Finite Group Databasebeta

← #44558386

Subgroups — #44558386

Order 32128.

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

normal subgroups only

This group has 644 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×#22548688C211yesmaximal, characteristic, complemented
2C2×#22548686C211yesmaximal, characteristic, complemented
3#30631615C212yesmaximal, complemented
4#30631614C211yesmaximal, characteristic, complemented
5#30631612C211yesmaximal, characteristic, complemented
6#30631596C212yesmaximal, complemented
7#30631595C212yesmaximal, complemented
8#30631595C212yesmaximal, complemented
9#30631595C211yesmaximal, characteristic, complemented
10#30631549C211yesmaximal, characteristic, complemented
11#392×C251C211yesmaximal, characteristic, complemented
12C22×D1004C2211yescharacteristic
13C22×C22⋊D251C2211yescharacteristic
14C22×C22⋊D251C2211yescharacteristic
15C22×C22⋊D251C2212yes—
16C22×C22⋊D251C2212yes—
17C22×C4×D251C2211yescharacteristic
18#22548688—24no—
19#22548686—24no—
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×C22⋊Dic251C2212yes—
23C2×C22⋊Dic251C2212yes—
24C2×C22⋊Dic251C2212yes—
25C2×(C2×D251)⋊C4C2211yescharacteristic
26C2×(C2×D251)⋊C4C2212yes—
27C2×(C2×D251)⋊C4C2212yes—
28C251⋊#95C2211yescharacteristic
29C251⋊#95C2212yes—
30C251⋊#95C2212yes—
31C251⋊#95C2212yes—
32C22×D4×C251C2211yescharacteristic
33C22×D4×C251C2212yes—
34C2×C22⋊C4×C251C2211yescharacteristic
35C2×C22⋊C4×C251C2212yes—
36#95×C251C2211yescharacteristic
37C23×D251C2311yescharacteristic
38C23×D251C2311yescharacteristic
39C2×D1004—22no—
40C2×D1004—42no—
41C2×C22⋊D251—24no—
42C2×C22⋊D251—22no—
43C2×C22⋊D251—22no—
44C2×C22⋊D251—42no—
45C2×C22⋊D251—42no—
46C2×C22⋊D251—42no—
47C2×C22⋊D251—42no—
48C2×C22⋊D251—42no—
49C2×C22⋊D251—42no—
50C2×C22⋊D251—42no—
51C22×Dic251C2312yes—
52C22×Dic251C2311yescharacteristic
53C22×Dic251C2312yes—
54C22×Dic251C2311yescharacteristic
55C2×C4×D251—42no—
56C2×C4×D251D414yescomplemented
57C4⋊Dic251—22no—
58Dic251⋊C4—22no—
59C22⋊Dic251—24no—
60C22⋊Dic251—42no—
61C22⋊Dic251—42no—
62(C2×D251)⋊C4—22no—
63(C2×D251)⋊C4—42no—
64(C2×D251)⋊C4—42no—
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×C251C2311yescharacteristic
73C22×C4×C251C2312yes—
74C22×C4×C251C2311yescharacteristic
75C22⋊C4×C251—22no—
76C22⋊C4×C251—42no—
77C22×D251—22no—
78C22×D251—21no—
79C22×D251—21no—
80C22×D251—22no—
81C22×D251—21no—
82C22×D251—21no—
83C22×D251—22no—
84C22×D251—21no—
85C22×D251—22no—
86C22×D251C2×D412yes—
87D1004—44no—
88C22⋊D251—44no—
89C22⋊D251—48no—
90C22⋊D251—44no—
91C22⋊D251—48no—
92C2×Dic251—22no—
93C2×Dic251—21no—
94C2×Dic251—22no—
95C2×Dic251—22no—
96C2×Dic251—22no—
97C2×Dic251—22no—
98C2×Dic251—22no—
99C2×Dic251—22no—
100C2×Dic251—22no—
101C2×Dic251C2×D412yes—
102C4×D251—44no—
103C23×C251—21no—
104C23×C251—22no—
105C23×C251—22no—
106C23×C251—22no—
107C23×C251—21no—
108C23×C251—22no—
109C23×C251—22no—
110C23×C251—22no—
111C23×C251—22no—
112C23×C251—22no—
113C23×C251—21no—
114C23×C251—22no—
115C23×C251C2411yescharacteristic
116D4×C251—44no—
117D4×C251—48no—
118C2×C4×C251—22no—
119C2×C4×C251—21no—
120C2×C4×C251—22no—
121C2×C4×C251—22no—
122C2×C4×C251—22no—
123C2×C4×C251—22no—
124C2×C4×C251C2×D412yes—
125C22×C251—42no—
126C22×C251—41no—
127C22×C251—42no—
128C22×C251—41no—
129C22×C251—41no—
130C22×C251—42no—
131C22×C251—42no—
132C22×C251—42no—
133C22×C251—42no—
134C22×C251—42no—
135C22×C251—42no—
136C22×C251—42no—
137C22×C251C22×D411yescharacteristic
138C22×C251H(5,𝔽2)12yes—
139C22×C251H(5,𝔽2)12yes—
140C22×C251H(5,𝔽2)11yescharacteristic
141C22×C251H(5,𝔽2)11yescharacteristic
142C2×D251—22no—
143C2×D251—22no—
144C2×D251—22no—
145C2×D251—42no—
146C2×D251—42no—
147C2×D251—41no—
148C2×D251—41no—
149C2×D251—41no—
150C2×D251—42no—
151C2×D251—42no—
152C4×C251—22no—
153C4×C251—41no—
154C4×C251—42no—
155Dic251—22no—
156Dic251—41no—
157Dic251—42no—
