Finite Group Databasebeta

← C151×#24328 #50329253

Subgroups — C151×#24328 #50329253

Order 38656.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#557×C151C211yesmaximal, characteristic, complemented
2C2×#474×C151C211yesmaximal, characteristic
3C2×C4×SD16×C151C211yesmaximal, characteristic
4#2538×C151C211yesmaximal, characteristic
5#2059×C151C212yesmaximal, complemented
6#2025×C151C212yesmaximal
7#1990×C151C211yesmaximal, characteristic, complemented
8#1984×C151C211yesmaximal, characteristic, complemented
9#1880×C151C212yesmaximal
10#1863×C151C212yesmaximal, complemented
11#1803×C151C211yesmaximal, characteristic, complemented
12C22×SD16×C151C2211yescharacteristic
13#557×C151—24no—
14#557×C151—24no—
15C2×C4⋊Q8×C151—21no—
16C2×C4⋊Q8×C151C2211yescharacteristic
17C2×#125×C151—22no—
18C2×#125×C151C2212yes—
19C2×C22⋊Q8×C151—21no—
20C2×C4×Q8×C151—21no—
21C2×C4×Q8×C151C2212yes—
22C2×C4×Q8×C151C2211yescharacteristic
23C2×C4×Q8×C151C2211yescharacteristic
24C2×C4×D4×C151C2211yescharacteristic
25#474×C151—24no—
26C4×SD16×C151—24no—
27C2×#107×C151C2212yes—
28C2×#106×C151C2211yescharacteristic
29C2×C4⋊C8×C151C2211yescharacteristic
30C2×#103×C151C2211yescharacteristic
31C2×#103×C151C2212yes—
32C2×#102×C151C2212yes—
33C2×#102×C151C2211yescharacteristic
34C2×C4×C8×C151C2211yescharacteristic
35#396×C151—22no—
36#390×C151C2212yes—
37#389×C151C2212yes—
38#384×C151C2211yescharacteristic
39#380×C151—21no—
40#340×C151C2212yes—
41#340×C151C2212yes—
42#340×C151C2212yes—
43#336×C151C2212yes—
44#336×C151C2212yes—
45C22×Q8×C151—21no—
46C22×Q8×C151D411yescharacteristic
47C22×Q8×C151C2311yescharacteristic
48C22×D4×C151C2311yescharacteristic
49C23×C4×C151—21no—
50C2×SD16×C151—22no—
51C2×SD16×C151—22no—
52C2×SD16×C151Q814yescomplemented
53C22×C8×C151C2311yescharacteristic
54C22×C8×C151C2312yes—
55C22×C8×C151C2311yescharacteristic
56C4⋊Q8×C151—28no—
57C4⋊Q8×C151—22no—
58C4⋊Q8×C151D414yes—
59#125×C151—216no—
60#125×C151—24no—
61#125×C151—24no—
62C22⋊Q8×C151—44no—
63C4×Q8×C151—28no—
64C4×Q8×C151—24no—
65C4×Q8×C151—22no—
66C4×Q8×C151—22no—
67C4×Q8×C151—22no—
68C4×Q8×C151—24no—
69C4×Q8×C151D414yes—
70C4×D4×C151—22no—
71C4×D4×C151—22no—
72C2×C4⋊C4×C151—22no—
73C2×C4⋊C4×C151—21no—
74C2×C4⋊C4×C151—22no—
75C2×C4⋊C4×C151—21no—
76C2×C4⋊C4×C151—22no—
77C2×C4⋊C4×C151—21no—
78C2×C4⋊C4×C151—22no—
79C2×C4⋊C4×C151—21no—
80C2×C4⋊C4×C151D411yescharacteristic
81C2×C4⋊C4×C151D412yes—
82C2×C4⋊C4×C151C2312yes—
83C2×C4⋊C4×C151C2311yescharacteristic
84C2×C4⋊C4×C151C2311yescharacteristic
85C2×C4⋊C4×C151C2312yes—
86C2×C22⋊C4×C151—22no—
87C2×C22⋊C4×C151—21no—
88C2×C22⋊C4×C151—22no—
89C2×C42×C151—22no—
90C2×C42×C151—21no—
91C2×C42×C151C2312yes—
92C2×C42×C151C2311yescharacteristic
93#107×C151—24no—
94#106×C151—22no—
95C4⋊C8×C151—22no—
96#103×C151—22no—
97#103×C151—24no—
98#102×C151—22no—
99#102×C151—24no—
100C4×C8×C151—22no—
101#95×C151—22no—
102C24×C151—21no—
103C2×Q8×C151—22no—
104C2×Q8×C151—24no—
105C2×Q8×C151—22no—
106C2×Q8×C151—22no—
107C2×Q8×C151—21no—
108C2×Q8×C151—21no—
109C2×Q8×C151—21no—
110C2×Q8×C151—42no—
111C2×Q8×C151—42no—
112C2×Q8×C151C2×D412yes—
113C2×Q8×C151C2×Q812yes—
