Finite Group Databasebeta

← #44558394

Subgroups — #44558394

Order 32128.

528 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548695C211yesmaximal, characteristic, complemented
2C2×#22548688C211yesmaximal, characteristic
3#30631622C211yesmaximal, characteristic, complemented
4#30631620C212yesmaximal, complemented
5#30631604C212yesmaximal, complemented
6#30631603C212yesmaximal, complemented
7#30631602C212yesmaximal, complemented
8#30631600C211yesmaximal, characteristic, complemented
9#30631599C211yesmaximal, characteristic, complemented
10#30631549C211yesmaximal, characteristic, complemented
11#399×C251C211yesmaximal, characteristic, complemented
12C22×Dic502C2211yescharacteristic
13C22×D1004C2211yescharacteristic
14C22×C22⋊D251C2211yescharacteristic
15C22×C22⋊D251C2211yescharacteristic
16C22×C4×D251C2211yescharacteristic
17#22548695—24no—
18#22548688—24no—
19C2×C4⋊Dic251C2212yes—
20C2×Dic251⋊C4C2211yescharacteristic
21C2×Dic251⋊C4C2212yes—
22C2×C22⋊Dic251C2212yes—
23C2×(C2×D251)⋊C4C2211yescharacteristic
24C2×(C2×D251)⋊C4C2212yes—
25C2×(C2×D251)⋊C4C2212yes—
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C251⋊#95C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C2×C4⋊C4×C251C2211yescharacteristic
33C2×C4⋊C4×C251C2212yes—
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2311yescharacteristic
38C23×D251C2311yescharacteristic
39C2×Dic502—22no—
40C2×Dic502—42no—
41C2×D1004—22no—
42C2×D1004—42no—
43C2×C22⋊D251—22no—
44C2×C22⋊D251—22no—
45C2×C22⋊D251—42no—
46C2×C22⋊D251—42no—
47C22×Dic251C2311yescharacteristic
48C22×Dic251C2312yes—
49C22×Dic251C2312yes—
50C22×Dic251C2311yescharacteristic
51C2×C4×D251—42no—
52C2×C4×D251D414yes—
53C4⋊Dic251—24no—
54Dic251⋊C4—22no—
55Dic251⋊C4—42no—
56C22⋊Dic251—42no—
57(C2×D251)⋊C4—24no—
58(C2×D251)⋊C4—22no—
59(C2×D251)⋊C4—42no—
60(C2×D251)⋊C4—42no—
61(C2×D251)⋊C4—42no—
62C24×C251C2311yescharacteristic
63C22×C4×C251C2311yescharacteristic
64C22×C4×C251C2312yes—
65C22×C4×C251C2312yes—
66C22×C4×C251C2311yescharacteristic
67C4⋊C4×C251—22no—
68C4⋊C4×C251—42no—
69C22⋊C4×C251—22no—
70C22⋊C4×C251—42no—
71C22×D251—22no—
72C22×D251—22no—
73C22×D251—21no—
74C22×D251—21no—
75C22×D251—21no—
76C22×D251—22no—
77C22×D251—22no—
78C22×D251—21no—
79C22×D251—21no—
80C22×D251C2×D412yes—
81Dic502—44no—
82D1004—44no—
83C22⋊D251—44no—
84C22⋊D251—44no—
85C2×Dic251—22no—
86C2×Dic251—21no—
87C2×Dic251—22no—
88C2×Dic251—22no—
89C2×Dic251—22no—
90C2×Dic251—22no—
91C2×Dic251—22no—
92C2×Dic251—22no—
93C2×Dic251—22no—
94C2×Dic251C2×D412yes—
95C4×D251—44no—
96C23×C251—21no—
97C23×C251—22no—
98C23×C251—22no—
99C23×C251—21no—
100C23×C251—21no—
101C23×C251C2411yescharacteristic
102C2×C4×C251—22no—
103C2×C4×C251—21no—
104C2×C4×C251—22no—
105C2×C4×C251—22no—
106C2×C4×C251—22no—
107C2×C4×C251—22no—
108C2×C4×C251—22no—
109C2×C4×C251—22no—
110C2×C4×C251—22no—
111C2×C4×C251C2×D412yes—
112C22×C251—41no—
113C22×C251—41no—
114C22×C251—41no—
115C22×C251—42no—
116C22×C251—42no—
117C22×C251C22×D411yescharacteristic
118C22×C251H(5,𝔽2)12yes—
119C22×C251H(5,𝔽2)12yes—
120C22×C2512−1+411yescharacteristic
121C22×C2512−1+411yescharacteristic
122C2×D251—22no—
123C2×D251—22no—
124C2×D251—22no—
125C2×D251—42no—
126C2×D251—42no—
127C2×D251—41no—
128C2×D251—41no—
129C2×D251—41no—
130C2×D251—42no—
131C2×D251—42no—
132C4×C251—22no—
133C4×C251—42no—
134C4×C251—41no—
135C4×C251—42no—
136Dic251—22no—
137Dic251—42no—
138Dic251—41no—
139Dic251—42no—
140C2×C251—81no—
141C2×C251#53411yescharacteristic
142C2×C251#53511yescharacteristic
