Finite Group Databasebeta

← #44558496

Subgroups — #44558496

Order 32128.

450 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30631623C211yesmaximal, characteristic, complemented
2#30631621C211yesmaximal, characteristic, complemented
3#30631614C211yesmaximal, characteristic, complemented
4#30631609C211yesmaximal, characteristic, complemented
5#30631609C211yesmaximal, characteristic, complemented
6#30631605C211yesmaximal, characteristic, complemented
7#30631605C211yesmaximal, characteristic, complemented
8#30631603C211yesmaximal, characteristic, complemented
9#30631603C211yesmaximal, characteristic, complemented
10#30631602C211yesmaximal, characteristic, complemented
11#30631596C211yesmaximal, characteristic
12#30631587C211yesmaximal, characteristic, complemented
13#30631579C211yesmaximal, characteristic, complemented
14#30631570C211yesmaximal, characteristic, complemented
15#397×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C2×C4⋊C4×C251C2211yescharacteristic
45C2×C22⋊C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48#95×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C23×D251C2311yescharacteristic
52C23×D251C2311yescharacteristic
53C2×D1004—41no—
54C2×D1004—41no—
55C2×D1004—41no—
56C2×C22⋊D251—22no—
57C2×C22⋊D251—41no—
58C2×C22⋊D251—41no—
59C2×C22⋊D251—41no—
60C2×C22⋊D251—41no—
61C2×C22⋊D251—41no—
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C4⋊Dic251—22no—
69C4⋊Dic251—41no—
70C4⋊Dic251—41no—
71Dic251⋊C4—41no—
72Dic251⋊C4—41no—
73Dic251⋊C4—41no—
74Dic251⋊C4—41no—
75C22⋊Dic251—41no—
76C22⋊Dic251—41no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—41no—
79(C2×D251)⋊C4—41no—
80(C2×D251)⋊C4—41no—
81(C2×D251)⋊C4—41no—
82(C2×D251)⋊C4—41no—
83(C2×D251)⋊C4—41no—
84(C2×D251)⋊C4—41no—
85(C2×D251)⋊C4—41no—
86(C2×D251)⋊C4—41no—
87C4×Dic251—41no—
88C24×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C4⋊C4×C251—41no—
96C22⋊C4×C251—41no—
97C22⋊C4×C251—41no—
98C22⋊C4×C251—41no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251—21no—
105C22×D251—21no—
106C22×D251—21no—
107C22×D251—21no—
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—21no—
113D1004—44no—
114C22⋊D251—44no—
115C22⋊D251—44no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251#4112yes—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—21no—
140C23×C251—21no—
141C23×C251C2411yescharacteristic
142C2×C4×C251—21no—
143C2×C4×C251—21no—
144C2×C4×C251—21no—
145C2×C4×C251—21no—
146C2×C4×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
156C2×C4×C251—21no—
157C2×C4×C251—21no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251C2×#4111yescharacteristic
168C22×C251H(5,𝔽2)11yescharacteristic
169C22×C251H(5,𝔽2)11yescharacteristic
170C22×C251H(5,𝔽2)11yescharacteristic
171C22×C251H(5,𝔽2)11yescharacteristic
172C22×C251H(5,𝔽2)11yescharacteristic
173C22×C2512−1+411yescharacteristic
174C2×D251—41no—
175C2×D251—41no—
176C2×D251—41no—
177C2×D251—41no—
178C2×D251—41no—
179C2×D251—41no—
180C2×D251—41no—
181C2×D251—41no—
182C2×D251—41no—
183C2×D251—41no—
184C2×D251—41no—
185C2×D251—41no—
186C2×D251—41no—
187C2×D251—41no—
188C4×C251—41no—
189C4×C251—41no—
190C4×C251—41no—
191C4×C251—41no—
192C4×C251—41no—
193C4×C251—41no—
194Dic251—22no—
195Dic251—41no—
196Dic251—41no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—81no—
201C2×C251#53811yescharacteristic
202C2×C251#54011yescharacteristic
203C2×C251#54211yescharacteristic
204C2×C251#56011yescharacteristic
205C2×C251#56011yescharacteristic
206C2×C251#56211yescharacteristic
207C2×C251#56211yescharacteristic
208D251—81no—
209D251—81no—
210C251#282911yescharacteristic, complemented, Sylow
211#2829—2511nomaximal, Sylow
212#399—2511no—
213#398—2511no—
214#397—2511no—
215#397D25111yescharacteristic, complemented
216#396—2511no—
217#396—2511no—
218#394—2511no—
219#394—2511no—
220#394—2511no—
221#394—2511no—
222#394—2511no—
223#392—2511no—
224#390—2511no—
225#388—2511no—
226#387—2511no—
227C22×D4—2511no—
228C22×D4—2511no—
229C22×D4—2511no—
230C2×C4⋊C4—2511no—
231C2×C4⋊C4—2511no—
232C2×C4⋊C4—2511no—
233C2×C4⋊C4—2511no—
