Finite Group Databasebeta

← #44553271

Subgroups — #44553271

Order 32128.

704 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D251C211yesmaximal, characteristic
2C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
3C2×(C22⋊D251)⋊C4C211yesmaximal, characteristic
4#30631584C211yesmaximal, characteristic, complemented
5#30631582C211yesmaximal, characteristic, complemented
6#30631562C211yesmaximal, characteristic, complemented
7#30631561C211yesmaximal, characteristic, complemented
8#30631558C211yesmaximal, characteristic, complemented
9#30631551C211yesmaximal, characteristic, complemented
10#30631545C211yesmaximal, characteristic, complemented
11#30631544C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic251C211yesmaximal, characteristic, complemented
13C2×C251⋊#95C211yesmaximal, characteristic, complemented
14#95×D251C211yesmaximal, characteristic, complemented
15#385×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C23×Dic251C2211yescharacteristic
18C23×Dic251C2211yescharacteristic
19C22×C4×D251C2211yescharacteristic
20C22×C4×D251C2211yescharacteristic
21C22×C4×D251C2211yescharacteristic
22C4×C22⋊D251—24no—
23(C22⋊D251)⋊C4—24no—
24(C22⋊D251)⋊C4C418yescomplemented
25C2×C4⋊Dic251C2211yescharacteristic
26C2×C4⋊Dic251C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×Dic251⋊C4C2211yescharacteristic
30C2×Dic251⋊C4C2211yescharacteristic
31C2×Dic251⋊C4C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×(C2×D251)⋊C4C2211yescharacteristic
34C2×(C2×D251)⋊C4C2211yescharacteristic
35C2×(C2×D251)⋊C4C2211yescharacteristic
36C2×C4×Dic251C2211yescharacteristic
37C2×C4×Dic251C2211yescharacteristic
38C2×C4×Dic251C2211yescharacteristic
39C251⋊#95C2211yescharacteristic
40C251⋊#95C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C251⋊#95C2211yescharacteristic
47C23×C4×C251C2211yescharacteristic
48C2×C4⋊C4×C251C2211yescharacteristic
49C2×C4⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53#95×C251C2211yescharacteristic
54C23×D251C2311yescharacteristic
55C2×C22⋊D251C2×C414yes—
56C2×C22⋊D251D414yes—
57C2×C22⋊D251Q814yes—
58C22×Dic251—22no—
59C22×Dic251—22no—
60C22×Dic251—41no—
61C22×Dic251—41no—
62C22×Dic251—41no—
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—22no—
74C2×C4×D251—22no—
75C2×C4×D251—41no—
76C2×C4×D251—41no—
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⋊C4—41no—
86Dic251⋊C4C2×C414yescomplemented
87C22⋊Dic251—22no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—22no—
90(C2×D251)⋊C4C2×C414yescomplemented
91C4×Dic251—22no—
92C4×Dic251—22no—
93C4×Dic251C2×C414yescomplemented
94C24×C251C2311yescharacteristic
95C22×C4×C251—22no—
96C22×C4×C251—22no—
97C22×C4×C251—41no—
98C22×C4×C251C2311yescharacteristic
99C22×C4×C251C2311yescharacteristic
100C22×C4×C251C2311yescharacteristic
101C22×C4×C251C2311yescharacteristic
102C22×C4×C251C2311yescharacteristic
103C22×C4×C251C2311yescharacteristic
104C4⋊C4×C251—22no—
105C4⋊C4×C251—41no—
106C22⋊C4×C251—22no—
107C22⋊C4×C251C2×C414yescomplemented
108C22×D251—21no—
109C22×D251—21no—
110C22×D251—21no—
111C22×D251—21no—
112C22×D251C22×C412yes—
113C22×D251C2×D412yes—
114C22×D251C2×Q812yes—
115C22⋊D251C4⋊C4116yescomplemented
116C2×Dic251—21no—
117C2×Dic251—21no—
118C2×Dic251—21no—
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—21no—
128C2×Dic251—21no—
129C2×Dic251—21no—
130C2×Dic251—21no—
