Finite Group Databasebeta

← #44558672

Subgroups — #44558672

Order 32128.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548705C211yesmaximal, characteristic, complemented
2C2×C42×D251C211yesmaximal, characteristic
3#30631584C211yesmaximal, characteristic, complemented
4#30631582C211yesmaximal, characteristic, complemented
5#30631574C211yesmaximal, characteristic, complemented
6#30631571C211yesmaximal, characteristic, complemented
7#30631570C211yesmaximal, characteristic, complemented
8#30631561C211yesmaximal, characteristic
9#30631561C211yesmaximal, characteristic
10#30631558C211yesmaximal, characteristic
11#30631558C211yesmaximal, characteristic
12#30631551C211yesmaximal, characteristic, complemented
13#30631546C211yesmaximal, characteristic
14#30631546C211yesmaximal, characteristic
15#387×C251C211yesmaximal, characteristic, complemented
16C22×C4×D251C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18C22×C4×D251C2211yescharacteristic
19#22548705—24no—
20C2×C4⋊Dic251C2211yescharacteristic
21C2×C4⋊Dic251C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊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
37C42×D251—24no—
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C42×C251C2211yescharacteristic
49#95×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C23×D251C2311yescharacteristic
54C22×Dic251C2311yescharacteristic
55C22×Dic251C2311yescharacteristic
56C22×Dic251C2311yescharacteristic
57C22×Dic251C2311yescharacteristic
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C2×C4×D251—22no—
62C2×C4×D251—22no—
63C2×C4×D251—22no—
64C2×C4×D251—22no—
65C2×C4×D251—22no—
66C2×C4×D251—22no—
67C2×C4×D251—22no—
68C2×C4×D251—22no—
69C2×C4×D251Q814yes—
70C4⋊Dic251—22no—
71C4⋊Dic251—22no—
72Dic251⋊C4—22no—
73Dic251⋊C4—22no—
74Dic251⋊C4—22no—
75Dic251⋊C4—22no—
76Dic251⋊C4—22no—
77Dic251⋊C4—22no—
78Dic251⋊C4—22no—
79Dic251⋊C4—22no—
80(C2×D251)⋊C4—22no—
81(C2×D251)⋊C4—22no—
82(C2×D251)⋊C4—22no—
83(C2×D251)⋊C4—22no—
84C4×Dic251—22no—
85C4×Dic251—22no—
86C4×Dic251D414yes—
87C22×C4×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—22no—
95C4⋊C4×C251—22no—
96C42×C251—22no—
97C22×D251—21no—
98C22×D251—21no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251C2×Q812yes—
102C22×D251#4112yes—
103C22×D251#4112yes—
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×Dic251C2×D412yes—
121C2×Dic251C2×D412yes—
122C2×Dic251C2×Q812yes—
123C2×Dic251#4112yes—
124C2×Dic251#4112yes—
125C4×D251—28no—
126C4×D251—28no—
127C4×D251—28no—
128C23×C251C2411yescharacteristic
129C2×C4×C251—21no—
130C2×C4×C251—21no—
131C2×C4×C251—21no—
132C2×C4×C251—21no—
133C2×C4×C251—21no—
134C2×C4×C251—21no—
135C2×C4×C251—21no—
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×C251C2×D412yes—
148C2×C4×C251C2×Q812yes—
149C2×C4×C251#4112yes—
150C22×C251C22×D411yescharacteristic
151C22×C251C22×Q811yescharacteristic
152C22×C251C2×#4111yescharacteristic
153C22×C251C2×#4111yescharacteristic
154C22×C251C2×#4111yescharacteristic
155C22×C251C2×#4111yescharacteristic
156C22×C251C2×#4111yescharacteristic
157C2×D251—22no—
158C2×D251—22no—
159C2×D251—22no—
160C2×D251—22no—
161C2×D251—22no—
162C2×D251—22no—
163C2×D251#12514yescomplemented
164C4×C251—22no—
165C4×C251—22no—
166C4×C251—22no—
167C4×C251—41no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251—41no—
171Dic251—22no—
172Dic251—22no—
173Dic251—41no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177Dic251C22⋊Q814yes—
178C2×C251C2×C22⋊Q811yescharacteristic
179C2×C251#52511yescharacteristic
180C2×C251#52511yescharacteristic
181C2×C251C2×#12511yescharacteristic
182C2×C251#52911yescharacteristic
183C2×C251#52911yescharacteristic
184C2×C251#53311yescharacteristic
185D251—24no—
186C251#285111yescharacteristic, complemented, Sylow
187#2851—2511nomaximal, Sylow
188C2×#125—2511no—
189C22×C42—2511no—
190#389—2511no—
191#389—2511no—
192#387—2511no—
193#387—2511no—
