Finite Group Databasebeta

← #44558404

Subgroups — #44558404

Order 32128.

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

normal subgroups only

This group has 630 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×#22548711C211yesmaximal, characteristic, complemented
2C2×#22548708C211yesmaximal, characteristic, complemented
3#30631605C211yesmaximal, characteristic, complemented
4#30631603C212yesmaximal, complemented
5#30631602C212yesmaximal, complemented
6#30631601C211yesmaximal, characteristic, complemented
7#30631596C211yesmaximal, characteristic, complemented
8#30631595C212yesmaximal, complemented
9#30631595C212yesmaximal, complemented
10#30631554C211yesmaximal, characteristic, complemented
11#392×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C22⋊D251C2212yes—
14C22×C22⋊D251C2212yes—
15C22×C22⋊D251C2212yes—
16#22548711—24no—
17#22548708—24no—
18C2×C4⋊Dic251C2211yescharacteristic
19C2×Dic251⋊C4C2211yescharacteristic
20C2×Dic251⋊C4C2211yescharacteristic
21C2×C22⋊Dic251C2211yescharacteristic
22C2×C22⋊Dic251C2211yescharacteristic
23C2×C22⋊Dic251C2212yes—
24C2×C22⋊Dic251C2212yes—
25C2×(C2×D251)⋊C4C2212yes—
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C2×C4×Dic251C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2211yescharacteristic
32C22×D4×C251C2211yescharacteristic
33C22×D4×C251C2212yes—
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2312yes—
38C2×D1004—22no—
39C2×D1004—42no—
40C2×C22⋊D251—24no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—42no—
43C2×C22⋊D251—42no—
44C2×C22⋊D251—42no—
45C2×C22⋊D251—42no—
46C2×C22⋊D251—42no—
47C2×C22⋊D251—42no—
48C2×C22⋊D251—42no—
49C22×Dic251C2312yes—
50C22×Dic251C2311yescharacteristic
51C22×Dic251C2311yescharacteristic
52C22×Dic251C2311yescharacteristic
53C22×Dic251C2311yescharacteristic
54C4⋊Dic251—22no—
55Dic251⋊C4—22no—
56Dic251⋊C4—22no—
57C22⋊Dic251—22no—
58C22⋊Dic251—22no—
59C22⋊Dic251—42no—
60C22⋊Dic251—42no—
61(C2×D251)⋊C4—42no—
62(C2×D251)⋊C4—42no—
63(C2×D251)⋊C4—42no—
64C4×Dic251D414yescomplemented
65C24×C251C2311yescharacteristic
66C24×C251C2312yes—
67C2×D4×C251—22no—
68C2×D4×C251—42no—
69C2×D4×C251—42no—
70C2×D4×C251—42no—
71C2×D4×C251—42no—
72C22×C4×C251C2312yes—
73C22×C4×C251C2311yescharacteristic
74C22×C4×C251C2311yescharacteristic
75C22⋊C4×C251—22no—
76C22⋊C4×C251—42no—
77C22×D251—22no—
78C22×D251—22no—
79C22×D251—22no—
80C22×D251—22no—
81C22×D251—22no—
82C22×D251—22no—
83C22×D251—22no—
84D1004—44no—
85C22⋊D251—48no—
86C22⋊D251—48no—
87C22⋊D251—48no—
88C2×Dic251—22no—
89C2×Dic251—21no—
90C2×Dic251—21no—
91C2×Dic251—21no—
92C2×Dic251—21no—
93C2×Dic251—21no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251—21no—
97C2×Dic251—22no—
98C2×Dic251—22no—
99C2×Dic251C2×D412yes—
100C2×Dic251C2×D412yes—
101C23×C251—21no—
102C23×C251—22no—
103C23×C251—22no—
104C23×C251—22no—
105C23×C251—21no—
106C23×C251—22no—
107C23×C251—22no—
108C23×C251—22no—
109C23×C251—22no—
110C23×C251—22no—
111C23×C251—22no—
112C23×C251—21no—
113C23×C251C2411yescharacteristic
114D4×C251—44no—
115D4×C251—48no—
116C2×C4×C251—22no—
117C2×C4×C251—21no—
118C2×C4×C251—22no—
119C2×C4×C251—22no—
120C2×C4×C251—22no—
121C2×C4×C251—22no—
122C2×C4×C251C2×D412yes—
123C22×C251—42no—
124C22×C251—41no—
125C22×C251—41no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251—42no—
129C22×C251—42no—
130C22×C251—42no—
131C22×C251—42no—
132C22×C251—42no—
133C22×C251—42no—
134C22×C251—42no—
135C22×C251C22×D411yescharacteristic
136C22×C251H(5,𝔽2)12yes—
137C22×C251H(5,𝔽2)12yes—
138C22×C251H(5,𝔽2)11yescharacteristic
139C22×C251H(5,𝔽2)11yescharacteristic
140C2×D251—42no—
141C2×D251—42no—
142C2×D251—42no—
143C2×D251—42no—
144C2×D251—42no—
145C2×D251—42no—
146C2×D251—42no—
147C4×C251—22no—
148C4×C251—41no—
149C4×C251—42no—
150Dic251—22no—
151Dic251—22no—
152Dic251—41no—
