Finite Group Databasebeta

← #44553406

Subgroups — #44553406

Order 32128.

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×#22548687C211yesmaximal, characteristic
2C2×C4×C22⋊D251C211yesmaximal, characteristic, complemented
3C2×#22548671C211yesmaximal, characteristic, complemented
4#30631590C211yesmaximal, characteristic, complemented
5#30631587C211yesmaximal, characteristic, complemented
6#30631587C211yesmaximal, characteristic, complemented
7#30631584C211yesmaximal, characteristic, complemented
8#30631581C211yesmaximal, characteristic, complemented
9#30631579C211yesmaximal, characteristic, complemented
10#30631563C211yesmaximal, characteristic, complemented
11#30631538C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic251C211yesmaximal, characteristic, complemented
13C4×C22⋊Dic251C211yesmaximal, characteristic, complemented
14#30631534C211yesmaximal, characteristic, complemented
15#386×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C23×Dic251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20#22548687C418yescomplemented
21C4×C22⋊D251—24no—
22#22548671—24no—
23C2×C4⋊Dic251C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×C22⋊Dic251C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×C4×Dic251C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C2×C4×Dic251C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C2×C4×Dic251C2211yescharacteristic
42C2×C4×Dic251C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C22×D4×C251C2211yescharacteristic
48C23×C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54C23×D251C2311yescharacteristic
55C2×C22⋊D251—22no—
56C2×C22⋊D251—22no—
57C2×C22⋊D251—41no—
58C2×C22⋊D251—41no—
59C2×C22⋊D251C2×C414yes—
60C2×C22⋊D251C2×C414yescomplemented
61C22×Dic251—22no—
62C22×Dic251—22no—
63C22×Dic251C2×C414yescomplemented
64C22×Dic251C2311yescharacteristic
65C22×Dic251C2311yescharacteristic
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C2×C4×D251—22no—
72C2×C4×D251—22no—
73C2×C4×D251—41no—
74C4⋊Dic251—41no—
75Dic251⋊C4—22no—
76Dic251⋊C4—41no—
77Dic251⋊C4C2×C414yescomplemented
78Dic251⋊C4D414yescomplemented
79C22⋊Dic251—22no—
80C22⋊Dic251—22no—
81C22⋊Dic251—22no—
82C22⋊Dic251—22no—
83C22⋊Dic251C2×C414yescomplemented
84(C2×D251)⋊C4—22no—
85(C2×D251)⋊C4—22no—
86(C2×D251)⋊C4—41no—
87(C2×D251)⋊C4C2×C414yescomplemented
88C4×Dic251—22no—
89C4×Dic251—22no—
90C4×Dic251—22no—
91C4×Dic251—22no—
92C4×Dic251—41no—
93C4×Dic251—41no—
94C24×C251C2311yescharacteristic
95C24×C251C2311yescharacteristic
96C2×D4×C251—41no—
97C2×D4×C251—41no—
98C2×D4×C251C2×C414yescomplemented
99C22×C4×C251—22no—
100C22×C4×C251—22no—
101C22×C4×C251—41no—
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C22⋊C4×C251—22no—
108C22⋊C4×C251—22no—
109C22⋊C4×C251—41no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251C22×C412yes—
117C22⋊D251—28no—
118C22⋊D251—44no—
119C2×Dic251—21no—
120C2×Dic251—21no—
121C2×Dic251—21no—
122C2×Dic251—21no—
123C2×Dic251—21no—
124C2×Dic251—21no—
125C2×Dic251—21no—
126C2×Dic251—21no—
127C2×Dic251—28no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251C22×C412yes—
135C2×Dic251C22×C412yes—
136C2×Dic251C22×C412yes—
137C2×Dic251C2×D412yes—
138C2×Dic251C2×D412yes—
139C2×Dic251#4112yes—
140C2×Dic251#4112yes—
141C4×D251—44no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251—21no—
