Finite Group Databasebeta

← #44558734

Subgroups — #44558734

Order 32128.

524 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548705C211yesmaximal, characteristic, complemented
2C2×#22548701C211yesmaximal, characteristic
3#30631591C211yesmaximal, characteristic, complemented
4#30631591C211yesmaximal, characteristic, complemented
5#30631587C211yesmaximal, characteristic
6#30631587C211yesmaximal, characteristic
7#30631581C211yesmaximal, characteristic, complemented
8#30631580C211yesmaximal, characteristic, complemented
9#30631579C211yesmaximal, characteristic, complemented
10#30631578C211yesmaximal, characteristic, complemented
11#30631577C211yesmaximal, characteristic, complemented
12#30631576C211yesmaximal, characteristic, complemented
13#30631575C211yesmaximal, characteristic, complemented
14C42×Dic251C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C22×Dic502C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18#22548705—24no—
19#22548701—24no—
20C2×C4⋊Dic251C2211yescharacteristic
21C2×Dic251⋊C4C2211yescharacteristic
22C2×Dic251⋊C4C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×C22⋊Dic251C2211yescharacteristic
26C2×C22⋊Dic251C2211yescharacteristic
27C2×C22⋊Dic251C2211yescharacteristic
28C2×(C2×D251)⋊C4C2211yescharacteristic
29C2×(C2×D251)⋊C4C2211yescharacteristic
30C2×(C2×D251)⋊C4C2211yescharacteristic
31C2×(C2×D251)⋊C4C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C2×C4×Dic251C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C2×C4×Dic251C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C2×C4⋊C4×C251C2211yescharacteristic
47C2×C22⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C42×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C23×D251C2311yescharacteristic
54C2×Dic502—22no—
55C2×Dic502—22no—
56C2×Dic502—22no—
57C2×C22⋊D251—22no—
58C2×C22⋊D251—22no—
59C2×C22⋊D251—22no—
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C4⋊Dic251—22no—
68Dic251⋊C4—22no—
69Dic251⋊C4—22no—
70Dic251⋊C4—22no—
71Dic251⋊C4—22no—
72C22⋊Dic251—22no—
73C22⋊Dic251—22no—
74C22⋊Dic251—22no—
75(C2×D251)⋊C4—22no—
76(C2×D251)⋊C4—22no—
77(C2×D251)⋊C4—22no—
78(C2×D251)⋊C4—22no—
79(C2×D251)⋊C4—22no—
80(C2×D251)⋊C4—22no—
81C4×Dic251—22no—
82C4×Dic251—22no—
83C4×Dic251—22no—
84C4×Dic251—22no—
85C4×Dic251—22no—
86C4×Dic251D414yes—
87C24×C251C2311yescharacteristic
88C22×C4×C251C2311yescharacteristic
89C22×C4×C251C2311yescharacteristic
90C22×C4×C251C2311yescharacteristic
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C4⋊C4×C251—22no—
95C22⋊C4×C251—22no—
96C22⋊C4×C251—22no—
97C22⋊C4×C251—22no—
98C42×C251—22no—
99C22×D251—21no—
100C22×D251—21no—
101C22×D251—21no—
102C22×D251—21no—
103C22×D251—21no—
104C22×D251—21no—
105C22×D251—21no—
106Dic502—44no—
107C22⋊D251—44no—
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×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251C2×D412yes—
124C2×Dic251C2×D412yes—
125C2×Dic251#4112yes—
126C2×Dic251#4112yes—
127C2×Dic251#4112yes—
128C2×Dic251#4112yes—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251C2411yescharacteristic
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×C251—21no—
147C2×C4×C251—21no—
148C2×C4×C251—21no—
149C2×C4×C251—21no—
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251C2×D412yes—
153C2×C4×C251#4112yes—
154C2×C4×C251#4112yes—
155C22×C251—41no—
156C22×C251—41no—
157C22×C251—41no—
158C22×C251—41no—
159C22×C251—41no—
160C22×C251—41no—
161C22×C251—41no—
162C22×C251C22×D411yescharacteristic
163C22×C251C2×#4111yescharacteristic
164C22×C251C2×#4111yescharacteristic
165C22×C251C2×#4111yescharacteristic
166C22×C251C2×#4111yescharacteristic
167C22×C251C2×#4111yescharacteristic
168C22×C251C2×#4111yescharacteristic
169C2×D251—41no—
170C2×D251—41no—
171C2×D251—41no—
172C2×D251—41no—
173C2×D251—41no—
174C2×D251—41no—
175C2×D251—41no—
176C4×C251—22no—
177C4×C251—22no—
178C4×C251—22no—
179C4×C251—41no—
180C4×C251—41no—
181C4×C251—41no—
182Dic251—22no—
183Dic251—22no—
184Dic251—22no—
185Dic251—41no—
186Dic251—41no—
187Dic251—41no—
188Dic251#12414yes—
189C2×C251—81no—
190C2×C251C2×#12411yescharacteristic
191C2×C251#52911yescharacteristic
192C2×C251#52911yescharacteristic
193C2×C251#52911yescharacteristic
194C2×C251#52911yescharacteristic
195C2×C251#53211yescharacteristic
196C2×C251#53211yescharacteristic
197D251—81no—