131C2×Dic251—44no—
132C2×Dic251—44no—
133C2×Dic251C22×C412yes—
134C2×Dic251C22×C412yes—
135C2×Dic251C22×C412yes—
136C2×Dic251C2×D412yes—
137C2×Dic251C2×Q812yes—
138C2×Dic251#4112yes—
139C4×D251—44no—
140C4×D251—44no—
141C4×D251—44no—
142C23×C251—21no—
143C23×C251—21no—
144C23×C251—21no—
145C23×C251—21no—
146C23×C251C22×C412yes—
147C23×C251C2×D412yes—
148C23×C251C2×Q812yes—
149C23×C251C2411yescharacteristic
150C2×C4×C251—21no—
151C2×C4×C251—21no—
152C2×C4×C251—21no—
153C2×C4×C251—21no—
154C2×C4×C251—21no—
155C2×C4×C251—21no—
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—44no—
166C2×C4×C251C22×C412yes—
167C2×C4×C251C22×C412yes—
168C2×C4×C251#4112yes—
169C22×C251—22no—
170C22×C251—22no—
171C22×C251—22no—
172C22×C251—22no—
173C22×C251—22no—
174C22×C251—22no—
175C22×C251C2×C4⋊C414yes—
176C22×C251C23×C411yescharacteristic
177C22×C251C22×D411yescharacteristic
178C22×C251C22×Q811yescharacteristic
179C22×C251C2×#4111yescharacteristic
180C22×C251C2×#4111yescharacteristic
181C22×C251H(5,𝔽2)11yescharacteristic
182C22×C251H(5,𝔽2)11yescharacteristic
183C2×D251—22no—
184C2×D251—22no—
185C2×D251—22no—
186C2×D251—22no—
187C2×D251—22no—
188C2×D251—22no—
189C2×D251C2×C4⋊C414yes—
190C4×C251—22no—
191C4×C251—22no—
192C4×C251—22no—
193C4×C251—41no—
194C4×C251—41no—
195C4×C251—41no—
196Dic251—22no—
197Dic251—22no—
198Dic251—22no—
199Dic251—41no—
200Dic251—41no—
201Dic251—41no—
202Dic251C2×C4⋊C414yes—
203C2×C251—24no—
204C2×C251C22×C4⋊C411yescharacteristic
205C2×C251C4×#4111yescharacteristic
206C2×C251#51811yescharacteristic
207C2×C251#54611yescharacteristic
208C2×C251#54611yescharacteristic
209C2×C251#55411yescharacteristic
210C2×C251#55411yescharacteristic
211D251—24no—
212C251#234911yescharacteristic, complemented, Sylow
213#2349—2511nomaximal, Sylow
214C2×C4×D4—2511no—
215C2×C4×D4—2511no—
216C2×C4×D4—2511no—
217#389—2511no—
218#389—2511no—
219#385—2511no—
220#385—2511no—
221#385—2511no—
222#385D25111yescharacteristic, complemented
223#382—2511no—
224#380—2511no—
225#380—2511no—
226C4×C4⋊C4—2511no—
227C2×#95—2511no—
228C2×#95—2511no—
229C22×D4—2511no—
230C23×C4—2511no—
231C23×C4—2511no—
232C23×C4—2511no—
233C23×C4—2511no—
234C23×C4—2511no—
235C23×C4C2×D25111yescharacteristic
236C4×D4—5024no—
237C4×D4—5024no—
238C4×D4—2518no—
239C2×C4⋊C4—2511no—
240C2×C4⋊C4—2511no—
241C2×C4⋊C4—2511no—
242C2×C4⋊C4—2511no—
243C2×C4⋊C4—2511no—
244C2×C4⋊C4—2511no—
245C2×C4⋊C4—2511no—
246C2×C4⋊C4C2×D25111yescharacteristic
247C2×C4⋊C4C2×D25111yescharacteristic
248C2×C22⋊C4—2511no—
249C2×C22⋊C4—2511no—
250C2×C22⋊C4—2511no—
251C2×C22⋊C4—2511no—
252C2×C22⋊C4C2×D25111yescharacteristic
253C2×C22⋊C4C2×D25111yescharacteristic
254C2×C42—2511no—
255C2×C42—2511no—
256C2×C42—2511no—
257#95—2511no—
258#95—2511no—
259#95—2511no—
260#95—2511no—
261#95—2511no—
262#95—2511no—
263#95—2511no—
264#95—2511no—
265#95C2×D25111yescharacteristic
266#95C2×D25111yescharacteristic
267C24—2511no—
268C24C22×D25111yescharacteristic
269C2×D4—2514no—
270C2×D4—2514no—
271C2×D4—2514no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—10041no—
276C22×C4—10041no—
277C22×C4—10041no—
278C22×C4—10041no—
279C22×C4—10041no—
280C22×C4—10041no—
281C22×C4—10041no—
282C22×C4—5022no—
283C22×C4—5022no—
284C22×C4—5022no—
285C22×C4—5022no—
286C22×C4—5022no—
287C22×C4—5022no—
288C22×C4—2514no—
289C22×C4—2514no—
290C22×C4—2511no—
291C22×C4—2511no—