114C2×Q8×C151#4112yes—
115C2×D4×C151—21no—
116C2×D4×C151—21no—
117C2×D4×C151—21no—
118C2×D4×C151—21no—
119C2×D4×C151—21no—
120C2×D4×C151C2×Q812yes—
121C22×C4×C151—21no—
122C22×C4×C151—21no—
123C22×C4×C151—22no—
124C22×C4×C151—21no—
125C22×C4×C151—22no—
126C22×C4×C151—42no—
127C22×C4×C151—42no—
128C22×C4×C151—42no—
129C22×C4×C151C2×D412yes—
130C22×C4×C151C2×D412yes—
131C22×C4×C151C2×D411yescharacteristic
132C22×C4×C151C2×D411yescharacteristic
133C22×C4×C151C2411yescharacteristic
134SD16×C151—28no—
135C2×C8×C151—21no—
136C2×C8×C151—21no—
137C2×C8×C151—21no—
138C2×C8×C151—22no—
139C2×C8×C151—22no—
140C2×C8×C151—21no—
141C2×C8×C151—21no—
142C2×C8×C151—22no—
143C2×C8×C151C2×Q812yes—
144C4⋊C4×C151—28no—
145C4⋊C4×C151—28no—
146C4⋊C4×C151—28no—
147C4⋊C4×C151—24no—
148C4⋊C4×C151—24no—
149C4⋊C4×C151—22no—
150C4⋊C4×C151—24no—
151C4⋊C4×C151—21no—
152C4⋊C4×C151—22no—
153C4⋊C4×C151—21no—
154C4⋊C4×C151—21no—
155C4⋊C4×C151—22no—
156C4⋊C4×C151—22no—
157C4⋊C4×C151—44no—
158C4⋊C4×C151—42no—
159C4⋊C4×C151—81no—
160C4⋊C4×C151C2×D412yes—
161C4⋊C4×C151C2×D414yes—
162C22⋊C4×C151—44no—
163C22⋊C4×C151—42no—
164C22⋊C4×C151—82no—
165C42×C151—28no—
166C42×C151—24no—
167C42×C151—22no—
168C42×C151—22no—
169C42×C151C2×D412yes—
170C42×C151C2×D412yes—
171C23×C151—22no—
172C23×C151—41no—
173C23×C151—41no—
174C23×C151—41no—
175C23×C151—41no—
176C23×C151—41no—
177C23×C151—41no—
178C23×C151C22×D411yescharacteristic
179Q8×C151—22no—
180Q8×C151—22no—
181Q8×C151—22no—
182Q8×C151—44no—
183Q8×C151—48no—
184Q8×C151C22⋊Q814yescomplemented
185D4×C151—22no—
186D4×C151—22no—
187D4×C151—41no—
188D4×C151—41no—
189C2×C4×C151—22no—
190C2×C4×C151—22no—
191C2×C4×C151—22no—
192C2×C4×C151—21no—
193C2×C4×C151—21no—
194C2×C4×C151—22no—
195C2×C4×C151—24no—
196C2×C4×C151—22no—
197C2×C4×C151—22no—
198C2×C4×C151—22no—
199C2×C4×C151—22no—
200C2×C4×C151—22no—
201C2×C4×C151—22no—
202C2×C4×C151—42no—
203C2×C4×C151—42no—
204C2×C4×C151—41no—
205C2×C4×C151—41no—
206C2×C4×C151—41no—
207C2×C4×C151—42no—
208C2×C4×C151—41no—
209C2×C4×C151—42no—
210C2×C4×C151—41no—
211C2×C4×C151—42no—
212C2×C4×C151—84no—
213C2×C4×C151#12012yes—
214C2×C4×C151#12012yes—
215C2×C4×C151#13512yes—
216C2×C4×C151AGL(1,ℤ/8)12yes—
217C2×C4×C151C22×D411yescharacteristic
218C2×C4×C151C22×D411yescharacteristic
219C2×C4×C151C22×Q811yescharacteristic
220C2×C4×C151C2×#4111yescharacteristic
221C2×C4×C1512−1+411yescharacteristic
222C2×C4×C1512−1+411yescharacteristic
223C8×C151—22no—
224C8×C151—41no—
225C8×C151—42no—
226C22×C151—42no—
227C22×C151—42no—
228C22×C151—42no—
229C22×C151—81no—
230C22×C151—81no—
231C22×C151—81no—
232C22×C151—81no—
233C22×C151C2×#12011yescharacteristic
234C22×C151#54811yescharacteristic
235C22×C151D4×Q811yescharacteristic
236C22×C151C2×#13511yescharacteristic
237C22×C151C2×AGL(1,ℤ/8)11yescharacteristic
238C22×C151#57711yescharacteristic
239C22×C151#57811yescharacteristic
240C4×C151—24no—
241C4×C151—22no—
242C4×C151—42no—
243C4×C151—44no—
244C4×C151—42no—
245C4×C151—42no—
246C4×C151—82no—
247C4×C151—81no—
248C4×C151#44914yes—
249C4×C151C2×C22⋊Q811yescharacteristic
250C4×C151#53611yescharacteristic
251C4×C151#54811yescharacteristic
