Finite Group Databasebeta

← #44558678

Subgroups — #44558678

Order 32128.

582 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548708C211yesmaximal, characteristic, complemented
2C22×C4×Dic251C211yesmaximal, characteristic, complemented
3#30631581C211yesmaximal, characteristic, complemented
4#30631580C211yesmaximal, characteristic, complemented
5#30631575C211yesmaximal, characteristic, complemented
6#30631574C211yesmaximal, characteristic, complemented
7#30631572C211yesmaximal, characteristic, complemented
8#30631569C211yesmaximal, characteristic, complemented
9#30631567C211yesmaximal, characteristic
10#30631567C211yesmaximal, characteristic
11#30631562C211yesmaximal, characteristic, complemented
12#30631560C211yesmaximal, characteristic, complemented
13#30631559C211yesmaximal, characteristic, complemented
14#30631554C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic
16C23×Dic251C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18#22548708—24no—
19C2×C4⋊Dic251C2211yescharacteristic
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×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C4×Dic251—24no—
32C2×C4×Dic251C2211yescharacteristic
33C2×C4×Dic251C2211yescharacteristic
34C2×C4×Dic251C2211yescharacteristic
35C2×C4×Dic251C2211yescharacteristic
36C251⋊#95C2211yescharacteristic
37C251⋊#95C2211yescharacteristic
38C251⋊#95C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C22×D4×C251C2211yescharacteristic
47C23×C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51#95×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C22×Dic251—22no—
54C22×Dic251—22no—
55C22×Dic251—22no—
56C22×Dic251—22no—
57C22×Dic251—22no—
58C22×Dic251—22no—
59C22×Dic251C2311yescharacteristic
60C22×Dic251C2311yescharacteristic
61C22×Dic251C2311yescharacteristic
62C22×Dic251C2311yescharacteristic
63C22×Dic251C2311yescharacteristic
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C4⋊Dic251—22no—
68C4⋊Dic251—22no—
69Dic251⋊C4—22no—
70Dic251⋊C4—22no—
71Dic251⋊C4—22no—
72Dic251⋊C4—22no—
73C22⋊Dic251—22no—
74C22⋊Dic251—22no—
75C22⋊Dic251—22no—
76C22⋊Dic251—22no—
77C22⋊Dic251—22no—
78C22⋊Dic251—22no—
79C4×Dic251—22no—
80C4×Dic251—22no—
81C4×Dic251—22no—
82C4×Dic251D414yescomplemented
83C24×C251C2311yescharacteristic
84C24×C251C2311yescharacteristic
85C2×D4×C251—22no—
86C2×D4×C251—22no—
87C2×D4×C251—22no—
88C22×C4×C251—22no—
89C22×C4×C251—22no—
90C22×C4×C251—22no—
91C22×C4×C251C2311yescharacteristic
92C22×C4×C251C2311yescharacteristic
93C22×C4×C251C2311yescharacteristic
94C22×C4×C251C2311yescharacteristic
95C22×C4×C251C2311yescharacteristic
96C22⋊C4×C251—22no—
97C22⋊C4×C251—22no—
98C22⋊C4×C251—22no—
99C2×Dic251—28no—
100C2×Dic251—28no—
101C2×Dic251—21no—
102C2×Dic251—21no—
103C2×Dic251—21no—
104C2×Dic251—21no—
105C2×Dic251—21no—
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×Dic251C2×D412yes—
122C2×Dic251C2×D412yes—
123C2×Dic251#4112yes—
124C2×Dic251#4112yes—
125C23×C251—21no—
126C23×C251—21no—
127C23×C251—21no—
128C23×C251—21no—
129C23×C251—21no—
130C23×C251—21no—
131C23×C251—21no—
132C23×C251—21no—
133C23×C251—21no—
134C23×C251—21no—
135C23×C251—21no—
136C23×C251#4112yes—
137C23×C251#4112yes—
138C23×C251#4112yes—
139C23×C251C2411yescharacteristic
140D4×C251—44no—
141C2×C4×C251—28no—
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×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251C2×D412yes—
156C2×C4×C251#4112yes—
157C22×C251—22no—
158C22×C251—22no—
159C22×C251—22no—
160C22×C251—22no—
161C22×C251—22no—
162C22×C251—22no—
163C22×C251—41no—
164C22×C251—41no—
165C22×C251—41no—
166C22×C251—41no—
167C22×C251—41no—
168C22×C251—41no—
169C22×C251—41no—
170C22×C251#12614yescomplemented
171C22×C251C22×D411yescharacteristic
172C22×C251C2×#4111yescharacteristic
173C22×C251C2×#4111yescharacteristic
174C22×C251C2×#4111yescharacteristic
175C22×C251C2×#4111yescharacteristic
176C22×C251C2×#4111yescharacteristic
177C22×C251C2×#4111yescharacteristic
178C4×C251—22no—
179C4×C251—22no—
180C4×C251—41no—
181C4×C251—41no—
182C4×C251—41no—
183Dic251—22no—
184Dic251—22no—
185Dic251—22no—
186Dic251—22no—
187Dic251—41no—
188Dic251—41no—
189Dic251—41no—
190Dic251—41no—
191C2×C251—24no—
192C2×C251—81no—
