Finite Group Databasebeta

← #45667359

Subgroups — #45667359

Order 32896.

688 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22551565C211yesmaximal, characteristic
2C2×C4×C22⋊D257C211yesmaximal, characteristic, complemented
3C2×#22551549C211yesmaximal, characteristic, complemented
4#30656639C211yesmaximal, characteristic, complemented
5#30656637C211yesmaximal, characteristic, complemented
6#30656637C211yesmaximal, characteristic, complemented
7#30656634C211yesmaximal, characteristic, complemented
8#30656630C211yesmaximal, characteristic, complemented
9#30656628C211yesmaximal, characteristic, complemented
10#30656616C211yesmaximal, characteristic, complemented
11#30656615C211yesmaximal, characteristic, complemented
12C4×#16861597C211yesmaximal, characteristic, complemented
13#30656584C211yesmaximal, characteristic, complemented
14#30656584C211yesmaximal, characteristic, complemented
15C4×C22⋊C4×C257C211yesmaximal, characteristic, complemented
16C22×D1028C2211yescharacteristic
17C22×C22⋊D257C2211yescharacteristic
18C22×C22⋊D257C2211yescharacteristic
19C22×C4×D257C2211yescharacteristic
20C22×C4×D257C2211yescharacteristic
21#22551565C418yescomplemented
22C4×C22⋊D257—24no—
23#22551549—24no—
24C2×C4⋊Dic257C2211yescharacteristic
25C2×#16861597C2211yescharacteristic
26C2×#16861597C2211yescharacteristic
27C2×#16861597C2211yescharacteristic
28C2×#16861597C2211yescharacteristic
29C2×C22⋊Dic257C2211yescharacteristic
30C2×C22⋊Dic257C2211yescharacteristic
31C2×#16861595C2211yescharacteristic
32C2×#16861595C2211yescharacteristic
33C2×#16861595C2211yescharacteristic
34C2×#16861595C2211yescharacteristic
35C2×#16861595C2211yescharacteristic
36C2×#16861595C2211yescharacteristic
37C2×#16861595C2211yescharacteristic
38C2×#16861595C2211yescharacteristic
39C2×C4×Dic257C2211yescharacteristic
40C2×C4×Dic257C2211yescharacteristic
41C2×C4×Dic257C2211yescharacteristic
42C2×C4×Dic257C2211yescharacteristic
43#22551499C2211yescharacteristic
44#22551499C2211yescharacteristic
45#22551499C2211yescharacteristic
46#22551499C2211yescharacteristic
47C23×C4×C257C2211yescharacteristic
48C2×C22⋊C4×C257C2211yescharacteristic
49C2×C22⋊C4×C257C2211yescharacteristic
50C2×C42×C257C2211yescharacteristic
51C2×C42×C257C2211yescharacteristic
52#95×C257C2211yescharacteristic
53#95×C257C2211yescharacteristic
54C23×D257C2311yescharacteristic
55C23×D257C2311yescharacteristic
56C2×D1028—41no—
57C2×D1028—41no—
58C2×D1028C2×C414yescomplemented
59C2×C22⋊D257—22no—
60C2×C22⋊D257—22no—
61C2×C22⋊D257—41no—
62C2×C22⋊D257—41no—
63C2×C22⋊D257C2×C414yes—
64C2×C22⋊D257C2×C414yescomplemented
65C22×Dic257C2311yescharacteristic
66C22×Dic257C2311yescharacteristic
67C22×Dic257C2311yescharacteristic
68C22×Dic257C2311yescharacteristic
69C22×Dic257C2311yescharacteristic
70C22×Dic257C2311yescharacteristic
71C2×C4×D257—22no—
72C2×C4×D257—22no—
73C2×C4×D257—22no—
74C2×C4×D257—22no—
75C2×C4×D257—41no—
