Finite Group Databasebeta

← #45667581

Subgroups — #45667581

Order 32896.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D257C211yesmaximal, characteristic
2C2×#22551549C211yesmaximal, characteristic, complemented
3#30656678C211yesmaximal, characteristic, complemented
4#30656673C211yesmaximal, characteristic, complemented
5#30656671C211yesmaximal, characteristic, complemented
6#30656666C211yesmaximal, characteristic, complemented
7#30656665C211yesmaximal, characteristic, complemented
8#30656664C211yesmaximal, characteristic, complemented
9#30656653C211yesmaximal, characteristic
10#30656652C211yesmaximal, characteristic
11C4×#16861597C211yesmaximal, characteristic, complemented
12C4×C4⋊Dic257C211yesmaximal, characteristic, complemented
13C4×#16861595C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic257C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C257C211yesmaximal, characteristic, complemented
16C22×C22⋊D257C2211yescharacteristic
17C22×C4×D257C2211yescharacteristic
18C4×C22⋊D257—24no—
19#22551549—24no—
20C2×C4⋊Dic257C2211yescharacteristic
21C2×C4⋊Dic257C2211yescharacteristic
22C2×#16861597C2211yescharacteristic
23C2×#16861597C2211yescharacteristic
24C2×#16861597C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×C22⋊Dic257C2211yescharacteristic
28C2×C22⋊Dic257C2211yescharacteristic
29C2×C22⋊Dic257C2211yescharacteristic
30C2×#16861595C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×C4×Dic257C2211yescharacteristic
36C2×C4×Dic257C2211yescharacteristic
37C2×C4×Dic257C2211yescharacteristic
38#22551499C2211yescharacteristic
39#22551499C2211yescharacteristic
40#22551499C2211yescharacteristic
41#22551499C2211yescharacteristic
42#22551499C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46C23×C4×C257C2211yescharacteristic
47C2×C22⋊C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C42×C257C2211yescharacteristic
50C2×C42×C257C2211yescharacteristic
51#95×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53C23×D257C2311yescharacteristic
54C2×C22⋊D257—22no—
55C2×C22⋊D257—22no—
56C2×C22⋊D257—22no—
57C22×Dic257C2311yescharacteristic
58C22×Dic257C2311yescharacteristic
59C22×Dic257C2311yescharacteristic
60C22×Dic257C2311yescharacteristic
61C22×Dic257C2311yescharacteristic
62C22×Dic257C2311yescharacteristic
63C22×Dic257C2311yescharacteristic
64C2×C4×D257—22no—
65C2×C4×D257—22no—
66C2×C4×D257D414yes—
67C4⋊Dic257—22no—
68C4⋊Dic257—22no—
69#16861597—22no—
70#16861597—22no—
71#16861597—22no—
72#16861597—22no—
73#16861597—22no—
74C22⋊Dic257—22no—
75C22⋊Dic257—22no—
76C22⋊Dic257—22no—
77#16861595—22no—
78#16861595—22no—
79#16861595—22no—
80#16861595—22no—
81#16861595—22no—
82C4×Dic257—22no—
83C4×Dic257—22no—
84C4×Dic257—22no—
85C24×C257C2311yescharacteristic
86C22×C4×C257—22no—
87C22×C4×C257—22no—
88C22×C4×C257—22no—
89C22×C4×C257C2311yescharacteristic
90C22×C4×C257C2311yescharacteristic
91C22×C4×C257C2311yescharacteristic
92C22×C4×C257C2311yescharacteristic
93C22×C4×C257C2311yescharacteristic
94C22×C4×C257C2311yescharacteristic
95C22⋊C4×C257—22no—
96C22⋊C4×C257—22no—
97C42×C257—22no—
98C42×C257—22no—
99C22×D257—21no—
100C22×D257—21no—
101C22×D257—21no—
102C22×D257—21no—
103C22×D257—21no—
104C22×D257—21no—
105C22×D257C2×D412yes—
106C22⋊D257—28no—
107C2×Dic257—21no—
108C2×Dic257—21no—
109C2×Dic257—21no—
