Finite Group Databasebeta

← #44553509

Subgroups — #44553509

Order 32128.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548698C211yesmaximal, characteristic
2#30631587C211yesmaximal, characteristic
3#30631584C211yesmaximal, characteristic, complemented
4#30631582C211yesmaximal, characteristic, complemented
5#30631581C211yesmaximal, characteristic, complemented
6#30631579C211yesmaximal, characteristic, complemented
7#30631577C211yesmaximal, characteristic, complemented
8#30631575C211yesmaximal, characteristic, complemented
9#30631574C211yesmaximal, characteristic, complemented
10#30631568C211yesmaximal, characteristic, complemented
11#30631563C211yesmaximal, characteristic
12#30631561C211yesmaximal, characteristic, complemented
13#30631558C211yesmaximal, characteristic, complemented
14#30631547C211yesmaximal, characteristic, complemented
15#380×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18#22548698C418yescomplemented
19C2×C4⋊Dic251C2211yescharacteristic
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C2×C4×Dic251C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C23×C4×C251C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52C23×D251C2311yescharacteristic
53C2×C22⋊D251—41no—
54C2×C22⋊D251—41no—
55C2×C22⋊D251C2×C414yes—
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C2×C4×D251—41no—
64C2×C4×D251—41no—
65C2×C4×D251—41no—
66C4⋊Dic251—41no—
67C4⋊Dic251—41no—
68Dic251⋊C4—41no—
69Dic251⋊C4—41no—
70Dic251⋊C4—41no—
71Dic251⋊C4C2×C414yescomplemented
72Dic251⋊C4C2×C414yescomplemented
73C22⋊Dic251—41no—
74C22⋊Dic251—41no—
75C22⋊Dic251C2×C414yescomplemented
76(C2×D251)⋊C4—41no—
77(C2×D251)⋊C4—41no—
78(C2×D251)⋊C4—41no—
79(C2×D251)⋊C4C2×C414yes—
80(C2×D251)⋊C4C2×C414yes—
81C4×Dic251—41no—
82C4×Dic251—41no—
83C4×Dic251—41no—
84C24×C251C2311yescharacteristic
85C22×C4×C251—41no—
86C22×C4×C251—41no—
87C22×C4×C251C2×C414yescomplemented
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—41no—
95C4⋊C4×C251—41no—
96C22⋊C4×C251—41no—
97C22⋊C4×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—44no—
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—21no—
124C2×Dic251C22×C412yes—
125C2×Dic251C22×C412yes—
126C2×Dic251C22×C412yes—
127C4×D251—44no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251C22×C412yes—
135C23×C251C2411yescharacteristic
136C2×C4×C251—21no—
137C2×C4×C251—21no—
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—44no—
153C2×C4×C251C22×C412yes—
154C2×C4×C251C22×C412yes—
155C22×C251—22no—
156C22×C251—22no—
157C22×C251—22no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251C23×C411yescharacteristic
163C22×C251H(5,𝔽2)11yescharacteristic
164C22×C251H(5,𝔽2)11yescharacteristic
165C22×C251H(5,𝔽2)11yescharacteristic
166C22×C251H(5,𝔽2)11yescharacteristic
167C22×C2512−1+411yescharacteristic
168C22×C2512−1+411yescharacteristic
169C2×D251—22no—
170C2×D251—22no—
171C2×D251—22no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—41no—
175C2×D251—41no—
176C4×C251—22no—
177C4×C251—22no—
178C4×C251—41no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182Dic251—22no—
183Dic251—22no—
184Dic251—22no—
185Dic251—41no—
186Dic251—41no—
187Dic251—41no—
188Dic251—41no—
189C2×C251—42no—
190C2×C251#51811yescharacteristic
191C2×C251#52011yescharacteristic
192C2×C251#52011yescharacteristic
193C2×C251#56011yescharacteristic
194C2×C251#56211yescharacteristic
195C2×C251#56211yescharacteristic
196C2×C251#56311yescharacteristic
197D251—42no—
198C251#237611yescharacteristic, complemented, Sylow
199#2376—2511nomaximal, Sylow
200C2×#123—2511no—
201#390—2511no—
202#389—2511no—
203#389—2511no—
204#388—2511no—
205#388—2511no—
206#388—2511no—
207#388—2511no—
208#387—2511no—
209#387—2511no—
210#386—2511no—
211#385—2511no—
212#385—2511no—
213#380—2511no—