76C2×C4×D257C2×C414yescomplemented
77C4⋊Dic257—41no—
78#16861597—22no—
79#16861597—41no—
80#16861597C2×C414yescomplemented
81#16861597D414yescomplemented
82C22⋊Dic257—22no—
83C22⋊Dic257—41no—
84#16861595—22no—
85#16861595—22no—
86#16861595—22no—
87#16861595—22no—
88#16861595—22no—
89#16861595—22no—
90#16861595—41no—
91#16861595C2×C414yescomplemented
92C4×Dic257—22no—
93C4×Dic257—22no—
94C4×Dic257—22no—
95C4×Dic257—41no—
96C24×C257C2311yescharacteristic
97C22×C4×C257—22no—
98C22×C4×C257—22no—
99C22×C4×C257—41no—
100C22×C4×C257C2311yescharacteristic
101C22×C4×C257C2311yescharacteristic
102C22×C4×C257C2311yescharacteristic
103C22×C4×C257C2311yescharacteristic
104C22×C4×C257C2311yescharacteristic
105C22×C4×C257C2311yescharacteristic
106C22⋊C4×C257—22no—
107C22⋊C4×C257C2×C414yescomplemented
108C42×C257—22no—
109C42×C257—41no—
110C22×D257—21no—
111C22×D257—21no—
112C22×D257—21no—
113C22×D257—21no—
114C22×D257—21no—
115C22×D257—21no—
116C22×D257—21no—
117C22×D257—21no—
118C22×D257—21no—
119C22×D257—21no—
120C22×D257—21no—
121C22×D257—21no—
122C22×D257C22×C412yes—
123C22×D257C22×C412yes—
124D1028—44no—
125C22⋊D257—28no—
126C22⋊D257—44no—
127C2×Dic257—21no—
128C2×Dic257—21no—
129C2×Dic257—21no—
130C2×Dic257—21no—
131C2×Dic257—21no—
132C2×Dic257—21no—
133C2×Dic257—21no—
134C2×Dic257—21no—
135C2×Dic257—21no—
136C2×Dic257—21no—
137C2×Dic257—21no—
138C2×Dic257—21no—
139C2×Dic257—21no—
140C2×Dic257C22×C412yes—
141C2×Dic257C22×C412yes—
142C2×Dic257C2×D412yes—
143C2×Dic257C2×D412yes—
144C2×Dic257#4112yes—
145C4×D257—28no—
146C4×D257—44no—
147C23×C257—21no—
148C23×C257—21no—
149C23×C257—21no—
150C23×C257—21no—
151C23×C257—21no—
152C23×C257—21no—
153C23×C257C22×C412yes—
154C23×C257C2411yescharacteristic
155C2×C4×C257—21no—
156C2×C4×C257—21no—
157C2×C4×C257—21no—
158C2×C4×C257—21no—
159C2×C4×C257—21no—
160C2×C4×C257—21no—
161C2×C4×C257—21no—
162C2×C4×C257—21no—
163C2×C4×C257—21no—
164C2×C4×C257—21no—
165C2×C4×C257—21no—
166C2×C4×C257—21no—
167C2×C4×C257—21no—
168C2×C4×C257—44no—
169C2×C4×C257C22×C412yes—
170C2×C4×C257C22×C412yes—
171C2×C4×C257C2×D412yes—
172C2×C4×C257#4112yes—
173C2×C4×C257#4112yes—
174C22×C257—22no—
175C22×C257—22no—
176C22×C257—22no—
177C22×C257—41no—
178C22×C257—41no—
179C22×C257—41no—
180C22×C257—41no—
181C22×C257C23×C411yescharacteristic
182C22×C257C22×D411yescharacteristic
183C22×C257C2×#4111yescharacteristic
184C22×C257C2×#4111yescharacteristic
185C22×C257C2×#4111yescharacteristic
186C22×C257H(5,𝔽2)11yescharacteristic
187C22×C257H(5,𝔽2)11yescharacteristic
188C2×D257—22no—
189C2×D257—22no—
190C2×D257—22no—
191C2×D257—22no—
192C2×D257—22no—
193C2×D257—22no—
194C2×D257—41no—
195C2×D257—41no—
196C2×D257—41no—
197C2×D257—41no—
198C2×D257—41no—
199C2×D257—41no—
200C2×D257—41no—
201C2×D257—41no—
202C4×C257—22no—
203C4×C257—22no—
204C4×C257—22no—
205C4×C257—22no—
206C4×C257—22no—
207C4×C257—41no—
208Dic257—22no—
209Dic257—22no—
