Finite Group Databasebeta

← #44558371

Subgroups — #44558371

Order 32128.

578 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#22548696C211yesmaximal, characteristic, complemented
2C2×#22548687C211yesmaximal, characteristic, complemented
3#30631628C211yesmaximal, characteristic, complemented
4#30631623C211yesmaximal, characteristic, complemented
5#30631611C211yesmaximal, characteristic, complemented
6#30631610C211yesmaximal, characteristic, complemented
7#30631603C211yesmaximal, characteristic, complemented
8#30631602C211yesmaximal, characteristic, complemented
9#30631602C211yesmaximal, characteristic, complemented
10#30631601C211yesmaximal, characteristic, complemented
11#30631601C211yesmaximal, characteristic, complemented
12#30631595C211yesmaximal, characteristic, complemented
13#30631595C211yesmaximal, characteristic, complemented
14#30631545C211yesmaximal, characteristic, complemented
15#392×C251C211yesmaximal, characteristic, complemented
16C22×C22⋊D251C2211yescharacteristic
17C22×C22⋊D251C2211yescharacteristic
18C22×C22⋊D251C2211yescharacteristic
19C23×Dic251C2211yescharacteristic
20#22548696—24no—
21#22548687—24no—
22C2×C4⋊Dic251C2211yescharacteristic
23C2×Dic251⋊C4C2211yescharacteristic
24C2×Dic251⋊C4C2211yescharacteristic
25C2×Dic251⋊C4C2211yescharacteristic
26C2×Dic251⋊C4C2211yescharacteristic
27C2×Dic251⋊C4C2211yescharacteristic
28C2×Dic251⋊C4C2211yescharacteristic
29C2×C22⋊Dic251C2211yescharacteristic
30C2×C22⋊Dic251C2211yescharacteristic
31C2×C22⋊Dic251C2211yescharacteristic
32C2×C22⋊Dic251C2211yescharacteristic
33C2×C22⋊Dic251C2211yescharacteristic
34C2×C22⋊Dic251C2211yescharacteristic
35C2×C22⋊Dic251C2211yescharacteristic
36C2×C22⋊Dic251C2211yescharacteristic
37C2×(C2×D251)⋊C4C2211yescharacteristic
38C2×(C2×D251)⋊C4C2211yescharacteristic
39C2×(C2×D251)⋊C4C2211yescharacteristic
40C2×(C2×D251)⋊C4C2211yescharacteristic
41C251⋊#95C2211yescharacteristic
42C251⋊#95C2211yescharacteristic
43C251⋊#95C2211yescharacteristic
44C251⋊#95C2211yescharacteristic
45C251⋊#95C2211yescharacteristic
46C22×D4×C251C2211yescharacteristic
47C22×D4×C251C2211yescharacteristic
48C22×D4×C251C2211yescharacteristic
49C2×C22⋊C4×C251C2211yescharacteristic
50C2×C22⋊C4×C251C2211yescharacteristic
51C2×C22⋊C4×C251C2211yescharacteristic
52#95×C251C2211yescharacteristic
53C23×D251C2311yescharacteristic
54C2×C22⋊D251—22no—
55C2×C22⋊D251—22no—
56C2×C22⋊D251—41no—
57C2×C22⋊D251—41no—
58C2×C22⋊D251—41no—
59C2×C22⋊D251—41no—
60C2×C22⋊D251—41no—
61C2×C22⋊D251—41no—
62C2×C22⋊D251—41no—
63C22×Dic251—41no—
64C22×Dic251—41no—
65C22×Dic251D414yescomplemented
66C22×Dic251C2311yescharacteristic
67C22×Dic251C2311yescharacteristic
68C22×Dic251C2311yescharacteristic
69C22×Dic251C2311yescharacteristic
70C22×Dic251C2311yescharacteristic
71C22×Dic251C2311yescharacteristic
72C22×Dic251C2311yescharacteristic
73C4⋊Dic251—41no—
74Dic251⋊C4—22no—
75Dic251⋊C4—22no—
76Dic251⋊C4—22no—
77Dic251⋊C4—41no—
78Dic251⋊C4—41no—
79Dic251⋊C4—41no—
80C22⋊Dic251—22no—
81C22⋊Dic251—22no—
82C22⋊Dic251—22no—
83C22⋊Dic251—22no—
84C22⋊Dic251—41no—
85C22⋊Dic251—41no—
86C22⋊Dic251—41no—
87C22⋊Dic251—41no—
88(C2×D251)⋊C4—22no—
89(C2×D251)⋊C4—41no—
90(C2×D251)⋊C4—41no—
91(C2×D251)⋊C4—41no—
92C24×C251C2311yescharacteristic
93C24×C251C2311yescharacteristic
94C24×C251C2311yescharacteristic
95C2×D4×C251—22no—
96C2×D4×C251—22no—
97C2×D4×C251—41no—
98C2×D4×C251—41no—
99C2×D4×C251—41no—
100C2×D4×C251—41no—
101C2×D4×C251—41no—
102C2×D4×C251—41no—
103C2×D4×C251—41no—