110C2×Dic257—21no—
111C2×Dic257—21no—
112C2×Dic257—21no—
113C2×Dic257—21no—
114C2×Dic257—21no—
115C2×Dic257—21no—
116C2×Dic257—21no—
117C2×Dic257—21no—
118C2×Dic257—21no—
119C2×Dic257—21no—
120C2×Dic257—21no—
121C2×Dic257—21no—
122C2×Dic257—21no—
123C2×Dic257—21no—
124C2×Dic257—21no—
125C2×Dic257C2×D412yes—
126C2×Dic257#4112yes—
127C2×Dic257#4112yes—
128C4×D257—28no—
129C23×C257—21no—
130C23×C257—21no—
131C23×C257—21no—
132C23×C257—21no—
133C23×C257—21no—
134C23×C257—21no—
135C23×C257#4112yes—
136C23×C257C2411yescharacteristic
137C2×C4×C257—21no—
138C2×C4×C257—21no—
139C2×C4×C257—21no—
140C2×C4×C257—21no—
141C2×C4×C257—21no—
142C2×C4×C257—21no—
143C2×C4×C257—21no—
144C2×C4×C257—21no—
145C2×C4×C257—21no—
146C2×C4×C257—21no—
147C2×C4×C257—21no—
148C2×C4×C257—21no—
149C2×C4×C257—21no—
150C2×C4×C257—21no—
151C2×C4×C257—44no—
152C2×C4×C257C2×D412yes—
153C2×C4×C257#4112yes—
154C2×C4×C257#4112yes—
155C2×C4×C257#4112yes—
156C22×C257—22no—
157C22×C257—22no—
158C22×C257—22no—
159C22×C257—41no—
160C22×C257—41no—
161C22×C257—41no—
162C22×C257—41no—
163C22×C257C22×D411yescharacteristic
164C22×C257C2×#4111yescharacteristic
165C22×C257C2×#4111yescharacteristic
166C22×C257C2×#4111yescharacteristic
167C22×C257C2×#4111yescharacteristic
168C22×C257C2×#4111yescharacteristic
169C22×C257C2×#4111yescharacteristic
170C2×D257—22no—
171C2×D257—22no—
172C2×D257—22no—
173C2×D257—41no—
174C2×D257—41no—
175C2×D257—41no—
176C2×D257—41no—
177C4×C257—22no—
178C4×C257—41no—
179C4×C257—41no—
180C4×C257—41no—
181C4×C257—41no—
182C4×C257#12314yes—
183Dic257—22no—
184Dic257—22no—
185Dic257—22no—
186Dic257—41no—
187Dic257—41no—
188Dic257—41no—
189Dic257—41no—
190C2×C257—42no—
191C2×C257C2×#12311yescharacteristic
192C2×C257#52511yescharacteristic
193C2×C257#52911yescharacteristic
194C2×C257#52911yescharacteristic
195C2×C257#52911yescharacteristic
196C2×C257#52911yescharacteristic
197C2×C257#53211yescharacteristic
198D257—42no—
199C257#239511yescharacteristic, complemented, Sylow
200#2395—2571nomaximal, Sylow
201C2×C4×D4—2571no—
202C2×#117—2571no—
203#400—2571no—
204#399—2571no—
205#398—2571no—
206#397—2571no—
207#397—2571no—
208#397—2571no—
209#394—2571no—
210#394—2571no—
211C4×C4⋊C4—2571no—
212C4×C4⋊C4—2571no—
213C4×C22⋊C4—2571no—
214C4×C22⋊C4—2571no—
215C4×C22⋊C4D25711yescharacteristic, complemented
216C22×D4—2571no—
217C23×C4—2571no—
218C23×C4C2×D25711yescharacteristic
219C4×D4—5144no—
220#117—5144no—
221C2×C4⋊C4—2571no—
222C2×C4⋊C4—2571no—
223C2×C4⋊C4—2571no—
224C2×C4⋊C4—2571no—
225C2×C4⋊C4—2571no—
226C2×C4⋊C4—2571no—
227C2×C4⋊C4—2571no—
228C2×C22⋊C4—2571no—
229C2×C22⋊C4—2571no—
230C2×C22⋊C4—2571no—
231C2×C22⋊C4—2571no—
232C2×C22⋊C4—2571no—
233C2×C22⋊C4—2571no—
234C2×C22⋊C4—2571no—
235C2×C22⋊C4—2571no—
236C2×C22⋊C4C2×D25711yescharacteristic
237C2×C22⋊C4C2×D25711yescharacteristic
238C2×C42—2571no—
239C2×C42—2571no—
240C2×C42—2571no—
241C2×C42C2×D25711yescharacteristic
242C2×C42C2×D25711yescharacteristic
243#95—2571no—
244#95—2571no—
245#95—2571no—
246#95—2571no—
247#95—2571no—
248#95—2571no—
249#95—2571no—
250#95—2571no—
251#95C2×D25711yescharacteristic
252#95C2×D25711yescharacteristic
253C24—2571no—
254C24C22×D25711yescharacteristic
255C2×D4—5142no—
256C2×D4—5142no—
257C2×D4—5142no—
258C22×C4—2571no—
259C22×C4—2571no—
260C22×C4—2571no—
261C22×C4—2574no—
262C22×C4—2571no—
