Finite Group Databasebeta

← #44553053

Subgroups — #44553053

Order 32128.

668 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×#22548688C211yesmaximal, characteristic
2C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic, complemented
3C2×#22548673C211yesmaximal, characteristic, complemented
4#30631589C211yesmaximal, characteristic, complemented
5#30631587C211yesmaximal, characteristic, complemented
6#30631587C211yesmaximal, characteristic, complemented
7#30631584C211yesmaximal, characteristic, complemented
8#30631577C211yesmaximal, characteristic, complemented
9#30631565C211yesmaximal, characteristic, complemented
10#30631564C211yesmaximal, characteristic, complemented
11#30631562C211yesmaximal, characteristic, complemented
12#30631561C211yesmaximal, characteristic, complemented
13#30631546C211yesmaximal, characteristic, complemented
14C4×Dic251⋊C4C211yesmaximal, characteristic, complemented
15#388×C251C211yesmaximal, characteristic, complemented
16C22×D1004C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C23×Dic251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22#22548688C418yescomplemented
23(C22⋊D251)⋊C4—24no—
24#22548673—24no—
25C2×C4⋊Dic251C2211yescharacteristic
26C2×C4⋊Dic251C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×Dic251⋊C4C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×(C2×D251)⋊C4C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×(C2×D251)⋊C4C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×(C2×D251)⋊C4C2211yescharacteristic
39C2×(C2×D251)⋊C4C2211yescharacteristic
40C2×C4×Dic251C2211yescharacteristic
41C2×C4×Dic251C2211yescharacteristic
42C2×C4×Dic251C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C2×C4⋊C4×C251C2211yescharacteristic
48C2×C22⋊C4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C42×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54C23×D251C2311yescharacteristic
55C23×D251C2311yescharacteristic
56C2×D1004—41no—
57C2×D1004—41no—
58C2×D1004C2×C414yescomplemented
59C2×C22⋊D251—22no—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—41no—
62C2×C22⋊D251—41no—
63C2×C22⋊D251C2×C414yes—
64C2×C22⋊D251C2×C414yescomplemented
65C22×Dic251—22no—
66C22×Dic251—41no—
67C22×Dic251—41no—
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C22×Dic251C2311yescharacteristic
74C2×C4×D251—22no—
75C2×C4×D251—22no—
76C2×C4×D251—22no—
77C2×C4×D251—41no—
78C2×C4×D251—41no—
79C2×C4×D251C2×C414yescomplemented
80C4⋊Dic251—22no—
81C4⋊Dic251—41no—
82Dic251⋊C4—22no—
83Dic251⋊C4—22no—
84Dic251⋊C4—41no—
85Dic251⋊C4C2×C414yescomplemented
86C22⋊Dic251—41no—
87(C2×D251)⋊C4—22no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4—41no—
91(C2×D251)⋊C4—41no—
92(C2×D251)⋊C4—41no—
93(C2×D251)⋊C4—41no—
94(C2×D251)⋊C4C2×C414yescomplemented
95C4×Dic251—22no—
96C4×Dic251—41no—
97C4×Dic251D414yescomplemented
98C24×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C22×C4×C251C2311yescharacteristic
105C4⋊C4×C251—22no—
106C22⋊C4×C251—22no—
107C22⋊C4×C251—41no—
108C22⋊C4×C251C2×C414yescomplemented
109C42×C251—41no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22×D251—21no—
119C22×D251—21no—
120C22×D251—21no—
121C22×D251—21no—
122C22×D251C22×C412yes—
123C22×D251C22×C412yes—
124D1004—44no—
125C22⋊D251—28no—
126C22⋊D251—44no—