252C4×C151D4×Q811yescharacteristic
253C2×C151—82no—
254C2×C151#240911yescharacteristic
255C2×C151C2×#44911yescharacteristic
256C2×C151#301311yescharacteristic
257C2×C151#336411yescharacteristic
258C2×C151#336811yescharacteristic
259C2×C151#338111yescharacteristic
260C2×C151#338611yescharacteristic
261#24328C15111yesmaximal, characteristic, direct product factor, Sylow
262C151#2432811yescharacteristic, direct product factor, Sylow
263C2×#557C2×C15111yescharacteristic, complemented
264C2×#474C2×C15111yescharacteristic
265C2×C4×SD16C2×C15111yescharacteristic
266#2538C2×C15111yescharacteristic
267#2059C2×C15112yescomplemented
268#2025C2×C15112yes—
269#1990C2×C15111yescharacteristic, complemented
270#1984C2×C15111yescharacteristic, complemented
271#1880C2×C15112yes—
272#1863C2×C15112yescomplemented
273#1803C2×C15111yescharacteristic, complemented
274C22×SD16C22×C15111yescharacteristic
275#557—24no—
276#557—24no—
277C2×C4⋊Q8—21no—
278C2×C4⋊Q8C22×C15111yescharacteristic
279C2×#125—22no—
280C2×#125C22×C15112yes—
281C2×C22⋊Q8—21no—
282C2×C4×Q8—21no—
283C2×C4×Q8C22×C15112yes—
284C2×C4×Q8C22×C15111yescharacteristic
285C2×C4×Q8C22×C15111yescharacteristic
286C2×C4×D4C22×C15111yescharacteristic
287#474—24no—
288C4×SD16—24no—
289C2×#107C22×C15112yes—
290C2×#106C22×C15111yescharacteristic
291C2×C4⋊C8C22×C15111yescharacteristic
292C2×#103C22×C15112yes—
293C2×#103C22×C15111yescharacteristic
294C2×#102C22×C15112yes—
295C2×#102C22×C15111yescharacteristic
296C2×C4×C8C22×C15111yescharacteristic
297#396—22no—
298#390C22×C15112yes—
299#389C22×C15112yes—
300#384C22×C15111yescharacteristic
301#380—21no—
302#340C22×C15112yes—
303#340C22×C15112yes—
304#340C22×C15112yes—
305#336C22×C15112yes—
306#336C22×C15112yes—
307C22×Q8—21no—
308C22×Q8D4×C15111yescharacteristic
309C22×Q8C23×C15111yescharacteristic
310C22×D4C23×C15111yescharacteristic
311C23×C4—21no—
312C2×SD16—22no—
313C2×SD16—22no—
314C2×SD16Q8×C15114yescomplemented
315C22×C8C23×C15111yescharacteristic
316C22×C8C23×C15112yes—
317C22×C8C23×C15111yescharacteristic
318C4⋊Q8—28no—
319C4⋊Q8—22no—
320C4⋊Q8D4×C15114yes—
321#125—216no—
322#125—24no—
323#125—24no—
324C22⋊Q8—44no—
325C4×Q8—28no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8D4×C15114yes—
332C4×D4—22no—
333C4×D4—22no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—21no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4D4×C15111yescharacteristic
343C2×C4⋊C4D4×C15112yes—
344C2×C4⋊C4C23×C15111yescharacteristic
345C2×C4⋊C4C23×C15112yes—
346C2×C4⋊C4C23×C15111yescharacteristic
347C2×C4⋊C4C23×C15112yes—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—22no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42C23×C15112yes—
354C2×C42C23×C15111yescharacteristic
355#107—24no—
356#106—22no—
357C4⋊C8—22no—
358#103—24no—
359#103—22no—
360#102—22no—
361#102—24no—
362C4×C8—22no—
363#95—22no—
364C24—21no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C2×D4×C15112yes—
375C2×Q8C2×Q8×C15112yes—
376C2×Q8#41×C15112yes—
377C2×D4—21no—
378C2×D4—21no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4C2×Q8×C15112yes—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—21no—