198C251#286111yescharacteristic, complemented, Sylow
199#2861—2511nomaximal, Sylow
200C2×#125—2511no—
201C2×#124—2511no—
202#391—2511no—
203#391—2511no—
204#390—2511no—
205#390—2511no—
206#388—2511no—
207#388—2511no—
208#388—2511no—
209#388—2511no—
210#388—2511no—
211#388—2511no—
212#388—2511no—
213#388D25111yescharacteristic, complemented
214C43—2511no—
215C22×Q8—2511no—
216C22×D4—2511no—
217#125—5024no—
218#124—5024no—
219C2×C4⋊C4—2511no—
220C2×C4⋊C4—2511no—
221C2×C4⋊C4—2511no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4C2×D25111yescharacteristic
225C2×C22⋊C4—2511no—
226C2×C22⋊C4—2511no—
227C2×C22⋊C4—2511no—
228C2×C22⋊C4—2511no—
229C2×C22⋊C4—2511no—
230C2×C22⋊C4—2511no—
231C2×C22⋊C4—2511no—
232C2×C22⋊C4—2511no—
233C2×C22⋊C4—2511no—
234C2×C22⋊C4C2×D25111yescharacteristic
235C2×C22⋊C4C2×D25111yescharacteristic
236C2×C22⋊C4C2×D25111yescharacteristic
237C2×C42—2511no—
238C2×C42—2511no—
239C2×C42—2511no—
240C2×C42—2511no—
241C2×C42—2511no—
242C2×C42—2511no—
243C2×C42C2×D25111yescharacteristic
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95—2511no—
249#95—2511no—
250#95C2×D25111yescharacteristic
251#95C2×D25111yescharacteristic
252C24—2511no—
253C24C22×D25111yescharacteristic
254C2×Q8—5022no—
255C2×Q8—5022no—
256C2×Q8—5022no—
257C2×D4—5022no—
258C2×D4—5022no—
259C2×D4—5022no—
260C22×C4—2511no—
261C22×C4—2511no—
262C22×C4—2511no—
263C22×C4—2511no—
264C22×C4—2511no—
265C22×C4—2511no—
266C22×C4—2511no—
267C22×C4C22×D25111yescharacteristic
268C22×C4C22×D25111yescharacteristic
269C22×C4C22×D25111yescharacteristic
270C22×C4C22×D25111yescharacteristic
271C22×C4C22×D25111yescharacteristic
272C22×C4C22×D25111yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—5022no—
275C4⋊C4—5022no—
276C4⋊C4—5022no—
277C4⋊C4—5022no—
278C4⋊C4—5022no—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—5022no—
283C22⋊C4—5022no—
284C22⋊C4—5022no—
285C22⋊C4—5022no—
286C22⋊C4—5022no—
287C22⋊C4—5022no—
288C22⋊C4—5022no—
289C22⋊C4—5022no—
290C22⋊C4—5022no—
291C42—22no—
292C42—5022no—
293C42—5022no—
294C42—5022no—
295C42—5022no—
296C42—5022no—
297C42—2514no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—5021no—
306C23—5021no—
307C23—5021no—
308C23—5021no—
309C23—5021no—
310C23—5021no—
311C23—5021no—
312C23C23×D25111yescharacteristic
313Q8—10044no—
314D4—10044no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—5021no—
331C2×C4—5021no—
332C2×C4—5021no—
333C2×C4—5021no—
334C2×C4—5021no—
335C2×C4—5021no—
336C2×C4—5021no—
337C2×C4—5021no—
338C2×C4—5021no—
339C2×C4—5021no—
340C2×C4—5021no—
341C2×C4—5021no—
342C2×C4—5021no—
343C2×C4—5021no—
344C2×C4—5021no—
345C2×C4—2512no—
346C2×C4—2512no—
347C2×C4—2512no—
348C2×C4—2512no—
349C2×C4—2512no—
350C2×C4—2512no—
351C2×C4D4×D25112yes—
352C2×C4#1686080912yes—
353C2×C4#1686081012yes—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—10041no—
362C22—10041no—
363C22—10041no—
364C22—10041no—
365C22—10041no—
366C22—10041no—
367C22—10041no—
368C22C2×D4×D25111yescharacteristic
369C22#41×D25111yescharacteristic
370C22#41×D25111yescharacteristic
371C22#41×D25111yescharacteristic
372C22#41×D25111yescharacteristic
373C22C2×#1686080911yescharacteristic
374C22C2×#1686081011yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—10041no—
382C4—10041no—
383C4—10041no—
384C4—5022no—
385C4—5022no—
386C4—5022no—
387C4—2514no—
388C2—81no—
389C2—20081no—
390C2#124×D25111yescharacteristic
391C2#3063221411yescharacteristic
392C2#3063221511yescharacteristic
393C2#3063221711yescharacteristic
394C2#3063222011yescharacteristic
395C2#3063223211yescharacteristic
396C2#3063223411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#388×C251C2
prime radicalC22×C22⋊D251C22
hypercenter#388D251
hypocenterC251#2861
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#22548705C2semidirect
3. #30631591C2semidirect
4. #30631591C2semidirect
7. #30631581C2semidirect
8. #30631580C2semidirect
9. #30631579C2semidirect
10. #30631578C2semidirect
11. #30631577C2semidirect
12. #30631576C2semidirect
13. #30631575C2semidirect
14. C42×Dic251C2semidirect
15. #388×C251C2semidirect
198. C251#2861semidirect
213. #388D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487052
C2×#225487012
#306315912
#306315872
#306315812
#306315802
#306315792
#306315782
#306315772
#306315762
#306315752
C42×Dic2512
#388×C2512
#2861251