Finite Group Databasebeta

← #44558694

Subgroups — #44558694

Order 32128.

688 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548698C211yesmaximal, characteristic, complemented
2C2×#22548687C211yesmaximal, characteristic
3C22×C4×Dic251C211yesmaximal, characteristic, complemented
4#30631587C212yesmaximal, complemented
5#30631582C212yesmaximal, complemented
6#30631581C212yesmaximal, complemented
7#30631577C212yesmaximal, complemented
8#30631567C211yesmaximal, characteristic, complemented
9#30631564C211yesmaximal, characteristic, complemented
10#30631545C211yesmaximal, characteristic, complemented
11C2×#123×C251C211yesmaximal, characteristic, complemented
12C22×C22⋊D251C2211yescharacteristic
13C22×C22⋊D251C2211yescharacteristic
14C23×Dic251C2211yescharacteristic
15C23×Dic251C2211yescharacteristic
16#22548698—24no—
17#22548687—24no—
18C2×Dic251⋊C4C2212yes—
19C2×Dic251⋊C4C2212yes—
20C2×Dic251⋊C4C2211yescharacteristic
21C2×C22⋊Dic251C2211yescharacteristic
22C2×C22⋊Dic251C2212yes—
23C2×C22⋊Dic251C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2212yes—
27C2×C4×Dic251—24no—
28C2×C4×Dic251C2212yes—
29C2×C4×Dic251C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C251⋊#95C2212yes—
33C22×D4×C251C2211yescharacteristic
34C23×C4×C251C2211yescharacteristic
35#123×C251—24no—
36C2×C4⋊C4×C251C2212yes—
37C2×C22⋊C4×C251C2211yescharacteristic
38C2×C22⋊C4×C251C2212yes—
39C23×D251C2311yescharacteristic
40C2×C22⋊D251—22no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—22no—
43C2×C22⋊D251—22no—
44C2×C22⋊D251—22no—
45C22×Dic251—24no—
46C22×Dic251—24no—
47C22×Dic251—22no—
48C22×Dic251D414yes—
49C22×Dic251C2311yescharacteristic
50C22×Dic251C2312yes—
51C22×Dic251C2311yescharacteristic
52C22×Dic251C2312yes—
53C22×Dic251C2311yescharacteristic
54Dic251⋊C4—22no—
55Dic251⋊C4—24no—
56Dic251⋊C4—24no—
57C22⋊Dic251—22no—
58C22⋊Dic251—24no—
59C22⋊Dic251—22no—
60(C2×D251)⋊C4—22no—
61(C2×D251)⋊C4—24no—
62(C2×D251)⋊C4—24no—
63C4×Dic251—24no—
64C4×Dic251—24no—
65C24×C251C2311yescharacteristic
66C24×C251C2311yescharacteristic
67C2×D4×C251—22no—
68C2×D4×C251—22no—
69C2×D4×C251—22no—
70C22×C4×C251—22no—
71C22×C4×C251—22no—
72C22×C4×C251D414yescomplemented
73C22×C4×C251C2312yes—
74C22×C4×C251C2311yescharacteristic
75C22×C4×C251C2312yes—
76C4⋊C4×C251—24no—
77C22⋊C4×C251—22no—
78C22⋊C4×C251—24no—
79C22×D251—21no—
80C22×D251—21no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—21no—
84C22⋊D251—44no—
85C22⋊D251—44no—
86C2×Dic251—28no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—28no—
91C2×Dic251—22no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251—21no—
97C2×Dic251C2×D412yes—
98C2×Dic251C2×D412yes—
99C2×Dic251#4114yes—
100C2×Dic251#4114yes—
101C23×C251—22no—
102C23×C251—22no—
103C23×C251—21no—
104C23×C251—21no—
105C23×C251—21no—
106C23×C251—22no—
107C23×C251—22no—
108C23×C251C2×D412yes—
109C23×C251C2×D412yes—
110C23×C251#4112yes—
111C23×C251C2411yescharacteristic
112D4×C251—44no—
113C2×C4×C251—28no—
114C2×C4×C251—22no—
115C2×C4×C251—22no—
116C2×C4×C251—22no—
117C2×C4×C251—21no—
118C2×C4×C251—21no—
119C2×C4×C251—21no—
120C2×C4×C251—22no—
121C2×C4×C251—22no—
122C2×C4×C251C2×D414yes—
123C22×C251—22no—
124C22×C251—22no—
125C22×C251—24no—
126C22×C251—24no—
127C22×C251—41no—
128C22×C251—41no—
129C22×C251—41no—
130C22×C251—42no—
131C22×C251—42no—
132C22×C251#12114yes—
133C22×C251C22×D411yescharacteristic
134C22×C251C22×D411yescharacteristic
135C22×C251C2×#4112yes—
136C22×C251C2×#4112yes—
137C22×C251C2×#4111yescharacteristic
138C2×D251—41no—
139C2×D251—41no—
140C2×D251—41no—
141C2×D251—42no—
142C2×D251—42no—
143C4×C251—24no—
144C4×C251—41no—
145C4×C251—42no—
146Dic251—22no—
147Dic251—24no—
148Dic251—41no—
149Dic251—42no—
150Dic251#12314yescomplemented
151C2×C251—24no—
152C2×C251—81no—
153C2×C251C2×#12111yescharacteristic
154C2×C251C2×#12311yescharacteristic