147C23×C251—21no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—21no—
154C23×C251C22×C412yes—
155C23×C251C22×C412yes—
156C23×C251C2411yescharacteristic
157D4×C251—44no—
158C2×C4×C251—21no—
159C2×C4×C251—21no—
160C2×C4×C251—21no—
161C2×C4×C251—21no—
162C2×C4×C251—21no—
163C2×C4×C251—21no—
164C2×C4×C251—21no—
165C2×C4×C251—21no—
166C2×C4×C251—21no—
167C2×C4×C251—21no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—44no—
171C2×C4×C251C22×C412yes—
172C2×C4×C251C2×D412yes—
173C2×C4×C251#4112yes—
174C22×C251—22no—
175C22×C251—22no—
176C22×C251—22no—
177C22×C251—22no—
178C22×C251—22no—
179C22×C251—22no—
180C22×C251—41no—
181C22×C251—41no—
182C22×C251—41no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251C23×C411yescharacteristic
189C22×C251C22×D411yescharacteristic
190C22×C251C2×#4111yescharacteristic
191C22×C251C2×#4111yescharacteristic
192C22×C251C2×#4111yescharacteristic
193C22×C251H(5,𝔽2)11yescharacteristic
194C22×C251H(5,𝔽2)11yescharacteristic
195C2×D251—22no—
196C2×D251—22no—
197C2×D251—22no—
198C2×D251—41no—
199C2×D251—41no—
200C2×D251—41no—
201C2×D251—41no—
202C4×C251—22no—
203C4×C251—22no—
204C4×C251—22no—
205C4×C251—41no—
206C4×C251—41no—
207Dic251—22no—
208Dic251—22no—
209Dic251—22no—
210Dic251—22no—
211Dic251—41no—
212Dic251—41no—
213Dic251C4×D414yes—
214C2×C251—42no—
215C2×C251—42no—
216C2×C251C2×C4×D411yescharacteristic
217C2×C251C4×#4111yescharacteristic
218C2×C251#51811yescharacteristic
219C2×C251#54611yescharacteristic
220C2×C251#55011yescharacteristic
221C2×C251#55311yescharacteristic
222C2×C251#55511yescharacteristic
223D251—42no—
224C251#236311yescharacteristic, complemented, Sylow
225#2363—2511nomaximal, Sylow
226C2×#121—2511no—
227C2×C4×D4—2511no—
228C2×#117—2511no—
229#390—2511no—
230#390—2511no—
231#390—2511no—
232#389—2511no—
233#388—2511no—
234#388—2511no—
235#386—2511no—
236#386D25111yescharacteristic, complemented
237C4×C4⋊C4—2511no—
238C4×C22⋊C4—2511no—
239C4×C22⋊C4—2511no—
240C4×C22⋊C4—2511no—
241C22×D4—2511no—
242C22×D4—2511no—
243C22×D4C2×D25111yescharacteristic
244C23×C4—2511no—
245C23×C4—2511no—
246C23×C4C2×D25111yescharacteristic
247#121—2518no—
248C4×D4—5024no—
249#117—5024no—
250C2×C4⋊C4—2511no—
251C2×C4⋊C4—2511no—
252C2×C4⋊C4—2511no—
253C2×C4⋊C4—2511no—
254C2×C4⋊C4—2511no—
255C2×C22⋊C4—2511no—
256C2×C22⋊C4—2511no—
257C2×C22⋊C4—2511no—
258C2×C22⋊C4—2511no—
259C2×C22⋊C4—2511no—
260C2×C22⋊C4—2511no—
261C2×C22⋊C4—2511no—
262C2×C22⋊C4—2511no—
263C2×C22⋊C4—2511no—
264C2×C22⋊C4C2×D25111yescharacteristic
265C2×C22⋊C4C2×D25111yescharacteristic
266C2×C22⋊C4C2×D25111yescharacteristic
267C2×C42—2511no—
268C2×C42—2511no—
269C2×C42—2511no—
270C2×C42—2511no—
271C2×C42—2511no—
272C2×C42—2511no—
273#95—2511no—
274#95—2511no—
275#95—2511no—
276#95—2511no—
277#95C2×D25111yescharacteristic
278#95C2×D25111yescharacteristic
279C24—2511no—
280C24C22×D25111yescharacteristic
281C24C22×D25111yescharacteristic
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—10041no—
285C2×D4—10041no—
286C2×D4—5022no—
287C2×D4—5022no—
288C2×D4—2514no—
289C2×D4—2514no—
290C2×D4C4×D25114yescomplemented
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—41no—
294C22×C4—10041no—
295C22×C4—5022no—
296C22×C4—5022no—
297C22×C4—5022no—
298C22×C4—5022no—
299C22×C4—2511no—
300C22×C4—2511no—
301C22×C4—2514no—
302C22×C4—2511no—
303C22×C4—2511no—