214#380D25111yescharacteristic, complemented
215C22×D4—2511no—
216C23×C4—2511no—
217C23×C4C2×D25111yescharacteristic
218#123—2518no—
219C2×C4⋊C4—2511no—
220C2×C4⋊C4—2511no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4—2511no—
226C2×C4⋊C4C2×D25111yescharacteristic
227C2×C4⋊C4C2×D25111yescharacteristic
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4—2511no—
233C2×C22⋊C4—2511no—
234C2×C22⋊C4—2511no—
235C2×C22⋊C4—2511no—
236C2×C22⋊C4C2×D25111yescharacteristic
237C2×C22⋊C4C2×D25111yescharacteristic
238C2×C42—2511no—
239C2×C42—2511no—
240C2×C42—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#95C2×D25111yescharacteristic
250#95C2×D25111yescharacteristic
251C24—2511no—
252C24C22×D25111yescharacteristic
253C2×D4—10041no—
254C2×D4—10041no—
255C2×D4—2514no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—10041no—
259C22×C4—10041no—
260C22×C4—10041no—
261C22×C4—2511no—
262C22×C4—2511no—
263C22×C4—2511no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4—2511no—
268C22×C4C4×D25114yescomplemented
269C22×C4C22×D25111yescharacteristic
270C22×C4C22×D25111yescharacteristic
271C22×C4C22×D25111yescharacteristic
272C22×C4C22×D25111yescharacteristic
273C22×C4C22×D25111yescharacteristic
274C22×C4C22×D25111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—10041no—
278C4⋊C4—10041no—
279C4⋊C4—10041no—
280C4⋊C4—10041no—
281C4⋊C4—10041no—
282C4⋊C4—2514no—
283C4⋊C4—2514no—
284C22⋊C4—41no—
285C22⋊C4—41no—
286C22⋊C4—10041no—
287C22⋊C4—10041no—
288C22⋊C4—10041no—
289C22⋊C4—10041no—
290C22⋊C4—10041no—
291C22⋊C4—2514no—
292C22⋊C4—2514no—
293C22⋊C4—2514no—
294C42—10041no—
295C42—10041no—
296C42—10041no—
297C23—21no—
298C23—21no—
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—
311C23C23×D25111yescharacteristic
312D4—10044no—
313C2×C4—21no—
314C2×C4—21no—
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—44no—
330C2×C4—10044no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
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—2512no—
350C2×C4—2512no—
351C2×C4—2512no—
352C2×C4C2×C4×D25112yes—
353C2×C4C2×C4×D25112yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—5022no—
366C22—5022no—
367C22—5022no—
368C22C22×C4×D25111yescharacteristic
369C22#2254876011yescharacteristic
370C22#2254876011yescharacteristic
371C22#2254876011yescharacteristic
372C22#2254876111yescharacteristic
373C22#2254876211yescharacteristic
374C22#2254876311yescharacteristic
375C4—22no—
376C4—22no—
377C4—41no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—10041no—
382C4—10041no—
383C4—10041no—
384C4—10041no—
385C4—5022no—
386C4—5022no—
387C4—5022no—
388C2—42no—
389C2—10042no—
390C2#3063216011yescharacteristic
391C2#3063216711yescharacteristic
392C2#3063216811yescharacteristic
393C2#3063242511yescharacteristic
394C2#3063243211yescharacteristic
395C2#3063243311yescharacteristic
396C2#3063243811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#380×C251C2
socle, cosocleC23×C251C24
prime radicalC22×C22⋊D251C22
hypercenter#380D251
hypocenterC251#2376
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #30631584C2semidirect
4. #30631582C2semidirect
5. #30631581C2semidirect
6. #30631579C2semidirect
7. #30631577C2semidirect
8. #30631575C2semidirect
9. #30631574C2semidirect
10. #30631568C2semidirect
12. #30631561C2semidirect
13. #30631558C2semidirect
14. #30631547C2semidirect
15. #380×C251C2semidirect
18. #22548698C4semidirect
71. Dic251⋊C4C2×C4semidirect
72. Dic251⋊C4C2×C4semidirect
75. C22⋊Dic251C2×C4semidirect
87. C22×C4×C251C2×C4semidirect
198. C251#2376semidirect
214. #380D251semidirect
268. C22×C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
#306315872
#306315842
#306315822
#306315812
#306315792
#306315772
#306315752
#306315742
#306315682
#306315632
#306315612
#306315582
#306315472
#380×C2512
#2376251