127C2×Dic251—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—21no—
132C2×Dic251—21no—
133C2×Dic251—21no—
134C2×Dic251—21no—
135C2×Dic251—21no—
136C2×Dic251—21no—
137C2×Dic251—21no—
138C2×Dic251—21no—
139C2×Dic251—21no—
140C2×Dic251—44no—
141C2×Dic251C22×C412yes—
142C2×Dic251C22×C412yes—
143C2×Dic251C2×D412yes—
144C2×Dic251C2×D412yes—
145C2×Dic251#4112yes—
146C4×D251—28no—
147C4×D251—44no—
148C23×C251—21no—
149C23×C251—21no—
150C23×C251—21no—
151C23×C251—21no—
152C23×C251—21no—
153C23×C251—21no—
154C23×C251C22×C412yes—
155C23×C251C2411yescharacteristic
156C2×C4×C251—21no—
157C2×C4×C251—21no—
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—21no—
171C2×C4×C251C22×C412yes—
172C2×C4×C251C22×C412yes—
173C2×C4×C251C2×D412yes—
174C22×C251—22no—
175C22×C251—22no—
176C22×C251—22no—
177C22×C251—41no—
178C22×C251—41no—
179C22×C251—41no—
180C22×C251—41no—
181C22×C251C23×C411yescharacteristic
182C22×C251C22×D411yescharacteristic
183C22×C251C2×#4111yescharacteristic
184C22×C251H(5,𝔽2)11yescharacteristic
185C22×C251H(5,𝔽2)11yescharacteristic
186C22×C251H(5,𝔽2)11yescharacteristic
187C22×C2512−1+411yescharacteristic
188C2×D251—22no—
189C2×D251—22no—
190C2×D251—22no—
191C2×D251—22no—
192C2×D251—22no—
193C2×D251—22no—
194C2×D251—41no—
195C2×D251—41no—
196C2×D251—41no—
197C2×D251—41no—
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—
207C4×C251—41no—
208Dic251—22no—
209Dic251—22no—
210Dic251—41no—
211Dic251—41no—
212Dic251—41no—
213Dic251C4×D414yes—
214C2×C251—42no—
215C2×C251C2×C4×D411yescharacteristic
216C2×C251#51811yescharacteristic
217C2×C251#52011yescharacteristic
218C2×C251#53511yescharacteristic
219C2×C251#53711yescharacteristic
220C2×C251#54011yescharacteristic
221C2×C251#54211yescharacteristic
222D251—42no—
223D251—42no—
224C251#232411yescharacteristic, complemented, Sylow
225#2324—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#388D25111yescharacteristic, complemented
235#386—2511no—
236#386—2511no—
237#385—2511no—
238#385—2511no—
239#380—2511no—
240C4×C4⋊C4—2511no—
241C22×D4—2511no—
242C22×D4—2511no—
243C22×D4—2511no—
244C23×C4—2511no—
245C23×C4—2511no—
246C23×C4—2511no—
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×C4⋊C4—2511no—
256C2×C4⋊C4C2×D25111yescharacteristic
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⋊C4—2511no—
265C2×C22⋊C4—2511no—
266C2×C22⋊C4C2×D25111yescharacteristic
267C2×C22⋊C4C2×D25111yescharacteristic
268C2×C22⋊C4C2×D25111yescharacteristic
269C2×C42—2511no—
270C2×C42—2511no—
271C2×C42—2511no—
272C2×C42C2×D25111yescharacteristic
273#95—2511no—
274#95—2511no—
275#95—2511no—
276#95—2511no—
277#95C2×D25111yescharacteristic
278#95C2×D25111yescharacteristic
279C24—2511no—
280C24—2511no—
281C24C22×D25111yescharacteristic
282C2×D4—10041no—
283C2×D4—10041no—
284C2×D4—10041no—
285C2×D4—10041no—
286C2×D4—5022no—
287C2×D4—5022no—
288C2×D4—2514no—
289C2×D4—2514no—
290C2×D4—2514no—
291C22×C4—10041no—
292C22×C4—10041no—
293C22×C4—10041no—
294C22×C4—10041no—
295C22×C4—5022no—
296C22×C4—5022no—
297C22×C4—5022no—
298C22×C4—5022no—
299C22×C4—2511no—
300C22×C4—2511no—
301C22×C4—2511no—
302C22×C4—2511no—
303C22×C4—2511no—
304C22×C4—2514no—
305C22×C4—2511no—
306C22×C4C22×D25111yescharacteristic
307C22×C4C22×D25111yescharacteristic
308C22×C4C22×D25111yescharacteristic
309C22×C4C22×D25111yescharacteristic
