Finite Group Databasebeta

← #44558304

Subgroups — #44558304

Order 32128.

470 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#30631615C211yesmaximal, characteristic, complemented
2#30631601C211yesmaximal, characteristic, complemented
3#30631601C211yesmaximal, characteristic, complemented
4#30631601C211yesmaximal, characteristic, complemented
5#30631601C211yesmaximal, characteristic, complemented
6#30631590C211yesmaximal, characteristic, complemented
7#30631590C211yesmaximal, characteristic, complemented
8#30631584C211yesmaximal, characteristic, complemented
9#30631581C211yesmaximal, characteristic, complemented
10#30631581C211yesmaximal, characteristic, complemented
11#30631580C211yesmaximal, characteristic, complemented
12#30631580C211yesmaximal, characteristic, complemented
13#30631575C211yesmaximal, characteristic, complemented
14#30631574C211yesmaximal, characteristic, complemented
15#392×C251C211yesmaximal, characteristic
16C2×C4⋊Dic251C2211yescharacteristic
17C2×C4⋊Dic251C2211yescharacteristic
18C2×Dic251⋊C4C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×C22⋊Dic251C2211yescharacteristic
23C2×C22⋊Dic251C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C22⋊Dic251C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C2×C4×Dic251C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C22×D4×C251C2211yescharacteristic
45C22×D4×C251C2211yescharacteristic
46C22×D4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51C22×Dic251C2311yescharacteristic
52C22×Dic251C2311yescharacteristic
53C22×Dic251C2311yescharacteristic
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C4⋊Dic251—41no—
60C4⋊Dic251—41no—
61Dic251⋊C4—22no—
62Dic251⋊C4—22no—
63Dic251⋊C4—41no—
64Dic251⋊C4—41no—
65C22⋊Dic251—22no—
66C22⋊Dic251—22no—
67C22⋊Dic251—22no—
68C22⋊Dic251—41no—
69C22⋊Dic251—41no—
70C22⋊Dic251—41no—
71C22⋊Dic251—41no—
72C22⋊Dic251—41no—
73C22⋊Dic251—41no—
74C22⋊Dic251—41no—
75C22⋊Dic251—41no—
76C22⋊Dic251—41no—
77C4×Dic251—41no—
78C4×Dic251—41no—
79C4×Dic251—41no—
80C24×C251C2311yescharacteristic
81C24×C251C2311yescharacteristic
82C24×C251C2311yescharacteristic
83C2×D4×C251—22no—
84C2×D4×C251—22no—
85C2×D4×C251—41no—
86C2×D4×C251—41no—
87C2×D4×C251—41no—
88C2×D4×C251—41no—
89C2×D4×C251—41no—
90C2×D4×C251—41no—
91C2×D4×C251—41no—
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C22×C4×C251C2311yescharacteristic
96C22⋊C4×C251—22no—
97C22⋊C4×C251—22no—
98C22⋊C4×C251—41no—
99C2×Dic251—21no—
100C2×Dic251—21no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
106C2×Dic251—21no—
107C2×Dic251—21no—
108C2×Dic251—21no—
109C2×Dic251—21no—
110C2×Dic251—21no—
111C2×Dic251—21no—
112C2×Dic251—21no—
113C2×Dic251—21no—
114C2×Dic251—21no—
115C2×Dic251—21no—
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251#4112yes—
123C23×C251—21no—
124C23×C251—21no—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251—21no—
137C23×C251—21no—
138C23×C251—21no—
139C23×C251—21no—
140C23×C251—21no—
141C23×C251—21no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251C2411yescharacteristic
145D4×C251—44no—
146D4×C251—44no—
147D4×C251—44no—
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#4112yes—
159C2×C4×C251#4112yes—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251—41no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251—41no—
171C22×C251—41no—
172C22×C251—41no—
173C22×C251—41no—
174C22×C251—41no—
175C22×C251—41no—
176C22×C251—41no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251C2×#4111yescharacteristic
182C22×C251C2×#4111yescharacteristic
183C22×C251C2×#4111yescharacteristic
184C22×C251H(5,𝔽2)11yescharacteristic
185C22×C251H(5,𝔽2)11yescharacteristic
186C22×C251H(5,𝔽2)11yescharacteristic
187C22×C251H(5,𝔽2)11yescharacteristic
188C4×C251—22no—
189C4×C251—22no—
190C4×C251—41no—
191C4×C251—41no—
192Dic251—22no—
193Dic251—41no—
194Dic251—41no—
195Dic251—41no—
196Dic251—41no—
197Dic251—41no—
198Dic251—41no—
199Dic251—41no—
200C2×C251—81no—