210Dic257—41no—
211Dic257—41no—
212Dic257—41no—
213Dic257C4×D414yes—
214C2×C257—42no—
215C2×C257C2×C4×D411yescharacteristic
216C2×C257C4×#4111yescharacteristic
217C2×C257#51811yescharacteristic
218C2×C257#54611yescharacteristic
219C2×C257#55011yescharacteristic
220C2×C257#55311yescharacteristic
221C2×C257#55511yescharacteristic
222D257—42no—
223D257—42no—
224C257#236311yescharacteristic, complemented, Sylow
225#2363—2571nomaximal, Sylow
226C2×#121—2571no—
227C2×C4×D4—2571no—
228C2×#117—2571no—
229#390—2571no—
230#390—2571no—
231#390—2571no—
232#389—2571no—
233#388—2571no—
234#388—2571no—
235#386—2571no—
236#386—2571no—
237C4×C4⋊C4—2571no—
238C4×C22⋊C4—2571no—
239C4×C22⋊C4—2571no—
240C4×C22⋊C4D25711yescharacteristic, complemented
241C22×D4—2571no—
242C22×D4—2571no—
243C22×D4—2571no—
244C23×C4—2571no—
245C23×C4—2571no—
246C23×C4C2×D25711yescharacteristic
247#121—2578no—
248C4×D4—5144no—
249#117—5144no—
250C2×C4⋊C4—2571no—
251C2×C4⋊C4—2571no—
252C2×C4⋊C4—2571no—
253C2×C4⋊C4—2571no—
254C2×C4⋊C4—2571no—
255C2×C22⋊C4—2571no—
256C2×C22⋊C4—2571no—
257C2×C22⋊C4—2571no—
258C2×C22⋊C4—2571no—
259C2×C22⋊C4—2571no—
260C2×C22⋊C4—2571no—
261C2×C22⋊C4—2571no—
262C2×C22⋊C4—2571no—
263C2×C22⋊C4—2571no—
264C2×C22⋊C4—2571no—
265C2×C22⋊C4C2×D25711yescharacteristic
266C2×C22⋊C4C2×D25711yescharacteristic
267C2×C42—2571no—
268C2×C42—2571no—
269C2×C42—2571no—
270C2×C42—2571no—
271C2×C42C2×D25711yescharacteristic
272C2×C42C2×D25711yescharacteristic
273#95—2571no—
274#95—2571no—
275#95—2571no—
276#95—2571no—
277#95C2×D25711yescharacteristic
278#95C2×D25711yescharacteristic
279C24—2571no—
280C24—2571no—
281C24C22×D25711yescharacteristic
282C2×D4—2574no—
283C2×D4—2574no—
284C2×D4—2574no—
285C2×D4—5142no—
286C2×D4—5142no—
287C2×D4—10281no—
288C2×D4—10281no—
289C2×D4—10281no—
290C2×D4—10281no—
291C22×C4—2571no—
292C22×C4—2571no—
293C22×C4—2571no—
294C22×C4—2574no—
295C22×C4—2571no—
296C22×C4—2571no—
297C22×C4—2571no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—5142no—
301C22×C4—5142no—
302C22×C4—5142no—
303C22×C4—5142no—
304C22×C4—41no—
305C22×C4—10281no—
306C22×C4C22×D25711yescharacteristic
307C22×C4C22×D25711yescharacteristic
308C22×C4C22×D25711yescharacteristic
309C22×C4C22×D25711yescharacteristic
310C22×C4C22×D25711yescharacteristic
311C22×C4C22×D25711yescharacteristic
312C4⋊C4—2574no—
313C4⋊C4—2574no—
314C4⋊C4—5142no—
315C4⋊C4—10281no—
316C4⋊C4—10281no—
317C22⋊C4—2574no—
318C22⋊C4—22no—
319C22⋊C4—5142no—
320C22⋊C4—5142no—
321C22⋊C4—5142no—
322C22⋊C4—5142no—
323C22⋊C4—5142no—
324C22⋊C4—5142no—
325C22⋊C4—5142no—
326C22⋊C4—10281no—
327C22⋊C4—10281no—
328C22⋊C4C4×D25714yescomplemented
329C42—22no—
330C42—5142no—
331C42—5142no—
332C42—5142no—
333C42—41no—
334C42—10281no—
335C23—2572no—
336C23—2572no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—5141no—
344C23—5141no—
345C23—5141no—
346C23—5141no—
347C23—5141no—
348C23—5141no—
349C23—5141no—
350C23—5141no—