155C2×C251#52511yescharacteristic
156C2×C251#52912yes—
157C2×C251#53212yes—
158D251—81no—
159C251#285411yescharacteristic, complemented, Sylow
160#2854—2511nomaximal, Sylow
161C2×#123—2511no—
162C2×#123D25111yescharacteristic, complemented
163C2×#121—2511no—
164C22×C42—2511no—
165#390—2512no—
166#389—2512no—
167#388—2512no—
168#388—2512no—
169#386—2511no—
170#386—2511no—
171#380—2511no—
172C22×D4—2511no—
173C22×D4—2511no—
174C22×D4C2×D25111yescharacteristic
175C23×C4—2511no—
176C23×C4—2511no—
177C23×C4C2×D25111yescharacteristic
178#123—24no—
179#123—5024no—
180#121—5024no—
181C2×C4⋊C4—2511no—
182C2×C4⋊C4—2512no—
183C2×C4⋊C4—2512no—
184C2×C4⋊C4C2×D25112yes—
185C2×C22⋊C4—2511no—
186C2×C22⋊C4—2511no—
187C2×C22⋊C4—2512no—
188C2×C22⋊C4—2511no—
189C2×C22⋊C4—2512no—
190C2×C22⋊C4—2512no—
191C2×C22⋊C4C2×D25111yescharacteristic
192C2×C22⋊C4C2×D25112yes—
193C2×C42—5024no—
194C2×C42—2512no—
195C2×C42—2512no—
196#95—2512no—
197#95—2512no—
198#95—2512no—
199C24—2511no—
200C24C22×D25111yescharacteristic
201C24C22×D25111yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—5022no—
206C2×D4—5022no—
207C2×D4—5022no—
208C2×D4—5022no—
209C2×D4—5024no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—5022no—
213C22×C4—5024no—
214C22×C4—5024no—
215C22×C4—2512no—
216C22×C4—2511no—
217C22×C4—2514no—
218C22×C4—2512no—
219C22×C4—2511no—
220C22×C4—2511no—
221C22×C4C22⋊D25114yescomplemented
222C22×C4C22×D25112yes—
223C22×C4C22×D25111yescharacteristic
224C22×C4C22×D25112yes—
225C4⋊C4—24no—
226C4⋊C4—5024no—
227C4⋊C4—5022no—
228C4⋊C4—5024no—
229C22⋊C4—22no—
230C22⋊C4—24no—
231C22⋊C4—5024no—
232C22⋊C4—5022no—
233C22⋊C4—5022no—
234C22⋊C4—5024no—
235C22⋊C4—5022no—
236C22⋊C4—5024no—
237C42—5024no—
238C42—5024no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—22no—
244C23—22no—
245C23—21no—
246C23—5021no—
247C23—5021no—
248C23—5022no—
249C23—5022no—
250C23—5021no—
251C23D4×D25112yes—
252C23C2×C22⋊D25112yes—
253C23#1686080912yes—
254C23C23×D25111yescharacteristic
255D4—44no—
256D4—10044no—
257D4—10044no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—28no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—21no—
267C2×C4—5021no—
268C2×C4—5022no—
269C2×C4—5021no—
270C2×C4—5022no—
271C2×C4—5022no—
272C2×C4—5021no—
273C2×C4—5028no—
274C2×C4—5022no—
275C2×C4—5022no—
276C2×C4—5022no—
277C2×C4—5028no—
278C2×C4—2512no—
279C2×C4—2514no—
280C2×C4—2512no—
281C2×C4—2514no—
282C2×C4C2×C22⋊D25114yes—
283C22—22no—
284C22—24no—
285C22—24no—
286C22—22no—
287C22—41no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—10041no—
293C22—10042no—
294C22—10042no—
295C22—10041no—
296C22—10041no—
297C22#2254868714yes—
298C22C2×D4×D25111yescharacteristic
299C22C22×C22⋊D25111yescharacteristic
300C22#41×D25112yes—
301C22#41×D25112yes—
302C22C2×#1686080911yescharacteristic
303C4—24no—
304C4—42no—
305C4—41no—
306C4—10042no—
307C4—10041no—
308C4—5024no—
309C4—5022no—
310C4—2514no—
311C2—24no—
312C2—81no—
313C2—20081no—
314C2C2×#2254868711yescharacteristic
315C2#123×D25111yescharacteristic
316C2#3063219511yescharacteristic
317C2#3063221412yes—
318C2#3063223312yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#123×C251C2
prime radicalC2×#22548687C2
hypercenterC2×#123D251
hypocenterC251#2854
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#22548698C2semidirect
3. C22×C4×Dic251C2semidirect
4. #30631587C2semidirect
5. #30631582C2semidirect
6. #30631581C2semidirect
7. #30631577C2semidirect
8. #30631567C2semidirect
9. #30631564C2semidirect
10. #30631545C2semidirect
11. C2×#123×C251C2semidirect
72. C22×C4×C251D4semidirect
150. Dic251#123semidirect
159. C251#2854semidirect
162. C2×#123D251semidirect
221. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
C2×#225486872
C22×C4×Dic2512
#306315872
#306315822
#306315812
#306315772
#306315672
#306315642
#306315452
C2×#123×C2512
#2854251