201C2×C251—81no—
202C2×C251—81no—
203C2×C251#53811yescharacteristic
204C2×C251#53811yescharacteristic
205C2×C251#54011yescharacteristic
206C2×C251#55311yescharacteristic
207C2×C251#55511yescharacteristic
208C2×C251#55911yescharacteristic
209C2×C251#56011yescharacteristic
210C251#280311yescharacteristic, complemented, Sylow
211#2803—2511nomaximal, Sylow
212#397—2511no—
213#394—2511no—
214#394—2511no—
215#394—2511no—
216#394—2511no—
217#392D25111yescharacteristic
218#390—2511no—
219#390—2511no—
220#389—2511no—
221#388—2511no—
222#388—2511no—
223#388—2511no—
224#388—2511no—
225#388—2511no—
226#387—2511no—
227C22×D4C2×D25111yescharacteristic
228C22×D4C2×D25111yescharacteristic
229C22×D4C2×D25111yescharacteristic
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×C22⋊C4—2511no—
237C2×C22⋊C4—2511no—
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⋊C4C2×D25111yescharacteristic
249C2×C22⋊C4C2×D25111yescharacteristic
250C2×C22⋊C4C2×D25111yescharacteristic
251C2×C42—2511no—
252C2×C42—2511no—
253C2×C42—2511no—
254#95—2511no—
255#95—2511no—
256#95—2511no—
257#95—2511no—
258#95—2511no—
259#95—2511no—
260#95—2511no—
261#95C2×D25111yescharacteristic
262C24C22×D25111yescharacteristic
263C24C22×D25111yescharacteristic
264C24C22×D25111yescharacteristic
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C22×C4—2511no—
275C22×C4—2511no—
276C22×C4—2511no—
277C22×C4—2511no—
278C22×C4—2511no—
279C22×C4—2511no—
280C22×C4—2511no—
281C22×C4—2511no—
282C22×C4C22×D25111yescharacteristic
283C22×C4C22×D25111yescharacteristic
284C22×C4C22×D25111yescharacteristic
285C22×C4C22×D25111yescharacteristic
286C4⋊C4—10041no—
287C4⋊C4—10041no—
288C4⋊C4—10041no—
289C4⋊C4—10041no—
290C4⋊C4—5022no—
291C4⋊C4—5022no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C22⋊C4—10041no—
296C22⋊C4—10041no—
297C22⋊C4—10041no—
298C22⋊C4—10041no—
299C22⋊C4—10041no—
300C22⋊C4—10041no—
301C22⋊C4—10041no—
302C22⋊C4—10041no—
303C22⋊C4—10041no—
304C22⋊C4—5022no—
305C22⋊C4—5022no—
306C22⋊C4—5022no—
307C42—10041no—
308C42—10041no—
309C42—10041no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—21no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23C23×D25111yescharacteristic
332D4—44no—
333D4—44no—
334D4—44no—
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—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—5021no—
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—2512no—
369C2×C4#1686080912yes—
370C2×C4#1686080912yes—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22#41×D25111yescharacteristic
393C22C2×#1686080911yescharacteristic
394C22C2×#1686080911yescharacteristic
395C22#2254876011yescharacteristic
396C22#2254876011yescharacteristic
397C22#2254876011yescharacteristic
398C22#2254876011yescharacteristic
399C4—22no—
400C4—22no—
401C4—41no—
402C4—41no—
403C4—10041no—
404C4—10041no—
405C4—10041no—
406C4—10041no—
407C4—10041no—
408C4—10041no—
409C4—10041no—
410C4—5022no—
411C2—81no—
412C2—81no—
413C2—81no—
414C2#3063227211yescharacteristic
415C2#3063227211yescharacteristic
416C2#3063228711yescharacteristic
417C2#3063238411yescharacteristic
418C2#3063240111yescharacteristic
419C2#3063242311yescharacteristic
420C2#3063242611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting, prime radical#392×C251C2
hypercenter#392D251
hypocenterC251#2803
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. #30631615C2semidirect
2. #30631601C2semidirect
3. #30631601C2semidirect
4. #30631601C2semidirect
5. #30631601C2semidirect
6. #30631590C2semidirect
7. #30631590C2semidirect
8. #30631584C2semidirect
9. #30631581C2semidirect
10. #30631581C2semidirect
11. #30631580C2semidirect
12. #30631580C2semidirect
13. #30631575C2semidirect
14. #30631574C2semidirect
210. C251#2803semidirect

Maximal subgroups

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

subgroupindex
#306316152
#306316012
#306315902
#306315842
#306315812
#306315802
#306315752
#306315742
#392×C2512
#2803251