Finite Group Databasebeta

← #44553046

Subgroups — #44553046

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×#22548693C211yesmaximal, characteristic
2C2×C4×Dic502C211yesmaximal, characteristic, complemented
3C2×#22548673C211yesmaximal, characteristic
4#30631591C211yesmaximal, characteristic, complemented
5#30631586C211yesmaximal, characteristic, complemented
6#30631584C211yesmaximal, characteristic, complemented
7#30631584C211yesmaximal, characteristic, complemented
8#30631579C211yesmaximal, characteristic
9#30631579C211yesmaximal, characteristic
10#30631578C211yesmaximal, characteristic
11#30631576C211yesmaximal, characteristic
12#30631568C211yesmaximal, characteristic, complemented
13#30631546C211yesmaximal, characteristic
14#30631538C211yesmaximal, characteristic, complemented
15#387×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C4×D251C2211yescharacteristic
18#22548693C418yescomplemented
19C4×Dic502—24no—
20#22548673—24no—
21C2×C4⋊Dic251C2211yescharacteristic
22C2×C4⋊Dic251C2211yescharacteristic
23C2×C4⋊Dic251C2211yescharacteristic
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×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C2×C4×Dic251C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C4⋊C4×C251C2211yescharacteristic
49C2×C42×C251C2211yescharacteristic
50#95×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54C23×D251C2311yescharacteristic
55C2×Dic502—22no—
56C2×Dic502—22no—
57C2×Dic502C2×C414yes—
58C22×Dic251C2311yescharacteristic
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C2×C4×D251—22no—
66C2×C4×D251—22no—
67C2×C4×D251C2×C414yes—
68C4⋊Dic251—22no—
69C4⋊Dic251—22no—
70C4⋊Dic251—41no—
71Dic251⋊C4—22no—
72Dic251⋊C4—22no—
73Dic251⋊C4—22no—
74Dic251⋊C4—41no—
75Dic251⋊C4C2×C414yes—
76Dic251⋊C4C2×C414yes—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—22no—
79(C2×D251)⋊C4—41no—
80(C2×D251)⋊C4—41no—
81(C2×D251)⋊C4C2×C414yes—
82(C2×D251)⋊C4C2×C414yes—
83C4×Dic251—22no—
84C4×Dic251—22no—
85C4×Dic251—41no—
86C4×Dic251—41no—
87C4×Dic251D414yes—
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C4⋊C4×C251—22no—
96C4⋊C4×C251C2×C414yes—
97C42×C251—22no—
98C22×D251—21no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251C22×C412yes—
105Dic502—28no—
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×Dic251—21no—
121C2×Dic251C22×C412yes—
122C2×Dic251C22×C412yes—
123C2×Dic251C22×C412yes—
124C2×Dic251C2×D412yes—
125C2×Dic251C2×D412yes—
126C2×Dic251#4112yes—
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×C251C22×C412yes—
147C2×C4×C251C22×C412yes—
148C2×C4×C251C22×C412yes—
149C2×C4×C251C2×D412yes—
150C22×C251C23×C411yescharacteristic
151C22×C251C22×D411yescharacteristic
152C22×C251C2×#4111yescharacteristic
153C22×C251H(5,𝔽2)11yescharacteristic
154C22×C2512−1+411yescharacteristic
155C22×C2512−1+411yescharacteristic
156C22×C2512−1+411yescharacteristic
157C2×D251—22no—
158C2×D251—22no—
159C2×D251—22no—
160C2×D251—41no—
161C2×D251—41no—
162C2×D251—41no—
163C2×D251—41no—
164C4×C251—22no—
165C4×C251—22no—
166C4×C251—22no—
167C4×C251—22no—
168C4×C251—41no—
169C4×C251—41no—
170C4×C251—41no—
171Dic251—22no—
172Dic251—22no—
173Dic251—22no—
174Dic251—41no—
175Dic251—41no—
176Dic251—41no—
177Dic251C4×D414yes—
178C2×C251C2×C4×D411yescharacteristic
179C2×C251#51911yescharacteristic
180C2×C251#52011yescharacteristic
181C2×C251#53611yescharacteristic
182C2×C251#53711yescharacteristic
183C2×C251#54111yescharacteristic
184C2×C251#54211yescharacteristic
185D251—42no—
186C251#232311yescharacteristic, complemented, Sylow
187#2323—2511nomaximal, Sylow
188C2×C22⋊Q8—2511no—
