Finite Group Databasebeta

← #44553299

Subgroups — #44553299

Order 32128.

588 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
2#30631578C211yesmaximal, characteristic, complemented
3#30631576C211yesmaximal, characteristic, complemented
4#30631571C211yesmaximal, characteristic, complemented
5#30631569C211yesmaximal, characteristic, complemented
6#30631566C211yesmaximal, characteristic
7#30631564C211yesmaximal, characteristic
8#30631554C211yesmaximal, characteristic, complemented
9#30631549C211yesmaximal, characteristic, complemented
10#30631545C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic251C211yesmaximal, characteristic, complemented
12C4×(C2×D251)⋊C4C211yesmaximal, characteristic, complemented
13#30631533C211yesmaximal, characteristic, complemented
14C2×C251⋊#95C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18C23×Dic251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20(C22⋊D251)⋊C4C418yescomplemented
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×C22⋊Dic251C2211yescharacteristic
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2211yescharacteristic
27C2×(C2×D251)⋊C4C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×C4×Dic251C2211yescharacteristic
31C2×C4×Dic251C2211yescharacteristic
32C2×C4×Dic251C2211yescharacteristic
33C251⋊#95C2211yescharacteristic
34C251⋊#95C2211yescharacteristic
35C251⋊#95C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C23×C4×C251C2211yescharacteristic
46C2×C22⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C42×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C2×C22⋊D251—22no—
54C2×C22⋊D251—22no—
55C2×C22⋊D251C2×C414yes—
56C22×Dic251—22no—
57C22×Dic251—22no—
58C22×Dic251—41no—
59C22×Dic251—41no—
60C22×Dic251—41no—
61C22×Dic251C2×C414yes—
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C2×C4×D251—41no—
70C2×C4×D251—41no—
71C2×C4×D251C2×C414yescomplemented
72Dic251⋊C4—41no—
73Dic251⋊C4—41no—
74Dic251⋊C4C2×C414yescomplemented
75C22⋊Dic251—22no—
76(C2×D251)⋊C4—22no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—41no—
79(C2×D251)⋊C4—41no—
80(C2×D251)⋊C4C2×C414yescomplemented
81C4×Dic251—22no—
82C4×Dic251—41no—
83C4×Dic251C2×C414yescomplemented
84C24×C251C2311yescharacteristic
85C22×C4×C251—22no—
86C22×C4×C251—22no—
87C22×C4×C251—41no—
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22⋊C4×C251—22no—
95C22⋊C4×C251C2×C414yes—
96C42×C251—22no—
97C42×C251—41no—
98C22×D251—21no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251C22×C412yes—
105C22⋊D251—28no—
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—21no—
123C2×Dic251—44no—
124C2×Dic251—44no—
125C2×Dic251C22×C412yes—
126C2×Dic251C22×C412yes—
127C2×Dic251C22×C412yes—
128C2×Dic251#4112yes—
129C4×D251—44no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251C22×C412yes—
135C23×C251#4112yes—
136C23×C251#4112yes—
137C23×C251C2411yescharacteristic
138C2×C4×C251—21no—
139C2×C4×C251—21no—
140C2×C4×C251—21no—
141C2×C4×C251—21no—
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—44no—
154C2×C4×C251C22×C412yes—
155C2×C4×C251C22×C412yes—
156C2×C4×C251#4112yes—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—22no—
160C22×C251—22no—
161C22×C251—22no—
162C22×C251—22no—
163C22×C251#11714yes—
164C22×C251C23×C411yescharacteristic
165C22×C251C2×#4111yescharacteristic
166C22×C251C2×#4111yescharacteristic
167C22×C251C2×#4111yescharacteristic
168C22×C251C2×#4111yescharacteristic
169C22×C251H(5,𝔽2)11yescharacteristic
170C22×C251H(5,𝔽2)11yescharacteristic
171C2×D251—22no—
172C2×D251—22no—
173C2×D251—22no—
174C2×D251—41no—
175C2×D251—41no—
176C2×D251—41no—
177C2×D251—41no—
178C4×C251—22no—
179C4×C251—22no—
180C4×C251—22no—
181C4×C251—41no—
182C4×C251—41no—
183C4×C251—41no—
184Dic251—22no—
185Dic251—22no—
186Dic251—22no—
187Dic251—22no—
188Dic251—41no—
189Dic251—41no—
190Dic251—41no—
191C2×C251—24no—
192C2×C251C2×#11711yescharacteristic
193C2×C251C4×#4111yescharacteristic
194C2×C251#51811yescharacteristic
195C2×C251#55111yescharacteristic
196C2×C251#55111yescharacteristic
197C2×C251#55311yescharacteristic
198C2×C251#55311yescharacteristic
199D251—42no—
200C251#235311yescharacteristic, complemented, Sylow
201#2353—2511nomaximal, Sylow
202C2×C4×D4—2511no—
203#388—2511no—
204#388—2511no—
205#387—2511no—
206#387—2511no—