158Dic251—42no—
159C2×C251—82no—
160C2×C251—81no—
161C2×C251#53411yescharacteristic
162C2×C251#53511yescharacteristic
163C2×C251#53511yescharacteristic
164C2×C251#56012yes—
165C2×C251#56012yes—
166D251—42no—
167D251—81no—
168C251#281511yescharacteristic, complemented, Sylow
169#2815—2511nomaximal, Sylow
170C2×#121—2511no—
171C2×#121—2511no—
172#397—2511no—
173#397—2512no—
174#397—2511no—
175#392—2511no—
176#392—2512no—
177#392—2512no—
178#392—2512no—
179#392D25111yescharacteristic, complemented
180#381—2511no—
181C22×D4—2512no—
182C22×D4—2512no—
183C22×D4—2511no—
184C22×D4—2511no—
185C22×D4—2511no—
186C22×D4C2×D25111yescharacteristic
187C22×D4C2×D25112yes—
188C23×C4—2511no—
189#121—5024no—
190#121—5024no—
191C2×C4⋊C4—2511no—
192C2×C4⋊C4—2511no—
193C2×C22⋊C4—2512no—
194C2×C22⋊C4—2512no—
195C2×C22⋊C4—2512no—
196C2×C22⋊C4—2512no—
197C2×C22⋊C4—2512no—
198C2×C22⋊C4—2511no—
199C2×C22⋊C4C2×D25111yescharacteristic
200C2×C22⋊C4C2×D25112yes—
201#95—2511no—
202#95—2512no—
203#95—2512no—
204#95—2512no—
205#95C2×D25111yescharacteristic
206C24—2511no—
207C24—2511no—
208C24C22×D25111yescharacteristic
209C24C22×D25112yes—
210C2×D4—22no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—10042no—
216C2×D4—10042no—
217C2×D4—10042no—
218C2×D4—10042no—
219C2×D4—10042no—
220C2×D4—10042no—
221C2×D4—10042no—
222C2×D4—10042no—
223C2×D4—5022no—
224C2×D4—5022no—
225C2×D4—5024no—
226C2×D4—5022no—
227C22×C4—10042no—
228C22×C4—2514no—
229C22×C4—2512no—
230C22×C4—2511no—
231C22×C4—2512no—
232C22×C4—2511no—
233C22×C4C22×D25112yes—
234C22×C4C22×D25111yescharacteristic
235C22×C4C22×D25111yescharacteristic
236C4⋊C4—5022no—
237C4⋊C4—5022no—
238C22⋊C4—22no—
239C22⋊C4—42no—
240C22⋊C4—10042no—
241C22⋊C4—10042no—
242C22⋊C4—10042no—
243C22⋊C4—10042no—
244C22⋊C4—5024no—
245C22⋊C4—5022no—
246C23—21no—
247C23—22no—
248C23—22no—
249C23—22no—
250C23—21no—
251C23—22no—
252C23—22no—
253C23—22no—
254C23—22no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—5021no—
259C23—5021no—
260C23—5022no—
261C23—5022no—
262C23—5021no—
263C23—5022no—
264C23—5022no—
265C23—5021no—
266C23—5021no—
267C23—2512no—
268C23C23×D25111yescharacteristic
269D4—44no—
270D4—48no—
271D4—10044no—
272D4—10048no—
273D4—10044no—
274D4—10048no—
275D4—10044no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—10044no—
283C2×C4—5021no—
284C2×C4—5022no—
285C2×C4—5022no—
286C2×C4—5022no—
287C2×C4—5022no—
288C2×C4—5022no—
289C2×C4—5022no—
290C2×C4—5022no—
291C2×C4—5022no—
292C2×C4—2512no—
293C2×C4D4×D25112yes—
294C22—42no—
295C22—41no—
296C22—41no—
297C22—42no—
298C22—42no—
299C22—41no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—42no—
306C22—10041no—
307C22—10041no—
308C22—10041no—
309C22—10042no—
310C22—10042no—
311C22—10042no—
312C22—10042no—
313C22—5022no—
314C22—5022no—
315C22—5022no—
316C22C2×D4×D25111yescharacteristic
317C22#2254876012yes—
318C22#2254876012yes—
319C22#2254876011yescharacteristic
320C22#2254876111yescharacteristic
321C4—22no—
322C4—41no—
323C4—42no—
324C4—10041no—
325C4—10042no—
326C4—10042no—
327C4—5022no—
328C2—82no—
329C2—81no—
330C2—20081no—
331C2—10042no—
332C2#3063224711yescharacteristic
333C2#3063225011yescharacteristic
334C2#3063225311yescharacteristic
335C2#3063242512yes—
336C2#3063242612yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2815
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548688C2semidirect
2. C2×#22548686C2semidirect
3. #30631615C2semidirect
4. #30631614C2semidirect
5. #30631612C2semidirect
6. #30631596C2semidirect
7. #30631595C2semidirect
8. #30631595C2semidirect
9. #30631595C2semidirect
10. #30631549C2semidirect
11. #392×C251C2semidirect
56. C2×C4×D251D4semidirect
168. C251#2815semidirect
179. #392D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486882
C2×#225486862
#306316152
#306316142
#306316122
#306315962
#306315952
#306315492
#392×C2512
#2815251