304C22×C4—2511no—
305C22×C4—2511no—
306C22×C4—2511no—
307C22×C4C22×D25111yescharacteristic
308C22×C4C22×D25111yescharacteristic
309C22×C4C22×D25111yescharacteristic
310C22×C4C22×D25111yescharacteristic
311C22×C4C22×D25111yescharacteristic
312C4⋊C4—10041no—
313C4⋊C4—10041no—
314C4⋊C4—5022no—
315C4⋊C4—2514no—
316C4⋊C4—2514no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—41no—
320C22⋊C4—10041no—
321C22⋊C4—5022no—
322C22⋊C4—5022no—
323C22⋊C4—5022no—
324C22⋊C4—5022no—
325C22⋊C4—5022no—
326C22⋊C4—5022no—
327C22⋊C4—2514no—
328C22⋊C4—2514no—
329C42—10041no—
330C42—10041no—
331C42—5022no—
332C42—5022no—
333C42—5022no—
334C42—5022no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—5021no—
348C23—5021no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—2512no—
354C23C2×C4×D25112yes—
355C23C2×C4×D25112yes—
356C23C23×D25111yescharacteristic
357D4—44no—
358D4—10044no—
359D4—5028no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—44no—
373C2×C4—10044no—
374C2×C4—5021no—
375C2×C4—5021no—
376C2×C4—5021no—
377C2×C4—5021no—
378C2×C4—5021no—
379C2×C4—5021no—
380C2×C4—5021no—
381C2×C4—5021no—
382C2×C4—5021no—
383C2×C4—5021no—
384C2×C4—5021no—
385C2×C4—5021no—
386C2×C4—5021no—
387C2×C4—5021no—
388C2×C4—5028no—
389C2×C4—2512no—
390C2×C4—2512no—
391C2×C4—2512no—
392C2×C4—2512no—
393C2×C4—2512no—
394C2×C4—2512no—
395C2×C4—2512no—
396C2×C4C2×C4×D25112yes—
397C2×C4D4×D25112yes—
398C2×C4#1686080912yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—10041no—
414C22—10041no—
415C22—10041no—
416C22—10041no—
417C22—5022no—
418C22—5022no—
419C22—5022no—
420C22C22×C4×D25111yescharacteristic
421C22C2×D4×D25111yescharacteristic
422C22#41×D25111yescharacteristic
423C22#41×D25111yescharacteristic
424C22C2×#1686080911yescharacteristic
425C22#2254876011yescharacteristic
426C22#2254876011yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—10041no—
433C4—10041no—
434C4—5022no—
435C4—5022no—
436C4—5022no—
437C4—5022no—
438C4—2514no—
439C2—42no—
440C2—42no—
441C2—10042no—
442C2C4×D4×D25111yescharacteristic
443C2C4×#1686080911yescharacteristic
444C2#3063216011yescharacteristic
445C2#3063232411yescharacteristic
446C2#3063235511yescharacteristic
447C2#3063238211yescharacteristic
448C2#3063240111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#386×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548687C2
hypercenter#386D251
hypocenterC251#2363
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×C22⋊D251C2semidirect
3. C2×#22548671C2semidirect
4. #30631590C2semidirect
5. #30631587C2semidirect
6. #30631587C2semidirect
7. #30631584C2semidirect
8. #30631581C2semidirect
9. #30631579C2semidirect
10. #30631563C2semidirect
11. #30631538C2semidirect
12. C22⋊C4×Dic251C2semidirect
13. C4×C22⋊Dic251C2semidirect
14. #30631534C2semidirect
15. #386×C251C2semidirect
20. #22548687C4semidirect
60. C2×C22⋊D251C2×C4semidirect
63. C22×Dic251C2×C4semidirect
77. Dic251⋊C4C2×C4semidirect
78. Dic251⋊C4D4semidirect
83. C22⋊Dic251C2×C4semidirect
87. (C2×D251)⋊C4C2×C4semidirect
98. C2×D4×C251C2×C4semidirect
224. C251#2363semidirect
236. #386D251semidirect
290. C2×D4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486872
C2×C4×C22⋊D2512
C2×#225486712
#306315902
#306315872
#306315842
#306315812
#306315792
#306315632
#306315382
C22⋊C4×Dic2512
C4×C22⋊Dic2512
#306315342
#386×C2512
#2363251