153Dic251—41no—
154Dic251—42no—
155C2×C251—82no—
156C2×C251—81no—
157C2×C251#53511yescharacteristic
158C2×C251#53511yescharacteristic
159C2×C251#53511yescharacteristic
160C2×C251#56012yes—
161C2×C251#56012yes—
162D251—82no—
163C251#281811yescharacteristic, complemented, Sylow
164#2818—2511nomaximal, Sylow
165C2×#127—2511no—
166C2×#126—2511no—
167#394—2511no—
168#394—2512no—
169#394—2512no—
170#394—2511no—
171#392—2512no—
172#392—2512no—
173#392—2511no—
174#392D25111yescharacteristic, complemented
175#383—2511no—
176C22×D4—2512no—
177C22×D4—2512no—
178C22×D4—2512no—
179C22×D4—2511no—
180C22×D4C2×D25111yescharacteristic
181C22×D4C2×D25112yes—
182#127—5024no—
183#126—5024no—
184C2×C4⋊C4—2511no—
185C2×C4⋊C4—2511no—
186C2×C4⋊C4—2511no—
187C2×C22⋊C4—2512no—
188C2×C22⋊C4—2512no—
189C2×C22⋊C4—2512no—
190C2×C22⋊C4—2512no—
191C2×C22⋊C4—2512no—
192C2×C22⋊C4—2511no—
193C2×C22⋊C4—2511no—
194C2×C22⋊C4C2×D25111yescharacteristic
195C2×C22⋊C4C2×D25112yes—
196C2×C42—2511no—
197#95—2511no—
198#95—2512no—
199#95—2512no—
200#95C2×D25111yescharacteristic
201C24—2512no—
202C24C22×D25111yescharacteristic
203C24C22×D25112yes—
204C2×D4—22no—
205C2×D4—42no—
206C2×D4—42no—
207C2×D4—42no—
208C2×D4—42no—
209C2×D4—10042no—
210C2×D4—10042no—
211C2×D4—10042no—
212C2×D4—10042no—
213C2×D4—10042no—
214C2×D4—10042no—
215C2×D4—10042no—
216C2×D4—10042no—
217C2×D4—5024no—
218C2×D4—5024no—
219C2×D4—5022no—
220C22×C4—2512no—
221C22×C4—2511no—
222C22×C4—2511no—
223C22×C4—2511no—
224C22×C4—2511no—
225C22×C4C22×D25112yes—
226C22×C4C22×D25111yescharacteristic
227C22×C4C22×D25111yescharacteristic
228C4⋊C4—5022no—
229C4⋊C4—5022no—
230C4⋊C4—5022no—
231C22⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—10042no—
234C22⋊C4—10042no—
235C22⋊C4—10042no—
236C22⋊C4—10042no—
237C22⋊C4—10042no—
238C22⋊C4—5022no—
239C22⋊C4—5022no—
240C42—2514no—
241C23—21no—
242C23—22no—
243C23—22no—
244C23—22no—
245C23—22no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—21no—
253C23—5022no—
254C23—5022no—
255C23—5022no—
256C23—5022no—
257C23—5022no—
258C23—5022no—
259C23—5022no—
260C23C23×D25111yescharacteristic
261D4—44no—
262D4—48no—
263D4—10044no—
264D4—10048no—
265D4—10048no—
266D4—10048no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—5021no—
274C2×C4—5021no—
275C2×C4—5022no—
276C2×C4—5021no—
277C2×C4—5022no—
278C2×C4—5022no—
279C2×C4—5021no—
280C2×C4—5022no—
281C2×C4—5022no—
282C2×C4—5021no—
283C2×C4—5021no—
284C2×C4—2512no—
285C2×C4—2512no—
286C2×C4D4×D25112yes—
287C22—42no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—41no—
293C22—42no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—42no—
298C22—42no—
299C22—10042no—
300C22—10042no—
301C22—10042no—
302C22—10042no—
303C22—10042no—
304C22—10042no—
305C22—10042no—
306C22C2×D4×D25111yescharacteristic
307C22#2254876012yes—
308C22#2254876012yes—
309C22#2254876011yescharacteristic
310C22#2254876111yescharacteristic
311C4—22no—
312C4—41no—
313C4—42no—
314C4—10041no—
315C4—10041no—
316C4—10042no—
317C4—5022no—
318C4—5022no—
319C2—82no—
320C2—81no—
321C2—20082no—
322C2#3063225011yescharacteristic
323C2#3063225211yescharacteristic
324C2#3063225311yescharacteristic
325C2#3063242512yes—
326C2#3063242612yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2818
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548711C2semidirect
2. C2×#22548708C2semidirect
3. #30631605C2semidirect
4. #30631603C2semidirect
5. #30631602C2semidirect
6. #30631601C2semidirect
7. #30631596C2semidirect
8. #30631595C2semidirect
9. #30631595C2semidirect
10. #30631554C2semidirect
11. #392×C251C2semidirect
64. C4×Dic251D4semidirect
163. C251#2818semidirect
174. #392D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487112
C2×#225487082
#306316052
#306316032
#306316022
#306316012
#306315962
#306315952
#306315542
#392×C2512
#2818251