104C22×C4×C251C2311yescharacteristic
105C22×C4×C251C2311yescharacteristic
106C22×C4×C251C2311yescharacteristic
107C22×C4×C251C2311yescharacteristic
108C22⋊C4×C251—41no—
109C22⋊C4×C251—41no—
110C22⋊C4×C251—41no—
111C22×D251—21no—
112C22×D251—21no—
113C22×D251—21no—
114C22×D251—21no—
115C22×D251—21no—
116C22×D251—21no—
117C22×D251—21no—
118C22⋊D251—44no—
119C22⋊D251—44no—
120C22⋊D251—44no—
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—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×Dic251C2×D412yes—
142C2×Dic251C2×D412yes—
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×C251—21no—
155C23×C251—21no—
156C23×C251—21no—
157C23×C251—21no—
158C23×C251—21no—
159C23×C251—21no—
160C23×C251—21no—
161C23×C251—21no—
162C23×C251—21no—
163C23×C251C2×D412yes—
164C23×C251C2411yescharacteristic
165D4×C251—44no—
166D4×C251—44no—
167D4×C251—44no—
168C2×C4×C251—21no—
169C2×C4×C251—21no—
170C2×C4×C251—21no—
171C2×C4×C251—21no—
172C2×C4×C251—21no—
173C2×C4×C251—21no—
174C2×C4×C251—21no—
175C2×C4×C251—21no—
176C2×C4×C251—21no—
177C2×C4×C251—21no—
178C2×C4×C251—21no—
179C2×C4×C251—21no—
180C22×C251—22no—
181C22×C251—22no—
182C22×C251—22no—
183C22×C251—41no—
184C22×C251—41no—
185C22×C251—41no—
186C22×C251—41no—
187C22×C251—41no—
188C22×C251—41no—
189C22×C251—41no—
190C22×C251—41no—
191C22×C251—41no—
192C22×C251—41no—
193C22×C251—41no—
194C22×C251—41no—
195C22×C251—41no—
196C22×C251—41no—
197C22×C251—41no—
198C22×C251—41no—
199C22×C251—41no—
200C22×C251—41no—
201C22×C251C22×D411yescharacteristic
202C22×C251H(5,𝔽2)11yescharacteristic
203C22×C251H(5,𝔽2)11yescharacteristic
204C22×C251H(5,𝔽2)11yescharacteristic
205C22×C251H(5,𝔽2)11yescharacteristic
206C22×C251H(5,𝔽2)11yescharacteristic
207C22×C2512−1+411yescharacteristic
208C2×D251—41no—
209C2×D251—41no—
210C2×D251—41no—
211C2×D251—41no—
212C2×D251—41no—
213C2×D251—41no—
214C2×D251—41no—
215C4×C251—41no—
216C4×C251—41no—
217C4×C251—41no—
218C4×C251—41no—
219Dic251—22no—
220Dic251—22no—
221Dic251—41no—
222Dic251—41no—
223Dic251—41no—
224Dic251—41no—
225Dic251—41no—
226C2×C251—42no—
227C2×C251—81no—
228C2×C251—81no—
229C2×C251#53411yescharacteristic
230C2×C251#53511yescharacteristic
231C2×C251#53711yescharacteristic
232C2×C251#56011yescharacteristic
233C2×C251#56011yescharacteristic
234C2×C251#56211yescharacteristic
235C2×C251#56211yescharacteristic
236D251—81no—
237C251#281311yescharacteristic, complemented, Sylow
238#2813—2511nomaximal, Sylow
239C2×#123—2511no—
240C2×#121—2511no—
241#400—2511no—
242#399—2511no—
243#396—2511no—
244#396—2511no—
245#394—2511no—
246#394—2511no—
247#394—2511no—
248#394—2511no—
249#394—2511no—
250#392—2511no—
251#392—2511no—
252#392D25111yescharacteristic, complemented
253#380—2511no—
254C22×D4—2511no—
255C22×D4—2511no—
256C22×D4—2511no—
257C22×D4C2×D25111yescharacteristic
258C22×D4C2×D25111yescharacteristic
259C22×D4C2×D25111yescharacteristic
260C23×C4—2511no—
261#123—5024no—
262#121—5024no—
263C2×C4⋊C4—2511no—
264C2×C4⋊C4—2511no—
265C2×C4⋊C4—2511no—
266C2×C4⋊C4—2511no—
267C2×C4⋊C4—2511no—
268C2×C4⋊C4—2511no—
269C2×C4⋊C4—2511no—
270C2×C22⋊C4—2511no—
271C2×C22⋊C4—2511no—
272C2×C22⋊C4—2511no—
273C2×C22⋊C4—2511no—
274C2×C22⋊C4—2511no—
275C2×C22⋊C4—2511no—
276C2×C22⋊C4—2511no—
277C2×C22⋊C4—2511no—
278C2×C22⋊C4—2511no—
279C2×C22⋊C4—2511no—
280C2×C22⋊C4—2511no—
281C2×C22⋊C4—2511no—
282C2×C22⋊C4C2×D25111yescharacteristic
283C2×C22⋊C4C2×D25111yescharacteristic
284C2×C22⋊C4C2×D25111yescharacteristic
285#95—2511no—
286#95—2511no—