143C2×C251#53611yescharacteristic
144C2×C251#56212yes—
145C2×C251#56212yes—
146D251—42no—
147D251—81no—
148C251#281611yescharacteristic, complemented, Sylow
149#2816—2511nomaximal, Sylow
150C2×C22⋊Q8—2511no—
151C2×#121—2511no—
152#399—2511no—
153#399D25111yescharacteristic, complemented
154#398—2512no—
155#394—2512no—
156#394—2512no—
157#394—2512no—
158#393—2511no—
159#393—2511no—
160#381—2511no—
161C22×Q8—2511no—
162C22×D4—2511no—
163C22×D4—2511no—
164C22×D4—2511no—
165C23×C4—2511no—
166C22⋊Q8—5024no—
167#121—5024no—
168C2×C4⋊C4—2512no—
169C2×C4⋊C4—2511no—
170C2×C4⋊C4—2512no—
171C2×C4⋊C4C2×D25111yescharacteristic
172C2×C4⋊C4C2×D25112yes—
173C2×C22⋊C4—2512no—
174C2×C22⋊C4—2512no—
175C2×C22⋊C4—2512no—
176C2×C22⋊C4—2512no—
177C2×C22⋊C4—2511no—
178C2×C22⋊C4—2512no—
179C2×C22⋊C4C2×D25111yescharacteristic
180C2×C22⋊C4C2×D25112yes—
181#95—2511no—
182#95—2512no—
183#95—2512no—
184#95—2512no—
185#95C2×D25111yescharacteristic
186C24—2511no—
187C24—2511no—
188C24C22×D25111yescharacteristic
189C2×Q8—10042no—
190C2×Q8—5022no—
191C2×D4—10042no—
192C2×D4—10042no—
193C2×D4—10042no—
194C2×D4—5022no—
195C2×D4—5022no—
196C2×D4—5022no—
197C22×C4—10042no—
198C22×C4—2514no—
199C22×C4—2512no—
200C22×C4—2512no—
201C22×C4—2511no—
202C22×C4—2511no—
203C22×C4C22×D25112yes—
204C22×C4C22×D25111yescharacteristic
205C22×C4C22×D25112yes—
206C22×C4C22×D25111yescharacteristic
207C4⋊C4—22no—
208C4⋊C4—42no—
209C4⋊C4—10042no—
210C4⋊C4—5022no—
211C4⋊C4—5024no—
212C22⋊C4—22no—
213C22⋊C4—42no—
214C22⋊C4—10042no—
215C22⋊C4—10042no—
216C22⋊C4—10042no—
217C22⋊C4—10042no—
218C22⋊C4—5024no—
219C22⋊C4—5022no—
220C23—21no—
221C23—22no—
222C23—22no—
223C23—21no—
224C23—21no—
225C23—5021no—
226C23—5022no—
227C23—5022no—
228C23—5021no—
229C23—5021no—
230C23—5022no—
231C23—5022no—
232C23—5021no—
233C23—5021no—
234C23—2512no—
235C23C23×D25111yescharacteristic
236Q8—10044no—
237D4—10044no—
238D4—10044no—
239D4—10044no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—10044no—
250C2×C4—5021no—
251C2×C4—5022no—
252C2×C4—5022no—
253C2×C4—5022no—
254C2×C4—5022no—
255C2×C4—5022no—
256C2×C4—5022no—
257C2×C4—5022no—
258C2×C4—5022no—
259C2×C4—2512no—
260C2×C4D4×D25112yes—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22—10041no—
267C22—10041no—
268C22—10041no—
269C22—10042no—
270C22—10042no—
271C22—10042no—
272C22—10042no—
273C22—5022no—
274C22—5022no—
275C22—5022no—
276C22C2×D4×D25111yescharacteristic
277C22#2254876012yes—
278C22#2254876112yes—
279C22#2254876211yescharacteristic
280C22#2254876311yescharacteristic
281C4—22no—
282C4—41no—
283C4—42no—
284C4—42no—
285C4—10042no—
286C4—10041no—
287C4—10042no—
288C4—5022no—
289C2—81no—
290C2—20081no—
291C2—10042no—
292C2#3063224811yescharacteristic
293C2#3063225011yescharacteristic
294C2#3063226011yescharacteristic
295C2#3063243312yes—
296C2#3063243412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#399×C251C2
prime radicalC2×#22548688C2
hypercenter#399D251
hypocenterC251#2816
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#22548695C2semidirect
3. #30631622C2semidirect
4. #30631620C2semidirect
5. #30631604C2semidirect
6. #30631603C2semidirect
7. #30631602C2semidirect
8. #30631600C2semidirect
9. #30631599C2semidirect
10. #30631549C2semidirect
11. #399×C251C2semidirect
148. C251#2816semidirect
153. #399D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486952
C2×#225486882
#306316222
#306316202
#306316042
#306316032
#306316022
#306316002
#306315992
#306315492
#399×C2512
#2816251