292C22×C4—2511no—
293C22×C4—2511no—
294C22×C4—2511no—
295C22×C4—2511no—
296C22×C4—2511no—
297C22×C4C22×D25111yescharacteristic
298C22×C4C22×D25111yescharacteristic
299C22×C4C22×D25111yescharacteristic
300C22×C4C22×D25111yescharacteristic
301C22×C4C22×D25111yescharacteristic
302C22×C4C22×D25111yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—10041no—
306C4⋊C4—10041no—
307C4⋊C4—10041no—
308C4⋊C4—5022no—
309C4⋊C4—5022no—
310C4⋊C4—5022no—
311C4⋊C4—2514no—
312C22⋊C4—22no—
313C22⋊C4—5022no—
314C22⋊C4—5022no—
315C22⋊C4—5022no—
316C22⋊C4—2514no—
317C22⋊C4C4×D25114yescomplemented
318C42—5022no—
319C42—5022no—
320C42—2514no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—5021no—
326C23—5021no—
327C23—5021no—
328C23—5021no—
329C23—2512no—
330C23—2512no—
331C23—2512no—
332C23C2×C4×D25112yes—
333C23D4×D25112yes—
334C23Q8×D25112yes—
335C23C23×D25111yescharacteristic
336D4—25116no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—10044no—
354C2×C4—10044no—
355C2×C4—10044no—
356C2×C4—10044no—
357C2×C4—10044no—
358C2×C4—5021no—
359C2×C4—5021no—
360C2×C4—5021no—
361C2×C4—5021no—
362C2×C4—5021no—
363C2×C4—5021no—
364C2×C4—5021no—
365C2×C4—5021no—
366C2×C4—5021no—
367C2×C4—5021no—
368C2×C4—5021no—
369C2×C4—5021no—
370C2×C4—5021no—
371C2×C4—5021no—
372C2×C4—5021no—
373C2×C4—2512no—
374C2×C4—2512no—
375C2×C4—2512no—
376C2×C4—2512no—
377C2×C4—2512no—
378C2×C4—2512no—
379C2×C4C2×C4×D25112yes—
380C2×C4C2×C4×D25112yes—
381C2×C4#1686080912yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—5022no—
389C22—5022no—
390C22—5022no—
391C22—5022no—
392C22—5022no—
393C22—5022no—
394C22—2514no—
395C22C4⋊C4×D25114yes—
396C22C22×C4×D25111yescharacteristic
397C22C2×D4×D25111yescharacteristic
398C22C2×Q8×D25111yescharacteristic
399C22#41×D25111yescharacteristic
400C22C2×#1686080911yescharacteristic
401C22#2254876011yescharacteristic
402C22#2254876111yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—10041no—
410C4—10041no—
411C4—10041no—
412C4—5022no—
413C4—5022no—
414C4—5022no—
415C4—2514no—
416C2—24no—
417C2—5024no—
418C2C2×C4⋊C4×D25111yescharacteristic
419C2C4×#1686080911yescharacteristic
420C2#3063215911yescharacteristic
421C2#3063232411yescharacteristic
422C2#3063232911yescharacteristic
423C2#3063239111yescharacteristic
424C2#3063239511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#385×C251C2
socle, cosocleC23×C251C24
prime radicalC22×C22⋊D251C22
hypercenter#385D251
hypocenterC251#2349
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #30631584C2semidirect
5. #30631582C2semidirect
6. #30631562C2semidirect
7. #30631561C2semidirect
8. #30631558C2semidirect
9. #30631551C2semidirect
10. #30631545C2semidirect
11. #30631544C2semidirect
12. C4⋊C4×Dic251C2semidirect
13. C2×C251⋊#95C2semidirect
14. #95×D251C2semidirect
15. #385×C251C2semidirect
24. (C22⋊D251)⋊C4C4semidirect
63. C22×Dic251C2×C4semidirect
79. C2×C4×D251C2×C4semidirect
86. Dic251⋊C4C2×C4semidirect
90. (C2×D251)⋊C4C2×C4semidirect
93. C4×Dic251C2×C4semidirect
107. C22⋊C4×C251C2×C4semidirect
115. C22⋊D251C4⋊C4semidirect
212. C251#2349semidirect
222. #385D251semidirect
317. C22⋊C4C4×D251semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D2512
C2×(C22⋊D251)⋊C42
#306315842
#306315822
#306315622
#306315612
#306315582
#306315512
#306315452
#306315442
C4⋊C4×Dic2512
C2×C251⋊#952
#95×D2512
#385×C2512
#2349251