263C22×C4—2571no—
264C22×C4—2571no—
265C22×C4—2571no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—5142no—
270C22×C4—5142no—
271C22×C4C22×D25711yescharacteristic
272C22×C4C22×D25711yescharacteristic
273C22×C4C22×D25711yescharacteristic
274C22×C4C22×D25711yescharacteristic
275C22×C4C22×D25711yescharacteristic
276C22×C4C22×D25711yescharacteristic
277C4⋊C4—5142no—
278C4⋊C4—5142no—
279C4⋊C4—5142no—
280C4⋊C4—5142no—
281C4⋊C4—5142no—
282C4⋊C4—5142no—
283C4⋊C4—5142no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—5142no—
287C22⋊C4—5142no—
288C22⋊C4—5142no—
289C22⋊C4—5142no—
290C22⋊C4—5142no—
291C22⋊C4—5142no—
292C22⋊C4—5142no—
293C22⋊C4—5142no—
294C42—22no—
295C42—22no—
296C42—5142no—
297C42—5142no—
298C42—5142no—
299C23—2572no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—5141no—
307C23—5141no—
308C23—5141no—
309C23—5141no—
310C23—5141no—
311C23—5141no—
312C23#1686161812yes—
313C23C23×D25711yescharacteristic
314D4—5148no—
315C2×C4—2572no—
316C2×C4—2572no—
317C2×C4—2572no—
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—21no—
331C2×C4—21no—
332C2×C4—5141no—
333C2×C4—5141no—
334C2×C4—5141no—
335C2×C4—5141no—
336C2×C4—5141no—
337C2×C4—5141no—
338C2×C4—5148no—
339C2×C4—5141no—
340C2×C4—5141no—
341C2×C4—5141no—
342C2×C4—5141no—
343C2×C4—5141no—
344C2×C4—5141no—
345C2×C4—5141no—
346C2×C4—5141no—
347C2×C4—5141no—
348C2×C4—5141no—
349C2×C4—5141no—
350C2×C4—5141no—
351C2×C4—44no—
352C2×C4D4×D25712yes—
353C2×C4#1686161712yes—
354C2×C4#1686161812yes—
355C2×C4#1686161812yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—5142no—
360C22—5142no—
361C22—5142no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—10281no—
367C22—10281no—
368C22—10281no—
369C22—10281no—
370C22C2×D4×D25711yescharacteristic
371C22#41×D25711yescharacteristic
372C22#41×D25711yescharacteristic
373C22C2×#1686161711yescharacteristic
374C22C2×#1686161811yescharacteristic
375C22C2×#1686161811yescharacteristic
376C22C2×#1686161811yescharacteristic
377C4—22no—
378C4—5142no—
379C4—5142no—
380C4—5142no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—10281no—
386C4—10281no—
387C4—10281no—
388C4—10281no—
389C4#2255157414yes—
390C2—42no—
391C2—10282no—
392C2C2×#2255157411yescharacteristic
393C2#3065724611yescharacteristic
394C2#3065726511yescharacteristic
395C2#3065726711yescharacteristic
396C2#3065726711yescharacteristic
397C2#3065727111yescharacteristic
398C2#3065728411yescharacteristic
subgroupG/H
centerC22×C4C22×D257
commutator, socle, cosocleC23×C257C24
FrattiniC23C23×D257
FittingC4×C22⋊C4×C257C2
prime radicalC22×C22⋊D257C22
hypercenterC4×C22⋊C4D257
hypocenterC257#2395
LCSG ▷ C23×C257 ▷ C257
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#22551549C2semidirect
3. #30656678C2semidirect
4. #30656673C2semidirect
5. #30656671C2semidirect
6. #30656666C2semidirect
7. #30656665C2semidirect
8. #30656664C2semidirect
11. C4×#16861597C2semidirect
12. C4×C4⋊Dic257C2semidirect
13. C4×#16861595C2semidirect
14. C4×C22⋊Dic257C2semidirect
15. C4×C22⋊C4×C257C2semidirect
199. C257#2395semidirect
215. C4×C22⋊C4D257semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D2572
C2×#225515492
#306566782
#306566732
#306566712
#306566662
#306566652
#306566642
#306566532
#306566522
C4×#168615972
C4×C4⋊Dic2572
C4×#168615952
C4×C22⋊Dic2572
C4×C22⋊C4×C2572
#2395257