189C2×C4×Q8—2511no—
190C2×#117—2511no—
191#391—2511no—
192#389—2511no—
193#389—2511no—
194#389—2511no—
195#388—2511no—
196#388—2511no—
197#388—2511no—
198#388—2511no—
199#387—2511no—
200#387D25111yescharacteristic, complemented
201#380—2511no—
202C4×C4⋊C4—2511no—
203C22×Q8—2511no—
204C23×C4—2511no—
205C22⋊Q8—2518no—
206C4×Q8—5024no—
207#117—5024no—
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×C22⋊C4—2511no—
224C2×C22⋊C4—2511no—
225C2×C42—2511no—
226C2×C42—2511no—
227C2×C42—2511no—
228C2×C42—2511no—
229C2×C42—2511no—
230C2×C42C2×D25111yescharacteristic
231#95—2511no—
232#95—2511no—
233#95—2511no—
234#95—2511no—
235#95—2511no—
236#95—2511no—
237#95C2×D25111yescharacteristic
238#95C2×D25111yescharacteristic
239#95C2×D25111yescharacteristic
240#95C2×D25111yescharacteristic
241C24—2511no—
242C2×Q8—5022no—
243C2×Q8—5022no—
244C2×Q8—2514no—
245C22×C4—5022no—
246C22×C4—5022no—
247C22×C4—2511no—
248C22×C4—2511no—
249C22×C4—2511no—
250C22×C4—2511no—
251C22×C4—2511no—
252C22×C4—2511no—
253C22×C4—2514no—
254C22×C4—2511no—
255C22×C4C22×D25111yescharacteristic
256C22×C4C22×D25111yescharacteristic
257C22×C4C22×D25111yescharacteristic
258C22×C4C22×D25111yescharacteristic
259C22×C4C22×D25111yescharacteristic
260C22×C4C22×D25111yescharacteristic
261C22×C4C22×D25111yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—10041no—
264C4⋊C4—10041no—
265C4⋊C4—5022no—
266C4⋊C4—5022no—
267C4⋊C4—5022no—
268C4⋊C4—5022no—
269C4⋊C4—5022no—
270C4⋊C4—2514no—
271C4⋊C4—2514no—
272C4⋊C4C4×D25114yes—
273C22⋊C4—10041no—
274C22⋊C4—10041no—
275C22⋊C4—5022no—
276C22⋊C4—5022no—
277C22⋊C4—2514no—
278C22⋊C4—2514no—
279C42—22no—
280C42—10041no—
281C42—10041no—
282C42—5022no—
283C42—5022no—
284C42—2514no—
285C23—5021no—
286C23—5021no—
287C23—5021no—
288C23—5021no—
289C23—5021no—
290C23—5021no—
291C23—2512no—
292C23C23×D25111yescharacteristic
293Q8—5028no—
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—21no—
311C2×C4—5028no—
312C2×C4—5021no—
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—2512no—
328C2×C4—2512no—
329C2×C4—2512no—
330C2×C4—2512no—
331C2×C4—2512no—
332C2×C4—2512no—
333C2×C4C2×C4×D25112yes—
334C2×C4C2×C4×D25112yes—
335C2×C4C2×C4×D25112yes—
336C2×C4D4×D25112yes—
337C22—10041no—
338C22—10041no—
339C22—10041no—
340C22—10041no—
341C22—5022no—
342C22—5022no—
343C22—5022no—
344C22C22×C4×D25111yescharacteristic
345C22C2×D4×D25111yescharacteristic
346C22#41×D25111yescharacteristic
347C22#2254876111yescharacteristic
348C22#2254876211yescharacteristic
349C22#2254876211yescharacteristic
350C22#2254876311yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—10041no—
359C4—10041no—
360C4—10041no—
361C4—5022no—
362C4—5022no—
363C4—5022no—
364C4—2514no—
365C2—10042no—
366C2C4×D4×D25111yescharacteristic
367C2#3063216311yescharacteristic
368C2#3063216711yescharacteristic
369C2#3063225911yescharacteristic
370C2#3063226711yescharacteristic
371C2#3063229611yescharacteristic
372C2#3063230311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#387×C251C2
socle, cosocleC23×C251C24
prime radicalC23×D251C23
hypercenter#387D251
hypocenterC251#2323
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic502C2semidirect
4. #30631591C2semidirect
5. #30631586C2semidirect
6. #30631584C2semidirect
7. #30631584C2semidirect
12. #30631568C2semidirect
14. #30631538C2semidirect
15. #387×C251C2semidirect
18. #22548693C4semidirect
186. C251#2323semidirect
200. #387D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486932
C2×C4×Dic5022
C2×#225486732
#306315912
#306315862
#306315842
#306315792
#306315782
#306315762
#306315682
#306315462
#306315382
#387×C2512
#2323251