193C2×C251#52511yescharacteristic
194C2×C251#52511yescharacteristic
195C2×C251#52511yescharacteristic
196C2×C251C2×#12611yescharacteristic
197C2×C251#52911yescharacteristic
198C2×C251#52911yescharacteristic
199C2×C251#52911yescharacteristic
200C251#285211yescharacteristic, complemented, Sylow
201#2852—2511nomaximal, Sylow
202C2×#126—2511no—
203C22×C42—2511no—
204#388—2511no—
205#388—2511no—
206#388—2511no—
207#387—2511no—
208#387—2511no—
209#387—2511no—
210#386—2511no—
211#386—2511no—
212#386D25111yescharacteristic
213#385—2511no—
214#385—2511no—
215#385—2511no—
216#383—2511no—
217C22×D4C2×D25111yescharacteristic
218C23×C4—2511no—
219C23×C4—2511no—
220C23×C4C2×D25111yescharacteristic
221#126—5024no—
222C2×C4⋊C4—2511no—
223C2×C4⋊C4—2511no—
224C2×C4⋊C4—2511no—
225C2×C4⋊C4—2511no—
226C2×C4⋊C4—2511no—
227C2×C4⋊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—5024no—
238C2×C42—2511no—
239C2×C42—2511no—
240C2×C42—2511no—
241C2×C42—2511no—
242#95—2511no—
243#95—2511no—
244#95—2511no—
245#95—2511no—
246#95—2511no—
247#95—2511no—
248#95—2511no—
249#95—2511no—
250#95—2511no—
251#95—2511no—
252#95C2×D25111yescharacteristic
253#95C2×D25111yescharacteristic
254C24C22×D25111yescharacteristic
255C24C22×D25111yescharacteristic
256C2×D4—22no—
257C2×D4—22no—
258C2×D4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—5022no—
263C22×C4—5022no—
264C22×C4—5022no—
265C22×C4—5022no—
266C22×C4—5022no—
267C22×C4—5022no—
268C22×C4—2511no—
269C22×C4—2511no—
270C22×C4—2511no—
271C22×C4—2511no—
272C22×C4—2511no—
273C22×C4—2511no—
274C22×C4—2511no—
275C22×C4—2511no—
276C22×C4C22×D25111yescharacteristic
277C22×C4C22×D25111yescharacteristic
278C22×C4C22×D25111yescharacteristic
279C22×C4C22×D25111yescharacteristic
280C22×C4C22×D25111yescharacteristic
281C4⋊C4—5022no—
282C4⋊C4—5022no—
283C4⋊C4—5022no—
284C4⋊C4—5022no—
285C4⋊C4—5022no—
286C4⋊C4—5022no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—5022no—
291C22⋊C4—5022no—
292C22⋊C4—5022no—
293C22⋊C4—5022no—
294C22⋊C4—5022no—
295C22⋊C4—5022no—
296C42—5022no—
297C42—5022no—
298C42—5022no—
299C42—2514no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23#1686080912yes—
312C23#1686080912yes—
313C23#1686081012yes—
314C23C23×D25111yescharacteristic
315D4—44no—
316C2×C4—28no—
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—5028no—
343C2×C4—5028no—
344C2×C4—5021no—
345C2×C4—5021no—
346C2×C4—5021no—
347C2×C4—5021no—
348C2×C4—5021no—
349C2×C4—5021no—
350C2×C4—5021no—
351C2×C4—5021no—
352C2×C4—2512no—
353C2×C4—2512no—
354C2×C4—2512no—
355C2×C4—2512no—
356C2×C4D4×D25112yes—
357C2×C4#1686080912yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#2254870814yescomplemented
372C22C2×D4×D25111yescharacteristic
373C22#41×D25111yescharacteristic
374C22#41×D25111yescharacteristic
375C22C2×#1686080911yescharacteristic
376C22C2×#1686080911yescharacteristic
377C22C2×#1686080911yescharacteristic
378C22C2×#1686081011yescharacteristic
379C4—22no—
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—10041no—
385C4—10041no—
386C4—10041no—
387C4—10041no—
388C4—5022no—
389C4—5022no—
390C4—5022no—
391C4—5022no—
392C2—24no—
393C2—81no—
394C2#3063219311yescharacteristic
395C2#3063219611yescharacteristic
396C2#3063219711yescharacteristic
397C2C2×#2254870811yescharacteristic
398C2#3063221411yescharacteristic
399C2#3063221711yescharacteristic
400C2#3063221811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#386×C251C2
prime radicalC22×D4×C251C22
hypercenter#386D251
hypocenterC251#2852
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
1. C2×#22548708C2semidirect
2. C22×C4×Dic251C2semidirect
3. #30631581C2semidirect
4. #30631580C2semidirect
5. #30631575C2semidirect
6. #30631574C2semidirect
7. #30631572C2semidirect
8. #30631569C2semidirect
11. #30631562C2semidirect
12. #30631560C2semidirect
13. #30631559C2semidirect
14. #30631554C2semidirect
82. C4×Dic251D4semidirect
170. C22×C251#126semidirect
200. C251#2852semidirect
371. C22#22548708semidirect

Maximal subgroups

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

subgroupindex
C2×#225487082
C22×C4×Dic2512
#306315812
#306315802
#306315752
#306315742
#306315722
#306315692
#306315672
#306315622
#306315602
#306315592
#306315542
#386×C2512
#2852251