic
295C4⋊C4—21no—
296C4⋊C4—5021no—
297C4⋊C4—5021no—
298C4⋊C4—5021no—
299C4⋊C4—5021no—
300C4⋊C4—5021no—
301C4⋊C4—5021no—
302C4⋊C4—5021no—
303C4⋊C4—5021no—
304C4⋊C4—5021no—
305C4⋊C4—2511no—
306C4⋊C4—2511no—
307C4⋊C4C22×D25111yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—5021no—
310C22⋊C4—5021no—
311C22⋊C4—5021no—
312C22⋊C4—2511no—
313C22⋊C4—2511no—
314C42—5021no—
315C42—5021no—
316C42—2511no—
317C23—2511no—
318C23D4×D25111yescharacteristic
319Q8—21no—
320Q8—21no—
321Q8—21no—
322Q8—21no—
323Q8—21no—
324Q8—42no—
325Q8—10042no—
326Q8—10042no—
327Q8—10042no—
328Q8—5021no—
329Q8—5021no—
330Q8—5021no—
331Q8—5021no—
332Q8—5021no—
333Q8—2512no—
334Q8D4×D25112yes—
335D4—2514no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—10042no—
344C2×C4—10042no—
345C2×C4—10042no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—2511no—
353C2×C4—2511no—
354C2×C4—2511no—
355C2×C4D4×D25111yescharacteristic
356C2×C4C23×D25111yescharacteristic
357C8—21no—
358C8—21no—
359C8—5021no—
360C8—2512no—
361C22—21no—
362C22—21no—
363C22—5021no—
364C22—5021no—
365C22—2512no—
366C22Q16×D25112yescomplemented
367C22C2×D4×D25111yescharacteristic
368C4—21no—
369C4—22no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—10041no—
374C4—10041no—
375C4—10041no—
376C4—5021no—
377C4—5022no—
378C4—5021no—
379C4—2512no—
380C4C2×D4×D25111yescharacteristic
381C4#2254876011yescharacteristic
382C2—22no—
383C2—5022no—
384C2#3063231911yescharacteristic
385C2C2×Q16×D25111yescharacteristic
386C2#3063250311yescharacteristic
subgroupG/H
centerC22C2×D4×D251
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
Fitting#451×C251C2
socleC22×C251C22×D4
prime radicalC2×C22⋊D251D4
hypercenter#451D251
hypocenterC251D4×Q16
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×#16860800C2semidirect
2. C2×Q16×D251C2semidirect
5. #30632065C2semidirect
6. #30631927C2semidirect
7. #30631926C2semidirect
8. #30631896C2semidirect
9. #30631895C2semidirect
10. #30631845C2semidirect
11. #30631844C2semidirect
12. #30631841C2semidirect
13. #30631779C2semidirect
15. #451×C251C2semidirect
84. #16860800D4semidirect
133. C22⋊D251Q16semidirect
169. C22×C251C2×Q16semidirect
173. C2×D251C2×Q16semidirect
187. Dic251C2×Q16semidirect
193. C251D4×Q16semidirect
207. #451D251semidirect
366. C22Q16×D251semidirect

Maximal subgroups

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

subgroupindex
C22×#168608002
C2×Q16×D2512
Q8×C22⋊D2512
#306323502
#306320652
#306319272
#306319262
#306318962
#306318952
#306318452
#306318442
#306318412
#306317792
(C22⋊D251)⋊C82
#451×C2512
D4×Q16251