194#387—2511no—
195#387D25111yescharacteristic, complemented
196#385—2511no—
197#385—2511no—
198#385—2511no—
199#385—2511no—
200#382—2511no—
201#380—2511no—
202#380—2511no—
203C23×C4—2511no—
204C23×C4—2511no—
205C23×C4—2511no—
206#125—5024no—
207C2×C4⋊C4—2511no—
208C2×C4⋊C4—2511no—
209C2×C4⋊C4—2511no—
210C2×C4⋊C4—2511no—
211C2×C4⋊C4—2511no—
212C2×C4⋊C4—2511no—
213C2×C4⋊C4—2511no—
214C2×C4⋊C4—2511no—
215C2×C4⋊C4—2511no—
216C2×C4⋊C4—2511no—
217C2×C4⋊C4C2×D25111yescharacteristic
218C2×C4⋊C4C2×D25111yescharacteristic
219C2×C22⋊C4—2511no—
220C2×C22⋊C4—2511no—
221C2×C22⋊C4—2511no—
222C2×C22⋊C4—2511no—
223C2×C42—5024no—
224C2×C42—2511no—
225C2×C42—2511no—
226C2×C42—2511no—
227C2×C42C2×D25111yescharacteristic
228#95—2511no—
229#95—2511no—
230#95—2511no—
231#95—2511no—
232#95—2511no—
233#95—2511no—
234#95—2511no—
235#95—2511no—
236#95C2×D25111yescharacteristic
237#95C2×D25111yescharacteristic
238#95C2×D25111yescharacteristic
239#95C2×D25111yescharacteristic
240C24—2511no—
241C22×C4—5022no—
242C22×C4—5022no—
243C22×C4—5022no—
244C22×C4—5022no—
245C22×C4—5022no—
246C22×C4—5022no—
247C22×C4—5022no—
248C22×C4—5022no—
249C22×C4—2511no—
250C22×C4—2511no—
251C22×C4—2511no—
252C22×C4—2511no—
253C22×C4—2511no—
254C22×C4—2511no—
255C22×C4—2514no—
256C22×C4—2511no—
257C22×C4C22×D25111yescharacteristic
258C22×C4C22×D25111yescharacteristic
259C22×C4C22×D25111yescharacteristic
260C22×C4C22×D25111yescharacteristic
261C22×C4C22×D25111yescharacteristic
262C22×C4C22×D25111yescharacteristic
263C22×C4C22×D25111yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—5022no—
267C4⋊C4—5022no—
268C4⋊C4—5022no—
269C4⋊C4—5022no—
270C4⋊C4—5022no—
271C4⋊C4—5022no—
272C4⋊C4—5022no—
273C4⋊C4—5022no—
274C4⋊C4—5022no—
275C4⋊C4—5022no—
276C22⋊C4—5022no—
277C22⋊C4—5022no—
278C22⋊C4—5022no—
279C22⋊C4—5022no—
280C42—22no—
281C42—5022no—
282C42—5022no—
283C42—2514no—
284C23—5021no—
285C23—5021no—
286C23—5021no—
287C23—5021no—
288C23—2512no—
289C23—2512no—
290C23—2512no—
291C23C23×D25111yescharacteristic
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—5028no—
311C2×C4—5028no—
312C2×C4—5028no—
313C2×C4—5021no—
314C2×C4—5021no—
315C2×C4—5021no—
316C2×C4—5021no—
317C2×C4—5021no—
318C2×C4—5021no—
319C2×C4—5021no—
320C2×C4—5021no—
321C2×C4—5021no—
322C2×C4—5021no—
323C2×C4—5021no—
324C2×C4—5021no—
325C2×C4—5021no—
326C2×C4—5021no—
327C2×C4—5021no—
328C2×C4—5021no—
329C2×C4—2512no—
330C2×C4—2512no—
331C2×C4—2512no—
332C2×C4—2512no—
333C2×C4—2512no—
334C2×C4D4×D25112yes—
335C2×C4Q8×D25112yes—
336C2×C4#1686081012yes—
337C22—5022no—
338C22—5022no—
339C22—5022no—
340C22—5022no—
341C22—5022no—
342C22—5022no—
343C22—2514no—
344C22C2×D4×D25111yescharacteristic
345C22C2×Q8×D25111yescharacteristic
346C22#41×D25111yescharacteristic
347C22#41×D25111yescharacteristic
348C22#41×D25111yescharacteristic
349C22#41×D25111yescharacteristic
350C22C2×#1686081011yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—10041no—
359C4—10041no—
360C4—10041no—
361C4—10041no—
362C4—5022no—
363C4—5022no—
364C4—2514no—
365C2—5024no—
366C2C22⋊Q8×D25111yescharacteristic
367C2#3063219511yescharacteristic
368C2#3063219611yescharacteristic
369C2#125×D25111yescharacteristic
370C2#3063221511yescharacteristic
371C2#3063222011yescharacteristic
372C2#3063224111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#387×C251C2
prime radicalC23×D251C23
hypercenter#387D251
hypocenterC251#2851
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#22548705C2semidirect
3. #30631584C2semidirect
4. #30631582C2semidirect
5. #30631574C2semidirect
6. #30631571C2semidirect
7. #30631570C2semidirect
12. #30631551C2semidirect
15. #387×C251C2semidirect
163. C2×D251#125semidirect
186. C251#2851semidirect
195. #387D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487052
C2×C42×D2512
#306315842
#306315822
#306315742
#306315712
#306315702
#306315612
#306315582
#306315512
#306315462
#387×C2512
#2851251