351C23—5141no—
352C23—5141no—
353C23—5141no—
354C23—5141no—
355C23C2×C4×D25712yes—
356C23C23×D25711yescharacteristic
357D4—5148no—
358D4—10284no—
359D4—10284no—
360C2×C4—2572no—
361C2×C4—2572no—
362C2×C4—2572no—
363C2×C4—2572no—
364C2×C4—2572no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—5148no—
379C2×C4—5141no—
380C2×C4—5141no—
381C2×C4—5141no—
382C2×C4—5141no—
383C2×C4—5141no—
384C2×C4—5141no—
385C2×C4—5141no—
386C2×C4—5141no—
387C2×C4—5141no—
388C2×C4—5141no—
389C2×C4—5141no—
390C2×C4—5141no—
391C2×C4—5141no—
392C2×C4—44no—
393C2×C4—10284no—
394C2×C4C2×C4×D25712yes—
395C2×C4C2×C4×D25712yes—
396C2×C4D4×D25712yes—
397C2×C4#1686161812yes—
398C2×C4#1686161812yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—5142no—
403C22—5142no—
404C22—5142no—
405C22—5142no—
406C22—5142no—
407C22—5142no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—10281no—
413C22—10281no—
414C22—10281no—
415C22—10281no—
416C22—10281no—
417C22—10281no—
418C22—10281no—
419C22—10281no—
420C22C22×C4×D25711yescharacteristic
421C22C2×D4×D25711yescharacteristic
422C22#41×D25711yescharacteristic
423C22C2×#1686161811yescharacteristic
424C22C2×#1686161811yescharacteristic
425C22#2255163711yescharacteristic
426C22#2255163811yescharacteristic
427C4—2574no—
428C4—22no—
429C4—22no—
430C4—22no—
431C4—22no—
432C4—22no—
433C4—5142no—
434C4—5142no—
435C4—41no—
436C4—10281no—
437C4—10281no—
438C4—10281no—
439C2—42no—
440C2—10282no—
441C2—10282no—
442C2C4×D4×D25711yescharacteristic
443C2C4×#1686161811yescharacteristic
444C2#3065720911yescharacteristic
445C2#3065737711yescharacteristic
446C2#3065740811yescharacteristic
447C2#3065743611yescharacteristic
448C2#3065745011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D257
commutatorC22×C257C23×C4
FittingC4×C22⋊C4×C257C2
socle, cosocleC23×C257C24
prime radicalC2×#22551565C2
hypercenterC4×C22⋊C4D257
hypocenterC257#2363
LCSG ▷ C22×C257 ▷ C257
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D257C2semidirect
3. C2×#22551549C2semidirect
4. #30656639C2semidirect
5. #30656637C2semidirect
6. #30656637C2semidirect
7. #30656634C2semidirect
8. #30656630C2semidirect
9. #30656628C2semidirect
10. #30656616C2semidirect
11. #30656615C2semidirect
12. C4×#16861597C2semidirect
13. #30656584C2semidirect
14. #30656584C2semidirect
15. C4×C22⋊C4×C257C2semidirect
21. #22551565C4semidirect
58. C2×D1028C2×C4semidirect
64. C2×C22⋊D257C2×C4semidirect
76. C2×C4×D257C2×C4semidirect
80. #16861597C2×C4semidirect
81. #16861597D4semidirect
91. #16861595C2×C4semidirect
107. C22⋊C4×C257C2×C4semidirect
224. C257#2363semidirect
240. C4×C22⋊C4D257semidirect
328. C22⋊C4C4×D257semidirect

Maximal subgroups

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

subgroupindex
C2×#225515652
C2×C4×C22⋊D2572
C2×#225515492
#306566392
#306566372
#306566342
#306566302
#306566282
#306566162
#306566152
C4×#168615972
#306565842
C4×C22⋊C4×C2572
#2363257