310C22×C4C22×D25111yescharacteristic
311C22×C4C22×D25111yescharacteristic
312C4⋊C4—22no—
313C4⋊C4—10041no—
314C4⋊C4—10041no—
315C4⋊C4—5022no—
316C4⋊C4—5022no—
317C4⋊C4—5022no—
318C4⋊C4—2514no—
319C22⋊C4—22no—
320C22⋊C4—41no—
321C22⋊C4—10041no—
322C22⋊C4—10041no—
323C22⋊C4—10041no—
324C22⋊C4—10041no—
325C22⋊C4—10041no—
326C22⋊C4—5022no—
327C22⋊C4—5022no—
328C22⋊C4—5022no—
329C22⋊C4—2514no—
330C22⋊C4C4×D25114yescomplemented
331C42—41no—
332C42—10041no—
333C42—5022no—
334C42—2514no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—5021no—
342C23—5021no—
343C23—5021no—
344C23—5021no—
345C23—5021no—
346C23—5021no—
347C23—5021no—
348C23—5021no—
349C23—5021no—
350C23—5021no—
351C23—5021no—
352C23—5021no—
353C23—2512no—
354C23—2512no—
355C23C2×C4×D25112yes—
356C23C23×D25111yescharacteristic
357D4—10044no—
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—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—10044no—
376C2×C4—10044no—
377C2×C4—5028no—
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—5021no—
389C2×C4—5021no—
390C2×C4—5021no—
391C2×C4—2512no—
392C2×C4—2512no—
393C2×C4—2512no—
394C2×C4—2512no—
395C2×C4—2512no—
396C2×C4C2×C4×D25112yes—
397C2×C4C2×C4×D25112yes—
398C2×C4D4×D25112yes—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—10041no—
407C22—10041no—
408C22—10041no—
409C22—10041no—
410C22—10041no—
411C22—10041no—
412C22—10041no—
413C22—10041no—
414C22—5022no—
415C22—5022no—
416C22—5022no—
417C22—5022no—
418C22—5022no—
419C22—5022no—
420C22C22×C4×D25111yescharacteristic
421C22C2×D4×D25111yescharacteristic
422C22#41×D25111yescharacteristic
423C22#2254876011yescharacteristic
424C22#2254876111yescharacteristic
425C22#2254876111yescharacteristic
426C22#2254876211yescharacteristic
427C4—22no—
428C4—22no—
429C4—22no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—10041no—
434C4—10041no—
435C4—10041no—
436C4—5022no—
437C4—5022no—
438C4—2514no—
439C2—42no—
440C2—10042no—
441C2—10042no—
442C2C4×D4×D25111yescharacteristic
443C2#3063215911yescharacteristic
444C2#3063216711yescharacteristic
445C2#3063225511yescharacteristic
446C2#3063226411yescharacteristic
447C2#3063228911yescharacteristic
448C2#3063230311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#388×C251C2
socle, cosocleC23×C251C24
prime radicalC2×#22548688C2
hypercenter#388D251
hypocenterC251#2324
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×(C22⋊D251)⋊C4C2semidirect
3. C2×#22548673C2semidirect
4. #30631589C2semidirect
5. #30631587C2semidirect
6. #30631587C2semidirect
7. #30631584C2semidirect
8. #30631577C2semidirect
9. #30631565C2semidirect
10. #30631564C2semidirect
11. #30631562C2semidirect
12. #30631561C2semidirect
13. #30631546C2semidirect
14. C4×Dic251⋊C4C2semidirect
15. #388×C251C2semidirect
22. #22548688C4semidirect
58. C2×D1004C2×C4semidirect
64. C2×C22⋊D251C2×C4semidirect
79. C2×C4×D251C2×C4semidirect
85. Dic251⋊C4C2×C4semidirect
94. (C2×D251)⋊C4C2×C4semidirect
97. C4×Dic251D4semidirect
108. C22⋊C4×C251C2×C4semidirect
224. C251#2324semidirect
234. #388D251semidirect
330. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486882
C2×(C22⋊D251)⋊C42
C2×#225486732
#306315892
#306315872
#306315842
#306315772
#306315652
#306315642
#306315622
#306315612
#306315462
C4×Dic251⋊C42
#388×C2512
#2324251