287#95—2511no—
288#95—2511no—
289#95—2511no—
290#95C2×D25111yescharacteristic
291C24—2511no—
292C24C22×D25111yescharacteristic
293C24C22×D25111yescharacteristic
294C24C22×D25111yescharacteristic
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—10041no—
305C2×D4—10041no—
306C2×D4—10041no—
307C2×D4—10041no—
308C2×D4—10041no—
309C2×D4—10041no—
310C2×D4—10041no—
311C2×D4—5022no—
312C2×D4—5022no—
313C22×C4—10041no—
314C22×C4—10041no—
315C22×C4—2514no—
316C22×C4—2511no—
317C22×C4—2511no—
318C22×C4—2511no—
319C22×C4—2511no—
320C22×C4—2511no—
321C22×C4—2511no—
322C22×C4—2511no—
323C22×C4C22×D25111yescharacteristic
324C22×C4C22×D25111yescharacteristic
325C22×C4C22×D25111yescharacteristic
326C22×C4C22×D25111yescharacteristic
327C4⋊C4—10041no—
328C4⋊C4—10041no—
329C4⋊C4—10041no—
330C4⋊C4—10041no—
331C4⋊C4—5022no—
332C4⋊C4—5022no—
333C4⋊C4—5022no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—10041no—
338C22⋊C4—10041no—
339C22⋊C4—10041no—
340C22⋊C4—10041no—
341C22⋊C4—10041no—
342C22⋊C4—10041no—
343C22⋊C4—10041no—
344C22⋊C4—5022no—
345C22⋊C4—5022no—
346C22⋊C4—5022no—
347C22⋊C4—5022no—
348C22⋊C4—5022no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—5021no—
370C23—5021no—
371C23—5021no—
372C23—5021no—
373C23—5021no—
374C23—5021no—
375C23—5021no—
376C23D4×D25112yes—
377C23C23×D25111yescharacteristic
378D4—44no—
379D4—44no—
380D4—44no—
381D4—10044no—
382D4—10044no—
383D4—10044no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—10044no—
397C2×C4—5021no—
398C2×C4—5021no—
399C2×C4—5021no—
400C2×C4—5021no—
401C2×C4—5021no—
402C2×C4—5021no—
403C2×C4—5021no—
404C2×C4—5021no—
405C2×C4—5021no—
406C2×C4—5021no—
407C2×C4—5021no—
408C2×C4—5021no—
409C2×C4—5021no—
410C2×C4—5021no—
411C2×C4—5021no—
412C2×C4—5021no—
413C2×C4—5021no—
414C2×C4—5021no—
415C2×C4—5021no—
416C2×C4—2512no—
417C2×C4—2512no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—10041no—
440C22—10041no—
441C22—10041no—
442C22—10041no—
443C22—10041no—
444C22—10041no—
445C22—10041no—
446C22C2×D4×D25111yescharacteristic
447C22#2254876011yescharacteristic
448C22#2254876011yescharacteristic
449C22#2254876011yescharacteristic
450C22#2254876011yescharacteristic
451C22#2254876011yescharacteristic
452C22#2254876211yescharacteristic
453C4—41no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—10041no—
458C4—10041no—
459C4—10041no—
460C4—10041no—
461C4—10041no—
462C4—5022no—
463C4—5022no—
464C2—42no—
465C2—81no—
466C2—81no—
467C2—20081no—
468C2#3063224711yescharacteristic
469C2#3063225211yescharacteristic
470C2#3063226411yescharacteristic
471C2#3063242411yescharacteristic
472C2#3063242611yescharacteristic
473C2#3063243211yescharacteristic
474C2#3063243211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#392×C251C2
hypercenter#392D251
hypocenterC251#2813
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#22548696C2semidirect
2. C2×#22548687C2semidirect
3. #30631628C2semidirect
4. #30631623C2semidirect
5. #30631611C2semidirect
6. #30631610C2semidirect
7. #30631603C2semidirect
8. #30631602C2semidirect
9. #30631602C2semidirect
10. #30631601C2semidirect
11. #30631601C2semidirect
12. #30631595C2semidirect
13. #30631595C2semidirect
14. #30631545C2semidirect
15. #392×C251C2semidirect
65. C22×Dic251D4semidirect
237. C251#2813semidirect
252. #392D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486962
C2×#225486872
#306316282
#306316232
#306316112
#306316102
#306316032
#306316022
#306316012
#306315952
#306315452
#392×C2512
#2813251