234C2×C4⋊C4—2511no—
235C2×C4⋊C4—2511no—
236C2×C4⋊C4—2511no—
237C2×C4⋊C4C2×D25111yescharacteristic
238C2×C22⋊C4—2511no—
239C2×C22⋊C4—2511no—
240C2×C22⋊C4—2511no—
241C2×C22⋊C4—2511no—
242C2×C22⋊C4—2511no—
243C2×C22⋊C4—2511no—
244C2×C22⋊C4—2511no—
245C2×C22⋊C4—2511no—
246C2×C22⋊C4—2511no—
247C2×C22⋊C4—2511no—
248C2×C22⋊C4—2511no—
249C2×C22⋊C4—2511no—
250C2×C22⋊C4C2×D25111yescharacteristic
251C2×C22⋊C4C2×D25111yescharacteristic
252C2×C22⋊C4C2×D25111yescharacteristic
253C2×C42—2511no—
254#95—2511no—
255#95—2511no—
256#95—2511no—
257#95—2511no—
258#95—2511no—
259#95C2×D25111yescharacteristic
260#95C2×D25111yescharacteristic
261#95C2×D25111yescharacteristic
262C24—2511no—
263C24—2511no—
264C24C22×D25111yescharacteristic
265C2×D4—10041no—
266C2×D4—10041no—
267C2×D4—10041no—
268C2×D4—10041no—
269C2×D4—10041no—
270C2×D4—10041no—
271C2×D4—10041no—
272C2×D4—10041no—
273C2×D4—5022no—
274C22×C4—2511no—
275C22×C4—2511no—
276C22×C4—2511no—
277C22×C4—2511no—
278C22×C4—2511no—
279C22×C4—2511no—
280C22×C4C22×D25111yescharacteristic
281C22×C4C22×D25111yescharacteristic
282C22×C4C22×D25111yescharacteristic
283C22×C4C22×D25111yescharacteristic
284C22×C4C22×D25111yescharacteristic
285C22×C4C22×D25111yescharacteristic
286C4⋊C4—41no—
287C4⋊C4—10041no—
288C4⋊C4—10041no—
289C4⋊C4—10041no—
290C4⋊C4—10041no—
291C4⋊C4—10041no—
292C4⋊C4—10041no—
293C4⋊C4—5022no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—10041no—
298C22⋊C4—10041no—
299C22⋊C4—10041no—
300C22⋊C4—10041no—
301C22⋊C4—10041no—
302C22⋊C4—10041no—
303C22⋊C4—10041no—
304C22⋊C4—10041no—
305C22⋊C4—10041no—
306C22⋊C4—10041no—
307C22⋊C4—10041no—
308C22⋊C4—5022no—
309C42—10041no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—5021no—
318C23—5021no—
319C23—5021no—
320C23—5021no—
321C23—5021no—
322C23—5021no—
323C23—5021no—
324C23—5021no—
325C23—5021no—
326C23—5021no—
327C23—5021no—
328C23—5021no—
329C23—5021no—
330C23—5021no—
331C23C23×D25111yescharacteristic
332D4—10044no—
333D4—10044no—
334D4—10044no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—5021no—
354C2×C4—5021no—
355C2×C4—5021no—
356C2×C4—5021no—
357C2×C4—5021no—
358C2×C4—5021no—
359C2×C4—5021no—
360C2×C4—5021no—
361C2×C4—5021no—
362C2×C4—5021no—
363C2×C4—5021no—
364C2×C4—5021no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—2512no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—10041no—
379C22—10041no—
380C22—10041no—
381C22—10041no—
382C22—10041no—
383C22—10041no—
384C22—10041no—
385C22—10041no—
386C22—10041no—
387C22—10041no—
388C22—10041no—
389C22—10041no—
390C22—10041no—
391C22—10041no—
392C22#41×D25111yescharacteristic
393C22#2254876011yescharacteristic
394C22#2254876011yescharacteristic
395C22#2254876111yescharacteristic
396C22#2254876111yescharacteristic
397C22#2254876111yescharacteristic
398C22#2254876211yescharacteristic
399C4—41no—
400C4—41no—
401C4—41no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—10041no—
406C4—10041no—
407C4—10041no—
408C4—10041no—
409C4—10041no—
410C4—5022no—
411C2—81no—
412C2—20081no—
413C2—20081no—
414C2#3063227611yescharacteristic
415C2#3063228711yescharacteristic
416C2#3063230311yescharacteristic
417C2#3063242711yescharacteristic
418C2#3063242711yescharacteristic
419C2#3063243111yescharacteristic
420C2#3063243411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#397×C251C2
prime radical#30631596C2
hypercenter#397D251
hypocenterC251#2829
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. #30631623C2semidirect
2. #30631621C2semidirect
3. #30631614C2semidirect
4. #30631609C2semidirect
5. #30631609C2semidirect
6. #30631605C2semidirect
7. #30631605C2semidirect
8. #30631603C2semidirect
9. #30631603C2semidirect
10. #30631602C2semidirect
12. #30631587C2semidirect
13. #30631579C2semidirect
14. #30631570C2semidirect
15. #397×C251C2semidirect
210. C251#2829semidirect
215. #397D251semidirect

Maximal subgroups

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

subgroupindex
#306316232
#306316212
#306316142
#306316092
#306316052
#306316032
#306316022
#306315962
#306315872
#306315792
#306315702
#397×C2512
#2829251