207#386—2511no—
208#386—2511no—
209#383—2511no—
210#381—2511no—
211#380—2511no—
212C4×C22⋊C4—2511no—
213C4×C22⋊C4—2511no—
214C4×C22⋊C4D25111yescharacteristic, complemented
215#376—2511no—
216C2×#95—2511no—
217C22×D4—2511no—
218C23×C4—2511no—
219C23×C4—2511no—
220C23×C4—2511no—
221C23×C4C2×D25111yescharacteristic
222C4×D4—2518no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4—2511no—
226C2×C22⋊C4—2511no—
227C2×C22⋊C4—2511no—
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4C2×D25111yescharacteristic
233C2×C22⋊C4C2×D25111yescharacteristic
234C2×C42—2511no—
235C2×C42—2511no—
236C2×C42—2511no—
237C2×C42C2×D25111yescharacteristic
238C2×C42C2×D25111yescharacteristic
239#95—2511no—
240#95—2511no—
241#95—2511no—
242#95—2511no—
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95—2511no—
249#95—2511no—
250#95—2511no—
251#95C2×D25111yescharacteristic
252#95C2×D25111yescharacteristic
253C24—2511no—
254C24C22×D25111yescharacteristic
255C2×D4—5022no—
256C2×D4—5022no—
257C2×D4—2514no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—10041no—
262C22×C4—10041no—
263C22×C4—10041no—
264C22×C4—10041no—
265C22×C4—10041no—
266C22×C4—5022no—
267C22×C4—5022no—
268C22×C4—2514no—
269C22×C4—2514no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4—2511no—
273C22×C4—2511no—
274C22×C4—2511no—
275C22×C4—2511no—
276C22×C4—2511no—
277C22×C4C22×D25111yescharacteristic
278C22×C4C22×D25111yescharacteristic
279C22×C4C22×D25111yescharacteristic
280C22×C4C22×D25111yescharacteristic
281C22×C4C22×D25111yescharacteristic
282C22×C4C22×D25111yescharacteristic
283C4⋊C4—10041no—
284C4⋊C4—10041no—
285C4⋊C4—2514no—
286C22⋊C4—22no—
287C22⋊C4—10041no—
288C22⋊C4—10041no—
289C22⋊C4—5022no—
290C22⋊C4—5022no—
291C22⋊C4—5022no—
292C22⋊C4—2514no—
293C22⋊C4C4×D25114yes—
294C42—22no—
295C42—41no—
296C42—10041no—
297C42—5022no—
298C42—2514no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—5021no—
304C23—5021no—
305C23—5021no—
306C23—5021no—
307C23—5021no—
308C23—5021no—
309C23—2512no—
310C23C2×C4×D25112yes—
311C23#1686081012yes—
312C23#1686081012yes—
313C23C23×D25111yescharacteristic
314D4—5028no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—44no—
331C2×C4—10044no—
332C2×C4—10044no—
333C2×C4—10044no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—2512no—
352C2×C4—2512no—
353C2×C4—2512no—
354C2×C4—2512no—
355C2×C4C2×C4×D25112yes—
356C2×C4C2×C4×D25112yes—
357C2×C4#1686080912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22—5022no—
369C22—5022no—
370C22—5022no—
371C22#2254867214yes—
372C22C22×C4×D25111yescharacteristic
373C22#41×D25111yescharacteristic
374C22C2×#1686080911yescharacteristic
375C22C2×#1686081011yescharacteristic
376C22C2×#1686081011yescharacteristic
377C22#2254876011yescharacteristic
378C22#2254876111yescharacteristic
379C4—22no—
380C4—22no—
381C4—22no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—10041no—
386C4—10041no—
387C4—10041no—
388C4—5022no—
389C4—5022no—
390C4—5022no—
391C4—5022no—
392C2—24no—
393C2—10042no—
394C2C2×#2254867211yescharacteristic
395C2C4×#1686080911yescharacteristic
396C2#3063215911yescharacteristic
397C2#3063236111yescharacteristic
398C2#3063236311yescharacteristic
399C2#3063238311yescharacteristic
400C2#3063238611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
FittingC4×C22⋊C4×C251C2
socle, cosocleC23×C251C24
prime radicalC22×C22⋊D251C22
hypercenterC4×C22⋊C4D251
hypocenterC251#2353
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #30631578C2semidirect
3. #30631576C2semidirect
4. #30631571C2semidirect
5. #30631569C2semidirect
8. #30631554C2semidirect
9. #30631549C2semidirect
10. #30631545C2semidirect
11. C22⋊C4×Dic251C2semidirect
12. C4×(C2×D251)⋊C4C2semidirect
13. #30631533C2semidirect
14. C2×C251⋊#95C2semidirect
15. C4×C22⋊C4×C251C2semidirect
20. (C22⋊D251)⋊C4C4semidirect
71. C2×C4×D251C2×C4semidirect
74. Dic251⋊C4C2×C4semidirect
80. (C2×D251)⋊C4C2×C4semidirect
83. C4×Dic251C2×C4semidirect
200. C251#2353semidirect
214. C4×C22⋊C4D251semidirect

Maximal subgroups

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

subgroupindex
C2×(C22⋊D251)⋊C42
#306315782
#306315762
#306315712
#306315692
#306315662
#306315642
#306315542
#306315492
#306315452
C22⋊C4×Dic2512
C4×(C2×D251)⋊C42
#306315332
C2×C251⋊#952
C4×C22⋊C4×C2512
#2353251