387C22×C4—22no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4—42no—
391C22×C4C2×D4×C15112yes—
392C22×C4C2×D4×C15112yes—
393C22×C4C2×D4×C15111yescharacteristic
394C22×C4C2×D4×C15111yescharacteristic
395C22×C4C24×C15111yescharacteristic
396SD16—28no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—22no—
405C2×C8C2×Q8×C15112yes—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—44no—
420C4⋊C4—42no—
421C4⋊C4—81no—
422C4⋊C4C2×D4×C15112yes—
423C4⋊C4C2×D4×C15114yes—
424C22⋊C4—44no—
425C22⋊C4—42no—
426C22⋊C4—82no—
427C42—28no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42C2×D4×C15112yes—
432C42C2×D4×C15112yes—
433C23—22no—
434C23—41no—
435C23—41no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23C22×D4×C15111yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—44no—
445Q8—48no—
446Q8C22⋊Q8×C15114yescomplemented
447D4—22no—
448D4—22no—
449D4—41no—
450D4—41no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4#120×C15112yes—
476C2×C4#120×C15112yes—
477C2×C4#135×C15112yes—
478C2×C4AGL(1,ℤ/8)×C15112yes—
479C2×C4C22×D4×C15111yescharacteristic
480C2×C4C22×D4×C15111yescharacteristic
481C2×C4C22×Q8×C15111yescharacteristic
482C2×C4C2×#41×C15111yescharacteristic
483C2×C42−1+4×C15111yescharacteristic
484C2×C42−1+4×C15111yescharacteristic
485C8—22no—
486C8—41no—
487C8—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22C2×#120×C15111yescharacteristic
496C22#548×C15111yescharacteristic
497C22D4×Q8×C15111yescharacteristic
498C22C2×#135×C15111yescharacteristic
499C22C2×AGL(1,ℤ/8)×C15111yescharacteristic
500C22#577×C15111yescharacteristic
501C22#578×C15111yescharacteristic
502C4—24no—
503C4—22no—
504C4—42no—
505C4—44no—
506C4—42no—
507C4—42no—
508C4—82no—
509C4—81no—
510C4#449×C15114yes—
511C4C2×C22⋊Q8×C15111yescharacteristic
512C4#536×C15111yescharacteristic
513C4#548×C15111yescharacteristic
514C4D4×Q8×C15111yescharacteristic
515C2—82no—
516C2#2409×C15111yescharacteristic
517C2C2×#449×C15111yescharacteristic
518C2#3013×C15111yescharacteristic
519C2#3364×C15111yescharacteristic
520C2#3368×C15111yescharacteristic
521C2#3381×C15111yescharacteristic
522C2#3386×C15111yescharacteristic
subgroupG/H
center, socleC23×C151C22×D4
commutator, Frattini, cosocleC22×C4C24×C151
prime radicalC22×D4×C151C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C4×Q8×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#557×C151C2semidirect
5. #2059×C151C2semidirect
7. #1990×C151C2semidirect
8. #1984×C151C2semidirect
10. #1863×C151C2semidirect
11. #1803×C151C2semidirect
52. C2×SD16×C151Q8semidirect
184. Q8×C151C22⋊Q8semidirect
261. #24328C151direct (N × H)
262. C151#24328direct (N × H)
263. C2×#557C2×C151semidirect
267. #2059C2×C151semidirect
269. #1990C2×C151semidirect
270. #1984C2×C151semidirect
272. #1863C2×C151semidirect
273. #1803C2×C151semidirect
314. C2×SD16Q8×C151semidirect
446. Q8C22⋊Q8×C151semidirect

Maximal subgroups

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

subgroupindex
C2×#557×C1512
C2×#474×C1512
C2×C4×SD16×C1512
#2538×C1512
#2059×C1512
#2025×C1512
#1990×C1512
#1984×C1512
#1880×C1512
#1863×C1512
#1803×C1512
#24328151